/ src / cluster_linearize.h
cluster_linearize.h
   1  // Copyright (c) The Bitcoin Core developers
   2  // Distributed under the MIT software license, see the accompanying
   3  // file COPYING or http://www.opensource.org/licenses/mit-license.php.
   4  
   5  #ifndef BITCOIN_CLUSTER_LINEARIZE_H
   6  #define BITCOIN_CLUSTER_LINEARIZE_H
   7  
   8  #include <algorithm>
   9  #include <cstdint>
  10  #include <numeric>
  11  #include <optional>
  12  #include <ranges>
  13  #include <utility>
  14  #include <vector>
  15  
  16  #include <attributes.h>
  17  #include <memusage.h>
  18  #include <random.h>
  19  #include <span.h>
  20  #include <util/feefrac.h>
  21  #include <util/vecdeque.h>
  22  
  23  namespace cluster_linearize {
  24  
  25  /** Data type to represent transaction indices in DepGraphs and the clusters they represent. */
  26  using DepGraphIndex = uint32_t;
  27  
  28  /** Data structure that holds a transaction graph's preprocessed data (fee, size, ancestors,
  29   *  descendants). */
  30  template<typename SetType>
  31  class DepGraph
  32  {
  33      /** Information about a single transaction. */
  34      struct Entry
  35      {
  36          /** Fee and size of transaction itself. */
  37          FeeFrac feerate;
  38          /** All ancestors of the transaction (including itself). */
  39          SetType ancestors;
  40          /** All descendants of the transaction (including itself). */
  41          SetType descendants;
  42  
  43          /** Equality operator (primarily for testing purposes). */
  44          friend bool operator==(const Entry&, const Entry&) noexcept = default;
  45  
  46          /** Construct an empty entry. */
  47          Entry() noexcept = default;
  48          /** Construct an entry with a given feerate, ancestor set, descendant set. */
  49          Entry(const FeeFrac& f, const SetType& a, const SetType& d) noexcept : feerate(f), ancestors(a), descendants(d) {}
  50      };
  51  
  52      /** Data for each transaction. */
  53      std::vector<Entry> entries;
  54  
  55      /** Which positions are used. */
  56      SetType m_used;
  57  
  58  public:
  59      /** Equality operator (primarily for testing purposes). */
  60      friend bool operator==(const DepGraph& a, const DepGraph& b) noexcept
  61      {
  62          if (a.m_used != b.m_used) return false;
  63          // Only compare the used positions within the entries vector.
  64          for (auto idx : a.m_used) {
  65              if (a.entries[idx] != b.entries[idx]) return false;
  66          }
  67          return true;
  68      }
  69  
  70      // Default constructors.
  71      DepGraph() noexcept = default;
  72      DepGraph(const DepGraph&) noexcept = default;
  73      DepGraph(DepGraph&&) noexcept = default;
  74      DepGraph& operator=(const DepGraph&) noexcept = default;
  75      DepGraph& operator=(DepGraph&&) noexcept = default;
  76  
  77      /** Construct a DepGraph object given another DepGraph and a mapping from old to new.
  78       *
  79       * @param depgraph   The original DepGraph that is being remapped.
  80       *
  81       * @param mapping    A span such that mapping[i] gives the position in the new DepGraph
  82       *                   for position i in the old depgraph. Its size must be equal to
  83       *                   depgraph.PositionRange(). The value of mapping[i] is ignored if
  84       *                   position i is a hole in depgraph (i.e., if !depgraph.Positions()[i]).
  85       *
  86       * @param pos_range  The PositionRange() for the new DepGraph. It must equal the largest
  87       *                   value in mapping for any used position in depgraph plus 1, or 0 if
  88       *                   depgraph.TxCount() == 0.
  89       *
  90       * Complexity: O(N^2) where N=depgraph.TxCount().
  91       */
  92      DepGraph(const DepGraph<SetType>& depgraph, std::span<const DepGraphIndex> mapping, DepGraphIndex pos_range) noexcept : entries(pos_range)
  93      {
  94          Assume(mapping.size() == depgraph.PositionRange());
  95          Assume((pos_range == 0) == (depgraph.TxCount() == 0));
  96          for (DepGraphIndex i : depgraph.Positions()) {
  97              auto new_idx = mapping[i];
  98              Assume(new_idx < pos_range);
  99              // Add transaction.
 100              entries[new_idx].ancestors = SetType::Singleton(new_idx);
 101              entries[new_idx].descendants = SetType::Singleton(new_idx);
 102              m_used.Set(new_idx);
 103              // Fill in fee and size.
 104              entries[new_idx].feerate = depgraph.entries[i].feerate;
 105          }
 106          for (DepGraphIndex i : depgraph.Positions()) {
 107              // Fill in dependencies by mapping direct parents.
 108              SetType parents;
 109              for (auto j : depgraph.GetReducedParents(i)) parents.Set(mapping[j]);
 110              AddDependencies(parents, mapping[i]);
 111          }
 112          // Verify that the provided pos_range was correct (no unused positions at the end).
 113          Assume(m_used.None() ? (pos_range == 0) : (pos_range == m_used.Last() + 1));
 114      }
 115  
 116      /** Get the set of transactions positions in use. Complexity: O(1). */
 117      const SetType& Positions() const noexcept { return m_used; }
 118      /** Get the range of positions in this DepGraph. All entries in Positions() are in [0, PositionRange() - 1]. */
 119      DepGraphIndex PositionRange() const noexcept { return entries.size(); }
 120      /** Get the number of transactions in the graph. Complexity: O(1). */
 121      auto TxCount() const noexcept { return m_used.Count(); }
 122      /** Get the feerate of a given transaction i. Complexity: O(1). */
 123      const FeeFrac& FeeRate(DepGraphIndex i) const noexcept { return entries[i].feerate; }
 124      /** Get the mutable feerate of a given transaction i. Complexity: O(1). */
 125      FeeFrac& FeeRate(DepGraphIndex i) noexcept { return entries[i].feerate; }
 126      /** Get the ancestors of a given transaction i. Complexity: O(1). */
 127      const SetType& Ancestors(DepGraphIndex i) const noexcept { return entries[i].ancestors; }
 128      /** Get the descendants of a given transaction i. Complexity: O(1). */
 129      const SetType& Descendants(DepGraphIndex i) const noexcept { return entries[i].descendants; }
 130  
 131      /** Add a new unconnected transaction to this transaction graph (in the first available
 132       *  position), and return its DepGraphIndex.
 133       *
 134       * Complexity: O(1) (amortized, due to resizing of backing vector).
 135       */
 136      DepGraphIndex AddTransaction(const FeeFrac& feefrac) noexcept
 137      {
 138          static constexpr auto ALL_POSITIONS = SetType::Fill(SetType::Size());
 139          auto available = ALL_POSITIONS - m_used;
 140          Assume(available.Any());
 141          DepGraphIndex new_idx = available.First();
 142          if (new_idx == entries.size()) {
 143              entries.emplace_back(feefrac, SetType::Singleton(new_idx), SetType::Singleton(new_idx));
 144          } else {
 145              entries[new_idx] = Entry(feefrac, SetType::Singleton(new_idx), SetType::Singleton(new_idx));
 146          }
 147          m_used.Set(new_idx);
 148          return new_idx;
 149      }
 150  
 151      /** Remove the specified positions from this DepGraph.
 152       *
 153       * The specified positions will no longer be part of Positions(), and dependencies with them are
 154       * removed. Note that due to DepGraph only tracking ancestors/descendants (and not direct
 155       * dependencies), if a parent is removed while a grandparent remains, the grandparent will
 156       * remain an ancestor.
 157       *
 158       * Complexity: O(N) where N=TxCount().
 159       */
 160      void RemoveTransactions(const SetType& del) noexcept
 161      {
 162          m_used -= del;
 163          // Remove now-unused trailing entries.
 164          while (!entries.empty() && !m_used[entries.size() - 1]) {
 165              entries.pop_back();
 166          }
 167          // Remove the deleted transactions from ancestors/descendants of other transactions. Note
 168          // that the deleted positions will retain old feerate and dependency information. This does
 169          // not matter as they will be overwritten by AddTransaction if they get used again.
 170          for (auto& entry : entries) {
 171              entry.ancestors &= m_used;
 172              entry.descendants &= m_used;
 173          }
 174      }
 175  
 176      /** Modify this transaction graph, adding multiple parents to a specified child.
 177       *
 178       * Complexity: O(N) where N=TxCount().
 179       */
 180      void AddDependencies(const SetType& parents, DepGraphIndex child) noexcept
 181      {
 182          Assume(m_used[child]);
 183          Assume(parents.IsSubsetOf(m_used));
 184          // Compute the ancestors of parents that are not already ancestors of child.
 185          SetType par_anc;
 186          for (auto par : parents - Ancestors(child)) {
 187              par_anc |= Ancestors(par);
 188          }
 189          par_anc -= Ancestors(child);
 190          // Bail out if there are no such ancestors.
 191          if (par_anc.None()) return;
 192          // To each such ancestor, add as descendants the descendants of the child.
 193          const auto& chl_des = entries[child].descendants;
 194          for (auto anc_of_par : par_anc) {
 195              entries[anc_of_par].descendants |= chl_des;
 196          }
 197          // To each descendant of the child, add those ancestors.
 198          for (auto dec_of_chl : Descendants(child)) {
 199              entries[dec_of_chl].ancestors |= par_anc;
 200          }
 201      }
 202  
 203      /** Compute the (reduced) set of parents of node i in this graph.
 204       *
 205       * This returns the minimal subset of the parents of i whose ancestors together equal all of
 206       * i's ancestors (unless i is part of a cycle of dependencies). Note that DepGraph does not
 207       * store the set of parents; this information is inferred from the ancestor sets.
 208       *
 209       * Complexity: O(N) where N=Ancestors(i).Count() (which is bounded by TxCount()).
 210       */
 211      SetType GetReducedParents(DepGraphIndex i) const noexcept
 212      {
 213          SetType parents = Ancestors(i);
 214          parents.Reset(i);
 215          for (auto parent : parents) {
 216              if (parents[parent]) {
 217                  parents -= Ancestors(parent);
 218                  parents.Set(parent);
 219              }
 220          }
 221          return parents;
 222      }
 223  
 224      /** Compute the (reduced) set of children of node i in this graph.
 225       *
 226       * This returns the minimal subset of the children of i whose descendants together equal all of
 227       * i's descendants (unless i is part of a cycle of dependencies). Note that DepGraph does not
 228       * store the set of children; this information is inferred from the descendant sets.
 229       *
 230       * Complexity: O(N) where N=Descendants(i).Count() (which is bounded by TxCount()).
 231       */
 232      SetType GetReducedChildren(DepGraphIndex i) const noexcept
 233      {
 234          SetType children = Descendants(i);
 235          children.Reset(i);
 236          for (auto child : children) {
 237              if (children[child]) {
 238                  children -= Descendants(child);
 239                  children.Set(child);
 240              }
 241          }
 242          return children;
 243      }
 244  
 245      /** Compute the aggregate feerate of a set of nodes in this graph.
 246       *
 247       * Complexity: O(N) where N=elems.Count().
 248       **/
 249      FeeFrac FeeRate(const SetType& elems) const noexcept
 250      {
 251          FeeFrac ret;
 252          for (auto pos : elems) ret += entries[pos].feerate;
 253          return ret;
 254      }
 255  
 256      /** Get the connected component within the subset "todo" that contains tx (which must be in
 257       *  todo).
 258       *
 259       * Two transactions are considered connected if they are both in `todo`, and one is an ancestor
 260       * of the other in the entire graph (so not just within `todo`), or transitively there is a
 261       * path of transactions connecting them. This does mean that if `todo` contains a transaction
 262       * and a grandparent, but misses the parent, they will still be part of the same component.
 263       *
 264       * Complexity: O(ret.Count()).
 265       */
 266      SetType GetConnectedComponent(const SetType& todo, DepGraphIndex tx) const noexcept
 267      {
 268          Assume(todo[tx]);
 269          Assume(todo.IsSubsetOf(m_used));
 270          auto to_add = SetType::Singleton(tx);
 271          SetType ret;
 272          do {
 273              SetType old = ret;
 274              for (auto add : to_add) {
 275                  ret |= Descendants(add);
 276                  ret |= Ancestors(add);
 277              }
 278              ret &= todo;
 279              to_add = ret - old;
 280          } while (to_add.Any());
 281          return ret;
 282      }
 283  
 284      /** Find some connected component within the subset "todo" of this graph.
 285       *
 286       * Specifically, this finds the connected component which contains the first transaction of
 287       * todo (if any).
 288       *
 289       * Complexity: O(ret.Count()).
 290       */
 291      SetType FindConnectedComponent(const SetType& todo) const noexcept
 292      {
 293          if (todo.None()) return todo;
 294          return GetConnectedComponent(todo, todo.First());
 295      }
 296  
 297      /** Determine if a subset is connected.
 298       *
 299       * Complexity: O(subset.Count()).
 300       */
 301      bool IsConnected(const SetType& subset) const noexcept
 302      {
 303          return FindConnectedComponent(subset) == subset;
 304      }
 305  
 306      /** Determine if this entire graph is connected.
 307       *
 308       * Complexity: O(TxCount()).
 309       */
 310      bool IsConnected() const noexcept { return IsConnected(m_used); }
 311  
 312      /** Append the entries of select to list in a topologically valid order.
 313       *
 314       * Complexity: O(select.Count() * log(select.Count())).
 315       */
 316      void AppendTopo(std::vector<DepGraphIndex>& list, const SetType& select) const noexcept
 317      {
 318          DepGraphIndex old_len = list.size();
 319          for (auto i : select) list.push_back(i);
 320          std::ranges::sort(std::span{list}.subspan(old_len), [&](DepGraphIndex a, DepGraphIndex b) noexcept {
 321              const auto a_anc_count = entries[a].ancestors.Count();
 322              const auto b_anc_count = entries[b].ancestors.Count();
 323              if (a_anc_count != b_anc_count) return a_anc_count < b_anc_count;
 324              return a < b;
 325          });
 326      }
 327  
 328      /** Check if this graph is acyclic. */
 329      bool IsAcyclic() const noexcept
 330      {
 331          for (auto i : Positions()) {
 332              if ((Ancestors(i) & Descendants(i)) != SetType::Singleton(i)) {
 333                  return false;
 334              }
 335          }
 336          return true;
 337      }
 338  
 339      unsigned CountDependencies() const noexcept
 340      {
 341          unsigned ret = 0;
 342          for (auto i : Positions()) {
 343              ret += GetReducedParents(i).Count();
 344          }
 345          return ret;
 346      }
 347  
 348      /** Reduce memory usage if possible. No observable effect. */
 349      void Compact() noexcept
 350      {
 351          entries.shrink_to_fit();
 352      }
 353  
 354      size_t DynamicMemoryUsage() const noexcept
 355      {
 356          return memusage::DynamicUsage(entries);
 357      }
 358  };
 359  
 360  /** A set of transactions together with their aggregate feerate. */
 361  template<typename SetType>
 362  struct SetInfo
 363  {
 364      /** The transactions in the set. */
 365      SetType transactions;
 366      /** Their combined fee and size. */
 367      FeeFrac feerate;
 368  
 369      /** Construct a SetInfo for the empty set. */
 370      SetInfo() noexcept = default;
 371  
 372      /** Construct a SetInfo for a specified set and feerate. */
 373      SetInfo(const SetType& txn, const FeeFrac& fr) noexcept : transactions(txn), feerate(fr) {}
 374  
 375      /** Construct a SetInfo for a given transaction in a depgraph. */
 376      explicit SetInfo(const DepGraph<SetType>& depgraph, DepGraphIndex pos) noexcept :
 377          transactions(SetType::Singleton(pos)), feerate(depgraph.FeeRate(pos)) {}
 378  
 379      /** Construct a SetInfo for a set of transactions in a depgraph. */
 380      explicit SetInfo(const DepGraph<SetType>& depgraph, const SetType& txn) noexcept :
 381          transactions(txn), feerate(depgraph.FeeRate(txn)) {}
 382  
 383      /** Add a transaction to this SetInfo (which must not yet be in it). */
 384      void Set(const DepGraph<SetType>& depgraph, DepGraphIndex pos) noexcept
 385      {
 386          Assume(!transactions[pos]);
 387          transactions.Set(pos);
 388          feerate += depgraph.FeeRate(pos);
 389      }
 390  
 391      /** Add the transactions of other to this SetInfo (no overlap allowed). */
 392      SetInfo& operator|=(const SetInfo& other) noexcept
 393      {
 394          Assume(!transactions.Overlaps(other.transactions));
 395          transactions |= other.transactions;
 396          feerate += other.feerate;
 397          return *this;
 398      }
 399  
 400      /** Remove the transactions of other from this SetInfo (which must be a subset). */
 401      SetInfo& operator-=(const SetInfo& other) noexcept
 402      {
 403          Assume(other.transactions.IsSubsetOf(transactions));
 404          transactions -= other.transactions;
 405          feerate -= other.feerate;
 406          return *this;
 407      }
 408  
 409      /** Compute the difference between this and other SetInfo (which must be a subset). */
 410      SetInfo operator-(const SetInfo& other) const noexcept
 411      {
 412          Assume(other.transactions.IsSubsetOf(transactions));
 413          return {transactions - other.transactions, feerate - other.feerate};
 414      }
 415  
 416      /** Swap two SetInfo objects. */
 417      friend void swap(SetInfo& a, SetInfo& b) noexcept
 418      {
 419          swap(a.transactions, b.transactions);
 420          swap(a.feerate, b.feerate);
 421      }
 422  
 423      /** Permit equality testing. */
 424      friend bool operator==(const SetInfo&, const SetInfo&) noexcept = default;
 425  };
 426  
 427  /** Compute the chunks of linearization as SetInfos. */
 428  template<typename SetType>
 429  std::vector<SetInfo<SetType>> ChunkLinearizationInfo(const DepGraph<SetType>& depgraph, std::span<const DepGraphIndex> linearization) noexcept
 430  {
 431      std::vector<SetInfo<SetType>> ret;
 432      for (DepGraphIndex i : linearization) {
 433          /** The new chunk to be added, initially a singleton. */
 434          SetInfo<SetType> new_chunk(depgraph, i);
 435          // As long as the new chunk has a higher feerate than the last chunk so far, absorb it.
 436          while (!ret.empty() && ByRatio{new_chunk.feerate} > ByRatio{ret.back().feerate}) {
 437              new_chunk |= ret.back();
 438              ret.pop_back();
 439          }
 440          // Actually move that new chunk into the chunking.
 441          ret.emplace_back(std::move(new_chunk));
 442      }
 443      return ret;
 444  }
 445  
 446  /** Compute the feerates of the chunks of linearization. Identical to ChunkLinearizationInfo, but
 447   *  only returns the chunk feerates, not the corresponding transaction sets. */
 448  template<typename SetType>
 449  std::vector<FeeFrac> ChunkLinearization(const DepGraph<SetType>& depgraph, std::span<const DepGraphIndex> linearization) noexcept
 450  {
 451      std::vector<FeeFrac> ret;
 452      for (DepGraphIndex i : linearization) {
 453          /** The new chunk to be added, initially a singleton. */
 454          auto new_chunk = depgraph.FeeRate(i);
 455          // As long as the new chunk has a higher feerate than the last chunk so far, absorb it.
 456          while (!ret.empty() && ByRatio{new_chunk} > ByRatio{ret.back()}) {
 457              new_chunk += ret.back();
 458              ret.pop_back();
 459          }
 460          // Actually move that new chunk into the chunking.
 461          ret.push_back(std::move(new_chunk));
 462      }
 463      return ret;
 464  }
 465  
 466  /** Concept for function objects that return std::strong_ordering when invoked with two Args. */
 467  template<typename F, typename Arg>
 468  concept StrongComparator =
 469      std::regular_invocable<F, Arg, Arg> &&
 470      std::is_same_v<std::invoke_result_t<F, Arg, Arg>, std::strong_ordering>;
 471  
 472  /** Simple default transaction ordering function for SpanningForestState::GetLinearization() and
 473   *  Linearize(), which just sorts by DepGraphIndex. */
 474  using IndexTxOrder = std::compare_three_way;
 475  
 476  /** A default cost model for SFL for SetType=BitSet<64>, based on benchmarks.
 477   *
 478   * The numbers here were obtained in February 2026 by:
 479   * - For a variety of machines:
 480   *   - Running a fixed collection of ~385000 clusters found through random generation and fuzzing,
 481   *     optimizing for difficulty of linearization.
 482   *     - Linearize each ~3000 times, with different random seeds. Sometimes without input
 483   *       linearization, sometimes with a bad one.
 484   *       - Gather cycle counts for each of the operations included in this cost model,
 485   *         broken down by their parameters.
 486   *   - Correct the data by subtracting the runtime of obtaining the cycle count.
 487   *   - Drop the 5% top and bottom samples from each cycle count dataset, and compute the average
 488   *     of the remaining samples.
 489   *   - For each operation, fit a least-squares linear function approximation through the samples.
 490   * - Rescale all machine expressions to make their total time match, as we only care about
 491   *   relative cost of each operation.
 492   * - Take the per-operation average of operation expressions across all machines, to construct
 493   *   expressions for an average machine.
 494   * - Approximate the result with integer coefficients. Each cost unit corresponds to somewhere
 495   *   between 0.5 ns and 2.5 ns, depending on the hardware.
 496   */
 497  class SFLDefaultCostModel
 498  {
 499      uint64_t m_cost{0};
 500  
 501  public:
 502      inline void InitializeBegin() noexcept {}
 503      inline void InitializeEnd(int num_txns, int num_deps) noexcept
 504      {
 505           // Cost of initialization.
 506           m_cost += 39 * num_txns;
 507           // Cost of producing linearization at the end.
 508           m_cost += 48 * num_txns + 4 * num_deps;
 509      }
 510      inline void GetLinearizationBegin() noexcept {}
 511      inline void GetLinearizationEnd(int num_txns, int num_deps) noexcept
 512      {
 513          // Note that we account for the cost of the final linearization at the beginning (see
 514          // InitializeEnd), because the cost budget decision needs to be made before calling
 515          // GetLinearization.
 516          // This function exists here to allow overriding it easily for benchmark purposes.
 517      }
 518      inline void MakeTopologicalBegin() noexcept {}
 519      inline void MakeTopologicalEnd(int num_chunks, int num_steps) noexcept
 520      {
 521          m_cost += 20 * num_chunks + 28 * num_steps;
 522      }
 523      inline void StartOptimizingBegin() noexcept {}
 524      inline void StartOptimizingEnd(int num_chunks) noexcept { m_cost += 13 * num_chunks; }
 525      inline void ActivateBegin() noexcept {}
 526      inline void ActivateEnd(int num_deps) noexcept { m_cost += 10 * num_deps + 1; }
 527      inline void DeactivateBegin() noexcept {}
 528      inline void DeactivateEnd(int num_deps) noexcept { m_cost += 11 * num_deps + 8; }
 529      inline void MergeChunksBegin() noexcept {}
 530      inline void MergeChunksMid(int num_txns) noexcept { m_cost += 2 * num_txns; }
 531      inline void MergeChunksEnd(int num_steps) noexcept { m_cost += 3 * num_steps + 5; }
 532      inline void PickMergeCandidateBegin() noexcept {}
 533      inline void PickMergeCandidateEnd(int num_steps) noexcept { m_cost += 8 * num_steps; }
 534      inline void PickChunkToOptimizeBegin() noexcept {}
 535      inline void PickChunkToOptimizeEnd(int num_steps) noexcept { m_cost += num_steps + 4; }
 536      inline void PickDependencyToSplitBegin() noexcept {}
 537      inline void PickDependencyToSplitEnd(int num_txns) noexcept { m_cost += 8 * num_txns + 9; }
 538      inline void StartMinimizingBegin() noexcept {}
 539      inline void StartMinimizingEnd(int num_chunks) noexcept { m_cost += 18 * num_chunks; }
 540      inline void MinimizeStepBegin() noexcept {}
 541      inline void MinimizeStepMid(int num_txns) noexcept { m_cost += 11 * num_txns + 11; }
 542      inline void MinimizeStepEnd(bool split) noexcept { m_cost += 17 * split + 7; }
 543  
 544      inline uint64_t GetCost() const noexcept { return m_cost; }
 545  };
 546  
 547  /** Class to represent the internal state of the spanning-forest linearization (SFL) algorithm.
 548   *
 549   * At all times, each dependency is marked as either "active" or "inactive". The subset of active
 550   * dependencies is the state of the SFL algorithm. The implementation maintains several other
 551   * values to speed up operations, but everything is ultimately a function of what that subset of
 552   * active dependencies is.
 553   *
 554   * Given such a subset, define a chunk as the set of transactions that are connected through active
 555   * dependencies (ignoring their parent/child direction). Thus, every state implies a particular
 556   * partitioning of the graph into chunks (including potential singletons). In the extreme, each
 557   * transaction may be in its own chunk, or in the other extreme all transactions may form a single
 558   * chunk. A chunk's feerate is its total fee divided by its total size.
 559   *
 560   * The algorithm consists of switching dependencies between active and inactive. The final
 561   * linearization that is produced at the end consists of these chunks, sorted from high to low
 562   * feerate, each individually sorted in an arbitrary but topological (= no child before parent)
 563   * way.
 564   *
 565   * We define four quality properties the state can have:
 566   *
 567   * - acyclic: The state is acyclic whenever no cycle of active dependencies exists within the
 568   *            graph, ignoring the parent/child direction. This is equivalent to saying that within
 569   *            each chunk the set of active dependencies form a tree, and thus the overall set of
 570   *            active dependencies in the graph form a spanning forest, giving the algorithm its
 571   *            name. Being acyclic is also equivalent to every chunk of N transactions having
 572   *            exactly N-1 active dependencies.
 573   *
 574   *            For example in a diamond graph, D->{B,C}->A, the 4 dependencies cannot be
 575   *            simultaneously active. If at least one is inactive, the state is acyclic.
 576   *
 577   *            The algorithm maintains an acyclic state at *all* times as an invariant. This implies
 578   *            that activating a dependency always corresponds to merging two chunks, and that
 579   *            deactivating one always corresponds to splitting two chunks.
 580   *
 581   * - topological: We say the state is topological whenever it is acyclic and no inactive dependency
 582   *                exists between two distinct chunks such that the child chunk has higher or equal
 583   *                feerate than the parent chunk.
 584   *
 585   *                The relevance is that whenever the state is topological, the produced output
 586   *                linearization will be topological too (i.e., not have children before parents).
 587   *                Note that the "or equal" part of the definition matters: if not, one can end up
 588   *                in a situation with mutually-dependent equal-feerate chunks that cannot be
 589   *                linearized. For example C->{A,B} and D->{A,B}, with C->A and D->B active. The AC
 590   *                chunk depends on DB through C->B, and the BD chunk depends on AC through D->A.
 591   *                Merging them into a single ABCD chunk fixes this.
 592   *
 593   *                The algorithm attempts to keep the state topological as much as possible, so it
 594   *                can be interrupted to produce an output whenever, but will sometimes need to
 595   *                temporarily deviate from it when improving the state.
 596   *
 597   * - optimal: For every active dependency, define its top and bottom set as the set of transactions
 598   *            in the chunks that would result if the dependency were deactivated; the top being the
 599   *            one with the dependency's parent, and the bottom being the one with the child. Note
 600   *            that due to acyclicity, every deactivation splits a chunk exactly in two.
 601   *
 602   *            We say the state is optimal whenever it is topological and it has no active
 603   *            dependency whose top feerate is strictly higher than its bottom feerate. The
 604   *            relevance is that it can be proven that whenever the state is optimal, the produced
 605   *            linearization will also be optimal (in the convexified feerate diagram sense). It can
 606   *            also be proven that for every graph at least one optimal state exists.
 607   *
 608   *            Note that it is possible for the SFL state to not be optimal, but the produced
 609   *            linearization to still be optimal. This happens when the chunks of a state are
 610   *            identical to those of an optimal state, but the exact set of active dependencies
 611   *            within a chunk differ in such a way that the state optimality condition is not
 612   *            satisfied. Thus, the state being optimal is more a "the eventual output is *known*
 613   *            to be optimal".
 614   *
 615   * - minimal: We say the state is minimal when it is:
 616   *            - acyclic
 617   *            - topological, except that inactive dependencies between equal-feerate chunks are
 618   *              allowed as long as they do not form a loop.
 619   *            - like optimal, no active dependencies whose top feerate is strictly higher than
 620   *              the bottom feerate are allowed.
 621   *            - no chunk contains a proper non-empty subset which includes all its own in-chunk
 622   *              dependencies of the same feerate as the chunk itself.
 623   *
 624   *            A minimal state effectively corresponds to an optimal state, where every chunk has
 625   *            been split into its minimal equal-feerate components.
 626   *
 627   *            The algorithm terminates whenever a minimal state is reached.
 628   *
 629   *
 630   * This leads to the following high-level algorithm:
 631   * - Start with all dependencies inactive, and thus all transactions in their own chunk. This is
 632   *   definitely acyclic.
 633   * - Activate dependencies (merging chunks) until the state is topological.
 634   * - Loop until optimal (no dependencies with higher-feerate top than bottom), or time runs out:
 635   *   - Deactivate a violating dependency, potentially making the state non-topological.
 636   *   - Activate other dependencies to make the state topological again.
 637   * - If there is time left and the state is optimal:
 638   *   - Attempt to split chunks into equal-feerate parts without mutual dependencies between them.
 639   *     When this succeeds, recurse into them.
 640   *   - If no such chunks can be found, the state is minimal.
 641   * - Output the chunks from high to low feerate, each internally sorted topologically.
 642   *
 643   * When merging, we always either:
 644   * - Merge upwards: merge a chunk with the lowest-feerate other chunk it depends on, among those
 645   *                  with lower or equal feerate than itself.
 646   * - Merge downwards: merge a chunk with the highest-feerate other chunk that depends on it, among
 647   *                    those with higher or equal feerate than itself.
 648   *
 649   * Using these strategies in the improvement loop above guarantees that the output linearization
 650   * after a deactivate + merge step is never worse or incomparable (in the convexified feerate
 651   * diagram sense) than the output linearization that would be produced before the step. With that,
 652   * we can refine the high-level algorithm to:
 653   * - Start with all dependencies inactive.
 654   * - Perform merges as described until none are possible anymore, making the state topological.
 655   * - Loop until optimal or time runs out:
 656   *   - Pick a dependency D to deactivate among those with higher feerate top than bottom.
 657   *   - Deactivate D, causing the chunk it is in to split into top T and bottom B.
 658   *   - Do an upwards merge of T, if possible. If so, repeat the same with the merged result.
 659   *   - Do a downwards merge of B, if possible. If so, repeat the same with the merged result.
 660   * - Split chunks further to obtain a minimal state, see below.
 661   * - Output the chunks from high to low feerate, each internally sorted topologically.
 662   *
 663   * Instead of performing merges arbitrarily to make the initial state topological, it is possible
 664   * to do so guided by an existing linearization. This has the advantage that the state's would-be
 665   * output linearization is immediately as good as the existing linearization it was based on:
 666   * - Start with all dependencies inactive.
 667   * - For each transaction t in the existing linearization:
 668   *   - Find the chunk C that transaction is in (which will be singleton).
 669   *   - Do an upwards merge of C, if possible. If so, repeat the same with the merged result.
 670   * No downwards merges are needed in this case.
 671   *
 672   * After reaching an optimal state, it can be transformed into a minimal state by attempting to
 673   * split chunks further into equal-feerate parts. To do so, pick a specific transaction in each
 674   * chunk (the pivot), and rerun the above split-then-merge procedure again:
 675   * - first, while pretending the pivot transaction has an infinitesimally higher (or lower) fee
 676   *   than it really has. If a split exists with the pivot in the top part (or bottom part), this
 677   *   will find it.
 678   * - if that fails to split, repeat while pretending the pivot transaction has an infinitesimally
 679   *   lower (or higher) fee. If a split exists with the pivot in the bottom part (or top part), this
 680   *   will find it.
 681   * - if either succeeds, repeat the procedure for the newly found chunks to split them further.
 682   *   If not, the chunk is already minimal.
 683   * If the chunk can be split into equal-feerate parts, then the pivot must exist in either the top
 684   * or bottom part of that potential split. By trying both with the same pivot, if a split exists,
 685   * it will be found.
 686   *
 687   * What remains to be specified are a number of heuristics:
 688   *
 689   * - How to decide which chunks to merge:
 690   *   - The merge upwards and downward rules specify that the lowest-feerate respectively
 691   *     highest-feerate candidate chunk is merged with, but if there are multiple equal-feerate
 692   *     candidates, a uniformly random one among them is picked.
 693   *
 694   * - How to decide what dependency to activate (when merging chunks):
 695   *   - After picking two chunks to be merged (see above), a uniformly random dependency between the
 696   *     two chunks is activated.
 697   *
 698   * - How to decide which chunk to find a dependency to split in:
 699   *   - A round-robin queue of chunks to improve is maintained. The initial ordering of this queue
 700   *     is uniformly randomly permuted.
 701   *
 702   * - How to decide what dependency to deactivate (when splitting chunks):
 703   *   - Inside the selected chunk (see above), among the dependencies whose top feerate is strictly
 704   *     higher than its bottom feerate in the selected chunk, if any, a uniformly random dependency
 705   *     is deactivated.
 706   *   - After every split, it is possible that the top and the bottom chunk merge with each other
 707   *     again in the merge sequence (through a top->bottom dependency, not through the deactivated
 708   *     one, which was bottom->top). Call this a self-merge. If a self-merge does not occur after
 709   *     a split, the resulting linearization is strictly improved (the area under the convexified
 710   *     feerate diagram increases by at least gain/2), while self-merges do not change it.
 711   *
 712   * - How to decide the exact output linearization:
 713   *   - When there are multiple equal-feerate chunks with no dependencies between them, pick the
 714   *     smallest one first. If there are multiple smallest ones, pick the one that contains the
 715   *     last transaction (according to the provided fallback order) last (note that this is not the
 716   *     same as picking the chunk with the first transaction first).
 717   *   - Within chunks, pick among all transactions without missing dependencies the one with the
 718   *     highest individual feerate. If there are multiple ones with the same individual feerate,
 719   *     pick the smallest first. If there are multiple with the same fee and size, pick the one
 720   *     that sorts first according to the fallback order first.
 721   */
 722  template<typename SetType, typename CostModel = SFLDefaultCostModel>
 723  class SpanningForestState
 724  {
 725  private:
 726      /** Internal RNG. */
 727      InsecureRandomContext m_rng;
 728  
 729      /** Data type to represent indexing into m_tx_data. */
 730      using TxIdx = DepGraphIndex;
 731      /** Data type to represent indexing into m_set_info. Use the smallest type possible to improve
 732       *  cache locality. */
 733      using SetIdx = std::conditional_t<(SetType::Size() <= 0xff),
 734                                        uint8_t,
 735                                        std::conditional_t<(SetType::Size() <= 0xffff),
 736                                                           uint16_t,
 737                                                           uint32_t>>;
 738      /** An invalid SetIdx. */
 739      static constexpr SetIdx INVALID_SET_IDX = SetIdx(-1);
 740  
 741      /** Structure with information about a single transaction. */
 742      struct TxData {
 743          /** The top set for every active child dependency this transaction has, indexed by child
 744           *  TxIdx. Only defined for indexes in active_children. */
 745          std::array<SetIdx, SetType::Size()> dep_top_idx;
 746          /** The set of parent transactions of this transaction. Immutable after construction. */
 747          SetType parents;
 748          /** The set of child transactions of this transaction. Immutable after construction. */
 749          SetType children;
 750          /** The set of child transactions reachable through an active dependency. */
 751          SetType active_children;
 752          /** Which chunk this transaction belongs to. */
 753          SetIdx chunk_idx;
 754      };
 755  
 756      /** The set of all TxIdx's of transactions in the cluster indexing into m_tx_data. */
 757      SetType m_transaction_idxs;
 758      /** The set of all chunk SetIdx's. This excludes the SetIdxs that refer to active
 759       *  dependencies' tops. */
 760      SetType m_chunk_idxs;
 761      /** The set of all SetIdx's that appear in m_suboptimal_chunks. Note that they do not need to
 762       *  be chunks: some of these sets may have been converted to a dependency's top set since being
 763       *  added to m_suboptimal_chunks. */
 764      SetType m_suboptimal_idxs;
 765      /** Information about each transaction (and chunks). Keeps the "holes" from DepGraph during
 766       *  construction. Indexed by TxIdx. */
 767      std::vector<TxData> m_tx_data;
 768      /** Information about each set (chunk, or active dependency top set). Indexed by SetIdx. */
 769      std::vector<SetInfo<SetType>> m_set_info;
 770      /** For each chunk, indexed by SetIdx, the set of out-of-chunk reachable transactions, in the
 771       *  upwards (.first) and downwards (.second) direction. */
 772      std::vector<std::pair<SetType, SetType>> m_reachable;
 773      /** A FIFO of chunk SetIdxs for chunks that may be improved still. */
 774      VecDeque<SetIdx> m_suboptimal_chunks;
 775      /** A FIFO of chunk indexes with a pivot transaction in them, and a flag to indicate their
 776       *  status:
 777       *  - bit 1: currently attempting to move the pivot down, rather than up.
 778       *  - bit 2: this is the second stage, so we have already tried moving the pivot in the other
 779       *           direction.
 780       */
 781      VecDeque<std::tuple<SetIdx, TxIdx, unsigned>> m_nonminimal_chunks;
 782  
 783      /** The DepGraph we are trying to linearize. */
 784      const DepGraph<SetType>& m_depgraph;
 785  
 786      /** Accounting for the cost of this computation. */
 787      CostModel m_cost;
 788  
 789      /** Pick a random transaction within a set (which must be non-empty). */
 790      TxIdx PickRandomTx(const SetType& tx_idxs) noexcept
 791      {
 792          Assume(tx_idxs.Any());
 793          unsigned pos = m_rng.randrange<unsigned>(tx_idxs.Count());
 794          for (auto tx_idx : tx_idxs) {
 795              if (pos == 0) return tx_idx;
 796              --pos;
 797          }
 798          Assume(false);
 799          return TxIdx(-1);
 800      }
 801  
 802      /** Find the set of out-of-chunk transactions reachable from tx_idxs, both in upwards and
 803       *  downwards direction. Only used by SanityCheck to verify the precomputed reachable sets in
 804       *  m_reachable that are maintained by Activate/Deactivate. */
 805      std::pair<SetType, SetType> GetReachable(const SetType& tx_idxs) const noexcept
 806      {
 807          SetType parents, children;
 808          for (auto tx_idx : tx_idxs) {
 809              const auto& tx_data = m_tx_data[tx_idx];
 810              parents |= tx_data.parents;
 811              children |= tx_data.children;
 812          }
 813          return {parents - tx_idxs, children - tx_idxs};
 814      }
 815  
 816      /** Make the inactive dependency from child to parent, which must not be in the same chunk
 817       *  already, active. Returns the merged chunk idx. */
 818      SetIdx Activate(TxIdx parent_idx, TxIdx child_idx) noexcept
 819      {
 820          m_cost.ActivateBegin();
 821          // Gather and check information about the parent and child transactions.
 822          auto& parent_data = m_tx_data[parent_idx];
 823          auto& child_data = m_tx_data[child_idx];
 824          Assume(parent_data.children[child_idx]);
 825          Assume(!parent_data.active_children[child_idx]);
 826          // Get the set index of the chunks the parent and child are currently in. The parent chunk
 827          // will become the top set of the newly activated dependency, while the child chunk will be
 828          // grown to become the merged chunk.
 829          auto parent_chunk_idx = parent_data.chunk_idx;
 830          auto child_chunk_idx = child_data.chunk_idx;
 831          Assume(parent_chunk_idx != child_chunk_idx);
 832          Assume(m_chunk_idxs[parent_chunk_idx]);
 833          Assume(m_chunk_idxs[child_chunk_idx]);
 834          auto& top_info = m_set_info[parent_chunk_idx];
 835          auto& bottom_info = m_set_info[child_chunk_idx];
 836  
 837          // Consider the following example:
 838          //
 839          //    A           A     There are two chunks, ABC and DEF, and the inactive E->C dependency
 840          //   / \         / \    is activated, resulting in a single chunk ABCDEF.
 841          //  B   C       B   C
 842          //      :  ==>      |   Dependency | top set before | top set after | change
 843          //  D   E       D   E   B->A       | AC             | ACDEF         | +DEF
 844          //   \ /         \ /    C->A       | AB             | AB            |
 845          //    F           F     F->D       | D              | D             |
 846          //                      F->E       | E              | ABCE          | +ABC
 847          //
 848          // The common pattern here is that any dependency which has the parent or child of the
 849          // dependency being activated (E->C here) in its top set, will have the opposite part added
 850          // to it. This is true for B->A and F->E, but not for C->A and F->D.
 851          //
 852          // Traverse the old parent chunk top_info (ABC in example), and add bottom_info (DEF) to
 853          // every dependency's top set which has the parent (C) in it. At the same time, change the
 854          // chunk_idx for each to be child_chunk_idx, which becomes the set for the merged chunk.
 855          for (auto tx_idx : top_info.transactions) {
 856              auto& tx_data = m_tx_data[tx_idx];
 857              tx_data.chunk_idx = child_chunk_idx;
 858              for (auto dep_child_idx : tx_data.active_children) {
 859                  auto& dep_top_info = m_set_info[tx_data.dep_top_idx[dep_child_idx]];
 860                  if (dep_top_info.transactions[parent_idx]) dep_top_info |= bottom_info;
 861              }
 862          }
 863          // Traverse the old child chunk bottom_info (DEF in example), and add top_info (ABC) to
 864          // every dependency's top set which has the child (E) in it.
 865          for (auto tx_idx : bottom_info.transactions) {
 866              auto& tx_data = m_tx_data[tx_idx];
 867              for (auto dep_child_idx : tx_data.active_children) {
 868                  auto& dep_top_info = m_set_info[tx_data.dep_top_idx[dep_child_idx]];
 869                  if (dep_top_info.transactions[child_idx]) dep_top_info |= top_info;
 870              }
 871          }
 872          // Merge top_info into bottom_info, which becomes the merged chunk.
 873          bottom_info |= top_info;
 874          // Compute merged sets of reachable transactions from the new chunk, based on the input
 875          // chunks' reachable sets.
 876          m_reachable[child_chunk_idx].first |= m_reachable[parent_chunk_idx].first;
 877          m_reachable[child_chunk_idx].second |= m_reachable[parent_chunk_idx].second;
 878          m_reachable[child_chunk_idx].first -= bottom_info.transactions;
 879          m_reachable[child_chunk_idx].second -= bottom_info.transactions;
 880          // Make parent chunk the set for the new active dependency.
 881          parent_data.dep_top_idx[child_idx] = parent_chunk_idx;
 882          parent_data.active_children.Set(child_idx);
 883          m_chunk_idxs.Reset(parent_chunk_idx);
 884          // Return the newly merged chunk.
 885          m_cost.ActivateEnd(/*num_deps=*/bottom_info.transactions.Count() - 1);
 886          return child_chunk_idx;
 887      }
 888  
 889      /** Make a specified active dependency inactive. Returns the created parent and child chunk
 890       *  indexes. */
 891      std::pair<SetIdx, SetIdx> Deactivate(TxIdx parent_idx, TxIdx child_idx) noexcept
 892      {
 893          m_cost.DeactivateBegin();
 894          // Gather and check information about the parent transactions.
 895          auto& parent_data = m_tx_data[parent_idx];
 896          Assume(parent_data.children[child_idx]);
 897          Assume(parent_data.active_children[child_idx]);
 898          // Get the top set of the active dependency (which will become the parent chunk) and the
 899          // chunk set the transactions are currently in (which will become the bottom chunk).
 900          auto parent_chunk_idx = parent_data.dep_top_idx[child_idx];
 901          auto child_chunk_idx = parent_data.chunk_idx;
 902          Assume(parent_chunk_idx != child_chunk_idx);
 903          Assume(m_chunk_idxs[child_chunk_idx]);
 904          Assume(!m_chunk_idxs[parent_chunk_idx]); // top set, not a chunk
 905          auto& top_info = m_set_info[parent_chunk_idx];
 906          auto& bottom_info = m_set_info[child_chunk_idx];
 907  
 908          // Remove the active dependency.
 909          parent_data.active_children.Reset(child_idx);
 910          m_chunk_idxs.Set(parent_chunk_idx);
 911          auto ntx = bottom_info.transactions.Count();
 912          // Subtract the top_info from the bottom_info, as it will become the child chunk.
 913          bottom_info -= top_info;
 914          // See the comment above in Activate(). We perform the opposite operations here, removing
 915          // instead of adding. Simultaneously, aggregate the top/bottom's union of parents/children.
 916          SetType top_parents, top_children;
 917          for (auto tx_idx : top_info.transactions) {
 918              auto& tx_data = m_tx_data[tx_idx];
 919              tx_data.chunk_idx = parent_chunk_idx;
 920              top_parents |= tx_data.parents;
 921              top_children |= tx_data.children;
 922              for (auto dep_child_idx : tx_data.active_children) {
 923                  auto& dep_top_info = m_set_info[tx_data.dep_top_idx[dep_child_idx]];
 924                  if (dep_top_info.transactions[parent_idx]) dep_top_info -= bottom_info;
 925              }
 926          }
 927          SetType bottom_parents, bottom_children;
 928          for (auto tx_idx : bottom_info.transactions) {
 929              auto& tx_data = m_tx_data[tx_idx];
 930              bottom_parents |= tx_data.parents;
 931              bottom_children |= tx_data.children;
 932              for (auto dep_child_idx : tx_data.active_children) {
 933                  auto& dep_top_info = m_set_info[tx_data.dep_top_idx[dep_child_idx]];
 934                  if (dep_top_info.transactions[child_idx]) dep_top_info -= top_info;
 935              }
 936          }
 937          // Compute the new sets of reachable transactions for each new chunk, based on the
 938          // top/bottom parents and children computed above.
 939          m_reachable[parent_chunk_idx].first = top_parents - top_info.transactions;
 940          m_reachable[parent_chunk_idx].second = top_children - top_info.transactions;
 941          m_reachable[child_chunk_idx].first = bottom_parents - bottom_info.transactions;
 942          m_reachable[child_chunk_idx].second = bottom_children - bottom_info.transactions;
 943          // Return the two new set idxs.
 944          m_cost.DeactivateEnd(/*num_deps=*/ntx - 1);
 945          return {parent_chunk_idx, child_chunk_idx};
 946      }
 947  
 948      /** Activate a dependency from the bottom set to the top set, which must exist. Return the
 949       *  index of the merged chunk. */
 950      SetIdx MergeChunks(SetIdx top_idx, SetIdx bottom_idx) noexcept
 951      {
 952          m_cost.MergeChunksBegin();
 953          Assume(m_chunk_idxs[top_idx]);
 954          Assume(m_chunk_idxs[bottom_idx]);
 955          auto& top_chunk_info = m_set_info[top_idx];
 956          auto& bottom_chunk_info = m_set_info[bottom_idx];
 957          // Count the number of dependencies between bottom_chunk and top_chunk.
 958          unsigned num_deps{0};
 959          for (auto tx_idx : top_chunk_info.transactions) {
 960              auto& tx_data = m_tx_data[tx_idx];
 961              num_deps += (tx_data.children & bottom_chunk_info.transactions).Count();
 962          }
 963          m_cost.MergeChunksMid(/*num_txns=*/top_chunk_info.transactions.Count());
 964          Assume(num_deps > 0);
 965          // Uniformly randomly pick one of them and activate it.
 966          unsigned pick = m_rng.randrange(num_deps);
 967          unsigned num_steps = 0;
 968          for (auto tx_idx : top_chunk_info.transactions) {
 969              ++num_steps;
 970              auto& tx_data = m_tx_data[tx_idx];
 971              auto intersect = tx_data.children & bottom_chunk_info.transactions;
 972              auto count = intersect.Count();
 973              if (pick < count) {
 974                  for (auto child_idx : intersect) {
 975                      if (pick == 0) {
 976                          m_cost.MergeChunksEnd(/*num_steps=*/num_steps);
 977                          return Activate(tx_idx, child_idx);
 978                      }
 979                      --pick;
 980                  }
 981                  Assume(false);
 982                  break;
 983              }
 984              pick -= count;
 985          }
 986          Assume(false);
 987          return INVALID_SET_IDX;
 988      }
 989  
 990      /** Activate a dependency from chunk_idx to merge_chunk_idx (if !DownWard), or a dependency
 991       *  from merge_chunk_idx to chunk_idx (if DownWard). Return the index of the merged chunk. */
 992      template<bool DownWard>
 993      SetIdx MergeChunksDirected(SetIdx chunk_idx, SetIdx merge_chunk_idx) noexcept
 994      {
 995          if constexpr (DownWard) {
 996              return MergeChunks(chunk_idx, merge_chunk_idx);
 997          } else {
 998              return MergeChunks(merge_chunk_idx, chunk_idx);
 999          }
1000      }
1001  
1002      /** Determine which chunk to merge chunk_idx with, or INVALID_SET_IDX if none. */
1003      template<bool DownWard>
1004      SetIdx PickMergeCandidate(SetIdx chunk_idx) noexcept
1005      {
1006          m_cost.PickMergeCandidateBegin();
1007          /** Information about the chunk. */
1008          Assume(m_chunk_idxs[chunk_idx]);
1009          auto& chunk_info = m_set_info[chunk_idx];
1010          // Iterate over all chunks reachable from this one. For those depended-on chunks,
1011          // remember the highest-feerate (if DownWard) or lowest-feerate (if !DownWard) one.
1012          // If multiple equal-feerate candidate chunks to merge with exist, pick a random one
1013          // among them.
1014  
1015          /** The minimum feerate (if downward) or maximum feerate (if upward) to consider when
1016           *  looking for candidate chunks to merge with. Initially, this is the original chunk's
1017           *  feerate, but is updated to be the current best candidate whenever one is found. */
1018          FeeFrac best_other_chunk_feerate = chunk_info.feerate;
1019          /** The chunk index for the best candidate chunk to merge with. INVALID_SET_IDX if none. */
1020          SetIdx best_other_chunk_idx = INVALID_SET_IDX;
1021          /** We generate random tiebreak values to pick between equal-feerate candidate chunks.
1022           *  This variable stores the tiebreak of the current best candidate. */
1023          uint64_t best_other_chunk_tiebreak{0};
1024  
1025          /** Which parent/child transactions we still need to process the chunks for. */
1026          auto todo = DownWard ? m_reachable[chunk_idx].second : m_reachable[chunk_idx].first;
1027          unsigned steps = 0;
1028          while (todo.Any()) {
1029              ++steps;
1030              // Find a chunk for a transaction in todo, and remove all its transactions from todo.
1031              auto reached_chunk_idx = m_tx_data[todo.First()].chunk_idx;
1032              auto& reached_chunk_info = m_set_info[reached_chunk_idx];
1033              todo -= reached_chunk_info.transactions;
1034              // See if it has an acceptable feerate.
1035              auto cmp = DownWard ? ByRatio{best_other_chunk_feerate} <=> ByRatio{reached_chunk_info.feerate}
1036                                  : ByRatio{reached_chunk_info.feerate} <=> ByRatio{best_other_chunk_feerate};
1037              if (cmp > 0) continue;
1038              uint64_t tiebreak = m_rng.rand64();
1039              if (cmp < 0 || tiebreak >= best_other_chunk_tiebreak) {
1040                  best_other_chunk_feerate = reached_chunk_info.feerate;
1041                  best_other_chunk_idx = reached_chunk_idx;
1042                  best_other_chunk_tiebreak = tiebreak;
1043              }
1044          }
1045          Assume(steps <= m_set_info.size());
1046  
1047          m_cost.PickMergeCandidateEnd(/*num_steps=*/steps);
1048          return best_other_chunk_idx;
1049      }
1050  
1051      /** Perform an upward or downward merge step, on the specified chunk. Returns the merged chunk,
1052       *  or INVALID_SET_IDX if no merge took place. */
1053      template<bool DownWard>
1054      SetIdx MergeStep(SetIdx chunk_idx) noexcept
1055      {
1056          auto merge_chunk_idx = PickMergeCandidate<DownWard>(chunk_idx);
1057          if (merge_chunk_idx == INVALID_SET_IDX) return INVALID_SET_IDX;
1058          chunk_idx = MergeChunksDirected<DownWard>(chunk_idx, merge_chunk_idx);
1059          Assume(chunk_idx != INVALID_SET_IDX);
1060          return chunk_idx;
1061      }
1062  
1063      /** Perform an upward or downward merge sequence on the specified chunk. */
1064      template<bool DownWard>
1065      void MergeSequence(SetIdx chunk_idx) noexcept
1066      {
1067          Assume(m_chunk_idxs[chunk_idx]);
1068          while (true) {
1069              auto merged_chunk_idx = MergeStep<DownWard>(chunk_idx);
1070              if (merged_chunk_idx == INVALID_SET_IDX) break;
1071              chunk_idx = merged_chunk_idx;
1072          }
1073          // Add the chunk to the queue of improvable chunks, if it wasn't already there.
1074          if (!m_suboptimal_idxs[chunk_idx]) {
1075              m_suboptimal_idxs.Set(chunk_idx);
1076              m_suboptimal_chunks.push_back(chunk_idx);
1077          }
1078      }
1079  
1080      /** Split a chunk, and then merge the resulting two chunks to make the graph topological
1081       *  again. */
1082      void Improve(TxIdx parent_idx, TxIdx child_idx) noexcept
1083      {
1084          // Deactivate the specified dependency, splitting it into two new chunks: a top containing
1085          // the parent, and a bottom containing the child. The top should have a higher feerate.
1086          auto [parent_chunk_idx, child_chunk_idx] = Deactivate(parent_idx, child_idx);
1087  
1088          // At this point we have exactly two chunks which may violate topology constraints (the
1089          // parent chunk and child chunk that were produced by deactivation). We can fix
1090          // these using just merge sequences, one upwards and one downwards, avoiding the need for a
1091          // full MakeTopological.
1092          const auto& parent_reachable = m_reachable[parent_chunk_idx].first;
1093          const auto& child_chunk_txn = m_set_info[child_chunk_idx].transactions;
1094          if (parent_reachable.Overlaps(child_chunk_txn)) {
1095              // The parent chunk has a dependency on a transaction in the child chunk. In this case,
1096              // the parent needs to merge back with the child chunk (a self-merge), and no other
1097              // merges are needed. Special-case this, so the overhead of PickMergeCandidate and
1098              // MergeSequence can be avoided.
1099  
1100              // In the self-merge, the roles reverse: the parent chunk (from the split) depends
1101              // on the child chunk, so child_chunk_idx is the "top" and parent_chunk_idx is the
1102              // "bottom" for MergeChunks.
1103              auto merged_chunk_idx = MergeChunks(child_chunk_idx, parent_chunk_idx);
1104              if (!m_suboptimal_idxs[merged_chunk_idx]) {
1105                  m_suboptimal_idxs.Set(merged_chunk_idx);
1106                  m_suboptimal_chunks.push_back(merged_chunk_idx);
1107              }
1108          } else {
1109              // Merge the top chunk with lower-feerate chunks it depends on.
1110              MergeSequence<false>(parent_chunk_idx);
1111              // Merge the bottom chunk with higher-feerate chunks that depend on it.
1112              MergeSequence<true>(child_chunk_idx);
1113          }
1114      }
1115  
1116      /** Determine the next chunk to optimize, or INVALID_SET_IDX if none. */
1117      SetIdx PickChunkToOptimize() noexcept
1118      {
1119          m_cost.PickChunkToOptimizeBegin();
1120          unsigned steps{0};
1121          while (!m_suboptimal_chunks.empty()) {
1122              ++steps;
1123              // Pop an entry from the potentially-suboptimal chunk queue.
1124              SetIdx chunk_idx = m_suboptimal_chunks.front();
1125              Assume(m_suboptimal_idxs[chunk_idx]);
1126              m_suboptimal_idxs.Reset(chunk_idx);
1127              m_suboptimal_chunks.pop_front();
1128              if (m_chunk_idxs[chunk_idx]) {
1129                  m_cost.PickChunkToOptimizeEnd(/*num_steps=*/steps);
1130                  return chunk_idx;
1131              }
1132              // If what was popped is not currently a chunk, continue. This may
1133              // happen when a split chunk merges in Improve() with one or more existing chunks that
1134              // are themselves on the suboptimal queue already.
1135          }
1136          m_cost.PickChunkToOptimizeEnd(/*num_steps=*/steps);
1137          return INVALID_SET_IDX;
1138      }
1139  
1140      /** Find a (parent, child) dependency to deactivate in chunk_idx, or (-1, -1) if none. */
1141      std::pair<TxIdx, TxIdx> PickDependencyToSplit(SetIdx chunk_idx) noexcept
1142      {
1143          m_cost.PickDependencyToSplitBegin();
1144          Assume(m_chunk_idxs[chunk_idx]);
1145          auto& chunk_info = m_set_info[chunk_idx];
1146  
1147          // Remember the best dependency {par, chl} seen so far.
1148          std::pair<TxIdx, TxIdx> candidate_dep = {TxIdx(-1), TxIdx(-1)};
1149          uint64_t candidate_tiebreak = 0;
1150          // Iterate over all transactions.
1151          for (auto tx_idx : chunk_info.transactions) {
1152              const auto& tx_data = m_tx_data[tx_idx];
1153              // Iterate over all active child dependencies of the transaction.
1154              for (auto child_idx : tx_data.active_children) {
1155                  auto& dep_top_info = m_set_info[tx_data.dep_top_idx[child_idx]];
1156                  // Skip if this dependency is ineligible (the top chunk that would be created
1157                  // does not have higher feerate than the chunk it is currently part of).
1158                  auto cmp = ByRatio{dep_top_info.feerate} <=> ByRatio{chunk_info.feerate};
1159                  if (cmp <= 0) continue;
1160                  // Generate a random tiebreak for this dependency, and reject it if its tiebreak
1161                  // is worse than the best so far. This means that among all eligible
1162                  // dependencies, a uniformly random one will be chosen.
1163                  uint64_t tiebreak = m_rng.rand64();
1164                  if (tiebreak < candidate_tiebreak) continue;
1165                  // Remember this as our (new) candidate dependency.
1166                  candidate_dep = {tx_idx, child_idx};
1167                  candidate_tiebreak = tiebreak;
1168              }
1169          }
1170          m_cost.PickDependencyToSplitEnd(/*num_txns=*/chunk_info.transactions.Count());
1171          return candidate_dep;
1172      }
1173  
1174  public:
1175      /** Construct a spanning forest for the given DepGraph, with every transaction in its own chunk
1176       *  (not topological). */
1177      explicit SpanningForestState(const DepGraph<SetType>& depgraph LIFETIMEBOUND, uint64_t rng_seed, const CostModel& cost = CostModel{}) noexcept :
1178          m_rng(rng_seed), m_depgraph(depgraph), m_cost(cost)
1179      {
1180          m_cost.InitializeBegin();
1181          m_transaction_idxs = depgraph.Positions();
1182          auto num_transactions = m_transaction_idxs.Count();
1183          m_tx_data.resize(depgraph.PositionRange());
1184          m_set_info.resize(num_transactions);
1185          m_reachable.resize(num_transactions);
1186          size_t num_chunks = 0;
1187          size_t num_deps = 0;
1188          for (auto tx_idx : m_transaction_idxs) {
1189              // Fill in transaction data.
1190              auto& tx_data = m_tx_data[tx_idx];
1191              tx_data.parents = depgraph.GetReducedParents(tx_idx);
1192              for (auto parent_idx : tx_data.parents) {
1193                  m_tx_data[parent_idx].children.Set(tx_idx);
1194              }
1195              num_deps += tx_data.parents.Count();
1196              // Create a singleton chunk for it.
1197              tx_data.chunk_idx = num_chunks;
1198              m_set_info[num_chunks++] = SetInfo(depgraph, tx_idx);
1199          }
1200          // Set the reachable transactions for each chunk to the transactions' parents and children.
1201          for (SetIdx chunk_idx = 0; chunk_idx < num_transactions; ++chunk_idx) {
1202              auto& tx_data = m_tx_data[m_set_info[chunk_idx].transactions.First()];
1203              m_reachable[chunk_idx].first = tx_data.parents;
1204              m_reachable[chunk_idx].second = tx_data.children;
1205          }
1206          Assume(num_chunks == num_transactions);
1207          // Mark all chunk sets as chunks.
1208          m_chunk_idxs = SetType::Fill(num_chunks);
1209          m_cost.InitializeEnd(/*num_txns=*/num_chunks, /*num_deps=*/num_deps);
1210      }
1211  
1212      /** Load an existing linearization. Must be called immediately after constructor. The result is
1213       *  topological if the linearization is valid. Otherwise, MakeTopological still needs to be
1214       *  called. */
1215      void LoadLinearization(std::span<const DepGraphIndex> old_linearization) noexcept
1216      {
1217          // Add transactions one by one, in order of existing linearization.
1218          for (DepGraphIndex tx_idx : old_linearization) {
1219              auto chunk_idx = m_tx_data[tx_idx].chunk_idx;
1220              // Merge the chunk upwards, as long as merging succeeds.
1221              while (true) {
1222                  chunk_idx = MergeStep<false>(chunk_idx);
1223                  if (chunk_idx == INVALID_SET_IDX) break;
1224              }
1225          }
1226      }
1227  
1228      /** Make state topological. Can be called after constructing, or after LoadLinearization. */
1229      void MakeTopological() noexcept
1230      {
1231          m_cost.MakeTopologicalBegin();
1232          Assume(m_suboptimal_chunks.empty());
1233          /** What direction to initially merge chunks in; one of the two directions is enough. This
1234           *  is sufficient because if a non-topological inactive dependency exists between two
1235           *  chunks, at least one of the two chunks will eventually be processed in a direction that
1236           *  discovers it - either the lower chunk tries upward, or the upper chunk tries downward.
1237           *  Chunks that are the result of the merging are always tried in both directions. */
1238          unsigned init_dir = m_rng.randbool();
1239          /** Which chunks are the result of merging, and thus need merge attempts in both
1240           *  directions. */
1241          SetType merged_chunks;
1242          // Mark chunks as suboptimal.
1243          m_suboptimal_idxs = m_chunk_idxs;
1244          for (auto chunk_idx : m_chunk_idxs) {
1245              m_suboptimal_chunks.emplace_back(chunk_idx);
1246              // Randomize the initial order of suboptimal chunks in the queue.
1247              SetIdx j = m_rng.randrange<SetIdx>(m_suboptimal_chunks.size());
1248              if (j != m_suboptimal_chunks.size() - 1) {
1249                  std::swap(m_suboptimal_chunks.back(), m_suboptimal_chunks[j]);
1250              }
1251          }
1252          unsigned chunks = m_chunk_idxs.Count();
1253          unsigned steps = 0;
1254          while (!m_suboptimal_chunks.empty()) {
1255              ++steps;
1256              // Pop an entry from the potentially-suboptimal chunk queue.
1257              SetIdx chunk_idx = m_suboptimal_chunks.front();
1258              m_suboptimal_chunks.pop_front();
1259              Assume(m_suboptimal_idxs[chunk_idx]);
1260              m_suboptimal_idxs.Reset(chunk_idx);
1261              // If what was popped is not currently a chunk, continue. This may
1262              // happen when it was merged with something else since being added.
1263              if (!m_chunk_idxs[chunk_idx]) continue;
1264              /** What direction(s) to attempt merging in. 1=up, 2=down, 3=both. */
1265              unsigned direction = merged_chunks[chunk_idx] ? 3 : init_dir + 1;
1266              int flip = m_rng.randbool();
1267              for (int i = 0; i < 2; ++i) {
1268                  if (i ^ flip) {
1269                      if (!(direction & 1)) continue;
1270                      // Attempt to merge the chunk upwards.
1271                      auto result_up = MergeStep<false>(chunk_idx);
1272                      if (result_up != INVALID_SET_IDX) {
1273                          if (!m_suboptimal_idxs[result_up]) {
1274                              m_suboptimal_idxs.Set(result_up);
1275                              m_suboptimal_chunks.push_back(result_up);
1276                          }
1277                          merged_chunks.Set(result_up);
1278                          break;
1279                      }
1280                  } else {
1281                      if (!(direction & 2)) continue;
1282                      // Attempt to merge the chunk downwards.
1283                      auto result_down = MergeStep<true>(chunk_idx);
1284                      if (result_down != INVALID_SET_IDX) {
1285                          if (!m_suboptimal_idxs[result_down]) {
1286                              m_suboptimal_idxs.Set(result_down);
1287                              m_suboptimal_chunks.push_back(result_down);
1288                          }
1289                          merged_chunks.Set(result_down);
1290                          break;
1291                      }
1292                  }
1293              }
1294          }
1295          m_cost.MakeTopologicalEnd(/*num_chunks=*/chunks, /*num_steps=*/steps);
1296      }
1297  
1298      /** Initialize the data structure for optimization. It must be topological already. */
1299      void StartOptimizing() noexcept
1300      {
1301          m_cost.StartOptimizingBegin();
1302          Assume(m_suboptimal_chunks.empty());
1303          // Mark chunks suboptimal.
1304          m_suboptimal_idxs = m_chunk_idxs;
1305          for (auto chunk_idx : m_chunk_idxs) {
1306              m_suboptimal_chunks.push_back(chunk_idx);
1307              // Randomize the initial order of suboptimal chunks in the queue.
1308              SetIdx j = m_rng.randrange<SetIdx>(m_suboptimal_chunks.size());
1309              if (j != m_suboptimal_chunks.size() - 1) {
1310                  std::swap(m_suboptimal_chunks.back(), m_suboptimal_chunks[j]);
1311              }
1312          }
1313          m_cost.StartOptimizingEnd(/*num_chunks=*/m_suboptimal_chunks.size());
1314      }
1315  
1316      /** Try to improve the forest. Returns false if it is optimal, true otherwise. */
1317      bool OptimizeStep() noexcept
1318      {
1319          auto chunk_idx = PickChunkToOptimize();
1320          if (chunk_idx == INVALID_SET_IDX) {
1321              // No improvable chunk was found, we are done.
1322              return false;
1323          }
1324          auto [parent_idx, child_idx] = PickDependencyToSplit(chunk_idx);
1325          if (parent_idx == TxIdx(-1)) {
1326              // Nothing to improve in chunk_idx. Need to continue with other chunks, if any.
1327              return !m_suboptimal_chunks.empty();
1328          }
1329          // Deactivate the found dependency and then make the state topological again with a
1330          // sequence of merges.
1331          Improve(parent_idx, child_idx);
1332          return true;
1333      }
1334  
1335      /** Initialize data structure for minimizing the chunks. Can only be called if state is known
1336       *  to be optimal. OptimizeStep() cannot be called anymore afterwards. */
1337      void StartMinimizing() noexcept
1338      {
1339          m_cost.StartMinimizingBegin();
1340          m_nonminimal_chunks.clear();
1341          m_nonminimal_chunks.reserve(m_transaction_idxs.Count());
1342          // Gather all chunks, and for each, add it with a random pivot in it, and a random initial
1343          // direction, to m_nonminimal_chunks.
1344          for (auto chunk_idx : m_chunk_idxs) {
1345              TxIdx pivot_idx = PickRandomTx(m_set_info[chunk_idx].transactions);
1346              m_nonminimal_chunks.emplace_back(chunk_idx, pivot_idx, m_rng.randbits<1>());
1347              // Randomize the initial order of nonminimal chunks in the queue.
1348              SetIdx j = m_rng.randrange<SetIdx>(m_nonminimal_chunks.size());
1349              if (j != m_nonminimal_chunks.size() - 1) {
1350                  std::swap(m_nonminimal_chunks.back(), m_nonminimal_chunks[j]);
1351              }
1352          }
1353          m_cost.StartMinimizingEnd(/*num_chunks=*/m_nonminimal_chunks.size());
1354      }
1355  
1356      /** Try to reduce a chunk's size. Returns false if all chunks are minimal, true otherwise. */
1357      bool MinimizeStep() noexcept
1358      {
1359          // If the queue of potentially-non-minimal chunks is empty, we are done.
1360          if (m_nonminimal_chunks.empty()) return false;
1361          m_cost.MinimizeStepBegin();
1362          // Pop an entry from the potentially-non-minimal chunk queue.
1363          auto [chunk_idx, pivot_idx, flags] = m_nonminimal_chunks.front();
1364          m_nonminimal_chunks.pop_front();
1365          auto& chunk_info = m_set_info[chunk_idx];
1366          /** Whether to move the pivot down rather than up. */
1367          bool move_pivot_down = flags & 1;
1368          /** Whether this is already the second stage. */
1369          bool second_stage = flags & 2;
1370  
1371          // Find a random dependency whose top and bottom set feerates are equal, and which has
1372          // pivot in bottom set (if move_pivot_down) or in top set (if !move_pivot_down).
1373          std::pair<TxIdx, TxIdx> candidate_dep;
1374          uint64_t candidate_tiebreak{0};
1375          bool have_any = false;
1376          // Iterate over all transactions.
1377          for (auto tx_idx : chunk_info.transactions) {
1378              const auto& tx_data = m_tx_data[tx_idx];
1379              // Iterate over all active child dependencies of the transaction.
1380              for (auto child_idx : tx_data.active_children) {
1381                  const auto& dep_top_info = m_set_info[tx_data.dep_top_idx[child_idx]];
1382                  // Skip if this dependency does not have equal top and bottom set feerates. Note
1383                  // that the top cannot have higher feerate than the bottom, or OptimizeSteps would
1384                  // have dealt with it.
1385                  if (ByRatio{dep_top_info.feerate} < ByRatio{chunk_info.feerate}) continue;
1386                  have_any = true;
1387                  // Skip if this dependency does not have pivot in the right place.
1388                  if (move_pivot_down == dep_top_info.transactions[pivot_idx]) continue;
1389                  // Remember this as our chosen dependency if it has a better tiebreak.
1390                  uint64_t tiebreak = m_rng.rand64() | 1;
1391                  if (tiebreak > candidate_tiebreak) {
1392                      candidate_tiebreak = tiebreak;
1393                      candidate_dep = {tx_idx, child_idx};
1394                  }
1395              }
1396          }
1397          m_cost.MinimizeStepMid(/*num_txns=*/chunk_info.transactions.Count());
1398          // If no dependencies have equal top and bottom set feerate, this chunk is minimal.
1399          if (!have_any) return true;
1400          // If all found dependencies have the pivot in the wrong place, try moving it in the other
1401          // direction. If this was the second stage already, we are done.
1402          if (candidate_tiebreak == 0) {
1403              // Switch to other direction, and to second phase.
1404              flags ^= 3;
1405              if (!second_stage) m_nonminimal_chunks.emplace_back(chunk_idx, pivot_idx, flags);
1406              return true;
1407          }
1408  
1409          // Otherwise, deactivate the dependency that was found.
1410          auto [parent_chunk_idx, child_chunk_idx] = Deactivate(candidate_dep.first, candidate_dep.second);
1411          // Determine if there is a dependency from the new bottom to the new top (opposite from the
1412          // dependency that was just deactivated).
1413          auto& parent_reachable = m_reachable[parent_chunk_idx].first;
1414          auto& child_chunk_txn = m_set_info[child_chunk_idx].transactions;
1415          if (parent_reachable.Overlaps(child_chunk_txn)) {
1416              // A self-merge is needed. Note that the child_chunk_idx is the top, and
1417              // parent_chunk_idx is the bottom, because we activate a dependency in the reverse
1418              // direction compared to the deactivation above.
1419              auto merged_chunk_idx = MergeChunks(child_chunk_idx, parent_chunk_idx);
1420              // Re-insert the chunk into the queue, in the same direction. Note that the chunk_idx
1421              // will have changed.
1422              m_nonminimal_chunks.emplace_back(merged_chunk_idx, pivot_idx, flags);
1423              m_cost.MinimizeStepEnd(/*split=*/false);
1424          } else {
1425              // No self-merge happens, and thus we have found a way to split the chunk. Create two
1426              // smaller chunks, and add them to the queue. The one that contains the current pivot
1427              // gets to continue with it in the same direction, to minimize the number of times we
1428              // alternate direction. If we were in the second phase already, the newly created chunk
1429              // inherits that too, because we know no split with the pivot on the other side is
1430              // possible already. The new chunk without the current pivot gets a new randomly-chosen
1431              // one.
1432              if (move_pivot_down) {
1433                  auto parent_pivot_idx = PickRandomTx(m_set_info[parent_chunk_idx].transactions);
1434                  m_nonminimal_chunks.emplace_back(parent_chunk_idx, parent_pivot_idx, m_rng.randbits<1>());
1435                  m_nonminimal_chunks.emplace_back(child_chunk_idx, pivot_idx, flags);
1436              } else {
1437                  auto child_pivot_idx = PickRandomTx(m_set_info[child_chunk_idx].transactions);
1438                  m_nonminimal_chunks.emplace_back(parent_chunk_idx, pivot_idx, flags);
1439                  m_nonminimal_chunks.emplace_back(child_chunk_idx, child_pivot_idx, m_rng.randbits<1>());
1440              }
1441              if (m_rng.randbool()) {
1442                  std::swap(m_nonminimal_chunks.back(), m_nonminimal_chunks[m_nonminimal_chunks.size() - 2]);
1443              }
1444              m_cost.MinimizeStepEnd(/*split=*/true);
1445          }
1446          return true;
1447      }
1448  
1449      /** Construct a topologically-valid linearization from the current forest state. Must be
1450       *  topological. fallback_order is a comparator that defines a strong order for DepGraphIndexes
1451       *  in this cluster, used to order equal-feerate transactions and chunks.
1452       *
1453       * Specifically, the resulting order consists of:
1454       * - The chunks of the current SFL state, sorted by (in decreasing order of priority):
1455       *   - topology (parents before children)
1456       *   - highest chunk feerate first
1457       *   - smallest chunk size first
1458       *   - the chunk with the lowest maximum transaction, by fallback_order, first
1459       * - The transactions within a chunk, sorted by (in decreasing order of priority):
1460       *   - topology (parents before children)
1461       *   - highest tx feerate first
1462       *   - smallest tx size first
1463       *   - the lowest transaction, by fallback_order, first
1464       */
1465      std::vector<DepGraphIndex> GetLinearization(const StrongComparator<DepGraphIndex> auto& fallback_order) noexcept
1466      {
1467          m_cost.GetLinearizationBegin();
1468          /** The output linearization. */
1469          std::vector<DepGraphIndex> ret;
1470          ret.reserve(m_set_info.size());
1471          /** A heap with all chunks (by set index) that can currently be included, sorted by
1472           *  chunk feerate (high to low), chunk size (small to large), and by least maximum element
1473           *  according to the fallback order (which is the second pair element). */
1474          std::vector<std::pair<SetIdx, TxIdx>> ready_chunks;
1475          /** For every chunk, indexed by SetIdx, the number of unmet dependencies the chunk has on
1476           *  other chunks (not including dependencies within the chunk itself). */
1477          std::vector<TxIdx> chunk_deps(m_set_info.size(), 0);
1478          /** For every transaction, indexed by TxIdx, the number of unmet dependencies the
1479           *  transaction has. */
1480          std::vector<TxIdx> tx_deps(m_tx_data.size(), 0);
1481          /** A heap with all transactions within the current chunk that can be included, sorted by
1482           *  tx feerate (high to low), tx size (small to large), and fallback order. */
1483          std::vector<TxIdx> ready_tx;
1484          // Populate chunk_deps and tx_deps.
1485          unsigned num_deps{0};
1486          for (TxIdx chl_idx : m_transaction_idxs) {
1487              const auto& chl_data = m_tx_data[chl_idx];
1488              tx_deps[chl_idx] = chl_data.parents.Count();
1489              num_deps += tx_deps[chl_idx];
1490              auto chl_chunk_idx = chl_data.chunk_idx;
1491              auto& chl_chunk_info = m_set_info[chl_chunk_idx];
1492              chunk_deps[chl_chunk_idx] += (chl_data.parents - chl_chunk_info.transactions).Count();
1493          }
1494          /** Function to compute the highest element of a chunk, by fallback_order. */
1495          auto max_fallback_fn = [&](SetIdx chunk_idx) noexcept {
1496              auto& chunk = m_set_info[chunk_idx].transactions;
1497              auto it = chunk.begin();
1498              DepGraphIndex ret = *it;
1499              ++it;
1500              while (it != chunk.end()) {
1501                  if (fallback_order(*it, ret) > 0) ret = *it;
1502                  ++it;
1503              }
1504              return ret;
1505          };
1506          /** Comparison function for the transaction heap. Note that it is a max-heap, so
1507           *  tx_cmp_fn(a, b) == true means "a appears after b in the linearization". */
1508          auto tx_cmp_fn = [&](const auto& a, const auto& b) noexcept {
1509              // Bail out for identical transactions.
1510              if (a == b) return false;
1511              // First sort by increasing transaction feerate.
1512              auto& a_feerate = m_depgraph.FeeRate(a);
1513              auto& b_feerate = m_depgraph.FeeRate(b);
1514              auto feerate_cmp = ByRatio{a_feerate} <=> ByRatio{b_feerate};
1515              if (feerate_cmp != 0) return feerate_cmp < 0;
1516              // Then by decreasing transaction size.
1517              if (a_feerate.size != b_feerate.size) {
1518                  return a_feerate.size > b_feerate.size;
1519              }
1520              // Tie-break by decreasing fallback_order.
1521              auto fallback_cmp = fallback_order(a, b);
1522              if (fallback_cmp != 0) return fallback_cmp > 0;
1523              // This should not be hit, because fallback_order defines a strong ordering.
1524              Assume(false);
1525              return a < b;
1526          };
1527          // Construct a heap with all chunks that have no out-of-chunk dependencies.
1528          /** Comparison function for the chunk heap. Note that it is a max-heap, so
1529           *  chunk_cmp_fn(a, b) == true means "a appears after b in the linearization". */
1530          auto chunk_cmp_fn = [&](const auto& a, const auto& b) noexcept {
1531              // Bail out for identical chunks.
1532              if (a.first == b.first) return false;
1533              // First sort by increasing chunk feerate.
1534              auto& chunk_feerate_a = m_set_info[a.first].feerate;
1535              auto& chunk_feerate_b = m_set_info[b.first].feerate;
1536              auto feerate_cmp = ByRatio{chunk_feerate_a} <=> ByRatio{chunk_feerate_b};
1537              if (feerate_cmp != 0) return feerate_cmp < 0;
1538              // Then by decreasing chunk size.
1539              if (chunk_feerate_a.size != chunk_feerate_b.size) {
1540                  return chunk_feerate_a.size > chunk_feerate_b.size;
1541              }
1542              // Tie-break by decreasing fallback_order.
1543              auto fallback_cmp = fallback_order(a.second, b.second);
1544              if (fallback_cmp != 0) return fallback_cmp > 0;
1545              // This should not be hit, because fallback_order defines a strong ordering.
1546              Assume(false);
1547              return a.second < b.second;
1548          };
1549          // Construct a heap with all chunks that have no out-of-chunk dependencies.
1550          for (SetIdx chunk_idx : m_chunk_idxs) {
1551              if (chunk_deps[chunk_idx] == 0) {
1552                  ready_chunks.emplace_back(chunk_idx, max_fallback_fn(chunk_idx));
1553              }
1554          }
1555          std::make_heap(ready_chunks.begin(), ready_chunks.end(), chunk_cmp_fn);
1556          // Pop chunks off the heap.
1557          while (!ready_chunks.empty()) {
1558              auto [chunk_idx, _rnd] = ready_chunks.front();
1559              std::pop_heap(ready_chunks.begin(), ready_chunks.end(), chunk_cmp_fn);
1560              ready_chunks.pop_back();
1561              Assume(chunk_deps[chunk_idx] == 0);
1562              const auto& chunk_txn = m_set_info[chunk_idx].transactions;
1563              // Build heap of all includable transactions in chunk.
1564              Assume(ready_tx.empty());
1565              for (TxIdx tx_idx : chunk_txn) {
1566                  if (tx_deps[tx_idx] == 0) ready_tx.push_back(tx_idx);
1567              }
1568              Assume(!ready_tx.empty());
1569              std::make_heap(ready_tx.begin(), ready_tx.end(), tx_cmp_fn);
1570              // Pick transactions from the ready heap, append them to linearization, and decrement
1571              // dependency counts.
1572              while (!ready_tx.empty()) {
1573                  // Pop an element from the tx_ready heap.
1574                  auto tx_idx = ready_tx.front();
1575                  std::pop_heap(ready_tx.begin(), ready_tx.end(), tx_cmp_fn);
1576                  ready_tx.pop_back();
1577                  // Append to linearization.
1578                  ret.push_back(tx_idx);
1579                  // Decrement dependency counts.
1580                  auto& tx_data = m_tx_data[tx_idx];
1581                  for (TxIdx chl_idx : tx_data.children) {
1582                      auto& chl_data = m_tx_data[chl_idx];
1583                      // Decrement tx dependency count.
1584                      Assume(tx_deps[chl_idx] > 0);
1585                      if (--tx_deps[chl_idx] == 0 && chunk_txn[chl_idx]) {
1586                          // Child tx has no dependencies left, and is in this chunk. Add it to the tx heap.
1587                          ready_tx.push_back(chl_idx);
1588                          std::push_heap(ready_tx.begin(), ready_tx.end(), tx_cmp_fn);
1589                      }
1590                      // Decrement chunk dependency count if this is out-of-chunk dependency.
1591                      if (chl_data.chunk_idx != chunk_idx) {
1592                          Assume(chunk_deps[chl_data.chunk_idx] > 0);
1593                          if (--chunk_deps[chl_data.chunk_idx] == 0) {
1594                              // Child chunk has no dependencies left. Add it to the chunk heap.
1595                              ready_chunks.emplace_back(chl_data.chunk_idx, max_fallback_fn(chl_data.chunk_idx));
1596                              std::push_heap(ready_chunks.begin(), ready_chunks.end(), chunk_cmp_fn);
1597                          }
1598                      }
1599                  }
1600              }
1601          }
1602          Assume(ret.size() == m_set_info.size());
1603          m_cost.GetLinearizationEnd(/*num_txns=*/m_set_info.size(), /*num_deps=*/num_deps);
1604          return ret;
1605      }
1606  
1607      /** Get the diagram for the current state, which must be topological. Test-only.
1608       *
1609       * The linearization produced by GetLinearization() is always at least as good (in the
1610       * CompareChunks() sense) as this diagram, but may be better.
1611       *
1612       * After an OptimizeStep(), the diagram will always be at least as good as before. Once
1613       * OptimizeStep() returns false, the diagram will be equivalent to that produced by
1614       * GetLinearization(), and optimal.
1615       *
1616       * After a MinimizeStep(), the diagram cannot change anymore (in the CompareChunks() sense),
1617       * but its number of segments can increase still. Once MinimizeStep() returns false, the number
1618       * of chunks of the produced linearization will match the number of segments in the diagram.
1619       */
1620      std::vector<FeeFrac> GetDiagram() const noexcept
1621      {
1622          std::vector<FeeFrac> ret;
1623          for (auto chunk_idx : m_chunk_idxs) {
1624              ret.push_back(m_set_info[chunk_idx].feerate);
1625          }
1626          std::ranges::sort(ret, std::greater<ByRatioNegSize<FeeFrac>>{});
1627          return ret;
1628      }
1629  
1630      /** Determine how much work was performed so far. */
1631      uint64_t GetCost() const noexcept { return m_cost.GetCost(); }
1632  
1633      /** Verify internal consistency of the data structure. */
1634      void SanityCheck() const
1635      {
1636          //
1637          // Verify dependency parent/child information, and build list of (active) dependencies.
1638          //
1639          std::vector<std::pair<TxIdx, TxIdx>> expected_dependencies;
1640          std::vector<std::pair<TxIdx, TxIdx>> all_dependencies;
1641          std::vector<std::pair<TxIdx, TxIdx>> active_dependencies;
1642          for (auto parent_idx : m_depgraph.Positions()) {
1643              for (auto child_idx : m_depgraph.GetReducedChildren(parent_idx)) {
1644                  expected_dependencies.emplace_back(parent_idx, child_idx);
1645              }
1646          }
1647          for (auto tx_idx : m_transaction_idxs) {
1648              for (auto child_idx : m_tx_data[tx_idx].children) {
1649                  all_dependencies.emplace_back(tx_idx, child_idx);
1650                  if (m_tx_data[tx_idx].active_children[child_idx]) {
1651                      active_dependencies.emplace_back(tx_idx, child_idx);
1652                  }
1653              }
1654          }
1655          std::ranges::sort(expected_dependencies);
1656          std::ranges::sort(all_dependencies);
1657          assert(expected_dependencies == all_dependencies);
1658  
1659          //
1660          // Verify the chunks against the list of active dependencies
1661          //
1662          SetType chunk_cover;
1663          for (auto chunk_idx : m_chunk_idxs) {
1664              const auto& chunk_info = m_set_info[chunk_idx];
1665              // Verify that transactions in the chunk point back to it. This guarantees
1666              // that chunks are non-overlapping.
1667              for (auto tx_idx : chunk_info.transactions) {
1668                  assert(m_tx_data[tx_idx].chunk_idx == chunk_idx);
1669              }
1670              assert(!chunk_cover.Overlaps(chunk_info.transactions));
1671              chunk_cover |= chunk_info.transactions;
1672              // Verify the chunk's transaction set: start from an arbitrary chunk transaction,
1673              // and for every active dependency, if it contains the parent or child, add the
1674              // other. It must have exactly N-1 active dependencies in it, guaranteeing it is
1675              // acyclic.
1676              assert(chunk_info.transactions.Any());
1677              SetType expected_chunk = SetType::Singleton(chunk_info.transactions.First());
1678              while (true) {
1679                  auto old = expected_chunk;
1680                  size_t active_dep_count{0};
1681                  for (const auto& [par, chl] : active_dependencies) {
1682                      if (expected_chunk[par] || expected_chunk[chl]) {
1683                          expected_chunk.Set(par);
1684                          expected_chunk.Set(chl);
1685                          ++active_dep_count;
1686                      }
1687                  }
1688                  if (old == expected_chunk) {
1689                      assert(expected_chunk.Count() == active_dep_count + 1);
1690                      break;
1691                  }
1692              }
1693              assert(chunk_info.transactions == expected_chunk);
1694              // Verify the chunk's feerate.
1695              assert(chunk_info.feerate == m_depgraph.FeeRate(chunk_info.transactions));
1696              // Verify the chunk's reachable transactions.
1697              assert(m_reachable[chunk_idx] == GetReachable(expected_chunk));
1698              // Verify that the chunk's reachable transactions don't include its own transactions.
1699              assert(!m_reachable[chunk_idx].first.Overlaps(chunk_info.transactions));
1700              assert(!m_reachable[chunk_idx].second.Overlaps(chunk_info.transactions));
1701          }
1702          // Verify that together, the chunks cover all transactions.
1703          assert(chunk_cover == m_depgraph.Positions());
1704  
1705          //
1706          // Verify transaction data.
1707          //
1708          assert(m_transaction_idxs == m_depgraph.Positions());
1709          for (auto tx_idx : m_transaction_idxs) {
1710              const auto& tx_data = m_tx_data[tx_idx];
1711              // Verify it has a valid chunk index, and that chunk includes this transaction.
1712              assert(m_chunk_idxs[tx_data.chunk_idx]);
1713              assert(m_set_info[tx_data.chunk_idx].transactions[tx_idx]);
1714              // Verify parents/children.
1715              assert(tx_data.parents == m_depgraph.GetReducedParents(tx_idx));
1716              assert(tx_data.children == m_depgraph.GetReducedChildren(tx_idx));
1717              // Verify active_children is a subset of children.
1718              assert(tx_data.active_children.IsSubsetOf(tx_data.children));
1719              // Verify each active child's dep_top_idx points to a valid non-chunk set.
1720              for (auto child_idx : tx_data.active_children) {
1721                  assert(tx_data.dep_top_idx[child_idx] < m_set_info.size());
1722                  assert(!m_chunk_idxs[tx_data.dep_top_idx[child_idx]]);
1723              }
1724          }
1725  
1726          //
1727          // Verify active dependencies' top sets.
1728          //
1729          for (const auto& [par_idx, chl_idx] : active_dependencies) {
1730              // Verify the top set's transactions: it must contain the parent, and for every
1731              // active dependency, except the chl_idx->par_idx dependency itself, if it contains the
1732              // parent or child, it must contain both. It must have exactly N-1 active dependencies
1733              // in it, guaranteeing it is acyclic.
1734              SetType expected_top = SetType::Singleton(par_idx);
1735              while (true) {
1736                  auto old = expected_top;
1737                  size_t active_dep_count{0};
1738                  for (const auto& [par2_idx, chl2_idx] : active_dependencies) {
1739                      if (par_idx == par2_idx && chl_idx == chl2_idx) continue;
1740                      if (expected_top[par2_idx] || expected_top[chl2_idx]) {
1741                          expected_top.Set(par2_idx);
1742                          expected_top.Set(chl2_idx);
1743                          ++active_dep_count;
1744                      }
1745                  }
1746                  if (old == expected_top) {
1747                      assert(expected_top.Count() == active_dep_count + 1);
1748                      break;
1749                  }
1750              }
1751              assert(!expected_top[chl_idx]);
1752              auto& dep_top_info = m_set_info[m_tx_data[par_idx].dep_top_idx[chl_idx]];
1753              assert(dep_top_info.transactions == expected_top);
1754              // Verify the top set's feerate.
1755              assert(dep_top_info.feerate == m_depgraph.FeeRate(dep_top_info.transactions));
1756          }
1757  
1758          //
1759          // Verify m_suboptimal_chunks.
1760          //
1761          SetType suboptimal_idxs;
1762          for (size_t i = 0; i < m_suboptimal_chunks.size(); ++i) {
1763              auto chunk_idx = m_suboptimal_chunks[i];
1764              assert(!suboptimal_idxs[chunk_idx]);
1765              suboptimal_idxs.Set(chunk_idx);
1766          }
1767          assert(m_suboptimal_idxs == suboptimal_idxs);
1768  
1769          //
1770          // Verify m_nonminimal_chunks.
1771          //
1772          SetType nonminimal_idxs;
1773          for (size_t i = 0; i < m_nonminimal_chunks.size(); ++i) {
1774              auto [chunk_idx, pivot, flags] = m_nonminimal_chunks[i];
1775              assert(m_tx_data[pivot].chunk_idx == chunk_idx);
1776              assert(!nonminimal_idxs[chunk_idx]);
1777              nonminimal_idxs.Set(chunk_idx);
1778          }
1779          assert(nonminimal_idxs.IsSubsetOf(m_chunk_idxs));
1780      }
1781  };
1782  
1783  /** Find or improve a linearization for a cluster.
1784   *
1785   * @param[in] depgraph            Dependency graph of the cluster to be linearized.
1786   * @param[in] max_cost            Upper bound on the amount of work that will be done.
1787   * @param[in] rng_seed            A random number seed to control search order. This prevents peers
1788   *                                from predicting exactly which clusters would be hard for us to
1789   *                                linearize.
1790   * @param[in] fallback_order      A comparator to order transactions, used to sort equal-feerate
1791   *                                chunks and transactions. See SpanningForestState::GetLinearization
1792   *                                for details.
1793   * @param[in] old_linearization   An existing linearization for the cluster, or empty.
1794   * @param[in] is_topological      (Only relevant if old_linearization is not empty) Whether
1795   *                                old_linearization is topologically valid.
1796   * @return                        A tuple of:
1797   *                                - The resulting linearization. It is guaranteed to be at least as
1798   *                                  good (in the feerate diagram sense) as old_linearization.
1799   *                                - A boolean indicating whether the result is guaranteed to be
1800   *                                  optimal with minimal chunks.
1801   *                                - How many optimization steps were actually performed.
1802   */
1803  template<typename SetType>
1804  std::tuple<std::vector<DepGraphIndex>, bool, uint64_t> Linearize(
1805      const DepGraph<SetType>& depgraph,
1806      uint64_t max_cost,
1807      uint64_t rng_seed,
1808      const StrongComparator<DepGraphIndex> auto& fallback_order,
1809      std::span<const DepGraphIndex> old_linearization = {},
1810      bool is_topological = true) noexcept
1811  {
1812      /** Initialize a spanning forest data structure for this cluster. */
1813      SpanningForestState forest(depgraph, rng_seed);
1814      if (!old_linearization.empty()) {
1815          forest.LoadLinearization(old_linearization);
1816          if (!is_topological) forest.MakeTopological();
1817      } else {
1818          forest.MakeTopological();
1819      }
1820      // Make improvement steps to it until we hit the max_iterations limit, or an optimal result
1821      // is found.
1822      if (forest.GetCost() < max_cost) {
1823          forest.StartOptimizing();
1824          do {
1825              if (!forest.OptimizeStep()) break;
1826          } while (forest.GetCost() < max_cost);
1827      }
1828      // Make chunk minimization steps until we hit the max_iterations limit, or all chunks are
1829      // minimal.
1830      bool optimal = false;
1831      if (forest.GetCost() < max_cost) {
1832          forest.StartMinimizing();
1833          do {
1834              if (!forest.MinimizeStep()) {
1835                  optimal = true;
1836                  break;
1837              }
1838          } while (forest.GetCost() < max_cost);
1839      }
1840      return {forest.GetLinearization(fallback_order), optimal, forest.GetCost()};
1841  }
1842  
1843  /** Improve a given linearization.
1844   *
1845   * @param[in]     depgraph       Dependency graph of the cluster being linearized.
1846   * @param[in,out] linearization  On input, an existing linearization for depgraph. On output, a
1847   *                               potentially better linearization for the same graph.
1848   *
1849   * Postlinearization guarantees:
1850   * - The resulting chunks are connected.
1851   * - If the input has a tree shape (either all transactions have at most one child, or all
1852   *   transactions have at most one parent), the result is optimal.
1853   * - Given a linearization L1 and a leaf transaction T in it. Let L2 be L1 with T moved to the end,
1854   *   optionally with its fee increased. Let L3 be the postlinearization of L2. L3 will be at least
1855   *   as good as L1. This means that replacing transactions with same-size higher-fee transactions
1856   *   will not worsen linearizations through a "drop conflicts, append new transactions,
1857   *   postlinearize" process.
1858   */
1859  template<typename SetType>
1860  void PostLinearize(const DepGraph<SetType>& depgraph, std::span<DepGraphIndex> linearization)
1861  {
1862      // This algorithm performs a number of passes (currently 2); the even ones operate from back to
1863      // front, the odd ones from front to back. Each results in an equal-or-better linearization
1864      // than the one started from.
1865      // - One pass in either direction guarantees that the resulting chunks are connected.
1866      // - Each direction corresponds to one shape of tree being linearized optimally (forward passes
1867      //   guarantee this for graphs where each transaction has at most one child; backward passes
1868      //   guarantee this for graphs where each transaction has at most one parent).
1869      // - Starting with a backward pass guarantees the moved-tree property.
1870      //
1871      // During an odd (forward) pass, the high-level operation is:
1872      // - Start with an empty list of groups L=[].
1873      // - For every transaction i in the old linearization, from front to back:
1874      //   - Append a new group C=[i], containing just i, to the back of L.
1875      //   - While L has at least one group before C, and the group immediately before C has feerate
1876      //     lower than C:
1877      //     - If C depends on P:
1878      //       - Merge P into C, making C the concatenation of P+C, continuing with the combined C.
1879      //     - Otherwise:
1880      //       - Swap P with C, continuing with the now-moved C.
1881      // - The output linearization is the concatenation of the groups in L.
1882      //
1883      // During even (backward) passes, i iterates from the back to the front of the existing
1884      // linearization, and new groups are prepended instead of appended to the list L. To enable
1885      // more code reuse, both passes append groups, but during even passes the meanings of
1886      // parent/child, and of high/low feerate are reversed, and the final concatenation is reversed
1887      // on output.
1888      //
1889      // In the implementation below, the groups are represented by singly-linked lists (pointing
1890      // from the back to the front), which are themselves organized in a singly-linked circular
1891      // list (each group pointing to its predecessor, with a special sentinel group at the front
1892      // that points back to the last group).
1893      //
1894      // Information about transaction t is stored in entries[t + 1], while the sentinel is in
1895      // entries[0].
1896  
1897      /** Index of the sentinel in the entries array below. */
1898      static constexpr DepGraphIndex SENTINEL{0};
1899      /** Indicator that a group has no previous transaction. */
1900      static constexpr DepGraphIndex NO_PREV_TX{0};
1901  
1902  
1903      /** Data structure per transaction entry. */
1904      struct TxEntry
1905      {
1906          /** The index of the previous transaction in this group; NO_PREV_TX if this is the first
1907           *  entry of a group. */
1908          DepGraphIndex prev_tx;
1909  
1910          // The fields below are only used for transactions that are the last one in a group
1911          // (referred to as tail transactions below).
1912  
1913          /** Index of the first transaction in this group, possibly itself. */
1914          DepGraphIndex first_tx;
1915          /** Index of the last transaction in the previous group. The first group (the sentinel)
1916           *  points back to the last group here, making it a singly-linked circular list. */
1917          DepGraphIndex prev_group;
1918          /** All transactions in the group. Empty for the sentinel. */
1919          SetType group;
1920          /** All dependencies of the group (descendants in even passes; ancestors in odd ones). */
1921          SetType deps;
1922          /** The combined fee/size of transactions in the group. Fee is negated in even passes. */
1923          FeeFrac feerate;
1924      };
1925  
1926      // As an example, consider the state corresponding to the linearization [1,0,3,2], with
1927      // groups [1,0,3] and [2], in an odd pass. The linked lists would be:
1928      //
1929      //                                        +-----+
1930      //                                 0<-P-- | 0 S | ---\     Legend:
1931      //                                        +-----+    |
1932      //                                           ^       |     - digit in box: entries index
1933      //             /--------------F---------+    G       |       (note: one more than tx value)
1934      //             v                         \   |       |     - S: sentinel group
1935      //          +-----+        +-----+        +-----+    |          (empty feerate)
1936      //   0<-P-- | 2   | <--P-- | 1   | <--P-- | 4 T |    |     - T: tail transaction, contains
1937      //          +-----+        +-----+        +-----+    |          fields beyond prev_tv.
1938      //                                           ^       |     - P: prev_tx reference
1939      //                                           G       G     - F: first_tx reference
1940      //                                           |       |     - G: prev_group reference
1941      //                                        +-----+    |
1942      //                                 0<-P-- | 3 T | <--/
1943      //                                        +-----+
1944      //                                         ^   |
1945      //                                         \-F-/
1946      //
1947      // During an even pass, the diagram above would correspond to linearization [2,3,0,1], with
1948      // groups [2] and [3,0,1].
1949  
1950      std::vector<TxEntry> entries(depgraph.PositionRange() + 1);
1951  
1952      // Perform two passes over the linearization.
1953      for (int pass = 0; pass < 2; ++pass) {
1954          int rev = !(pass & 1);
1955          // Construct a sentinel group, identifying the start of the list.
1956          entries[SENTINEL].prev_group = SENTINEL;
1957          Assume(entries[SENTINEL].feerate.IsEmpty());
1958  
1959          // Iterate over all elements in the existing linearization.
1960          for (DepGraphIndex i = 0; i < linearization.size(); ++i) {
1961              // Even passes are from back to front; odd passes from front to back.
1962              DepGraphIndex idx = linearization[rev ? linearization.size() - 1 - i : i];
1963              // Construct a new group containing just idx. In even passes, the meaning of
1964              // parent/child and high/low feerate are swapped.
1965              DepGraphIndex cur_group = idx + 1;
1966              entries[cur_group].group = SetType::Singleton(idx);
1967              entries[cur_group].deps = rev ? depgraph.Descendants(idx): depgraph.Ancestors(idx);
1968              entries[cur_group].feerate = depgraph.FeeRate(idx);
1969              if (rev) entries[cur_group].feerate.fee = -entries[cur_group].feerate.fee;
1970              entries[cur_group].prev_tx = NO_PREV_TX; // No previous transaction in group.
1971              entries[cur_group].first_tx = cur_group; // Transaction itself is first of group.
1972              // Insert the new group at the back of the groups linked list.
1973              entries[cur_group].prev_group = entries[SENTINEL].prev_group;
1974              entries[SENTINEL].prev_group = cur_group;
1975  
1976              // Start merge/swap cycle.
1977              DepGraphIndex next_group = SENTINEL; // We inserted at the end, so next group is sentinel.
1978              DepGraphIndex prev_group = entries[cur_group].prev_group;
1979              // Continue as long as the current group has higher feerate than the previous one.
1980              while (ByRatio{entries[cur_group].feerate} > ByRatio{entries[prev_group].feerate}) {
1981                  // prev_group/cur_group/next_group refer to (the last transactions of) 3
1982                  // consecutive entries in groups list.
1983                  Assume(cur_group == entries[next_group].prev_group);
1984                  Assume(prev_group == entries[cur_group].prev_group);
1985                  // The sentinel has empty feerate, which is neither higher or lower than other
1986                  // feerates. Thus, the while loop we are in here guarantees that cur_group and
1987                  // prev_group are not the sentinel.
1988                  Assume(cur_group != SENTINEL);
1989                  Assume(prev_group != SENTINEL);
1990                  if (entries[cur_group].deps.Overlaps(entries[prev_group].group)) {
1991                      // There is a dependency between cur_group and prev_group; merge prev_group
1992                      // into cur_group. The group/deps/feerate fields of prev_group remain unchanged
1993                      // but become unused.
1994                      entries[cur_group].group |= entries[prev_group].group;
1995                      entries[cur_group].deps |= entries[prev_group].deps;
1996                      entries[cur_group].feerate += entries[prev_group].feerate;
1997                      // Make the first of the current group point to the tail of the previous group.
1998                      entries[entries[cur_group].first_tx].prev_tx = prev_group;
1999                      // The first of the previous group becomes the first of the newly-merged group.
2000                      entries[cur_group].first_tx = entries[prev_group].first_tx;
2001                      // The previous group becomes whatever group was before the former one.
2002                      prev_group = entries[prev_group].prev_group;
2003                      entries[cur_group].prev_group = prev_group;
2004                  } else {
2005                      // There is no dependency between cur_group and prev_group; swap them.
2006                      DepGraphIndex preprev_group = entries[prev_group].prev_group;
2007                      // If PP, P, C, N were the old preprev, prev, cur, next groups, then the new
2008                      // layout becomes [PP, C, P, N]. Update prev_groups to reflect that order.
2009                      entries[next_group].prev_group = prev_group;
2010                      entries[prev_group].prev_group = cur_group;
2011                      entries[cur_group].prev_group = preprev_group;
2012                      // The current group remains the same, but the groups before/after it have
2013                      // changed.
2014                      next_group = prev_group;
2015                      prev_group = preprev_group;
2016                  }
2017              }
2018          }
2019  
2020          // Convert the entries back to linearization (overwriting the existing one).
2021          DepGraphIndex cur_group = entries[0].prev_group;
2022          DepGraphIndex done = 0;
2023          while (cur_group != SENTINEL) {
2024              DepGraphIndex cur_tx = cur_group;
2025              // Traverse the transactions of cur_group (from back to front), and write them in the
2026              // same order during odd passes, and reversed (front to back) in even passes.
2027              if (rev) {
2028                  do {
2029                      *(linearization.begin() + (done++)) = cur_tx - 1;
2030                      cur_tx = entries[cur_tx].prev_tx;
2031                  } while (cur_tx != NO_PREV_TX);
2032              } else {
2033                  do {
2034                      *(linearization.end() - (++done)) = cur_tx - 1;
2035                      cur_tx = entries[cur_tx].prev_tx;
2036                  } while (cur_tx != NO_PREV_TX);
2037              }
2038              cur_group = entries[cur_group].prev_group;
2039          }
2040          Assume(done == linearization.size());
2041      }
2042  }
2043  
2044  } // namespace cluster_linearize
2045  
2046  #endif // BITCOIN_CLUSTER_LINEARIZE_H