Warshall-Floyd
(graph/warshall_floyd.hpp)

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

Overview

Warshall-Floyd computes shortest paths between every pair of vertices. It keeps a distance matrix and tries each vertex k as an intermediate point, improving dist[i][j] with dist[i][k] + dist[k][j].

Use it when N is small enough for $O(N^3)$ time and you need many shortest path queries after preprocessing. It can handle negative edge costs, but if a negative cycle exists, shortest distances involving that cycle are not well-defined.

For one-source shortest paths on larger graphs, use Dijkstra or Bellman-Ford.

Graph Orientation

Direction is respected. warshall_floyd works on directed graphs as written, and also on undirected graphs built with add_edge.

How to Use It

There are two entry points:

warshall_floyd(g) builds the initial matrix from a Graph<T>.
warshall_floyd(dist) starts from a matrix you prepared yourself and runs the full $O(N^3)$ Floyd-Warshall transition on it.

For a custom matrix, initialize dist[i][i] = 0, unreachable entries to inf, and direct edge costs to their minimum values.

This second overload is useful when your initial distances do not come directly from Graph<T>, for example when you already have a dense matrix or want to set some distances manually before running all-pairs shortest paths.

After running the algorithm, dist[s][t] is the shortest distance from s to t, or inf if t is unreachable from s.

Use has_negative_cycle(dist) after the relaxation. It checks whether some dist[i][i] became negative.

Adding an Edge

After computing an all-pairs distance matrix, adding one edge can be applied in $O(N^2)$.

The input matrix must already be the correct all-pairs shortest distance matrix for the graph before adding the edge.

For a new directed edge from -> to with cost cost, every improved shortest path has the form:

i -> ... -> from -> to -> ... -> j

So the function checks:

dist[i][j] = min(dist[i][j], dist[i][from] + cost + dist[to][j])

Use warshall_floyd_add_directed_edge(dist, from, to, cost) for a directed edge and warshall_floyd_add_undirected_edge(dist, u, v, cost) for an undirected edge. Adding a parallel edge with a smaller cost is the same operation.

These functions modify dist in place and return true if at least one entry changed.

Functions

Function	Signature	Description	Complexity
`warshall_floyd`	`template <class T> std::vector<std::vector<T>> warshall_floyd(const Graph<T>& g, T inf = std::numeric_limits<T>::max() / T(4))`	Builds and relaxes the distance matrix from a graph.	$O(N^3)$
`warshall_floyd`	`template <class T> std::vector<std::vector<T>> warshall_floyd(std::vector<std::vector<T>> dist, T inf = std::numeric_limits<T>::max() / T(4))`	Runs the full Floyd-Warshall transition on a matrix you initialized yourself.	$O(N^3)$
`warshall_floyd_add_directed_edge`	`template <class T> bool warshall_floyd_add_directed_edge(std::vector<std::vector<T>>& dist, int from, int to, T cost, T inf = std::numeric_limits<T>::max() / T(4))`	Adds one directed edge to an already-computed distance matrix.	$O(N^2)$
`warshall_floyd_add_undirected_edge`	`template <class T> bool warshall_floyd_add_undirected_edge(std::vector<std::vector<T>>& dist, int u, int v, T cost, T inf = std::numeric_limits<T>::max() / T(4))`	Adds one undirected edge to an already-computed distance matrix.	$O(N^2)$
`has_negative_cycle`	`template <class T> bool has_negative_cycle(const std::vector<std::vector<T>>& dist)`	Checks whether any diagonal entry is negative.	$O(N)$

Example

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

int main() {
    m1une::graph::Graph<long long> g(3);
    g.add_directed_edge(0, 1, 5);
    g.add_directed_edge(1, 2, 7);
    g.add_directed_edge(0, 2, 20);

    auto dist = m1une::graph::warshall_floyd(g);
    std::cout << dist[0][2] << "\n";  // 12

    m1une::graph::warshall_floyd_add_directed_edge(dist, 0, 2, 4LL);
    std::cout << dist[0][2] << "\n";  // 4
}

Depends on

Graph (graph/graph.hpp)

Required by

Verified with

Code

#ifndef M1UNE_GRAPH_WARSHALL_FLOYD_HPP
#define M1UNE_GRAPH_WARSHALL_FLOYD_HPP 1

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

#include "graph.hpp"

namespace m1une {
namespace graph {

template <class T>
std::vector<std::vector<T>> warshall_floyd(std::vector<std::vector<T>> dist,
                                           T inf = std::numeric_limits<T>::max() / T(4)) {
    int n = int(dist.size());
    for (int k = 0; k < n; k++) {
        for (int i = 0; i < n; i++) {
            if (dist[i][k] == inf) continue;
            for (int j = 0; j < n; j++) {
                if (dist[k][j] == inf) continue;
                T nd = dist[i][k] + dist[k][j];
                if (nd < dist[i][j]) dist[i][j] = nd;
            }
        }
    }
    return dist;
}

template <class T>
std::vector<std::vector<T>> warshall_floyd(const Graph<T>& g, T inf = std::numeric_limits<T>::max() / T(4)) {
    int n = g.size();
    std::vector<std::vector<T>> dist(n, std::vector<T>(n, inf));
    for (int i = 0; i < n; i++) dist[i][i] = T(0);
    for (int v = 0; v < n; v++) {
        for (const auto& e : g[v]) {
            if (!e.alive) continue;
            if (e.cost < dist[e.from][e.to]) dist[e.from][e.to] = e.cost;
        }
    }
    return warshall_floyd(std::move(dist), inf);
}

template <class T>
bool warshall_floyd_add_directed_edge(std::vector<std::vector<T>>& dist, int from, int to, T cost,
                                      T inf = std::numeric_limits<T>::max() / T(4)) {
    int n = int(dist.size());
    assert(0 <= from && from < n);
    assert(0 <= to && to < n);

    std::vector<T> to_from(n), from_to(n);
    for (int i = 0; i < n; i++) {
        to_from[i] = dist[i][from];
        from_to[i] = dist[to][i];
    }

    bool updated = false;
    for (int i = 0; i < n; i++) {
        if (to_from[i] == inf) continue;
        for (int j = 0; j < n; j++) {
            if (from_to[j] == inf) continue;
            T nd = to_from[i] + cost + from_to[j];
            if (nd < dist[i][j]) {
                dist[i][j] = nd;
                updated = true;
            }
        }
    }
    return updated;
}

template <class T>
bool warshall_floyd_add_undirected_edge(std::vector<std::vector<T>>& dist, int u, int v, T cost,
                                        T inf = std::numeric_limits<T>::max() / T(4)) {
    int n = int(dist.size());
    assert(0 <= u && u < n);
    assert(0 <= v && v < n);

    std::vector<T> to_u(n), from_u(n), to_v(n), from_v(n);
    for (int i = 0; i < n; i++) {
        to_u[i] = dist[i][u];
        from_u[i] = dist[u][i];
        to_v[i] = dist[i][v];
        from_v[i] = dist[v][i];
    }

    bool updated = false;
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            if (to_u[i] != inf && from_v[j] != inf) {
                T nd = to_u[i] + cost + from_v[j];
                if (nd < dist[i][j]) {
                    dist[i][j] = nd;
                    updated = true;
                }
            }
            if (to_v[i] != inf && from_u[j] != inf) {
                T nd = to_v[i] + cost + from_u[j];
                if (nd < dist[i][j]) {
                    dist[i][j] = nd;
                    updated = true;
                }
            }
        }
    }
    return updated;
}

template <class T>
bool has_negative_cycle(const std::vector<std::vector<T>>& dist) {
    int n = int(dist.size());
    for (int i = 0; i < n; i++) {
        if (dist[i][i] < T(0)) return true;
    }
    return false;
}

}  // namespace graph
}  // namespace m1une

#endif  // M1UNE_GRAPH_WARSHALL_FLOYD_HPP

#line 1 "graph/warshall_floyd.hpp"



#include <cassert>
#include <limits>
#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 10 "graph/warshall_floyd.hpp"

namespace m1une {
namespace graph {

template <class T>
std::vector<std::vector<T>> warshall_floyd(std::vector<std::vector<T>> dist,
                                           T inf = std::numeric_limits<T>::max() / T(4)) {
    int n = int(dist.size());
    for (int k = 0; k < n; k++) {
        for (int i = 0; i < n; i++) {
            if (dist[i][k] == inf) continue;
            for (int j = 0; j < n; j++) {
                if (dist[k][j] == inf) continue;
                T nd = dist[i][k] + dist[k][j];
                if (nd < dist[i][j]) dist[i][j] = nd;
            }
        }
    }
    return dist;
}

template <class T>
std::vector<std::vector<T>> warshall_floyd(const Graph<T>& g, T inf = std::numeric_limits<T>::max() / T(4)) {
    int n = g.size();
    std::vector<std::vector<T>> dist(n, std::vector<T>(n, inf));
    for (int i = 0; i < n; i++) dist[i][i] = T(0);
    for (int v = 0; v < n; v++) {
        for (const auto& e : g[v]) {
            if (!e.alive) continue;
            if (e.cost < dist[e.from][e.to]) dist[e.from][e.to] = e.cost;
        }
    }
    return warshall_floyd(std::move(dist), inf);
}

template <class T>
bool warshall_floyd_add_directed_edge(std::vector<std::vector<T>>& dist, int from, int to, T cost,
                                      T inf = std::numeric_limits<T>::max() / T(4)) {
    int n = int(dist.size());
    assert(0 <= from && from < n);
    assert(0 <= to && to < n);

    std::vector<T> to_from(n), from_to(n);
    for (int i = 0; i < n; i++) {
        to_from[i] = dist[i][from];
        from_to[i] = dist[to][i];
    }

    bool updated = false;
    for (int i = 0; i < n; i++) {
        if (to_from[i] == inf) continue;
        for (int j = 0; j < n; j++) {
            if (from_to[j] == inf) continue;
            T nd = to_from[i] + cost + from_to[j];
            if (nd < dist[i][j]) {
                dist[i][j] = nd;
                updated = true;
            }
        }
    }
    return updated;
}

template <class T>
bool warshall_floyd_add_undirected_edge(std::vector<std::vector<T>>& dist, int u, int v, T cost,
                                        T inf = std::numeric_limits<T>::max() / T(4)) {
    int n = int(dist.size());
    assert(0 <= u && u < n);
    assert(0 <= v && v < n);

    std::vector<T> to_u(n), from_u(n), to_v(n), from_v(n);
    for (int i = 0; i < n; i++) {
        to_u[i] = dist[i][u];
        from_u[i] = dist[u][i];
        to_v[i] = dist[i][v];
        from_v[i] = dist[v][i];
    }

    bool updated = false;
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            if (to_u[i] != inf && from_v[j] != inf) {
                T nd = to_u[i] + cost + from_v[j];
                if (nd < dist[i][j]) {
                    dist[i][j] = nd;
                    updated = true;
                }
            }
            if (to_v[i] != inf && from_u[j] != inf) {
                T nd = to_v[i] + cost + from_u[j];
                if (nd < dist[i][j]) {
                    dist[i][j] = nd;
                    updated = true;
                }
            }
        }
    }
    return updated;
}

template <class T>
bool has_negative_cycle(const std::vector<std::vector<T>>& dist) {
    int n = int(dist.size());
    for (int i = 0; i < n; i++) {
        if (dist[i][i] < T(0)) return true;
    }
    return false;
}

}  // namespace graph
}  // namespace m1une

m1une's library

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

Warshall-Floyd
(graph/warshall_floyd.hpp)

Overview

Graph Orientation

How to Use It

Adding an Edge

Functions

Example

Depends on

Required by

Verified with

Code

Warshall-Floyd (graph/warshall_floyd.hpp)

Overview

Graph Orientation

How to Use It

Adding an Edge

Functions

Example

Depends on

Required by

Verified with

Code

Warshall-Floyd
(graph/warshall_floyd.hpp)