st-Numbering
(graph/st_numbering.hpp)

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

Overview

st_numbering(graph, source, sink) finds a bipolar numbering of an undirected graph. It returns a rank p[v] for every vertex such that:

p is a permutation of 0, ..., N - 1;
p[source] = 0 and p[sink] = N - 1;
every other vertex has an active neighbor of smaller rank and an active neighbor of larger rank.

Orienting every edge from smaller to larger rank then gives, for every vertex, a directed path from source through that vertex to sink. If no such numbering exists, the function returns an empty vector.

Graph Requirements

Build the undirected graph with Graph<T>::add_edge. The graph must contain at least two vertices, and source and sink must be distinct valid vertices.

Parallel edges are supported. Inactive edges and self-loops are ignored. The algorithm is iterative and does not mutate the graph.

API

template <class T>
std::vector<int> st_numbering(
    const Graph<T>& graph,
    int source,
    int sink
);

Function	Description	Complexity
`st_numbering(graph, source, sink)`	Returns vertex ranks, or an empty vector when impossible.	`O(N + M)` time and `O(N)` auxiliary memory

Here M is the number of active undirected edges. The returned vector is indexed by vertex, not by rank: result[v] is the number assigned to v.

Example

#include "graph/graph.hpp"
#include "graph/st_numbering.hpp"

#include <iostream>

int main() {
    m1une::graph::Graph<> graph(4);
    graph.add_edge(0, 1);
    graph.add_edge(1, 2);
    graph.add_edge(2, 3);

    auto rank = m1une::graph::st_numbering(graph, 0, 3);
    for (int vertex = 0; vertex < graph.size(); vertex++) {
        std::cout << rank[vertex] << "\n";
    }
}

Depends on

Graph (graph/graph.hpp)

Required by

Verified with

Code

#ifndef M1UNE_GRAPH_ST_NUMBERING_HPP
#define M1UNE_GRAPH_ST_NUMBERING_HPP 1

#include <cassert>
#include <vector>

#include "graph.hpp"

namespace m1une {
namespace graph {

// Returns ranks p with p[source] = 0 and p[sink] = n - 1 such that every
// other vertex has neighbors of both smaller and larger rank. Returns an empty
// vector when no such numbering exists.
template <class T>
std::vector<int> st_numbering(
    const Graph<T>& graph,
    int source,
    int sink
) {
    const int n = graph.size();
    assert(0 < n);
    assert(0 <= source && source < n);
    assert(0 <= sink && sink < n);
    assert(source != sink);

#ifndef NDEBUG
    std::vector<int> incidence_count(graph.edge_count(), 0);
    for (int vertex = 0; vertex < n; vertex++) {
        for (const Edge<T>& edge : graph[vertex]) {
            if (!edge.alive) continue;
            assert(0 <= edge.id && edge.id < graph.edge_count());
            incidence_count[edge.id]++;
        }
    }
    for (int edge_id = 0; edge_id < graph.edge_count(); edge_id++) {
        if (graph.is_edge_alive(edge_id)) {
            assert(incidence_count[edge_id] == 2);
        }
    }
#endif

    std::vector<int> parent(n, -1);
    std::vector<int> preorder(n, -1);
    std::vector<int> low_vertex(n, -1);
    std::vector<int> next_edge(n, 0);
    std::vector<int> traversal;
    traversal.reserve(n);

    preorder[source] = 0;
    low_vertex[source] = source;
    traversal.push_back(source);
    preorder[sink] = 1;
    low_vertex[sink] = sink;
    traversal.push_back(sink);

    std::vector<int> stack(1, sink);
    while (!stack.empty()) {
        const int vertex = stack.back();
        if (next_edge[vertex] < int(graph[vertex].size())) {
            const Edge<T>& edge = graph[vertex][next_edge[vertex]++];
            if (!edge.alive || edge.to == vertex) continue;
            const int to = edge.to;
            if (preorder[to] == -1) {
                parent[to] = vertex;
                preorder[to] = int(traversal.size());
                low_vertex[to] = to;
                traversal.push_back(to);
                stack.push_back(to);
            } else if (preorder[to] < preorder[low_vertex[vertex]]) {
                low_vertex[vertex] = to;
            }
            continue;
        }

        stack.pop_back();
        const int parent_vertex = parent[vertex];
        if (parent_vertex != -1 &&
            preorder[low_vertex[vertex]] <
                preorder[low_vertex[parent_vertex]]) {
            low_vertex[parent_vertex] = low_vertex[vertex];
        }
    }
    if (int(traversal.size()) != n) return {};

    std::vector<int> next(n, -1);
    std::vector<int> previous(n, -1);
    std::vector<int> sign(n, 0);
    next[source] = sink;
    previous[sink] = source;
    sign[source] = -1;

    for (int index = 2; index < n; index++) {
        const int vertex = traversal[index];
        const int parent_vertex = parent[vertex];
        assert(parent_vertex != -1);
        if (sign[low_vertex[vertex]] == -1) {
            const int before = previous[parent_vertex];
            if (before == -1) return {};
            next[before] = vertex;
            next[vertex] = parent_vertex;
            previous[vertex] = before;
            previous[parent_vertex] = vertex;
            sign[parent_vertex] = 1;
        } else {
            const int after = next[parent_vertex];
            if (after == -1) return {};
            next[parent_vertex] = vertex;
            next[vertex] = after;
            previous[vertex] = parent_vertex;
            previous[after] = vertex;
            sign[parent_vertex] = -1;
        }
    }

    std::vector<int> order;
    order.reserve(n);
    int vertex = source;
    while (vertex != -1 && int(order.size()) <= n) {
        order.push_back(vertex);
        if (vertex == sink) break;
        vertex = next[vertex];
    }
    if (int(order.size()) != n || order.back() != sink) return {};

    std::vector<int> rank(n, -1);
    for (int index = 0; index < n; index++) rank[order[index]] = index;

    for (int index = 0; index < n; index++) {
        const int current = order[index];
        bool has_smaller = false;
        bool has_larger = false;
        for (const Edge<T>& edge : graph[current]) {
            if (!edge.alive || edge.to == current) continue;
            has_smaller = has_smaller || rank[edge.to] < index;
            has_larger = has_larger || index < rank[edge.to];
        }
        if (index > 0 && !has_smaller) return {};
        if (index + 1 < n && !has_larger) return {};
    }
    return rank;
}

}  // namespace graph
}  // namespace m1une

#endif  // M1UNE_GRAPH_ST_NUMBERING_HPP

#line 1 "graph/st_numbering.hpp"



#include <cassert>
#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 8 "graph/st_numbering.hpp"

namespace m1une {
namespace graph {

// Returns ranks p with p[source] = 0 and p[sink] = n - 1 such that every
// other vertex has neighbors of both smaller and larger rank. Returns an empty
// vector when no such numbering exists.
template <class T>
std::vector<int> st_numbering(
    const Graph<T>& graph,
    int source,
    int sink
) {
    const int n = graph.size();
    assert(0 < n);
    assert(0 <= source && source < n);
    assert(0 <= sink && sink < n);
    assert(source != sink);

#ifndef NDEBUG
    std::vector<int> incidence_count(graph.edge_count(), 0);
    for (int vertex = 0; vertex < n; vertex++) {
        for (const Edge<T>& edge : graph[vertex]) {
            if (!edge.alive) continue;
            assert(0 <= edge.id && edge.id < graph.edge_count());
            incidence_count[edge.id]++;
        }
    }
    for (int edge_id = 0; edge_id < graph.edge_count(); edge_id++) {
        if (graph.is_edge_alive(edge_id)) {
            assert(incidence_count[edge_id] == 2);
        }
    }
#endif

    std::vector<int> parent(n, -1);
    std::vector<int> preorder(n, -1);
    std::vector<int> low_vertex(n, -1);
    std::vector<int> next_edge(n, 0);
    std::vector<int> traversal;
    traversal.reserve(n);

    preorder[source] = 0;
    low_vertex[source] = source;
    traversal.push_back(source);
    preorder[sink] = 1;
    low_vertex[sink] = sink;
    traversal.push_back(sink);

    std::vector<int> stack(1, sink);
    while (!stack.empty()) {
        const int vertex = stack.back();
        if (next_edge[vertex] < int(graph[vertex].size())) {
            const Edge<T>& edge = graph[vertex][next_edge[vertex]++];
            if (!edge.alive || edge.to == vertex) continue;
            const int to = edge.to;
            if (preorder[to] == -1) {
                parent[to] = vertex;
                preorder[to] = int(traversal.size());
                low_vertex[to] = to;
                traversal.push_back(to);
                stack.push_back(to);
            } else if (preorder[to] < preorder[low_vertex[vertex]]) {
                low_vertex[vertex] = to;
            }
            continue;
        }

        stack.pop_back();
        const int parent_vertex = parent[vertex];
        if (parent_vertex != -1 &&
            preorder[low_vertex[vertex]] <
                preorder[low_vertex[parent_vertex]]) {
            low_vertex[parent_vertex] = low_vertex[vertex];
        }
    }
    if (int(traversal.size()) != n) return {};

    std::vector<int> next(n, -1);
    std::vector<int> previous(n, -1);
    std::vector<int> sign(n, 0);
    next[source] = sink;
    previous[sink] = source;
    sign[source] = -1;

    for (int index = 2; index < n; index++) {
        const int vertex = traversal[index];
        const int parent_vertex = parent[vertex];
        assert(parent_vertex != -1);
        if (sign[low_vertex[vertex]] == -1) {
            const int before = previous[parent_vertex];
            if (before == -1) return {};
            next[before] = vertex;
            next[vertex] = parent_vertex;
            previous[vertex] = before;
            previous[parent_vertex] = vertex;
            sign[parent_vertex] = 1;
        } else {
            const int after = next[parent_vertex];
            if (after == -1) return {};
            next[parent_vertex] = vertex;
            next[vertex] = after;
            previous[vertex] = parent_vertex;
            previous[after] = vertex;
            sign[parent_vertex] = -1;
        }
    }

    std::vector<int> order;
    order.reserve(n);
    int vertex = source;
    while (vertex != -1 && int(order.size()) <= n) {
        order.push_back(vertex);
        if (vertex == sink) break;
        vertex = next[vertex];
    }
    if (int(order.size()) != n || order.back() != sink) return {};

    std::vector<int> rank(n, -1);
    for (int index = 0; index < n; index++) rank[order[index]] = index;

    for (int index = 0; index < n; index++) {
        const int current = order[index];
        bool has_smaller = false;
        bool has_larger = false;
        for (const Edge<T>& edge : graph[current]) {
            if (!edge.alive || edge.to == current) continue;
            has_smaller = has_smaller || rank[edge.to] < index;
            has_larger = has_larger || index < rank[edge.to];
        }
        if (index > 0 && !has_smaller) return {};
        if (index + 1 < n && !has_larger) return {};
    }
    return rank;
}

}  // namespace graph
}  // namespace m1une

m1une's library

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

st-Numbering
(graph/st_numbering.hpp)

Overview

Graph Requirements

API

Example

Depends on

Required by

Verified with

Code

st-Numbering (graph/st_numbering.hpp)

Overview

Graph Requirements

API

Example

Depends on

Required by

Verified with

Code

st-Numbering
(graph/st_numbering.hpp)