K-Shortest Walk
(graph/k_shortest_walk.hpp)

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Last update: 2026-08-13 01:41:40+09:00
Include: #include "graph/k_shortest_walk.hpp"

Overview

k_shortest_walk lists the lengths of the shortest walks from one vertex to another in nondecreasing order. A walk may repeat vertices and edges, and two different walks are counted separately even when their lengths are equal.

The implementation uses a shortest-path tree toward the target and persistently melded heaps of Eppstein sidetracks. It is intended for large sparse or dense directed graphs with non-negative edge costs.

Requirements and Behavior

T must support construction from integers, addition, subtraction, equality, and ordering.
Every alive edge cost must be non-negative.
Costs of all relevant walks and intermediate sums must fit in T and remain below inf.
Inactive edges are ignored.
If s == t, the empty walk of length zero is the first answer.
If fewer than k walks exist, the returned vector is shorter than k. No sentinel value is appended.

Function

Function	Signature	Description	Complexity
`k_shortest_walk`	`template <class T> std::vector<T> k_shortest_walk(const Graph<T>& g, int s, int t, int k, T inf = std::numeric_limits<T>::max() / T(4))`	Returns up to `k` walk lengths from `s` to `t` in nondecreasing order.	$O((N+M)\log N + M\log M + K\log K)$ time and $O(N+M+N\log M+K)$ memory.

Here M counts alive adjacency entries, so an undirected edge stored by Graph::add_edge contributes two entries. The graph itself is not modified.

Example

#include "graph/graph.hpp"
#include "graph/k_shortest_walk.hpp"
#include <iostream>

int main() {
    m1une::graph::Graph<long long> g(3);
    g.add_directed_edge(0, 1, 2);
    g.add_directed_edge(1, 2, 3);
    g.add_directed_edge(0, 2, 8);
    g.add_directed_edge(1, 1, 1);

    auto lengths = m1une::graph::k_shortest_walk(g, 0, 2, 4);
    for (long long length : lengths) std::cout << length << "\n";
    // 5, 6, 7, 8
}

Depends on

Graph (graph/graph.hpp)

Required by

Verified with

Code

#ifndef M1UNE_GRAPH_K_SHORTEST_WALK_HPP
#define M1UNE_GRAPH_K_SHORTEST_WALK_HPP 1

#include <cassert>
#include <functional>
#include <limits>
#include <queue>
#include <utility>
#include <vector>

#include "graph.hpp"

namespace m1une {
namespace graph {

namespace internal {

template <class T>
class KShortestWalkHeap {
    struct Node {
        T key;
        int to;
        int left;
        int right;
        int rank;
    };

    std::vector<Node> _nodes;

    int rank(int root) const {
        return root == -1 ? 0 : _nodes[root].rank;
    }

   public:
    int make_node(T key, int to) {
        int result = int(_nodes.size());
        _nodes.push_back(Node{key, to, -1, -1, 1});
        return result;
    }

    int meld_mutable(int first, int second) {
        if (first == -1) return second;
        if (second == -1) return first;
        if (_nodes[second].key < _nodes[first].key) std::swap(first, second);
        _nodes[first].right = meld_mutable(_nodes[first].right, second);
        if (rank(_nodes[first].left) < rank(_nodes[first].right)) {
            std::swap(_nodes[first].left, _nodes[first].right);
        }
        _nodes[first].rank = rank(_nodes[first].right) + 1;
        return first;
    }

    int meld_persistent(int first, int second) {
        if (first == -1) return second;
        if (second == -1) return first;
        if (_nodes[second].key < _nodes[first].key) std::swap(first, second);
        int result = int(_nodes.size());
        _nodes.push_back(_nodes[first]);
        _nodes[result].right = meld_persistent(_nodes[result].right, second);
        if (rank(_nodes[result].left) < rank(_nodes[result].right)) {
            std::swap(_nodes[result].left, _nodes[result].right);
        }
        _nodes[result].rank = rank(_nodes[result].right) + 1;
        return result;
    }

    const Node& operator[](int index) const {
        return _nodes[index];
    }
};

}  // namespace internal

template <class T>
std::vector<T> k_shortest_walk(
    const Graph<T>& g,
    int s,
    int t,
    int k,
    T inf = std::numeric_limits<T>::max() / T(4)
) {
    int n = g.size();
    assert(0 <= s && s < n);
    assert(0 <= t && t < n);
    assert(0 <= k);
    if (k == 0) return {};

    struct ReverseEdge {
        int from;
        int index;
        T cost;
    };
    std::vector<std::vector<ReverseEdge>> reverse_graph(n);
    for (int from = 0; from < n; from++) {
        for (int index = 0; index < int(g[from].size()); index++) {
            const auto& edge = g[from][index];
            if (!edge.alive) continue;
            assert(T(0) <= edge.cost);
            reverse_graph[edge.to].push_back(ReverseEdge{from, index, edge.cost});
        }
    }

    std::vector<T> dist(n, inf);
    std::vector<int> tree_edge(n, -1);
    std::vector<int> order;
    order.reserve(n);
    using QueueEntry = std::pair<T, int>;
    std::priority_queue<QueueEntry, std::vector<QueueEntry>, std::greater<QueueEntry>> queue;
    dist[t] = T(0);
    queue.emplace(T(0), t);
    while (!queue.empty()) {
        auto [current_dist, vertex] = queue.top();
        queue.pop();
        if (dist[vertex] != current_dist) continue;
        order.push_back(vertex);
        for (const auto& edge : reverse_graph[vertex]) {
            T next_dist = current_dist + edge.cost;
            if (dist[edge.from] <= next_dist) continue;
            dist[edge.from] = next_dist;
            tree_edge[edge.from] = edge.index;
            queue.emplace(next_dist, edge.from);
        }
    }
    if (dist[s] == inf) return {};

    internal::KShortestWalkHeap<T> heap_pool;
    std::vector<int> local_heap(n, -1);
    for (int vertex : order) {
        for (int index = 0; index < int(g[vertex].size()); index++) {
            const auto& edge = g[vertex][index];
            if (!edge.alive || dist[edge.to] == inf || index == tree_edge[vertex]) continue;
            T extra = edge.cost + dist[edge.to] - dist[vertex];
            assert(T(0) <= extra);
            int node = heap_pool.make_node(extra, edge.to);
            local_heap[vertex] = heap_pool.meld_mutable(local_heap[vertex], node);
        }
    }

    std::vector<int> path_heap(n, -1);
    for (int vertex : order) {
        int inherited = -1;
        if (tree_edge[vertex] != -1) inherited = path_heap[g[vertex][tree_edge[vertex]].to];
        path_heap[vertex] = heap_pool.meld_persistent(inherited, local_heap[vertex]);
    }

    std::vector<T> result;
    result.reserve(k);
    result.push_back(dist[s]);
    std::priority_queue<QueueEntry, std::vector<QueueEntry>, std::greater<QueueEntry>> candidates;
    if (path_heap[s] != -1) {
        candidates.emplace(dist[s] + heap_pool[path_heap[s]].key, path_heap[s]);
    }
    while (int(result.size()) < k && !candidates.empty()) {
        auto [cost, node_index] = candidates.top();
        candidates.pop();
        result.push_back(cost);
        const auto& node = heap_pool[node_index];
        if (node.left != -1) {
            candidates.emplace(cost - node.key + heap_pool[node.left].key, node.left);
        }
        if (node.right != -1) {
            candidates.emplace(cost - node.key + heap_pool[node.right].key, node.right);
        }
        int next_heap = path_heap[node.to];
        if (next_heap != -1) {
            candidates.emplace(cost + heap_pool[next_heap].key, next_heap);
        }
    }
    return result;
}

}  // namespace graph
}  // namespace m1une

#endif  // M1UNE_GRAPH_K_SHORTEST_WALK_HPP

#line 1 "graph/k_shortest_walk.hpp"



#include <cassert>
#include <functional>
#include <limits>
#include <queue>
#include <utility>
#include <vector>

#line 1 "graph/graph.hpp"



#include <array>
#line 8 "graph/graph.hpp"

namespace m1une {
namespace graph {

template <class T = int>
struct Edge {
    using cost_type = T;

    int from;
    int to;
    T cost;
    int id;
    bool alive;

    Edge() : from(-1), to(-1), cost(T()), id(-1), alive(true) {}
    Edge(int from_, int to_, T cost_ = T(1), int id_ = -1, bool alive_ = true)
        : from(from_), to(to_), cost(cost_), id(id_), alive(alive_) {}

    int other(int v) const {
        assert(v == from || v == to);
        return from ^ to ^ v;
    }
};

template <class T = int>
struct Graph {
    using edge_type = Edge<T>;
    using cost_type = T;

   private:
    struct EdgePositions {
        std::array<std::pair<int, int>, 2> value{};
        int size = 0;

        void push_back(std::pair<int, int> position) {
            assert(size < 2);
            value[size++] = position;
        }
    };

    int _n;
    int _edge_count;
    std::vector<std::vector<edge_type>> _g;
    std::vector<EdgePositions> _edge_positions;

   public:
    Graph() : _n(0), _edge_count(0) {}
    explicit Graph(int n) : _n(n), _edge_count(0), _g(n) {
        assert(0 <= n);
    }

    int size() const {
        return _n;
    }

    bool empty() const {
        return _n == 0;
    }

    int edge_count() const {
        return _edge_count;
    }

    int add_vertex() {
        _g.emplace_back();
        return _n++;
    }

    int add_directed_edge(int from, int to, T cost = T(1)) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        int id = _edge_count++;
        int idx = int(_g[from].size());
        _g[from].push_back(edge_type(from, to, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({from, idx});
        return id;
    }

    int add_edge(int u, int v, T cost = T(1)) {
        assert(0 <= u && u < _n);
        assert(0 <= v && v < _n);
        int id = _edge_count++;
        int u_idx = int(_g[u].size());
        _g[u].push_back(edge_type(u, v, cost, id));
        int v_idx = int(_g[v].size());
        _g[v].push_back(edge_type(v, u, cost, id));
        _edge_positions.emplace_back();
        _edge_positions.back().push_back({u, u_idx});
        _edge_positions.back().push_back({v, v_idx});
        return id;
    }

    void set_edge_alive(int id, bool alive) {
        assert(0 <= id && id < _edge_count);
        for (int i = 0; i < _edge_positions[id].size; ++i) {
            auto [v, idx] = _edge_positions[id].value[i];
            _g[v][idx].alive = alive;
        }
    }

    void erase_edge(int id) {
        set_edge_alive(id, false);
    }

    void revive_edge(int id) {
        set_edge_alive(id, true);
    }

    bool is_edge_alive(int id) const {
        assert(0 <= id && id < _edge_count);
        assert(_edge_positions[id].size != 0);
        auto [v, idx] = _edge_positions[id].value[0];
        return _g[v][idx].alive;
    }

    const std::vector<edge_type>& operator[](int v) const {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    std::vector<edge_type>& operator[](int v) {
        assert(0 <= v && v < _n);
        return _g[v];
    }

    const std::vector<std::vector<edge_type>>& adjacency() const {
        return _g;
    }

    std::vector<std::vector<edge_type>>& adjacency() {
        return _g;
    }

    std::vector<edge_type> edges(bool include_inactive = false) const {
        std::vector<edge_type> result;
        result.reserve(_edge_count);
        std::vector<char> used(_edge_count, false);
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                if (!include_inactive && !e.alive) continue;
                if (0 <= e.id && e.id < _edge_count) {
                    if (used[e.id]) continue;
                    used[e.id] = true;
                }
                result.push_back(e);
            }
        }
        return result;
    }

    Graph reversed() const {
        Graph result(_n);
        result._edge_count = _edge_count;
        result._edge_positions.assign(_edge_count, {});
        for (int v = 0; v < _n; v++) {
            for (const auto& e : _g[v]) {
                int idx = int(result._g[e.to].size());
                result._g[e.to].push_back(edge_type(e.to, e.from, e.cost, e.id, e.alive));
                if (0 <= e.id && e.id < _edge_count) result._edge_positions[e.id].push_back({e.to, idx});
            }
        }
        return result;
    }
};

}  // namespace graph
}  // namespace m1une


#line 12 "graph/k_shortest_walk.hpp"

namespace m1une {
namespace graph {

namespace internal {

template <class T>
class KShortestWalkHeap {
    struct Node {
        T key;
        int to;
        int left;
        int right;
        int rank;
    };

    std::vector<Node> _nodes;

    int rank(int root) const {
        return root == -1 ? 0 : _nodes[root].rank;
    }

   public:
    int make_node(T key, int to) {
        int result = int(_nodes.size());
        _nodes.push_back(Node{key, to, -1, -1, 1});
        return result;
    }

    int meld_mutable(int first, int second) {
        if (first == -1) return second;
        if (second == -1) return first;
        if (_nodes[second].key < _nodes[first].key) std::swap(first, second);
        _nodes[first].right = meld_mutable(_nodes[first].right, second);
        if (rank(_nodes[first].left) < rank(_nodes[first].right)) {
            std::swap(_nodes[first].left, _nodes[first].right);
        }
        _nodes[first].rank = rank(_nodes[first].right) + 1;
        return first;
    }

    int meld_persistent(int first, int second) {
        if (first == -1) return second;
        if (second == -1) return first;
        if (_nodes[second].key < _nodes[first].key) std::swap(first, second);
        int result = int(_nodes.size());
        _nodes.push_back(_nodes[first]);
        _nodes[result].right = meld_persistent(_nodes[result].right, second);
        if (rank(_nodes[result].left) < rank(_nodes[result].right)) {
            std::swap(_nodes[result].left, _nodes[result].right);
        }
        _nodes[result].rank = rank(_nodes[result].right) + 1;
        return result;
    }

    const Node& operator[](int index) const {
        return _nodes[index];
    }
};

}  // namespace internal

template <class T>
std::vector<T> k_shortest_walk(
    const Graph<T>& g,
    int s,
    int t,
    int k,
    T inf = std::numeric_limits<T>::max() / T(4)
) {
    int n = g.size();
    assert(0 <= s && s < n);
    assert(0 <= t && t < n);
    assert(0 <= k);
    if (k == 0) return {};

    struct ReverseEdge {
        int from;
        int index;
        T cost;
    };
    std::vector<std::vector<ReverseEdge>> reverse_graph(n);
    for (int from = 0; from < n; from++) {
        for (int index = 0; index < int(g[from].size()); index++) {
            const auto& edge = g[from][index];
            if (!edge.alive) continue;
            assert(T(0) <= edge.cost);
            reverse_graph[edge.to].push_back(ReverseEdge{from, index, edge.cost});
        }
    }

    std::vector<T> dist(n, inf);
    std::vector<int> tree_edge(n, -1);
    std::vector<int> order;
    order.reserve(n);
    using QueueEntry = std::pair<T, int>;
    std::priority_queue<QueueEntry, std::vector<QueueEntry>, std::greater<QueueEntry>> queue;
    dist[t] = T(0);
    queue.emplace(T(0), t);
    while (!queue.empty()) {
        auto [current_dist, vertex] = queue.top();
        queue.pop();
        if (dist[vertex] != current_dist) continue;
        order.push_back(vertex);
        for (const auto& edge : reverse_graph[vertex]) {
            T next_dist = current_dist + edge.cost;
            if (dist[edge.from] <= next_dist) continue;
            dist[edge.from] = next_dist;
            tree_edge[edge.from] = edge.index;
            queue.emplace(next_dist, edge.from);
        }
    }
    if (dist[s] == inf) return {};

    internal::KShortestWalkHeap<T> heap_pool;
    std::vector<int> local_heap(n, -1);
    for (int vertex : order) {
        for (int index = 0; index < int(g[vertex].size()); index++) {
            const auto& edge = g[vertex][index];
            if (!edge.alive || dist[edge.to] == inf || index == tree_edge[vertex]) continue;
            T extra = edge.cost + dist[edge.to] - dist[vertex];
            assert(T(0) <= extra);
            int node = heap_pool.make_node(extra, edge.to);
            local_heap[vertex] = heap_pool.meld_mutable(local_heap[vertex], node);
        }
    }

    std::vector<int> path_heap(n, -1);
    for (int vertex : order) {
        int inherited = -1;
        if (tree_edge[vertex] != -1) inherited = path_heap[g[vertex][tree_edge[vertex]].to];
        path_heap[vertex] = heap_pool.meld_persistent(inherited, local_heap[vertex]);
    }

    std::vector<T> result;
    result.reserve(k);
    result.push_back(dist[s]);
    std::priority_queue<QueueEntry, std::vector<QueueEntry>, std::greater<QueueEntry>> candidates;
    if (path_heap[s] != -1) {
        candidates.emplace(dist[s] + heap_pool[path_heap[s]].key, path_heap[s]);
    }
    while (int(result.size()) < k && !candidates.empty()) {
        auto [cost, node_index] = candidates.top();
        candidates.pop();
        result.push_back(cost);
        const auto& node = heap_pool[node_index];
        if (node.left != -1) {
            candidates.emplace(cost - node.key + heap_pool[node.left].key, node.left);
        }
        if (node.right != -1) {
            candidates.emplace(cost - node.key + heap_pool[node.right].key, node.right);
        }
        int next_heap = path_heap[node.to];
        if (next_heap != -1) {
            candidates.emplace(cost + heap_pool[next_heap].key, next_heap);
        }
    }
    return result;
}

}  // namespace graph
}  // namespace m1une

m1une's library

This documentation is automatically generated by online-judge-tools/verification-helper

K-Shortest Walk
(graph/k_shortest_walk.hpp)

Overview

Requirements and Behavior

Function

Example

Depends on

Required by

Verified with

Code

K-Shortest Walk (graph/k_shortest_walk.hpp)

Overview

Requirements and Behavior

Function

Example

Depends on

Required by

Verified with

Code

K-Shortest Walk
(graph/k_shortest_walk.hpp)