Range Edge Graph
(graph/range_edge_graph.hpp)

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• Last update: 2024-01-08 13:32:33+09:00
• Include: #include "graph/range_edge_graph.hpp"

Description

セグメント木と同様の構造を用いることで，効率的に区間に辺を張ることができる．

空間計算量: $O(n + m\log n)$, $m$ は張った辺の数

Operations

• RangeEdgeGraph(int n)
  • グラフを $n$ 頂点で初期化する
  • 時間計算量: $O(n)$
• void add_edge(int l1, int r1, int l2, int r2, T w)
  • 区間 $[l_1, r_1)$ から区間 $[l_2, r_2)$ に重み $w$ の辺を張る
  • 時間計算量: $O(\log n)$
• vector<T> dist(int s)
  • 始点 $s$ から各頂点への最短距離を Dijkstra 法で求める
  • 時間計算量: $O((n + m\log n) \log (n + m))$

Note

セグメント木ではなく sparse table の構造を用いて区間に辺を張ることもできる．この場合空間計算量は $O(n\log n + m)$，時間計算量は構築 $O(n\log n)$，辺の追加 $O(1)$，Dijkstra 法 $O((n\log n + m) \log (n\log n + m)))$ となる．特に辺が多いときに高速であるという利点があるが，空間計算量が大きくなるので注意が必要である．Sqrt tree の構造を用いることもできるが狂気である．

Reference

• 区間に辺を貼る一般的なテクニック
• 区間に辺を張るテクの実装例（Dijkstra法セット）と使用例

Depends on

• Shortest Path Algorithms (graph/shortest_path.hpp)

Code

#pragma once
#include <bit>
#include <vector>

#include "shortest_path.hpp"

template <typename T>
class RangeEdgeGraph {
   public:
    RangeEdgeGraph() = default;
    explicit RangeEdgeGraph(int n)
        : size(std::bit_ceil((unsigned int)n)), G(4 * size) {
        for (int i = 1; i < size; ++i) {
            int l = 2 * i, r = 2 * i + 1;
            G[i].emplace_back(l, 0);
            G[i].emplace_back(r, 0);
            G[l + 2 * size].emplace_back(i + 2 * size, 0);
            G[r + 2 * size].emplace_back(i + 2 * size, 0);
        }
        for (int i = size; i < 2 * size; ++i)
            G[i].emplace_back(i + 2 * size, 0);
    }

    void add_edge(int l1, int r1, int l2, int r2, T w) {
        int idx = G.size();
        for (l1 += size, r1 += size; l1 < r1; l1 >>= 1, r1 >>= 1) {
            if (l1 & 1) G[l1 + 2 * size].emplace_back(idx, 0), ++l1;
            if (r1 & 1) --r1, G[r1 + 2 * size].emplace_back(idx, 0);
        }
        std::vector<std::pair<int, T>> node;
        for (l2 += size, r2 += size; l2 < r2; l2 >>= 1, r2 >>= 1) {
            if (l2 & 1) node.emplace_back(l2++, w);
            if (r2 & 1) node.emplace_back(--r2, w);
        }
        G.push_back(node);
    }

    std::vector<T> dist(int s) const {
        auto dist = dijkstra(G, s + size);
        return std::vector<T>(dist.begin() + size, dist.begin() + 2 * size);
    }

   private:
    int size;
    std::vector<std::vector<std::pair<int, T>>> G;
};

/*
Implementation with a sparse table

template <typename T>
class RangeEdgeGraph {
   public:
    RangeEdgeGraph() = default;
    explicit RangeEdgeGraph(int n)
        : n(n), b(std::bit_width((unsigned int)n)), G(2 * n * b) {
        for (int i = 1; i < b; ++i) {
            for (int j = 0; j + (1 << i) <= n; ++j) {
                int k = n * i + j;
                int l = n * (i - 1) + j;
                int r = l + (1 << (i - 1));
                G[k].emplace_back(l, 0);
                G[k].emplace_back(r, 0);
                G[n * b + l].emplace_back(n * b + k, 0);
                G[n * b + r].emplace_back(n * b + k, 0);
            }
        }
        for (int j = 0; j < n; ++j) G[j].emplace_back(n * b + j, 0);
    }

    void add_edge(int l1, int r1, int l2, int r2, T w) {
        int idx = G.size();
        std::vector<std::pair<int, T>> node;
        int i = std::bit_width((unsigned int)(r1 - l1)) - 1;
        G[n * b + n * i + l1].emplace_back(idx, 0);
        G[n * b + n * i + r1 - (1 << i)].emplace_back(idx, 0);

        i = std::bit_width((unsigned int)(r2 - l2)) - 1;
        node.emplace_back(n * i + l2, w);
        node.emplace_back(n * i + r2 - (1 << i), w);
        G.push_back(node);
    }

    std::vector<T> dist(int s) const {
        auto dist = dijkstra(G, s);
        return std::vector<T>(dist.begin(), dist.begin() + n);
    }

   private:
    int n, b;
    std::vector<std::vector<std::pair<int, T>>> G;
};

*/

#line 2 "graph/range_edge_graph.hpp"
#include <bit>
#include <vector>

#line 2 "graph/shortest_path.hpp"
#include <limits>
#include <queue>
#include <tuple>
#include <utility>
#line 7 "graph/shortest_path.hpp"

/*
 * Bellman-Ford Algorithm
 */
template <typename T>
std::vector<T> bellman_ford(const std::vector<std::tuple<int, int, T>>& G,
                            int V, int s) {
    constexpr T INF = std::numeric_limits<T>::max();
    std::vector<T> dist(V, INF);
    dist[s] = 0;
    for (int i = 0; i < V; ++i) {
        for (auto& [s, t, w] : G) {
            if (dist[s] != INF && dist[t] > dist[s] + w) {
                dist[t] = dist[s] + w;
                if (i == V - 1) return {};
            }
        }
    }
    return dist;
}

/*
 * Floyd-Warshall Algorithm
 */
template <typename T>
void floyd_warshall(std::vector<std::vector<T>>& dist) {
    const int V = dist.size();
    for (int k = 0; k < V; ++k) {
        for (int i = 0; i < V; ++i) {
            for (int j = 0; j < V; ++j) {
                dist[i][j] = std::min(dist[i][j], dist[i][k] + dist[k][j]);
            }
        }
    }
}

/*
 * Dijkstra's Algorithm
 */
template <typename T>
std::vector<T> dijkstra(const std::vector<std::vector<std::pair<int, T>>>& G,
                        int s) {
    std::vector<T> dist(G.size(), std::numeric_limits<T>::max());
    dist[s] = 0;
    using P = std::pair<T, int>;
    std::priority_queue<P, std::vector<P>, std::greater<P>> pq;
    pq.emplace(0, s);

    while (!pq.empty()) {
        auto [d, v] = pq.top();
        pq.pop();
        if (dist[v] < d) continue;
        for (auto& [u, w] : G[v]) {
            if (dist[u] > d + w) {
                dist[u] = d + w;
                pq.emplace(dist[u], u);
            }
        }
    }

    return dist;
}

template <typename T>
std::pair<std::vector<T>, std::vector<int>> shortest_path_tree(
    const std::vector<std::vector<std::pair<int, T>>>& G, int s) {
    std::vector<T> dist(G.size(), std::numeric_limits<T>::max());
    std::vector<int> par(G.size(), -1);
    dist[s] = 0;
    using P = std::pair<T, int>;
    std::priority_queue<P, std::vector<P>, std::greater<P>> pq;
    pq.emplace(0, s);

    while (!pq.empty()) {
        auto [d, v] = pq.top();
        pq.pop();
        if (dist[v] < d) continue;
        for (auto& [u, w] : G[v]) {
            if (dist[u] > d + w) {
                dist[u] = d + w;
                par[u] = v;
                pq.emplace(dist[u], u);
            }
        }
    }

    return {dist, par};
}

/*
 * Breadth-First Search
 */
std::vector<int> bfs(const std::vector<std::vector<int>>& G, int s) {
    std::vector<int> dist(G.size(), -1);
    dist[s] = 0;
    std::queue<int> que;
    que.push(s);

    while (!que.empty()) {
        int v = que.front();
        que.pop();
        for (int u : G[v]) {
            if (dist[u] == -1) {
                dist[u] = dist[v] + 1;
                que.push(u);
            }
        }
    }

    return dist;
}

/*
 * Dial's Algorithm
 */
std::vector<int> dial(const std::vector<std::vector<std::pair<int, int>>>& G,
                      int s, int w) {
    std::vector<int> dist(G.size(), std::numeric_limits<int>::max());
    dist[s] = 0;
    std::vector<std::vector<int>> buckets(w * G.size(), std::vector<int>());
    buckets[0].push_back(s);

    for (int d = 0; d < (int)buckets.size(); ++d) {
        while (!buckets[d].empty()) {
            int v = buckets[d].back();
            buckets[d].pop_back();
            if (dist[v] < d) continue;
            for (auto& [u, w] : G[v]) {
                if (dist[u] > d + w) {
                    dist[u] = d + w;
                    buckets[dist[u]].push_back(u);
                }
            }
        }
    }
    return dist;
}
#line 6 "graph/range_edge_graph.hpp"

template <typename T>
class RangeEdgeGraph {
   public:
    RangeEdgeGraph() = default;
    explicit RangeEdgeGraph(int n)
        : size(std::bit_ceil((unsigned int)n)), G(4 * size) {
        for (int i = 1; i < size; ++i) {
            int l = 2 * i, r = 2 * i + 1;
            G[i].emplace_back(l, 0);
            G[i].emplace_back(r, 0);
            G[l + 2 * size].emplace_back(i + 2 * size, 0);
            G[r + 2 * size].emplace_back(i + 2 * size, 0);
        }
        for (int i = size; i < 2 * size; ++i)
            G[i].emplace_back(i + 2 * size, 0);
    }

    void add_edge(int l1, int r1, int l2, int r2, T w) {
        int idx = G.size();
        for (l1 += size, r1 += size; l1 < r1; l1 >>= 1, r1 >>= 1) {
            if (l1 & 1) G[l1 + 2 * size].emplace_back(idx, 0), ++l1;
            if (r1 & 1) --r1, G[r1 + 2 * size].emplace_back(idx, 0);
        }
        std::vector<std::pair<int, T>> node;
        for (l2 += size, r2 += size; l2 < r2; l2 >>= 1, r2 >>= 1) {
            if (l2 & 1) node.emplace_back(l2++, w);
            if (r2 & 1) node.emplace_back(--r2, w);
        }
        G.push_back(node);
    }

    std::vector<T> dist(int s) const {
        auto dist = dijkstra(G, s + size);
        return std::vector<T>(dist.begin() + size, dist.begin() + 2 * size);
    }

   private:
    int size;
    std::vector<std::vector<std::pair<int, T>>> G;
};

/*
Implementation with a sparse table

template <typename T>
class RangeEdgeGraph {
   public:
    RangeEdgeGraph() = default;
    explicit RangeEdgeGraph(int n)
        : n(n), b(std::bit_width((unsigned int)n)), G(2 * n * b) {
        for (int i = 1; i < b; ++i) {
            for (int j = 0; j + (1 << i) <= n; ++j) {
                int k = n * i + j;
                int l = n * (i - 1) + j;
                int r = l + (1 << (i - 1));
                G[k].emplace_back(l, 0);
                G[k].emplace_back(r, 0);
                G[n * b + l].emplace_back(n * b + k, 0);
                G[n * b + r].emplace_back(n * b + k, 0);
            }
        }
        for (int j = 0; j < n; ++j) G[j].emplace_back(n * b + j, 0);
    }

    void add_edge(int l1, int r1, int l2, int r2, T w) {
        int idx = G.size();
        std::vector<std::pair<int, T>> node;
        int i = std::bit_width((unsigned int)(r1 - l1)) - 1;
        G[n * b + n * i + l1].emplace_back(idx, 0);
        G[n * b + n * i + r1 - (1 << i)].emplace_back(idx, 0);

        i = std::bit_width((unsigned int)(r2 - l2)) - 1;
        node.emplace_back(n * i + l2, w);
        node.emplace_back(n * i + r2 - (1 << i), w);
        G.push_back(node);
    }

    std::vector<T> dist(int s) const {
        auto dist = dijkstra(G, s);
        return std::vector<T>(dist.begin(), dist.begin() + n);
    }

   private:
    int n, b;
    std::vector<std::vector<std::pair<int, T>>> G;
};

*/

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