cluster_linearize.h

Go to the documentation of this file.

// 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 {
 
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;
 
    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, Span<const ClusterIndex> mapping, ClusterIndex pos_range) noexcept : entries(pos_range)
    {
        Assume(mapping.size() == depgraph.PositionRange());
        Assume((pos_range == 0) == (depgraph.TxCount() == 0));
        for (ClusterIndex 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 (ClusterIndex 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; }
    ClusterIndex PositionRange() const noexcept { return entries.size(); }
    auto TxCount() const noexcept { return m_used.Count(); }
    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
    {
        static constexpr auto ALL_POSITIONS = SetType::Fill(SetType::Size());
        auto available = ALL_POSITIONS - m_used;
        Assume(available.Any());
        ClusterIndex 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, ClusterIndex 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(ClusterIndex 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(ClusterIndex 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 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(m_used); }
 
    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)) {}
 
    void Set(const DepGraph<SetType>& depgraph, ClusterIndex 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;
    }
 
    [[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{depgraph.Positions()},
        m_ancestor_set_feerates(depgraph.PositionRange())
    {
        // Precompute ancestor-set feerates.
        for (ClusterIndex i : m_depgraph.Positions()) {
            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();
    }
 
    ClusterIndex NumRemaining() const noexcept
    {
        return m_todo.Count();
    }
 
    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;
    std::vector<ClusterIndex> m_sorted_to_original;
    std::vector<ClusterIndex> m_original_to_sorted;
    DepGraph<SetType> m_sorted_depgraph;
    SetType m_todo;
 
    SetType SortedToOriginal(const SetType& arg) const noexcept
    {
        SetType ret;
        for (auto pos : arg) ret.Set(m_sorted_to_original[pos]);
        return ret;
    }
 
    SetType OriginalToSorted(const SetType& arg) const noexcept
    {
        SetType ret;
        for (auto pos : arg) ret.Set(m_original_to_sorted[pos]);
        return ret;
    }
 
public:
    SearchCandidateFinder(const DepGraph<SetType>& depgraph, uint64_t rng_seed) noexcept :
        m_rng(rng_seed),
        m_sorted_to_original(depgraph.TxCount()),
        m_original_to_sorted(depgraph.PositionRange())
    {
        // Determine reordering mapping, by sorting by decreasing feerate. Unused positions are
        // not included, as they will never be looked up anyway.
        ClusterIndex sorted_pos{0};
        for (auto i : depgraph.Positions()) {
            m_sorted_to_original[sorted_pos++] = i;
        }
        std::sort(m_sorted_to_original.begin(), m_sorted_to_original.end(), [&](auto a, auto b) {
            auto feerate_cmp = depgraph.FeeRate(a) <=> depgraph.FeeRate(b);
            if (feerate_cmp == 0) return a < b;
            return feerate_cmp > 0;
        });
        // Compute reverse mapping.
        for (ClusterIndex i = 0; i < m_sorted_to_original.size(); ++i) {
            m_original_to_sorted[m_sorted_to_original[i]] = i;
        }
        // Compute reordered dependency graph.
        m_sorted_depgraph = DepGraph(depgraph, m_original_to_sorted, m_sorted_to_original.size());
        m_todo = m_sorted_depgraph.Positions();
    }
 
    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());
 
        // Convert the provided best to internal sorted indices.
        best.transactions = OriginalToSorted(best.transactions);
 
        struct WorkItem
        {
            SetInfo<SetType> inc;
            SetType und;
            FeeFrac pot_feerate;
 
            WorkItem(SetInfo<SetType>&& i, SetType&& u, FeeFrac&& p_f) noexcept :
                inc(std::move(i)), und(std::move(u)), pot_feerate(std::move(p_f))
            {
                Assume(pot_feerate.IsEmpty() == inc.feerate.IsEmpty());
            }
 
            void Swap(WorkItem& other) noexcept
            {
                swap(inc, other.inc);
                swap(und, other.und);
                swap(pot_feerate, other.pot_feerate);
            }
        };
 
        VecDeque<WorkItem> queue;
        queue.reserve(std::max<size_t>(256, 2 * m_todo.Count()));
 
        // Create initial entries per connected component of m_todo. While clusters themselves are
        // generally connected, this is not necessarily true after some parts have already been
        // removed from m_todo. Without this, effort can be wasted on searching "inc" sets that
        // span multiple components.
        auto to_cover = m_todo;
        do {
            auto component = m_sorted_depgraph.FindConnectedComponent(to_cover);
            to_cover -= component;
            // If best is not provided, set it to the first component, 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_sorted_depgraph, component);
            queue.emplace_back(/*inc=*/SetInfo<SetType>{},
                               /*und=*/std::move(component),
                               /*pot_feerate=*/FeeFrac{});
        } while (to_cover.Any());
 
        uint64_t iterations_left = max_iterations;
 
        SetType imp = m_todo;
        while (imp.Any()) {
            ClusterIndex check = imp.Last();
            if (m_sorted_depgraph.FeeRate(check) >> best.feerate) break;
            imp.Reset(check);
        }
 
        auto add_fn = [&](SetInfo<SetType> inc, SetType und) noexcept {
            auto pot = inc;
            if (!inc.feerate.IsEmpty()) {
                // Add entries to pot. We iterate over all undecided transactions whose feerate is
                // higher than best. While undecided transactions of lower feerate may improve pot,
                // the resulting pot feerate cannot possibly exceed best's (and this item will be
                // skipped in split_fn anyway).
                for (auto pos : imp & und) {
                    // Determine if adding transaction pos to pot (ignoring topology) would improve
                    // it. If not, we're done updating pot. This relies on the fact that
                    // m_sorted_depgraph, and thus the transactions iterated over, are in decreasing
                    // individual feerate order.
                    if (!(m_sorted_depgraph.FeeRate(pos) >> pot.feerate)) break;
                    pot.Set(m_sorted_depgraph, pos);
                }
 
                // The "jump ahead" optimization: whenever pot has a topologically-valid subset,
                // that subset can be added to inc. Any subset of (pot - inc) has the property that
                // its feerate exceeds that of any set compatible with this work item (superset of
                // inc, subset of (inc | und)). Thus, if T is a topological subset of pot, and B is
                // the best topologically-valid set compatible with this work item, and (T - B) is
                // non-empty, then (T | B) is better than B and also topological. This is in
                // contradiction with the assumption that B is best. Thus, (T - B) must be empty,
                // or T must be a subset of B.
                //
                // See https://delvingbitcoin.org/t/how-to-linearize-your-cluster/303 section 2.4.
                const auto init_inc = inc.transactions;
                for (auto pos : pot.transactions - inc.transactions) {
                    // If the transaction's ancestors are a subset of pot, we can add it together
                    // with its ancestors to inc. Just update the transactions here; the feerate
                    // update happens below.
                    auto anc_todo = m_sorted_depgraph.Ancestors(pos) & m_todo;
                    if (anc_todo.IsSubsetOf(pot.transactions)) inc.transactions |= anc_todo;
                }
                // Finally update und and inc's feerate to account for the added transactions.
                und -= inc.transactions;
                inc.feerate += m_sorted_depgraph.FeeRate(inc.transactions - init_inc);
 
                // If inc's feerate is better than best's, remember it as our new best.
                if (inc.feerate > best.feerate) {
                    best = inc;
                    // See if we can remove any entries from imp now.
                    while (imp.Any()) {
                        ClusterIndex check = imp.Last();
                        if (m_sorted_depgraph.FeeRate(check) >> best.feerate) break;
                        imp.Reset(check);
                    }
                }
 
                // If no potential transactions exist beyond the already included ones, no
                // improvement is possible anymore.
                if (pot.feerate.size == inc.feerate.size) return;
                // At this point und must be non-empty. If it were empty then pot would equal inc.
                Assume(und.Any());
            } else {
                Assume(inc.transactions.None());
                // If inc is empty, we just 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(/*inc=*/std::move(inc),
                               /*und=*/std::move(und),
                               /*pot_feerate=*/std::move(pot.feerate));
        };
 
        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));
            // If pot is empty, then so is inc.
            Assume(elem.inc.feerate.IsEmpty() == elem.pot_feerate.IsEmpty());
 
            const ClusterIndex first = elem.und.First();
            if (!elem.inc.feerate.IsEmpty()) {
                // If no undecided transactions remain with feerate higher than best, this entry
                // cannot be improved beyond best.
                if (!elem.und.Overlaps(imp)) return;
                // We can ignore any queue item whose potential feerate isn't better than the best
                // seen so far.
                if (elem.pot_feerate <= best.feerate) return;
            } else {
                // In case inc is empty use a simpler alternative check.
                if (m_sorted_depgraph.FeeRate(first) <= best.feerate) return;
            }
 
            // Decide which transaction to split on. Splitting is how new work items are added, and
            // how progress is made. One split transaction is chosen among the queue item's
            // undecided ones, and:
            // - A work item is (potentially) added with that transaction plus its remaining
            //   descendants excluded (removed from the und set).
            // - A work item is (potentially) added with that transaction plus its remaining
            //   ancestors included (added to the inc set).
            //
            // To decide what to split on, consider the undecided ancestors of the highest
            // individual feerate undecided transaction. Pick the one which reduces the search space
            // most. Let I(t) be the size of the undecided set after including t, and E(t) the size
            // of the undecided set after excluding t. Then choose the split transaction t such
            // that 2^I(t) + 2^E(t) is minimal, tie-breaking by highest individual feerate for t.
            ClusterIndex split = 0;
            const auto select = elem.und & m_sorted_depgraph.Ancestors(first);
            Assume(select.Any());
            std::optional<std::pair<ClusterIndex, ClusterIndex>> split_counts;
            for (auto t : select) {
                // Call max = max(I(t), E(t)) and min = min(I(t), E(t)). Let counts = {max,min}.
                // Sorting by the tuple counts is equivalent to sorting by 2^I(t) + 2^E(t). This
                // expression is equal to 2^max + 2^min = 2^max * (1 + 1/2^(max - min)). The second
                // factor (1 + 1/2^(max - min)) there is in (1,2]. Thus increasing max will always
                // increase it, even when min decreases. Because of this, we can first sort by max.
                std::pair<ClusterIndex, ClusterIndex> counts{
                    (elem.und - m_sorted_depgraph.Ancestors(t)).Count(),
                    (elem.und - m_sorted_depgraph.Descendants(t)).Count()};
                if (counts.first < counts.second) std::swap(counts.first, counts.second);
                // Remember the t with the lowest counts.
                if (!split_counts.has_value() || counts < *split_counts) {
                    split = t;
                    split_counts = counts;
                }
            }
            // Since there was at least one transaction in select, we must always find one.
            Assume(split_counts.has_value());
 
            // Add a work item corresponding to exclusion of the split transaction.
            const auto& desc = m_sorted_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_sorted_depgraph.Ancestors(split) & m_todo;
            add_fn(/*inc=*/elem.inc.Add(m_sorted_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 (converted to the original transaction indices), and the
        // number of iterations performed.
        best.transactions = SortedToOriginal(best.transactions);
        return {std::move(best), max_iterations - iterations_left};
    }
 
    void MarkDone(const SetType& done) noexcept
    {
        const auto done_sorted = OriginalToSorted(done);
        Assume(done_sorted.Any());
        Assume(done_sorted.IsSubsetOf(m_todo));
        m_todo -= done_sorted;
    }
};
 
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);
    std::optional<SearchCandidateFinder<SetType>> src_finder;
    linearization.reserve(depgraph.TxCount());
    bool optimal = true;
 
    // Treat the initialization of SearchCandidateFinder as taking N^2/64 (rounded up) iterations
    // (largely due to the cost of constructing the internal sorted-by-feerate DepGraph inside
    // SearchCandidateFinder), a rough approximation based on benchmark. If we don't have that
    // many, don't start it.
    uint64_t start_iterations = (uint64_t{depgraph.TxCount()} * depgraph.TxCount() + 63) / 64;
    if (iterations_left > start_iterations) {
        iterations_left -= start_iterations;
        src_finder.emplace(depgraph, rng_seed);
    }
 
    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;
 
        uint64_t iterations_done_now = 0;
        uint64_t max_iterations_now = 0;
        if (src_finder) {
            // Treat the invocation of SearchCandidateFinder::FindCandidateSet() as costing N/4
            // up-front (rounded up) iterations (largely due to the cost of connected-component
            // splitting), a rough approximation based on benchmarks.
            uint64_t base_iterations = (anc_finder.NumRemaining() + 3) / 4;
            if (iterations_left > base_iterations) {
                // Invoke bounded search to update best, with up to half of our remaining
                // iterations as limit.
                iterations_left -= base_iterations;
                max_iterations_now = (iterations_left + 1) / 2;
                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;
        if (src_finder) 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(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 (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