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 <numeric>
#include <optional>
#include <stdint.h>
#include <vector>
#include <utility>
 
#include <random.h>
#include <span.h>
#include <util/feefrac.h>
#include <util/vecdeque.h>
 
namespace cluster_linearize {
 
template<typename SetType>
using Cluster = std::vector<std::pair<FeeFrac, SetType>>;
 
using ClusterIndex = 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;
 
public:
    friend bool operator==(const DepGraph&, const DepGraph&) noexcept = default;
 
    // 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;
 
    explicit DepGraph(ClusterIndex ntx) noexcept
    {
        Assume(ntx <= SetType::Size());
        entries.resize(ntx);
        for (ClusterIndex i = 0; i < ntx; ++i) {
            entries[i].ancestors = SetType::Singleton(i);
            entries[i].descendants = SetType::Singleton(i);
        }
    }
 
    explicit DepGraph(const Cluster<SetType>& cluster) noexcept : entries(cluster.size())
    {
        for (ClusterIndex i = 0; i < cluster.size(); ++i) {
            // Fill in fee and size.
            entries[i].feerate = cluster[i].first;
            // Fill in direct parents as ancestors.
            entries[i].ancestors = cluster[i].second;
            // Make sure transactions are ancestors of themselves.
            entries[i].ancestors.Set(i);
        }
 
        // Propagate ancestor information.
        for (ClusterIndex i = 0; i < entries.size(); ++i) {
            // At this point, entries[a].ancestors[b] is true iff b is an ancestor of a and there
            // is a path from a to b through the subgraph consisting of {a, b} union
            // {0, 1, ..., (i-1)}.
            SetType to_merge = entries[i].ancestors;
            for (ClusterIndex j = 0; j < entries.size(); ++j) {
                if (entries[j].ancestors[i]) {
                    entries[j].ancestors |= to_merge;
                }
            }
        }
 
        // Fill in descendant information by transposing the ancestor information.
        for (ClusterIndex i = 0; i < entries.size(); ++i) {
            for (auto j : entries[i].ancestors) {
                entries[j].descendants.Set(i);
            }
        }
    }
 
    auto TxCount() const noexcept { return entries.size(); }
    const FeeFrac& FeeRate(ClusterIndex i) const noexcept { return entries[i].feerate; }
    FeeFrac& FeeRate(ClusterIndex i) noexcept { return entries[i].feerate; }
    const SetType& Ancestors(ClusterIndex i) const noexcept { return entries[i].ancestors; }
    const SetType& Descendants(ClusterIndex i) const noexcept { return entries[i].descendants; }
 
    ClusterIndex AddTransaction(const FeeFrac& feefrac) noexcept
    {
        Assume(TxCount() < SetType::Size());
        ClusterIndex new_idx = TxCount();
        entries.emplace_back(feefrac, SetType::Singleton(new_idx), SetType::Singleton(new_idx));
        return new_idx;
    }
 
    void AddDependency(ClusterIndex parent, ClusterIndex child) noexcept
    {
        // Bail out if dependency is already implied.
        if (entries[child].ancestors[parent]) return;
        // To each ancestor of the parent, add as descendants the descendants of the child.
        const auto& chl_des = entries[child].descendants;
        for (auto anc_of_par : Ancestors(parent)) {
            entries[anc_of_par].descendants |= chl_des;
        }
        // To each descendant of the child, add as ancestors the ancestors of the parent.
        const auto& par_anc = entries[parent].ancestors;
        for (auto dec_of_chl : Descendants(child)) {
            entries[dec_of_chl].ancestors |= par_anc;
        }
    }
 
    FeeFrac FeeRate(const SetType& elems) const noexcept
    {
        FeeFrac ret;
        for (auto pos : elems) ret += entries[pos].feerate;
        return ret;
    }
 
    SetType FindConnectedComponent(const SetType& todo) const noexcept
    {
        if (todo.None()) return todo;
        auto to_add = SetType::Singleton(todo.First());
        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;
    }
 
    bool IsConnected(const SetType& subset) const noexcept
    {
        return FindConnectedComponent(subset) == subset;
    }
 
    bool IsConnected() const noexcept { return IsConnected(SetType::Fill(TxCount())); }
 
    void AppendTopo(std::vector<ClusterIndex>& list, const SetType& select) const noexcept
    {
        ClusterIndex old_len = list.size();
        for (auto i : select) list.push_back(i);
        std::sort(list.begin() + old_len, list.end(), [&](ClusterIndex a, ClusterIndex 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;
        });
    }
};
 
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, ClusterIndex 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)) {}
 
    SetInfo& operator|=(const SetInfo& other) noexcept
    {
        Assume(!transactions.Overlaps(other.transactions));
        transactions |= other.transactions;
        feerate += other.feerate;
        return *this;
    }
 
    [[nodiscard]] SetInfo Add(const DepGraph<SetType>& depgraph, const SetType& txn) const noexcept
    {
        return {transactions | txn, feerate + depgraph.FeeRate(txn - transactions)};
    }
 
    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<FeeFrac> ChunkLinearization(const DepGraph<SetType>& depgraph, Span<const ClusterIndex> linearization) noexcept
{
    std::vector<FeeFrac> ret;
    for (ClusterIndex 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 LinearizationChunking
{
    const DepGraph<SetType>& m_depgraph;
 
    Span<const ClusterIndex> m_linearization;
 
    std::vector<SetInfo<SetType>> m_chunks;
 
    ClusterIndex m_chunks_skip{0};
 
    SetType m_todo;
 
    void BuildChunks() noexcept
    {
        // Caller must clear m_chunks.
        Assume(m_chunks.empty());
 
        // Chop off the initial part of m_linearization that is already done.
        while (!m_linearization.empty() && !m_todo[m_linearization.front()]) {
            m_linearization = m_linearization.subspan(1);
        }
 
        // Iterate over the remaining entries in m_linearization. This is effectively the same
        // algorithm as ChunkLinearization, but supports skipping parts of the linearization and
        // keeps track of the sets themselves instead of just their feerates.
        for (auto idx : m_linearization) {
            if (!m_todo[idx]) continue;
            // Start with an initial chunk containing just element idx.
            SetInfo add(m_depgraph, idx);
            // Absorb existing final chunks into add while they have lower feerate.
            while (!m_chunks.empty() && add.feerate >> m_chunks.back().feerate) {
                add |= m_chunks.back();
                m_chunks.pop_back();
            }
            // Remember new chunk.
            m_chunks.push_back(std::move(add));
        }
    }
 
public:
    explicit LinearizationChunking(const DepGraph<SetType>& depgraph LIFETIMEBOUND, Span<const ClusterIndex> lin LIFETIMEBOUND) noexcept :
        m_depgraph(depgraph), m_linearization(lin)
    {
        // Mark everything in lin as todo still.
        for (auto i : m_linearization) m_todo.Set(i);
        // Compute the initial chunking.
        m_chunks.reserve(depgraph.TxCount());
        BuildChunks();
    }
 
    ClusterIndex NumChunksLeft() const noexcept { return m_chunks.size() - m_chunks_skip; }
 
    const SetInfo<SetType>& GetChunk(ClusterIndex n) const noexcept
    {
        Assume(n + m_chunks_skip < m_chunks.size());
        return m_chunks[n + m_chunks_skip];
    }
 
    void MarkDone(SetType subset) noexcept
    {
        Assume(subset.Any());
        Assume(subset.IsSubsetOf(m_todo));
        m_todo -= subset;
        if (GetChunk(0).transactions == subset) {
            // If the newly done transactions exactly match the first chunk of the remainder of
            // the linearization, we do not need to rechunk; just remember to skip one
            // additional chunk.
            ++m_chunks_skip;
            // With subset marked done, some prefix of m_linearization will be done now. How long
            // that prefix is depends on how many done elements were interspersed with subset,
            // but at least as many transactions as there are in subset.
            m_linearization = m_linearization.subspan(subset.Count());
        } else {
            // Otherwise rechunk what remains of m_linearization.
            m_chunks.clear();
            m_chunks_skip = 0;
            BuildChunks();
        }
    }
 
    SetInfo<SetType> IntersectPrefixes(const SetInfo<SetType>& subset) const noexcept
    {
        Assume(subset.transactions.IsSubsetOf(m_todo));
        SetInfo<SetType> accumulator;
        // Iterate over all chunks of the remaining linearization.
        for (ClusterIndex i = 0; i < NumChunksLeft(); ++i) {
            // Find what (if any) intersection the chunk has with subset.
            const SetType to_add = GetChunk(i).transactions & subset.transactions;
            if (to_add.Any()) {
                // If adding that to accumulator makes us hit all of subset, we are done as no
                // shorter intersection with higher/equal feerate exists.
                accumulator.transactions |= to_add;
                if (accumulator.transactions == subset.transactions) break;
                // Otherwise update the accumulator feerate.
                accumulator.feerate += m_depgraph.FeeRate(to_add);
                // If that does result in something better, or something with the same feerate but
                // smaller, return that. Even if a longer, higher-feerate intersection exists, it
                // does not hurt to return the shorter one (the remainder of the longer intersection
                // will generally be found in the next call to Intersect, but even if not, it is not
                // required for the improvement guarantee this function makes).
                if (!(accumulator.feerate << subset.feerate)) return accumulator;
            }
        }
        return subset;
    }
};
 
template<typename SetType>
class AncestorCandidateFinder
{
    const DepGraph<SetType>& m_depgraph;
    SetType m_todo;
    std::vector<FeeFrac> m_ancestor_set_feerates;
 
public:
    AncestorCandidateFinder(const DepGraph<SetType>& depgraph LIFETIMEBOUND) noexcept :
        m_depgraph(depgraph),
        m_todo{SetType::Fill(depgraph.TxCount())},
        m_ancestor_set_feerates(depgraph.TxCount())
    {
        // Precompute ancestor-set feerates.
        for (ClusterIndex i = 0; i < depgraph.TxCount(); ++i) {
            SetType anc_to_add = m_depgraph.Ancestors(i);
            FeeFrac anc_feerate;
            // Reuse accumulated feerate from first ancestor, if usable.
            Assume(anc_to_add.Any());
            ClusterIndex first = anc_to_add.First();
            if (first < i) {
                anc_feerate = m_ancestor_set_feerates[first];
                Assume(!anc_feerate.IsEmpty());
                anc_to_add -= m_depgraph.Ancestors(first);
            }
            // Add in other ancestors (which necessarily include i itself).
            Assume(anc_to_add[i]);
            anc_feerate += m_depgraph.FeeRate(anc_to_add);
            // Store the result.
            m_ancestor_set_feerates[i] = anc_feerate;
        }
    }
 
    void MarkDone(SetType select) noexcept
    {
        Assume(select.Any());
        Assume(select.IsSubsetOf(m_todo));
        m_todo -= select;
        for (auto i : select) {
            auto feerate = m_depgraph.FeeRate(i);
            for (auto j : m_depgraph.Descendants(i) & m_todo) {
                m_ancestor_set_feerates[j] -= feerate;
            }
        }
    }
 
    bool AllDone() const noexcept
    {
        return m_todo.None();
    }
 
    SetInfo<SetType> FindCandidateSet() const noexcept
    {
        Assume(!AllDone());
        std::optional<ClusterIndex> best;
        for (auto i : m_todo) {
            if (best.has_value()) {
                Assume(!m_ancestor_set_feerates[i].IsEmpty());
                if (!(m_ancestor_set_feerates[i] > m_ancestor_set_feerates[*best])) continue;
            }
            best = i;
        }
        Assume(best.has_value());
        return {m_depgraph.Ancestors(*best) & m_todo, m_ancestor_set_feerates[*best]};
    }
};
 
template<typename SetType>
class SearchCandidateFinder
{
    InsecureRandomContext m_rng;
    const DepGraph<SetType>& m_depgraph;
    SetType m_todo;
 
public:
    SearchCandidateFinder(const DepGraph<SetType>& depgraph LIFETIMEBOUND, uint64_t rng_seed) noexcept :
        m_rng(rng_seed),
        m_depgraph(depgraph),
        m_todo(SetType::Fill(depgraph.TxCount())) {}
 
    bool AllDone() const noexcept
    {
        return m_todo.None();
    }
 
    std::pair<SetInfo<SetType>, uint64_t> FindCandidateSet(uint64_t max_iterations, SetInfo<SetType> best) noexcept
    {
        Assume(!AllDone());
 
        struct WorkItem
        {
            SetInfo<SetType> inc;
            SetType und;
 
            WorkItem(SetInfo<SetType>&& i, SetType&& u) noexcept :
                inc(std::move(i)), und(std::move(u)) {}
 
            void Swap(WorkItem& other) noexcept
            {
                swap(inc, other.inc);
                swap(und, other.und);
            }
        };
 
        VecDeque<WorkItem> queue;
        queue.reserve(std::max<size_t>(256, 2 * m_todo.Count()));
 
        // Create an initial entry with m_todo as undecided. Also use it as best if not provided,
        // so that during the work processing loop below, and during the add_fn/split_fn calls, we
        // do not need to deal with the best=empty case.
        if (best.feerate.IsEmpty()) best = SetInfo(m_depgraph, m_todo);
        queue.emplace_back(SetInfo<SetType>{}, SetType{m_todo});
 
        uint64_t iterations_left = max_iterations;
 
        auto add_fn = [&](SetInfo<SetType> inc, SetType und) noexcept {
            if (!inc.feerate.IsEmpty()) {
                // If inc's feerate is better than best's, remember it as our new best.
                if (inc.feerate > best.feerate) {
                    best = inc;
                }
            } else {
                Assume(inc.transactions.None());
            }
 
            // Make sure there are undecided transactions left to split on.
            if (und.None()) return;
 
            // Actually construct a new work item on the queue. Due to the switch to DFS when queue
            // space runs out (see below), we know that no reallocation of the queue should ever
            // occur.
            Assume(queue.size() < queue.capacity());
            queue.emplace_back(std::move(inc), std::move(und));
        };
 
        auto split_fn = [&](WorkItem&& elem) noexcept {
            // Any queue element must have undecided transactions left, otherwise there is nothing
            // to explore anymore.
            Assume(elem.und.Any());
            // The included and undecided set are all subsets of m_todo.
            Assume(elem.inc.transactions.IsSubsetOf(m_todo) && elem.und.IsSubsetOf(m_todo));
            // Included transactions cannot be undecided.
            Assume(!elem.inc.transactions.Overlaps(elem.und));
 
            // Pick the first undecided transaction as the one to split on.
            const ClusterIndex split = elem.und.First();
 
            // Add a work item corresponding to exclusion of the split transaction.
            const auto& desc = m_depgraph.Descendants(split);
            add_fn(/*inc=*/elem.inc,
                   /*und=*/elem.und - desc);
 
            // Add a work item corresponding to inclusion of the split transaction.
            const auto anc = m_depgraph.Ancestors(split) & m_todo;
            add_fn(/*inc=*/elem.inc.Add(m_depgraph, anc),
                   /*und=*/elem.und - anc);
 
            // Account for the performed split.
            --iterations_left;
        };
 
        // Work processing loop.
        //
        // New work items are always added at the back of the queue, but items to process use a
        // hybrid approach where they can be taken from the front or the back.
        //
        // Depth-first search (DFS) corresponds to always taking from the back of the queue. This
        // is very memory-efficient (linear in the number of transactions). Breadth-first search
        // (BFS) corresponds to always taking from the front, which potentially uses more memory
        // (up to exponential in the transaction count), but seems to work better in practice.
        //
        // The approach here combines the two: use BFS (plus random swapping) until the queue grows
        // too large, at which point we temporarily switch to DFS until the size shrinks again.
        while (!queue.empty()) {
            // Randomly swap the first two items to randomize the search order.
            if (queue.size() > 1 && m_rng.randbool()) {
                queue[0].Swap(queue[1]);
            }
 
            // Processing the first queue item, and then using DFS for everything it gives rise to,
            // may increase the queue size by the number of undecided elements in there, minus 1
            // for the first queue item being removed. Thus, only when that pushes the queue over
            // its capacity can we not process from the front (BFS), and should we use DFS.
            while (queue.size() - 1 + queue.front().und.Count() > queue.capacity()) {
                if (!iterations_left) break;
                auto elem = queue.back();
                queue.pop_back();
                split_fn(std::move(elem));
            }
 
            // Process one entry from the front of the queue (BFS exploration)
            if (!iterations_left) break;
            auto elem = queue.front();
            queue.pop_front();
            split_fn(std::move(elem));
        }
 
        // Return the found best set and the number of iterations performed.
        return {std::move(best), max_iterations - iterations_left};
    }
 
    void MarkDone(const SetType& done) noexcept
    {
        Assume(done.Any());
        Assume(done.IsSubsetOf(m_todo));
        m_todo -= done;
    }
};
 
template<typename SetType>
std::pair<std::vector<ClusterIndex>, bool> Linearize(const DepGraph<SetType>& depgraph, uint64_t max_iterations, uint64_t rng_seed, Span<const ClusterIndex> old_linearization = {}) noexcept
{
    Assume(old_linearization.empty() || old_linearization.size() == depgraph.TxCount());
    if (depgraph.TxCount() == 0) return {{}, true};
 
    uint64_t iterations_left = max_iterations;
    std::vector<ClusterIndex> linearization;
 
    AncestorCandidateFinder anc_finder(depgraph);
    SearchCandidateFinder src_finder(depgraph, rng_seed);
    linearization.reserve(depgraph.TxCount());
    bool optimal = true;
 
    LinearizationChunking old_chunking(depgraph, old_linearization);
 
    while (true) {
        // Find the highest-feerate prefix of the remainder of old_linearization.
        SetInfo<SetType> best_prefix;
        if (old_chunking.NumChunksLeft()) best_prefix = old_chunking.GetChunk(0);
 
        // Then initialize best to be either the best remaining ancestor set, or the first chunk.
        auto best = anc_finder.FindCandidateSet();
        if (!best_prefix.feerate.IsEmpty() && best_prefix.feerate >= best.feerate) best = best_prefix;
 
        // Invoke bounded search to update best, with up to half of our remaining iterations as
        // limit.
        uint64_t max_iterations_now = (iterations_left + 1) / 2;
        uint64_t iterations_done_now = 0;
        std::tie(best, iterations_done_now) = src_finder.FindCandidateSet(max_iterations_now, best);
        iterations_left -= iterations_done_now;
 
        if (iterations_done_now == max_iterations_now) {
            optimal = false;
            // If the search result is not (guaranteed to be) optimal, run intersections to make
            // sure we don't pick something that makes us unable to reach further diagram points
            // of the old linearization.
            if (old_chunking.NumChunksLeft() > 0) {
                best = old_chunking.IntersectPrefixes(best);
            }
        }
 
        // Add to output in topological order.
        depgraph.AppendTopo(linearization, best.transactions);
 
        // Update state to reflect best is no longer to be linearized.
        anc_finder.MarkDone(best.transactions);
        if (anc_finder.AllDone()) break;
        src_finder.MarkDone(best.transactions);
        if (old_chunking.NumChunksLeft() > 0) {
            old_chunking.MarkDone(best.transactions);
        }
    }
 
    return {std::move(linearization), optimal};
}
 
template<typename SetType>
void PostLinearize(const DepGraph<SetType>& depgraph, Span<ClusterIndex> 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 ClusterIndex SENTINEL{0};
    static constexpr ClusterIndex NO_PREV_TX{0};
 
 
    struct TxEntry
    {
        ClusterIndex prev_tx;
 
        // The fields below are only used for transactions that are the last one in a group
        // (referred to as tail transactions below).
 
        ClusterIndex first_tx;
        ClusterIndex 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(linearization.size() + 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 (ClusterIndex i = 0; i < linearization.size(); ++i) {
            // Even passes are from back to front; odd passes from front to back.
            ClusterIndex 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.
            ClusterIndex 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.
            ClusterIndex next_group = SENTINEL; // We inserted at the end, so next group is sentinel.
            ClusterIndex 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.
                    ClusterIndex 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).
        ClusterIndex cur_group = entries[0].prev_group;
        ClusterIndex done = 0;
        while (cur_group != SENTINEL) {
            ClusterIndex 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());
    }
}
 
template<typename SetType>
std::vector<ClusterIndex> MergeLinearizations(const DepGraph<SetType>& depgraph, Span<const ClusterIndex> lin1, Span<const ClusterIndex> lin2)
{
    Assume(lin1.size() == depgraph.TxCount());
    Assume(lin2.size() == depgraph.TxCount());
 
    LinearizationChunking chunking1(depgraph, lin1), chunking2(depgraph, lin2);
    std::vector<ClusterIndex> ret;
    if (depgraph.TxCount() == 0) return ret;
    ret.reserve(depgraph.TxCount());
 
    while (true) {
        // As long as we are not done, both linearizations must have chunks left.
        Assume(chunking1.NumChunksLeft() > 0);
        Assume(chunking2.NumChunksLeft() > 0);
        // Find the set to output by taking the best remaining chunk, and then intersecting it with
        // prefixes of remaining chunks of the other linearization.
        SetInfo<SetType> best;
        const auto& lin1_firstchunk = chunking1.GetChunk(0);
        const auto& lin2_firstchunk = chunking2.GetChunk(0);
        if (lin2_firstchunk.feerate >> lin1_firstchunk.feerate) {
            best = chunking1.IntersectPrefixes(lin2_firstchunk);
        } else {
            best = chunking2.IntersectPrefixes(lin1_firstchunk);
        }
        // Append the result to the output and mark it as done.
        depgraph.AppendTopo(ret, best.transactions);
        chunking1.MarkDone(best.transactions);
        if (chunking1.NumChunksLeft() == 0) break;
        chunking2.MarkDone(best.transactions);
    }
 
    Assume(ret.size() == depgraph.TxCount());
    return ret;
}
 
} // namespace cluster_linearize
 
#endif // BITCOIN_CLUSTER_LINEARIZE_H