Bellman-Ford
(graph/bellman_ford.hpp)

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

Overview

Bellman-Ford shortest paths for graphs that may contain negative edge costs. It also marks vertices reachable from a negative cycle that is reachable from the source set.

The algorithm relaxes every edge up to N times. If a distance can still be improved on the N-th relaxation, that improvement must come from a negative cycle. Every vertex reachable from such a cycle is marked as negative.

Use Bellman-Ford when negative edge costs are possible. If all edge costs are non-negative, Dijkstra is much faster.

Graph Orientation

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

How to Use It

Call bellman_ford(g, s) for one source, or bellman_ford(g, sources) for multiple starting vertices.

The result contains these members:

Member	Type / Signature	Meaning
`dist`	`std::vector<T>`	`dist[v]` is the best finite distance found from the source set, or `inf` if unreachable.
`parent`	`std::vector<int>`	`parent[v]` is the previous vertex on the stored path, or `-1`.
`parent_edge`	`std::vector<int>`	`parent_edge[v]` is the edge id used to enter `v`, or `-1`.
`negative`	`std::vector<bool>`	`negative[v]` is whether `v` is reachable from a reachable negative cycle.
`inf`	`T`	The unreachable-distance sentinel used by this run.
`has_negative_cycle`	`bool`	Whether any reachable vertex is affected by a negative cycle.
`reachable`	`bool reachable(int v) const`	Returns whether `v` was reached at all.
`affected_by_negative_cycle`	`bool affected_by_negative_cycle(int v) const`	Returns `negative[v]`.
`path`	`std::vector<int> path(int t) const`	Restores a path to `t`. Requires that `t` is reachable and not negative.

If negative[v] is true, the shortest distance to v is not well-defined because it can be made arbitrarily small.

Functions

Function	Signature	Description	Complexity
`bellman_ford`	`template <class T> BellmanFordResult<T> bellman_ford(const Graph<T>& g, int s, T inf = std::numeric_limits<T>::max() / T(4))`	Runs from one source.	$O(N \cdot M)$
`bellman_ford`	`template <class T> BellmanFordResult<T> bellman_ford(const Graph<T>& g, const std::vector<int>& sources, T inf = std::numeric_limits<T>::max() / T(4))`	Runs from multiple sources.	$O(N \cdot M)$

Example

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

int main() {
    m1une::graph::Graph<long long> g(4);
    g.add_directed_edge(0, 1, 1);
    g.add_directed_edge(1, 2, -3);
    g.add_directed_edge(2, 1, 1);
    g.add_directed_edge(2, 3, 1);

    auto res = m1une::graph::bellman_ford(g, 0);
    std::cout << res.has_negative_cycle << "\n";        // 1
    std::cout << res.affected_by_negative_cycle(3) << "\n";  // 1
}

Depends on

Graph (graph/graph.hpp)

Required by

Verified with

Code

#ifndef M1UNE_GRAPH_BELLMAN_FORD_HPP
#define M1UNE_GRAPH_BELLMAN_FORD_HPP 1

#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <vector>

#include "graph.hpp"

namespace m1une {
namespace graph {

template <class T>
struct BellmanFordResult {
    std::vector<T> dist;
    std::vector<int> parent;
    std::vector<int> parent_edge;
    std::vector<bool> negative;
    T inf;
    bool has_negative_cycle;

    bool reachable(int v) const {
        assert(0 <= v && v < int(dist.size()));
        return dist[v] != inf;
    }

    bool affected_by_negative_cycle(int v) const {
        assert(0 <= v && v < int(negative.size()));
        return negative[v];
    }

    std::vector<int> path(int t) const {
        assert(reachable(t));
        assert(!affected_by_negative_cycle(t));
        std::vector<int> result;
        for (int v = t; v != -1; v = parent[v]) result.push_back(v);
        std::reverse(result.begin(), result.end());
        return result;
    }
};

template <class T>
BellmanFordResult<T> bellman_ford(const Graph<T>& g, const std::vector<int>& sources,
                                  T inf = std::numeric_limits<T>::max() / T(4)) {
    int n = g.size();
    BellmanFordResult<T> result;
    result.dist.assign(n, inf);
    result.parent.assign(n, -1);
    result.parent_edge.assign(n, -1);
    result.negative.assign(n, false);
    result.inf = inf;
    result.has_negative_cycle = false;

    for (int s : sources) {
        assert(0 <= s && s < n);
        result.dist[s] = T(0);
    }

    std::vector<int> relaxed_vertices;
    for (int iter = 0; iter < n; iter++) {
        bool updated = false;
        for (int v = 0; v < n; v++) {
            if (result.dist[v] == inf) continue;
            for (const auto& e : g[v]) {
                if (!e.alive) continue;
                T nd = result.dist[v] + e.cost;
                if (result.dist[e.to] <= nd) continue;
                result.dist[e.to] = nd;
                result.parent[e.to] = v;
                result.parent_edge[e.to] = e.id;
                updated = true;
                if (iter == n - 1) relaxed_vertices.push_back(e.to);
            }
        }
        if (!updated) break;
    }

    std::queue<int> que;
    for (int v : relaxed_vertices) {
        if (result.negative[v]) continue;
        result.negative[v] = true;
        que.push(v);
    }
    while (!que.empty()) {
        int v = que.front();
        que.pop();
        for (const auto& e : g[v]) {
            if (!e.alive) continue;
            if (result.negative[e.to]) continue;
            result.negative[e.to] = true;
            que.push(e.to);
        }
    }

    for (bool x : result.negative) result.has_negative_cycle = result.has_negative_cycle || x;
    return result;
}

template <class T>
BellmanFordResult<T> bellman_ford(const Graph<T>& g, int s, T inf = std::numeric_limits<T>::max() / T(4)) {
    return bellman_ford(g, std::vector<int>{s}, inf);
}

}  // namespace graph
}  // namespace m1une

#endif  // M1UNE_GRAPH_BELLMAN_FORD_HPP

#line 1 "graph/bellman_ford.hpp"



#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <vector>

#line 1 "graph/graph.hpp"



#include <array>
#line 6 "graph/graph.hpp"
#include <utility>
#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 11 "graph/bellman_ford.hpp"

namespace m1une {
namespace graph {

template <class T>
struct BellmanFordResult {
    std::vector<T> dist;
    std::vector<int> parent;
    std::vector<int> parent_edge;
    std::vector<bool> negative;
    T inf;
    bool has_negative_cycle;

    bool reachable(int v) const {
        assert(0 <= v && v < int(dist.size()));
        return dist[v] != inf;
    }

    bool affected_by_negative_cycle(int v) const {
        assert(0 <= v && v < int(negative.size()));
        return negative[v];
    }

    std::vector<int> path(int t) const {
        assert(reachable(t));
        assert(!affected_by_negative_cycle(t));
        std::vector<int> result;
        for (int v = t; v != -1; v = parent[v]) result.push_back(v);
        std::reverse(result.begin(), result.end());
        return result;
    }
};

template <class T>
BellmanFordResult<T> bellman_ford(const Graph<T>& g, const std::vector<int>& sources,
                                  T inf = std::numeric_limits<T>::max() / T(4)) {
    int n = g.size();
    BellmanFordResult<T> result;
    result.dist.assign(n, inf);
    result.parent.assign(n, -1);
    result.parent_edge.assign(n, -1);
    result.negative.assign(n, false);
    result.inf = inf;
    result.has_negative_cycle = false;

    for (int s : sources) {
        assert(0 <= s && s < n);
        result.dist[s] = T(0);
    }

    std::vector<int> relaxed_vertices;
    for (int iter = 0; iter < n; iter++) {
        bool updated = false;
        for (int v = 0; v < n; v++) {
            if (result.dist[v] == inf) continue;
            for (const auto& e : g[v]) {
                if (!e.alive) continue;
                T nd = result.dist[v] + e.cost;
                if (result.dist[e.to] <= nd) continue;
                result.dist[e.to] = nd;
                result.parent[e.to] = v;
                result.parent_edge[e.to] = e.id;
                updated = true;
                if (iter == n - 1) relaxed_vertices.push_back(e.to);
            }
        }
        if (!updated) break;
    }

    std::queue<int> que;
    for (int v : relaxed_vertices) {
        if (result.negative[v]) continue;
        result.negative[v] = true;
        que.push(v);
    }
    while (!que.empty()) {
        int v = que.front();
        que.pop();
        for (const auto& e : g[v]) {
            if (!e.alive) continue;
            if (result.negative[e.to]) continue;
            result.negative[e.to] = true;
            que.push(e.to);
        }
    }

    for (bool x : result.negative) result.has_negative_cycle = result.has_negative_cycle || x;
    return result;
}

template <class T>
BellmanFordResult<T> bellman_ford(const Graph<T>& g, int s, T inf = std::numeric_limits<T>::max() / T(4)) {
    return bellman_ford(g, std::vector<int>{s}, inf);
}

}  // namespace graph
}  // namespace m1une

m1une's library

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Bellman-Ford
(graph/bellman_ford.hpp)

Overview

Graph Orientation

How to Use It

Functions

Example

Depends on

Required by

Verified with

Code

Bellman-Ford (graph/bellman_ford.hpp)

Overview

Graph Orientation

How to Use It

Functions

Example

Depends on

Required by

Verified with

Code

Bellman-Ford
(graph/bellman_ford.hpp)