cluster_linearize.h

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// Copyright (c) The Bitcoin Core developers
// Distributed under the MIT software license, see the accompanying
// file COPYING or http://www.opensource.org/licenses/mit-license.php.
 
#ifndef BITCOIN_CLUSTER_LINEARIZE_H
#define BITCOIN_CLUSTER_LINEARIZE_H
 
#include <algorithm>
#include <cstdint>
#include <numeric>
#include <optional>
#include <utility>
#include <vector>
 
#include <memusage.h>
#include <random.h>
#include <span.h>
#include <util/feefrac.h>
#include <util/vecdeque.h>
 
namespace cluster_linearize {
 
using DepGraphIndex = uint32_t;
 
template<typename SetType>
class DepGraph
{
    struct Entry
    {
        FeeFrac feerate;
        SetType ancestors;
        SetType descendants;
 
        friend bool operator==(const Entry&, const Entry&) noexcept = default;
 
        Entry() noexcept = default;
        Entry(const FeeFrac& f, const SetType& a, const SetType& d) noexcept : feerate(f), ancestors(a), descendants(d) {}
    };
 
    std::vector<Entry> entries;
 
    SetType m_used;
 
public:
    friend bool operator==(const DepGraph& a, const DepGraph& b) noexcept
    {
        if (a.m_used != b.m_used) return false;
        // Only compare the used positions within the entries vector.
        for (auto idx : a.m_used) {
            if (a.entries[idx] != b.entries[idx]) return false;
        }
        return true;
    }
 
    // Default constructors.
    DepGraph() noexcept = default;
    DepGraph(const DepGraph&) noexcept = default;
    DepGraph(DepGraph&&) noexcept = default;
    DepGraph& operator=(const DepGraph&) noexcept = default;
    DepGraph& operator=(DepGraph&&) noexcept = default;
 
    DepGraph(const DepGraph<SetType>& depgraph, std::span<const DepGraphIndex> mapping, DepGraphIndex pos_range) noexcept : entries(pos_range)
    {
        Assume(mapping.size() == depgraph.PositionRange());
        Assume((pos_range == 0) == (depgraph.TxCount() == 0));
        for (DepGraphIndex i : depgraph.Positions()) {
            auto new_idx = mapping[i];
            Assume(new_idx < pos_range);
            // Add transaction.
            entries[new_idx].ancestors = SetType::Singleton(new_idx);
            entries[new_idx].descendants = SetType::Singleton(new_idx);
            m_used.Set(new_idx);
            // Fill in fee and size.
            entries[new_idx].feerate = depgraph.entries[i].feerate;
        }
        for (DepGraphIndex i : depgraph.Positions()) {
            // Fill in dependencies by mapping direct parents.
            SetType parents;
            for (auto j : depgraph.GetReducedParents(i)) parents.Set(mapping[j]);
            AddDependencies(parents, mapping[i]);
        }
        // Verify that the provided pos_range was correct (no unused positions at the end).
        Assume(m_used.None() ? (pos_range == 0) : (pos_range == m_used.Last() + 1));
    }
 
    const SetType& Positions() const noexcept { return m_used; }
    DepGraphIndex PositionRange() const noexcept { return entries.size(); }
    auto TxCount() const noexcept { return m_used.Count(); }
    const FeeFrac& FeeRate(DepGraphIndex i) const noexcept { return entries[i].feerate; }
    FeeFrac& FeeRate(DepGraphIndex i) noexcept { return entries[i].feerate; }
    const SetType& Ancestors(DepGraphIndex i) const noexcept { return entries[i].ancestors; }
    const SetType& Descendants(DepGraphIndex i) const noexcept { return entries[i].descendants; }
 
    DepGraphIndex AddTransaction(const FeeFrac& feefrac) noexcept
    {
        static constexpr auto ALL_POSITIONS = SetType::Fill(SetType::Size());
        auto available = ALL_POSITIONS - m_used;
        Assume(available.Any());
        DepGraphIndex new_idx = available.First();
        if (new_idx == entries.size()) {
            entries.emplace_back(feefrac, SetType::Singleton(new_idx), SetType::Singleton(new_idx));
        } else {
            entries[new_idx] = Entry(feefrac, SetType::Singleton(new_idx), SetType::Singleton(new_idx));
        }
        m_used.Set(new_idx);
        return new_idx;
    }
 
    void RemoveTransactions(const SetType& del) noexcept
    {
        m_used -= del;
        // Remove now-unused trailing entries.
        while (!entries.empty() && !m_used[entries.size() - 1]) {
            entries.pop_back();
        }
        // Remove the deleted transactions from ancestors/descendants of other transactions. Note
        // that the deleted positions will retain old feerate and dependency information. This does
        // not matter as they will be overwritten by AddTransaction if they get used again.
        for (auto& entry : entries) {
            entry.ancestors &= m_used;
            entry.descendants &= m_used;
        }
    }
 
    void AddDependencies(const SetType& parents, DepGraphIndex child) noexcept
    {
        Assume(m_used[child]);
        Assume(parents.IsSubsetOf(m_used));
        // Compute the ancestors of parents that are not already ancestors of child.
        SetType par_anc;
        for (auto par : parents - Ancestors(child)) {
            par_anc |= Ancestors(par);
        }
        par_anc -= Ancestors(child);
        // Bail out if there are no such ancestors.
        if (par_anc.None()) return;
        // To each such ancestor, add as descendants the descendants of the child.
        const auto& chl_des = entries[child].descendants;
        for (auto anc_of_par : par_anc) {
            entries[anc_of_par].descendants |= chl_des;
        }
        // To each descendant of the child, add those ancestors.
        for (auto dec_of_chl : Descendants(child)) {
            entries[dec_of_chl].ancestors |= par_anc;
        }
    }
 
    SetType GetReducedParents(DepGraphIndex i) const noexcept
    {
        SetType parents = Ancestors(i);
        parents.Reset(i);
        for (auto parent : parents) {
            if (parents[parent]) {
                parents -= Ancestors(parent);
                parents.Set(parent);
            }
        }
        return parents;
    }
 
    SetType GetReducedChildren(DepGraphIndex i) const noexcept
    {
        SetType children = Descendants(i);
        children.Reset(i);
        for (auto child : children) {
            if (children[child]) {
                children -= Descendants(child);
                children.Set(child);
            }
        }
        return children;
    }
 
    FeeFrac FeeRate(const SetType& elems) const noexcept
    {
        FeeFrac ret;
        for (auto pos : elems) ret += entries[pos].feerate;
        return ret;
    }
 
    SetType GetConnectedComponent(const SetType& todo, DepGraphIndex tx) const noexcept
    {
        Assume(todo[tx]);
        Assume(todo.IsSubsetOf(m_used));
        auto to_add = SetType::Singleton(tx);
        SetType ret;
        do {
            SetType old = ret;
            for (auto add : to_add) {
                ret |= Descendants(add);
                ret |= Ancestors(add);
            }
            ret &= todo;
            to_add = ret - old;
        } while (to_add.Any());
        return ret;
    }
 
    SetType FindConnectedComponent(const SetType& todo) const noexcept
    {
        if (todo.None()) return todo;
        return GetConnectedComponent(todo, todo.First());
    }
 
    bool IsConnected(const SetType& subset) const noexcept
    {
        return FindConnectedComponent(subset) == subset;
    }
 
    bool IsConnected() const noexcept { return IsConnected(m_used); }
 
    void AppendTopo(std::vector<DepGraphIndex>& list, const SetType& select) const noexcept
    {
        DepGraphIndex old_len = list.size();
        for (auto i : select) list.push_back(i);
        std::sort(list.begin() + old_len, list.end(), [&](DepGraphIndex a, DepGraphIndex b) noexcept {
            const auto a_anc_count = entries[a].ancestors.Count();
            const auto b_anc_count = entries[b].ancestors.Count();
            if (a_anc_count != b_anc_count) return a_anc_count < b_anc_count;
            return a < b;
        });
    }
 
    bool IsAcyclic() const noexcept
    {
        for (auto i : Positions()) {
            if ((Ancestors(i) & Descendants(i)) != SetType::Singleton(i)) {
                return false;
            }
        }
        return true;
    }
 
    unsigned CountDependencies() const noexcept
    {
        unsigned ret = 0;
        for (auto i : Positions()) {
            ret += GetReducedParents(i).Count();
        }
        return ret;
    }
 
    void Compact() noexcept
    {
        entries.shrink_to_fit();
    }
 
    size_t DynamicMemoryUsage() const noexcept
    {
        return memusage::DynamicUsage(entries);
    }
};
 
template<typename SetType>
struct SetInfo
{
    SetType transactions;
    FeeFrac feerate;
 
    SetInfo() noexcept = default;
 
    SetInfo(const SetType& txn, const FeeFrac& fr) noexcept : transactions(txn), feerate(fr) {}
 
    explicit SetInfo(const DepGraph<SetType>& depgraph, DepGraphIndex pos) noexcept :
        transactions(SetType::Singleton(pos)), feerate(depgraph.FeeRate(pos)) {}
 
    explicit SetInfo(const DepGraph<SetType>& depgraph, const SetType& txn) noexcept :
        transactions(txn), feerate(depgraph.FeeRate(txn)) {}
 
    void Set(const DepGraph<SetType>& depgraph, DepGraphIndex pos) noexcept
    {
        Assume(!transactions[pos]);
        transactions.Set(pos);
        feerate += depgraph.FeeRate(pos);
    }
 
    SetInfo& operator|=(const SetInfo& other) noexcept
    {
        Assume(!transactions.Overlaps(other.transactions));
        transactions |= other.transactions;
        feerate += other.feerate;
        return *this;
    }
 
    SetInfo& operator-=(const SetInfo& other) noexcept
    {
        Assume(other.transactions.IsSubsetOf(transactions));
        transactions -= other.transactions;
        feerate -= other.feerate;
        return *this;
    }
 
    SetInfo operator-(const SetInfo& other) const noexcept
    {
        Assume(other.transactions.IsSubsetOf(transactions));
        return {transactions - other.transactions, feerate - other.feerate};
    }
 
    friend void swap(SetInfo& a, SetInfo& b) noexcept
    {
        swap(a.transactions, b.transactions);
        swap(a.feerate, b.feerate);
    }
 
    friend bool operator==(const SetInfo&, const SetInfo&) noexcept = default;
};
 
template<typename SetType>
std::vector<SetInfo<SetType>> ChunkLinearizationInfo(const DepGraph<SetType>& depgraph, std::span<const DepGraphIndex> linearization) noexcept
{
    std::vector<SetInfo<SetType>> ret;
    for (DepGraphIndex i : linearization) {
        SetInfo<SetType> new_chunk(depgraph, i);
        // As long as the new chunk has a higher feerate than the last chunk so far, absorb it.
        while (!ret.empty() && new_chunk.feerate >> ret.back().feerate) {
            new_chunk |= ret.back();
            ret.pop_back();
        }
        // Actually move that new chunk into the chunking.
        ret.emplace_back(std::move(new_chunk));
    }
    return ret;
}
 
template<typename SetType>
std::vector<FeeFrac> ChunkLinearization(const DepGraph<SetType>& depgraph, std::span<const DepGraphIndex> linearization) noexcept
{
    std::vector<FeeFrac> ret;
    for (DepGraphIndex i : linearization) {
        auto new_chunk = depgraph.FeeRate(i);
        // As long as the new chunk has a higher feerate than the last chunk so far, absorb it.
        while (!ret.empty() && new_chunk >> ret.back()) {
            new_chunk += ret.back();
            ret.pop_back();
        }
        // Actually move that new chunk into the chunking.
        ret.push_back(std::move(new_chunk));
    }
    return ret;
}
 
template<typename SetType>
class SpanningForestState
{
private:
    InsecureRandomContext m_rng;
 
    using TxIdx = uint32_t;
    using DepIdx = uint32_t;
 
    struct TxData {
        std::vector<DepIdx> child_deps;
        SetType parents;
        SetType children;
        TxIdx chunk_rep;
        SetInfo<SetType> chunk_setinfo;
    };
 
    struct DepData {
        bool active;
        TxIdx parent, child;
        SetInfo<SetType> top_setinfo;
    };
 
    SetType m_transaction_idxs;
    std::vector<TxData> m_tx_data;
    std::vector<DepData> m_dep_data;
    VecDeque<TxIdx> m_suboptimal_chunks;
    VecDeque<std::tuple<TxIdx, TxIdx, unsigned>> m_nonminimal_chunks;
 
    uint64_t m_cost{0};
 
    TxIdx PickRandomTx(const SetType& tx_idxs) noexcept
    {
        Assume(tx_idxs.Any());
        unsigned pos = m_rng.randrange<unsigned>(tx_idxs.Count());
        for (auto tx_idx : tx_idxs) {
            if (pos == 0) return tx_idx;
            --pos;
        }
        Assume(false);
        return TxIdx(-1);
    }
 
    template<bool Subtract>
    void UpdateChunk(const SetType& chunk, TxIdx query, TxIdx chunk_rep, const SetInfo<SetType>& dep_change) noexcept
    {
        // Iterate over all the chunk's transactions.
        for (auto tx_idx : chunk) {
            auto& tx_data = m_tx_data[tx_idx];
            // Update the chunk representative.
            tx_data.chunk_rep = chunk_rep;
            // Iterate over all active dependencies with tx_idx as parent. Combined with the outer
            // loop this iterates over all internal active dependencies of the chunk.
            auto child_deps = std::span{tx_data.child_deps};
            for (auto dep_idx : child_deps) {
                auto& dep_entry = m_dep_data[dep_idx];
                Assume(dep_entry.parent == tx_idx);
                // Skip inactive dependencies.
                if (!dep_entry.active) continue;
                // If this dependency's top_setinfo contains query, update it to add/remove
                // dep_change.
                if (dep_entry.top_setinfo.transactions[query]) {
                    if constexpr (Subtract) {
                        dep_entry.top_setinfo -= dep_change;
                    } else {
                        dep_entry.top_setinfo |= dep_change;
                    }
                }
            }
        }
    }
 
    TxIdx Activate(DepIdx dep_idx) noexcept
    {
        auto& dep_data = m_dep_data[dep_idx];
        Assume(!dep_data.active);
        auto& child_tx_data = m_tx_data[dep_data.child];
        auto& parent_tx_data = m_tx_data[dep_data.parent];
 
        // Gather information about the parent and child chunks.
        Assume(parent_tx_data.chunk_rep != child_tx_data.chunk_rep);
        auto& par_chunk_data = m_tx_data[parent_tx_data.chunk_rep];
        auto& chl_chunk_data = m_tx_data[child_tx_data.chunk_rep];
        TxIdx top_rep = parent_tx_data.chunk_rep;
        auto top_part = par_chunk_data.chunk_setinfo;
        auto bottom_part = chl_chunk_data.chunk_setinfo;
        // Update the parent chunk to also contain the child.
        par_chunk_data.chunk_setinfo |= bottom_part;
        m_cost += par_chunk_data.chunk_setinfo.transactions.Count();
 
        // Consider the following example:
        //
        //    A           A     There are two chunks, ABC and DEF, and the inactive E->C dependency
        //   / \         / \    is activated, resulting in a single chunk ABCDEF.
        //  B   C       B   C
        //      :  ==>      |   Dependency | top set before | top set after | change
        //  D   E       D   E   B->A       | AC             | ACDEF         | +DEF
        //   \ /         \ /    C->A       | AB             | AB            |
        //    F           F     F->D       | D              | D             |
        //                      F->E       | E              | ABCE          | +ABC
        //
        // The common pattern here is that any dependency which has the parent or child of the
        // dependency being activated (E->C here) in its top set, will have the opposite part added
        // to it. This is true for B->A and F->E, but not for C->A and F->D.
        //
        // Let UpdateChunk traverse the old parent chunk top_part (ABC in example), and add
        // bottom_part (DEF) to every dependency's top_set which has the parent (C) in it. The
        // representative of each of these transactions was already top_rep, so that is not being
        // changed here.
        UpdateChunk<false>(/*chunk=*/top_part.transactions, /*query=*/dep_data.parent,
                           /*chunk_rep=*/top_rep, /*dep_change=*/bottom_part);
        // Let UpdateChunk traverse the old child chunk bottom_part (DEF in example), and add
        // top_part (ABC) to every dependency's top_set which has the child (E) in it. At the same
        // time, change the representative of each of these transactions to be top_rep, which
        // becomes the representative for the merged chunk.
        UpdateChunk<false>(/*chunk=*/bottom_part.transactions, /*query=*/dep_data.child,
                           /*chunk_rep=*/top_rep, /*dep_change=*/top_part);
        // Make active.
        dep_data.active = true;
        dep_data.top_setinfo = top_part;
        return top_rep;
    }
 
    void Deactivate(DepIdx dep_idx) noexcept
    {
        auto& dep_data = m_dep_data[dep_idx];
        Assume(dep_data.active);
        auto& parent_tx_data = m_tx_data[dep_data.parent];
        // Make inactive.
        dep_data.active = false;
        // Update representatives.
        auto& chunk_data = m_tx_data[parent_tx_data.chunk_rep];
        m_cost += chunk_data.chunk_setinfo.transactions.Count();
        auto top_part = dep_data.top_setinfo;
        auto bottom_part = chunk_data.chunk_setinfo - top_part;
        TxIdx bottom_rep = dep_data.child;
        auto& bottom_chunk_data = m_tx_data[bottom_rep];
        bottom_chunk_data.chunk_setinfo = bottom_part;
        TxIdx top_rep = dep_data.parent;
        auto& top_chunk_data = m_tx_data[top_rep];
        top_chunk_data.chunk_setinfo = top_part;
 
        // See the comment above in Activate(). We perform the opposite operations here,
        // removing instead of adding.
        //
        // Let UpdateChunk traverse the old parent chunk top_part, and remove bottom_part from
        // every dependency's top_set which has the parent in it. At the same time, change the
        // representative of each of these transactions to be top_rep.
        UpdateChunk<true>(/*chunk=*/top_part.transactions, /*query=*/dep_data.parent,
                          /*chunk_rep=*/top_rep, /*dep_change=*/bottom_part);
        // Let UpdateChunk traverse the old child chunk bottom_part, and remove top_part from every
        // dependency's top_set which has the child in it. At the same time, change the
        // representative of each of these transactions to be bottom_rep.
        UpdateChunk<true>(/*chunk=*/bottom_part.transactions, /*query=*/dep_data.child,
                          /*chunk_rep=*/bottom_rep, /*dep_change=*/top_part);
    }
 
    TxIdx MergeChunks(TxIdx top_rep, TxIdx bottom_rep) noexcept
    {
        auto& top_chunk = m_tx_data[top_rep];
        Assume(top_chunk.chunk_rep == top_rep);
        auto& bottom_chunk = m_tx_data[bottom_rep];
        Assume(bottom_chunk.chunk_rep == bottom_rep);
        // Count the number of dependencies between bottom_chunk and top_chunk.
        TxIdx num_deps{0};
        for (auto tx : top_chunk.chunk_setinfo.transactions) {
            auto& tx_data = m_tx_data[tx];
            num_deps += (tx_data.children & bottom_chunk.chunk_setinfo.transactions).Count();
        }
        if (num_deps == 0) return TxIdx(-1);
        // Uniformly randomly pick one of them and activate it.
        TxIdx pick = m_rng.randrange(num_deps);
        for (auto tx : top_chunk.chunk_setinfo.transactions) {
            auto& tx_data = m_tx_data[tx];
            auto intersect = tx_data.children & bottom_chunk.chunk_setinfo.transactions;
            auto count = intersect.Count();
            if (pick < count) {
                for (auto dep : tx_data.child_deps) {
                    auto& dep_data = m_dep_data[dep];
                    if (bottom_chunk.chunk_setinfo.transactions[dep_data.child]) {
                        if (pick == 0) return Activate(dep);
                        --pick;
                    }
                }
                break;
            }
            pick -= count;
        }
        Assume(false);
        return TxIdx(-1);
    }
 
    template<bool DownWard>
    TxIdx MergeStep(TxIdx chunk_rep) noexcept
    {
        auto& chunk_data = m_tx_data[chunk_rep];
        SetType chunk_txn = chunk_data.chunk_setinfo.transactions;
        // Iterate over all transactions in the chunk, figuring out which other chunk each
        // depends on, but only testing each other chunk once. For those depended-on chunks,
        // remember the highest-feerate (if DownWard) or lowest-feerate (if !DownWard) one.
        // If multiple equal-feerate candidate chunks to merge with exist, pick a random one
        // among them.
 
        SetType explored = chunk_txn;
        FeeFrac best_other_chunk_feerate = chunk_data.chunk_setinfo.feerate;
        TxIdx best_other_chunk_rep = TxIdx(-1);
        uint64_t best_other_chunk_tiebreak{0};
        for (auto tx : chunk_txn) {
            auto& tx_data = m_tx_data[tx];
            auto newly_reached = (DownWard ? tx_data.children : tx_data.parents) - explored;
            explored |= newly_reached;
            while (newly_reached.Any()) {
                // Find a chunk inside newly_reached, and remove it from newly_reached.
                auto reached_chunk_rep = m_tx_data[newly_reached.First()].chunk_rep;
                auto& reached_chunk = m_tx_data[reached_chunk_rep].chunk_setinfo;
                newly_reached -= reached_chunk.transactions;
                // See if it has an acceptable feerate.
                auto cmp = DownWard ? FeeRateCompare(best_other_chunk_feerate, reached_chunk.feerate)
                                    : FeeRateCompare(reached_chunk.feerate, best_other_chunk_feerate);
                if (cmp > 0) continue;
                uint64_t tiebreak = m_rng.rand64();
                if (cmp < 0 || tiebreak >= best_other_chunk_tiebreak) {
                    best_other_chunk_feerate = reached_chunk.feerate;
                    best_other_chunk_rep = reached_chunk_rep;
                    best_other_chunk_tiebreak = tiebreak;
                }
            }
        }
        // Stop if there are no candidate chunks to merge with.
        if (best_other_chunk_rep == TxIdx(-1)) return TxIdx(-1);
        if constexpr (DownWard) {
            chunk_rep = MergeChunks(chunk_rep, best_other_chunk_rep);
        } else {
            chunk_rep = MergeChunks(best_other_chunk_rep, chunk_rep);
        }
        Assume(chunk_rep != TxIdx(-1));
        return chunk_rep;
    }
 
 
    template<bool DownWard>
    void MergeSequence(TxIdx tx_idx) noexcept
    {
        auto chunk_rep = m_tx_data[tx_idx].chunk_rep;
        while (true) {
            auto merged_rep = MergeStep<DownWard>(chunk_rep);
            if (merged_rep == TxIdx(-1)) break;
            chunk_rep = merged_rep;
        }
        // Add the chunk to the queue of improvable chunks.
        m_suboptimal_chunks.push_back(chunk_rep);
    }
 
    void Improve(DepIdx dep_idx) noexcept
    {
        auto& dep_data = m_dep_data[dep_idx];
        Assume(dep_data.active);
        // Deactivate the specified dependency, splitting it into two new chunks: a top containing
        // the parent, and a bottom containing the child. The top should have a higher feerate.
        Deactivate(dep_idx);
 
        // At this point we have exactly two chunks which may violate topology constraints (the
        // parent chunk and child chunk that were produced by deactivating dep_idx). We can fix
        // these using just merge sequences, one upwards and one downwards, avoiding the need for a
        // full MakeTopological.
 
        // Merge the top chunk with lower-feerate chunks it depends on (which may be the bottom it
        // was just split from, or other pre-existing chunks).
        MergeSequence<false>(dep_data.parent);
        // Merge the bottom chunk with higher-feerate chunks that depend on it.
        MergeSequence<true>(dep_data.child);
    }
 
public:
    explicit SpanningForestState(const DepGraph<SetType>& depgraph, uint64_t rng_seed) noexcept : m_rng(rng_seed)
    {
        m_transaction_idxs = depgraph.Positions();
        auto num_transactions = m_transaction_idxs.Count();
        m_tx_data.resize(depgraph.PositionRange());
        // Reserve the maximum number of (reserved) dependencies the cluster can have, so
        // m_dep_data won't need any reallocations during construction. For a cluster with N
        // transactions, the worst case consists of two sets of transactions, the parents and the
        // children, where each child depends on each parent and nothing else. For even N, both
        // sets can be sized N/2, which means N^2/4 dependencies. For odd N, one can be (N + 1)/2
        // and the other can be (N - 1)/2, meaning (N^2 - 1)/4 dependencies. Because N^2 is odd in
        // this case, N^2/4 (with rounding-down division) is the correct value in both cases.
        m_dep_data.reserve((num_transactions * num_transactions) / 4);
        for (auto tx : m_transaction_idxs) {
            // Fill in transaction data.
            auto& tx_data = m_tx_data[tx];
            tx_data.chunk_rep = tx;
            tx_data.chunk_setinfo.transactions = SetType::Singleton(tx);
            tx_data.chunk_setinfo.feerate = depgraph.FeeRate(tx);
            // Add its dependencies.
            SetType parents = depgraph.GetReducedParents(tx);
            for (auto par : parents) {
                auto& par_tx_data = m_tx_data[par];
                auto dep_idx = m_dep_data.size();
                // Construct new dependency.
                auto& dep = m_dep_data.emplace_back();
                dep.active = false;
                dep.parent = par;
                dep.child = tx;
                // Add it as parent of the child.
                tx_data.parents.Set(par);
                // Add it as child of the parent.
                par_tx_data.child_deps.push_back(dep_idx);
                par_tx_data.children.Set(tx);
            }
        }
    }
 
    void LoadLinearization(std::span<const DepGraphIndex> old_linearization) noexcept
    {
        // Add transactions one by one, in order of existing linearization.
        for (DepGraphIndex tx : old_linearization) {
            auto chunk_rep = m_tx_data[tx].chunk_rep;
            // Merge the chunk upwards, as long as merging succeeds.
            while (true) {
                chunk_rep = MergeStep<false>(chunk_rep);
                if (chunk_rep == TxIdx(-1)) break;
            }
        }
    }
 
    void MakeTopological() noexcept
    {
        for (auto tx : m_transaction_idxs) {
            auto& tx_data = m_tx_data[tx];
            if (tx_data.chunk_rep == tx) {
                m_suboptimal_chunks.emplace_back(tx);
                // Randomize the initial order of suboptimal chunks in the queue.
                TxIdx j = m_rng.randrange<TxIdx>(m_suboptimal_chunks.size());
                if (j != m_suboptimal_chunks.size() - 1) {
                    std::swap(m_suboptimal_chunks.back(), m_suboptimal_chunks[j]);
                }
            }
        }
        while (!m_suboptimal_chunks.empty()) {
            // Pop an entry from the potentially-suboptimal chunk queue.
            TxIdx chunk = m_suboptimal_chunks.front();
            m_suboptimal_chunks.pop_front();
            auto& chunk_data = m_tx_data[chunk];
            // If what was popped is not currently a chunk representative, continue. This may
            // happen when it was merged with something else since being added.
            if (chunk_data.chunk_rep != chunk) continue;
            int flip = m_rng.randbool();
            for (int i = 0; i < 2; ++i) {
                if (i ^ flip) {
                    // Attempt to merge the chunk upwards.
                    auto result_up = MergeStep<false>(chunk);
                    if (result_up != TxIdx(-1)) {
                        m_suboptimal_chunks.push_back(result_up);
                        break;
                    }
                } else {
                    // Attempt to merge the chunk downwards.
                    auto result_down = MergeStep<true>(chunk);
                    if (result_down != TxIdx(-1)) {
                        m_suboptimal_chunks.push_back(result_down);
                        break;
                    }
                }
            }
        }
    }
 
    void StartOptimizing() noexcept
    {
        // Mark chunks suboptimal.
        for (auto tx : m_transaction_idxs) {
            auto& tx_data = m_tx_data[tx];
            if (tx_data.chunk_rep == tx) {
                m_suboptimal_chunks.push_back(tx);
                // Randomize the initial order of suboptimal chunks in the queue.
                TxIdx j = m_rng.randrange<TxIdx>(m_suboptimal_chunks.size());
                if (j != m_suboptimal_chunks.size() - 1) {
                    std::swap(m_suboptimal_chunks.back(), m_suboptimal_chunks[j]);
                }
            }
        }
    }
 
    bool OptimizeStep() noexcept
    {
        while (!m_suboptimal_chunks.empty()) {
            // Pop an entry from the potentially-suboptimal chunk queue.
            TxIdx chunk = m_suboptimal_chunks.front();
            m_suboptimal_chunks.pop_front();
            auto& chunk_data = m_tx_data[chunk];
            // If what was popped is not currently a chunk representative, continue. This may
            // happen when a split chunk merges in Improve() with one or more existing chunks that
            // are themselves on the suboptimal queue already.
            if (chunk_data.chunk_rep != chunk) continue;
            // Remember the best dependency seen so far.
            DepIdx candidate_dep = DepIdx(-1);
            uint64_t candidate_tiebreak = 0;
            // Iterate over all transactions.
            for (auto tx : chunk_data.chunk_setinfo.transactions) {
                const auto& tx_data = m_tx_data[tx];
                // Iterate over all active child dependencies of the transaction.
                const auto children = std::span{tx_data.child_deps};
                for (DepIdx dep_idx : children) {
                    const auto& dep_data = m_dep_data[dep_idx];
                    if (!dep_data.active) continue;
                    // Skip if this dependency is ineligible (the top chunk that would be created
                    // does not have higher feerate than the chunk it is currently part of).
                    auto cmp = FeeRateCompare(dep_data.top_setinfo.feerate, chunk_data.chunk_setinfo.feerate);
                    if (cmp <= 0) continue;
                    // Generate a random tiebreak for this dependency, and reject it if its tiebreak
                    // is worse than the best so far. This means that among all eligible
                    // dependencies, a uniformly random one will be chosen.
                    uint64_t tiebreak = m_rng.rand64();
                    if (tiebreak < candidate_tiebreak) continue;
                    // Remember this as our (new) candidate dependency.
                    candidate_dep = dep_idx;
                    candidate_tiebreak = tiebreak;
                }
            }
            // If a candidate with positive gain was found, deactivate it and then make the state
            // topological again with a sequence of merges.
            if (candidate_dep != DepIdx(-1)) Improve(candidate_dep);
            // Stop processing for now, even if nothing was activated, as the loop above may have
            // had a nontrivial cost.
            return !m_suboptimal_chunks.empty();
        }
        // No improvable chunk was found, we are done.
        return false;
    }
 
    void StartMinimizing() noexcept
    {
        m_nonminimal_chunks.clear();
        m_nonminimal_chunks.reserve(m_transaction_idxs.Count());
        // Gather all chunks, and for each, add it with a random pivot in it, and a random initial
        // direction, to m_nonminimal_chunks.
        for (auto tx : m_transaction_idxs) {
            auto& tx_data = m_tx_data[tx];
            if (tx_data.chunk_rep == tx) {
                TxIdx pivot_idx = PickRandomTx(tx_data.chunk_setinfo.transactions);
                m_nonminimal_chunks.emplace_back(tx, pivot_idx, m_rng.randbits<1>());
                // Randomize the initial order of nonminimal chunks in the queue.
                TxIdx j = m_rng.randrange<TxIdx>(m_nonminimal_chunks.size());
                if (j != m_nonminimal_chunks.size() - 1) {
                    std::swap(m_nonminimal_chunks.back(), m_nonminimal_chunks[j]);
                }
            }
        }
    }
 
    bool MinimizeStep() noexcept
    {
        // If the queue of potentially-non-minimal chunks is empty, we are done.
        if (m_nonminimal_chunks.empty()) return false;
        // Pop an entry from the potentially-non-minimal chunk queue.
        auto [chunk_rep, pivot_idx, flags] = m_nonminimal_chunks.front();
        m_nonminimal_chunks.pop_front();
        auto& chunk_data = m_tx_data[chunk_rep];
        Assume(chunk_data.chunk_rep == chunk_rep);
        bool move_pivot_down = flags & 1;
        bool second_stage = flags & 2;
 
        // Find a random dependency whose top and bottom set feerates are equal, and which has
        // pivot in bottom set (if move_pivot_down) or in top set (if !move_pivot_down).
        DepIdx candidate_dep = DepIdx(-1);
        uint64_t candidate_tiebreak{0};
        bool have_any = false;
        // Iterate over all transactions.
        for (auto tx_idx : chunk_data.chunk_setinfo.transactions) {
            const auto& tx_data = m_tx_data[tx_idx];
            // Iterate over all active child dependencies of the transaction.
            for (auto dep_idx : tx_data.child_deps) {
                auto& dep_data = m_dep_data[dep_idx];
                // Skip inactive child dependencies.
                if (!dep_data.active) continue;
                // Skip if this dependency does not have equal top and bottom set feerates. Note
                // that the top cannot have higher feerate than the bottom, or OptimizeSteps would
                // have dealt with it.
                if (dep_data.top_setinfo.feerate << chunk_data.chunk_setinfo.feerate) continue;
                have_any = true;
                // Skip if this dependency does not have pivot in the right place.
                if (move_pivot_down == dep_data.top_setinfo.transactions[pivot_idx]) continue;
                // Remember this as our chosen dependency if it has a better tiebreak.
                uint64_t tiebreak = m_rng.rand64() | 1;
                if (tiebreak > candidate_tiebreak) {
                    candidate_tiebreak = tiebreak;
                    candidate_dep = dep_idx;
                }
            }
        }
        // If no dependencies have equal top and bottom set feerate, this chunk is minimal.
        if (!have_any) return true;
        // If all found dependencies have the pivot in the wrong place, try moving it in the other
        // direction. If this was the second stage already, we are done.
        if (candidate_tiebreak == 0) {
            // Switch to other direction, and to second phase.
            flags ^= 3;
            if (!second_stage) m_nonminimal_chunks.emplace_back(chunk_rep, pivot_idx, flags);
            return true;
        }
 
        // Otherwise, deactivate the dependency that was found.
        Deactivate(candidate_dep);
        auto& dep_data = m_dep_data[candidate_dep];
        auto parent_chunk_rep = m_tx_data[dep_data.parent].chunk_rep;
        auto child_chunk_rep = m_tx_data[dep_data.child].chunk_rep;
        // Try to activate a dependency between the new bottom and the new top (opposite from the
        // dependency that was just deactivated).
        auto merged_chunk_rep = MergeChunks(child_chunk_rep, parent_chunk_rep);
        if (merged_chunk_rep != TxIdx(-1)) {
            // A self-merge happened.
            // Re-insert the chunk into the queue, in the same direction. Note that the chunk_rep
            // will have changed.
            m_nonminimal_chunks.emplace_back(merged_chunk_rep, pivot_idx, flags);
        } else {
            // No self-merge happens, and thus we have found a way to split the chunk. Create two
            // smaller chunks, and add them to the queue. The one that contains the current pivot
            // gets to continue with it in the same direction, to minimize the number of times we
            // alternate direction. If we were in the second phase already, the newly created chunk
            // inherits that too, because we know no split with the pivot on the other side is
            // possible already. The new chunk without the current pivot gets a new randomly-chosen
            // one.
            if (move_pivot_down) {
                auto parent_pivot_idx = PickRandomTx(m_tx_data[parent_chunk_rep].chunk_setinfo.transactions);
                m_nonminimal_chunks.emplace_back(parent_chunk_rep, parent_pivot_idx, m_rng.randbits<1>());
                m_nonminimal_chunks.emplace_back(child_chunk_rep, pivot_idx, flags);
            } else {
                auto child_pivot_idx = PickRandomTx(m_tx_data[child_chunk_rep].chunk_setinfo.transactions);
                m_nonminimal_chunks.emplace_back(parent_chunk_rep, pivot_idx, flags);
                m_nonminimal_chunks.emplace_back(child_chunk_rep, child_pivot_idx, m_rng.randbits<1>());
            }
            if (m_rng.randbool()) {
                std::swap(m_nonminimal_chunks.back(), m_nonminimal_chunks[m_nonminimal_chunks.size() - 2]);
            }
        }
        return true;
    }
 
    std::vector<DepGraphIndex> GetLinearization() noexcept
    {
        std::vector<DepGraphIndex> ret;
        ret.reserve(m_transaction_idxs.Count());
        std::vector<std::pair<TxIdx, uint64_t>> ready_chunks;
        std::vector<std::pair<TxIdx, TxIdx>> chunk_deps(m_tx_data.size(), {0, 0});
        SetType chunk_reps;
        std::vector<TxIdx> ready_tx;
        // Populate chunk_deps[c] with the number of {out-of-chunk dependencies, dependencies} the
        // child has.
        for (TxIdx chl_idx : m_transaction_idxs) {
            const auto& chl_data = m_tx_data[chl_idx];
            chunk_deps[chl_idx].second = chl_data.parents.Count();
            auto chl_chunk_rep = chl_data.chunk_rep;
            chunk_reps.Set(chl_chunk_rep);
            for (auto par_idx : chl_data.parents) {
                auto par_chunk_rep = m_tx_data[par_idx].chunk_rep;
                chunk_deps[chl_chunk_rep].first += (par_chunk_rep != chl_chunk_rep);
            }
        }
        // Construct a heap with all chunks that have no out-of-chunk dependencies.
        auto chunk_cmp_fn = [&](const std::pair<TxIdx, uint64_t>& a, const std::pair<TxIdx, uint64_t>& b) noexcept {
            auto& chunk_a = m_tx_data[a.first];
            auto& chunk_b = m_tx_data[b.first];
            Assume(chunk_a.chunk_rep == a.first);
            Assume(chunk_b.chunk_rep == b.first);
            // First sort by chunk feerate.
            if (chunk_a.chunk_setinfo.feerate != chunk_b.chunk_setinfo.feerate) {
                return chunk_a.chunk_setinfo.feerate < chunk_b.chunk_setinfo.feerate;
            }
            // Tie-break randomly.
            if (a.second != b.second) return a.second < b.second;
            // Lastly, tie-break by chunk representative.
            return a.first < b.first;
        };
        for (TxIdx chunk_rep : chunk_reps) {
            if (chunk_deps[chunk_rep].first == 0) ready_chunks.emplace_back(chunk_rep, m_rng.rand64());
        }
        std::make_heap(ready_chunks.begin(), ready_chunks.end(), chunk_cmp_fn);
        // Pop chunks off the heap, highest-feerate ones first.
        while (!ready_chunks.empty()) {
            auto [chunk_rep, _rnd] = ready_chunks.front();
            std::pop_heap(ready_chunks.begin(), ready_chunks.end(), chunk_cmp_fn);
            ready_chunks.pop_back();
            Assume(m_tx_data[chunk_rep].chunk_rep == chunk_rep);
            Assume(chunk_deps[chunk_rep].first == 0);
            const auto& chunk_txn = m_tx_data[chunk_rep].chunk_setinfo.transactions;
            // Build heap of all includable transactions in chunk.
            for (TxIdx tx_idx : chunk_txn) {
                if (chunk_deps[tx_idx].second == 0) {
                    ready_tx.push_back(tx_idx);
                }
            }
            Assume(!ready_tx.empty());
            // Pick transactions from the ready queue, append them to linearization, and decrement
            // dependency counts.
            while (!ready_tx.empty()) {
                // Move a random queue element to the back.
                auto pos = m_rng.randrange(ready_tx.size());
                if (pos != ready_tx.size() - 1) std::swap(ready_tx.back(), ready_tx[pos]);
                // Pop from the back.
                auto tx_idx = ready_tx.back();
                Assume(chunk_txn[tx_idx]);
                ready_tx.pop_back();
                // Append to linearization.
                ret.push_back(tx_idx);
                // Decrement dependency counts.
                auto& tx_data = m_tx_data[tx_idx];
                for (TxIdx chl_idx : tx_data.children) {
                    auto& chl_data = m_tx_data[chl_idx];
                    // Decrement tx dependency count.
                    Assume(chunk_deps[chl_idx].second > 0);
                    if (--chunk_deps[chl_idx].second == 0 && chunk_txn[chl_idx]) {
                        // Child tx has no dependencies left, and is in this chunk. Add it to the tx queue.
                        ready_tx.push_back(chl_idx);
                    }
                    // Decrement chunk dependency count if this is out-of-chunk dependency.
                    if (chl_data.chunk_rep != chunk_rep) {
                        Assume(chunk_deps[chl_data.chunk_rep].first > 0);
                        if (--chunk_deps[chl_data.chunk_rep].first == 0) {
                            // Child chunk has no dependencies left. Add it to the chunk heap.
                            ready_chunks.emplace_back(chl_data.chunk_rep, m_rng.rand64());
                            std::push_heap(ready_chunks.begin(), ready_chunks.end(), chunk_cmp_fn);
                        }
                    }
                }
            }
        }
        Assume(ret.size() == m_transaction_idxs.Count());
        return ret;
    }
 
    std::vector<FeeFrac> GetDiagram() const noexcept
    {
        std::vector<FeeFrac> ret;
        for (auto tx : m_transaction_idxs) {
            if (m_tx_data[tx].chunk_rep == tx) {
                ret.push_back(m_tx_data[tx].chunk_setinfo.feerate);
            }
        }
        std::sort(ret.begin(), ret.end(), std::greater{});
        return ret;
    }
 
    uint64_t GetCost() const noexcept { return m_cost; }
 
    void SanityCheck(const DepGraph<SetType>& depgraph) const
    {
        //
        // Verify dependency parent/child information, and build list of (active) dependencies.
        //
        std::vector<std::pair<TxIdx, TxIdx>> expected_dependencies;
        std::vector<std::tuple<TxIdx, TxIdx, DepIdx>> all_dependencies;
        std::vector<std::tuple<TxIdx, TxIdx, DepIdx>> active_dependencies;
        for (auto parent_idx : depgraph.Positions()) {
            for (auto child_idx : depgraph.GetReducedChildren(parent_idx)) {
                expected_dependencies.emplace_back(parent_idx, child_idx);
            }
        }
        for (DepIdx dep_idx = 0; dep_idx < m_dep_data.size(); ++dep_idx) {
            const auto& dep_data = m_dep_data[dep_idx];
            all_dependencies.emplace_back(dep_data.parent, dep_data.child, dep_idx);
            // Also add to active_dependencies if it is active.
            if (m_dep_data[dep_idx].active) {
                active_dependencies.emplace_back(dep_data.parent, dep_data.child, dep_idx);
            }
        }
        std::sort(expected_dependencies.begin(), expected_dependencies.end());
        std::sort(all_dependencies.begin(), all_dependencies.end());
        assert(expected_dependencies.size() == all_dependencies.size());
        for (size_t i = 0; i < expected_dependencies.size(); ++i) {
            assert(expected_dependencies[i] ==
                   std::make_pair(std::get<0>(all_dependencies[i]),
                                  std::get<1>(all_dependencies[i])));
        }
 
        //
        // Verify the chunks against the list of active dependencies
        //
        for (auto tx_idx: depgraph.Positions()) {
            // Only process chunks for now.
            if (m_tx_data[tx_idx].chunk_rep == tx_idx) {
                const auto& chunk_data = m_tx_data[tx_idx];
                // Verify that transactions in the chunk point back to it. This guarantees
                // that chunks are non-overlapping.
                for (auto chunk_tx : chunk_data.chunk_setinfo.transactions) {
                    assert(m_tx_data[chunk_tx].chunk_rep == tx_idx);
                }
                // Verify the chunk's transaction set: it must contain the representative, and for
                // every active dependency, if it contains the parent or child, it must contain
                // both. It must have exactly N-1 active dependencies in it, guaranteeing it is
                // acyclic.
                SetType expected_chunk = SetType::Singleton(tx_idx);
                while (true) {
                    auto old = expected_chunk;
                    size_t active_dep_count{0};
                    for (const auto& [par, chl, _dep] : active_dependencies) {
                        if (expected_chunk[par] || expected_chunk[chl]) {
                            expected_chunk.Set(par);
                            expected_chunk.Set(chl);
                            ++active_dep_count;
                        }
                    }
                    if (old == expected_chunk) {
                        assert(expected_chunk.Count() == active_dep_count + 1);
                        break;
                    }
                }
                assert(chunk_data.chunk_setinfo.transactions == expected_chunk);
                // Verify the chunk's feerate.
                assert(chunk_data.chunk_setinfo.feerate ==
                       depgraph.FeeRate(chunk_data.chunk_setinfo.transactions));
            }
        }
 
        //
        // Verify other transaction data.
        //
        assert(m_transaction_idxs == depgraph.Positions());
        for (auto tx_idx : m_transaction_idxs) {
            const auto& tx_data = m_tx_data[tx_idx];
            // Verify it has a valid chunk representative, and that chunk includes this
            // transaction.
            assert(m_tx_data[tx_data.chunk_rep].chunk_rep == tx_data.chunk_rep);
            assert(m_tx_data[tx_data.chunk_rep].chunk_setinfo.transactions[tx_idx]);
            // Verify parents/children.
            assert(tx_data.parents == depgraph.GetReducedParents(tx_idx));
            assert(tx_data.children == depgraph.GetReducedChildren(tx_idx));
            // Verify list of child dependencies.
            std::vector<DepIdx> expected_child_deps;
            for (const auto& [par_idx, chl_idx, dep_idx] : all_dependencies) {
                if (tx_idx == par_idx) {
                    assert(tx_data.children[chl_idx]);
                    expected_child_deps.push_back(dep_idx);
                }
            }
            std::sort(expected_child_deps.begin(), expected_child_deps.end());
            auto child_deps_copy = tx_data.child_deps;
            std::sort(child_deps_copy.begin(), child_deps_copy.end());
            assert(expected_child_deps == child_deps_copy);
        }
 
        //
        // Verify active dependencies' top_setinfo.
        //
        for (const auto& [par_idx, chl_idx, dep_idx] : active_dependencies) {
            const auto& dep_data = m_dep_data[dep_idx];
            // Verify the top_info's transactions: it must contain the parent, and for every
            // active dependency, except dep_idx itself, if it contains the parent or child, it
            // must contain both.
            SetType expected_top = SetType::Singleton(par_idx);
            while (true) {
                auto old = expected_top;
                for (const auto& [par2_idx, chl2_idx, dep2_idx] : active_dependencies) {
                    if (dep2_idx != dep_idx && (expected_top[par2_idx] || expected_top[chl2_idx])) {
                        expected_top.Set(par2_idx);
                        expected_top.Set(chl2_idx);
                    }
                }
                if (old == expected_top) break;
            }
            assert(!expected_top[chl_idx]);
            assert(dep_data.top_setinfo.transactions == expected_top);
            // Verify the top_info's feerate.
            assert(dep_data.top_setinfo.feerate ==
                   depgraph.FeeRate(dep_data.top_setinfo.transactions));
        }
 
        //
        // Verify m_suboptimal_chunks.
        //
        for (size_t i = 0; i < m_suboptimal_chunks.size(); ++i) {
            auto tx_idx = m_suboptimal_chunks[i];
            assert(m_transaction_idxs[tx_idx]);
        }
 
        //
        // Verify m_nonminimal_chunks.
        //
        SetType nonminimal_reps;
        for (size_t i = 0; i < m_nonminimal_chunks.size(); ++i) {
            auto [chunk_rep, pivot, flags] = m_nonminimal_chunks[i];
            assert(m_tx_data[chunk_rep].chunk_rep == chunk_rep);
            assert(m_tx_data[pivot].chunk_rep == chunk_rep);
            assert(!nonminimal_reps[chunk_rep]);
            nonminimal_reps.Set(chunk_rep);
        }
        assert(nonminimal_reps.IsSubsetOf(m_transaction_idxs));
    }
};
 
template<typename SetType>
std::tuple<std::vector<DepGraphIndex>, bool, uint64_t> Linearize(const DepGraph<SetType>& depgraph, uint64_t max_iterations, uint64_t rng_seed, std::span<const DepGraphIndex> old_linearization = {}, bool is_topological = true) noexcept
{
    SpanningForestState forest(depgraph, rng_seed);
    if (!old_linearization.empty()) {
        forest.LoadLinearization(old_linearization);
        if (!is_topological) forest.MakeTopological();
    } else {
        forest.MakeTopological();
    }
    // Make improvement steps to it until we hit the max_iterations limit, or an optimal result
    // is found.
    if (forest.GetCost() < max_iterations) {
        forest.StartOptimizing();
        do {
            if (!forest.OptimizeStep()) break;
        } while (forest.GetCost() < max_iterations);
    }
    // Make chunk minimization steps until we hit the max_iterations limit, or all chunks are
    // minimal.
    bool optimal = false;
    if (forest.GetCost() < max_iterations) {
        forest.StartMinimizing();
        do {
            if (!forest.MinimizeStep()) {
                optimal = true;
                break;
            }
        } while (forest.GetCost() < max_iterations);
    }
    return {forest.GetLinearization(), optimal, forest.GetCost()};
}
 
template<typename SetType>
void PostLinearize(const DepGraph<SetType>& depgraph, std::span<DepGraphIndex> linearization)
{
    // This algorithm performs a number of passes (currently 2); the even ones operate from back to
    // front, the odd ones from front to back. Each results in an equal-or-better linearization
    // than the one started from.
    // - One pass in either direction guarantees that the resulting chunks are connected.
    // - Each direction corresponds to one shape of tree being linearized optimally (forward passes
    //   guarantee this for graphs where each transaction has at most one child; backward passes
    //   guarantee this for graphs where each transaction has at most one parent).
    // - Starting with a backward pass guarantees the moved-tree property.
    //
    // During an odd (forward) pass, the high-level operation is:
    // - Start with an empty list of groups L=[].
    // - For every transaction i in the old linearization, from front to back:
    //   - Append a new group C=[i], containing just i, to the back of L.
    //   - While L has at least one group before C, and the group immediately before C has feerate
    //     lower than C:
    //     - If C depends on P:
    //       - Merge P into C, making C the concatenation of P+C, continuing with the combined C.
    //     - Otherwise:
    //       - Swap P with C, continuing with the now-moved C.
    // - The output linearization is the concatenation of the groups in L.
    //
    // During even (backward) passes, i iterates from the back to the front of the existing
    // linearization, and new groups are prepended instead of appended to the list L. To enable
    // more code reuse, both passes append groups, but during even passes the meanings of
    // parent/child, and of high/low feerate are reversed, and the final concatenation is reversed
    // on output.
    //
    // In the implementation below, the groups are represented by singly-linked lists (pointing
    // from the back to the front), which are themselves organized in a singly-linked circular
    // list (each group pointing to its predecessor, with a special sentinel group at the front
    // that points back to the last group).
    //
    // Information about transaction t is stored in entries[t + 1], while the sentinel is in
    // entries[0].
 
    static constexpr DepGraphIndex SENTINEL{0};
    static constexpr DepGraphIndex NO_PREV_TX{0};
 
 
    struct TxEntry
    {
        DepGraphIndex prev_tx;
 
        // The fields below are only used for transactions that are the last one in a group
        // (referred to as tail transactions below).
 
        DepGraphIndex first_tx;
        DepGraphIndex prev_group;
        SetType group;
        SetType deps;
        FeeFrac feerate;
    };
 
    // As an example, consider the state corresponding to the linearization [1,0,3,2], with
    // groups [1,0,3] and [2], in an odd pass. The linked lists would be:
    //
    //                                        +-----+
    //                                 0<-P-- | 0 S | ---\     Legend:
    //                                        +-----+    |
    //                                           ^       |     - digit in box: entries index
    //             /--------------F---------+    G       |       (note: one more than tx value)
    //             v                         \   |       |     - S: sentinel group
    //          +-----+        +-----+        +-----+    |          (empty feerate)
    //   0<-P-- | 2   | <--P-- | 1   | <--P-- | 4 T |    |     - T: tail transaction, contains
    //          +-----+        +-----+        +-----+    |          fields beyond prev_tv.
    //                                           ^       |     - P: prev_tx reference
    //                                           G       G     - F: first_tx reference
    //                                           |       |     - G: prev_group reference
    //                                        +-----+    |
    //                                 0<-P-- | 3 T | <--/
    //                                        +-----+
    //                                         ^   |
    //                                         \-F-/
    //
    // During an even pass, the diagram above would correspond to linearization [2,3,0,1], with
    // groups [2] and [3,0,1].
 
    std::vector<TxEntry> entries(depgraph.PositionRange() + 1);
 
    // Perform two passes over the linearization.
    for (int pass = 0; pass < 2; ++pass) {
        int rev = !(pass & 1);
        // Construct a sentinel group, identifying the start of the list.
        entries[SENTINEL].prev_group = SENTINEL;
        Assume(entries[SENTINEL].feerate.IsEmpty());
 
        // Iterate over all elements in the existing linearization.
        for (DepGraphIndex i = 0; i < linearization.size(); ++i) {
            // Even passes are from back to front; odd passes from front to back.
            DepGraphIndex idx = linearization[rev ? linearization.size() - 1 - i : i];
            // Construct a new group containing just idx. In even passes, the meaning of
            // parent/child and high/low feerate are swapped.
            DepGraphIndex cur_group = idx + 1;
            entries[cur_group].group = SetType::Singleton(idx);
            entries[cur_group].deps = rev ? depgraph.Descendants(idx): depgraph.Ancestors(idx);
            entries[cur_group].feerate = depgraph.FeeRate(idx);
            if (rev) entries[cur_group].feerate.fee = -entries[cur_group].feerate.fee;
            entries[cur_group].prev_tx = NO_PREV_TX; // No previous transaction in group.
            entries[cur_group].first_tx = cur_group; // Transaction itself is first of group.
            // Insert the new group at the back of the groups linked list.
            entries[cur_group].prev_group = entries[SENTINEL].prev_group;
            entries[SENTINEL].prev_group = cur_group;
 
            // Start merge/swap cycle.
            DepGraphIndex next_group = SENTINEL; // We inserted at the end, so next group is sentinel.
            DepGraphIndex prev_group = entries[cur_group].prev_group;
            // Continue as long as the current group has higher feerate than the previous one.
            while (entries[cur_group].feerate >> entries[prev_group].feerate) {
                // prev_group/cur_group/next_group refer to (the last transactions of) 3
                // consecutive entries in groups list.
                Assume(cur_group == entries[next_group].prev_group);
                Assume(prev_group == entries[cur_group].prev_group);
                // The sentinel has empty feerate, which is neither higher or lower than other
                // feerates. Thus, the while loop we are in here guarantees that cur_group and
                // prev_group are not the sentinel.
                Assume(cur_group != SENTINEL);
                Assume(prev_group != SENTINEL);
                if (entries[cur_group].deps.Overlaps(entries[prev_group].group)) {
                    // There is a dependency between cur_group and prev_group; merge prev_group
                    // into cur_group. The group/deps/feerate fields of prev_group remain unchanged
                    // but become unused.
                    entries[cur_group].group |= entries[prev_group].group;
                    entries[cur_group].deps |= entries[prev_group].deps;
                    entries[cur_group].feerate += entries[prev_group].feerate;
                    // Make the first of the current group point to the tail of the previous group.
                    entries[entries[cur_group].first_tx].prev_tx = prev_group;
                    // The first of the previous group becomes the first of the newly-merged group.
                    entries[cur_group].first_tx = entries[prev_group].first_tx;
                    // The previous group becomes whatever group was before the former one.
                    prev_group = entries[prev_group].prev_group;
                    entries[cur_group].prev_group = prev_group;
                } else {
                    // There is no dependency between cur_group and prev_group; swap them.
                    DepGraphIndex preprev_group = entries[prev_group].prev_group;
                    // If PP, P, C, N were the old preprev, prev, cur, next groups, then the new
                    // layout becomes [PP, C, P, N]. Update prev_groups to reflect that order.
                    entries[next_group].prev_group = prev_group;
                    entries[prev_group].prev_group = cur_group;
                    entries[cur_group].prev_group = preprev_group;
                    // The current group remains the same, but the groups before/after it have
                    // changed.
                    next_group = prev_group;
                    prev_group = preprev_group;
                }
            }
        }
 
        // Convert the entries back to linearization (overwriting the existing one).
        DepGraphIndex cur_group = entries[0].prev_group;
        DepGraphIndex done = 0;
        while (cur_group != SENTINEL) {
            DepGraphIndex cur_tx = cur_group;
            // Traverse the transactions of cur_group (from back to front), and write them in the
            // same order during odd passes, and reversed (front to back) in even passes.
            if (rev) {
                do {
                    *(linearization.begin() + (done++)) = cur_tx - 1;
                    cur_tx = entries[cur_tx].prev_tx;
                } while (cur_tx != NO_PREV_TX);
            } else {
                do {
                    *(linearization.end() - (++done)) = cur_tx - 1;
                    cur_tx = entries[cur_tx].prev_tx;
                } while (cur_tx != NO_PREV_TX);
            }
            cur_group = entries[cur_group].prev_group;
        }
        Assume(done == linearization.size());
    }
}
 
} // namespace cluster_linearize
 
#endif // BITCOIN_CLUSTER_LINEARIZE_H