algorithm

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

View the Project on GitHub drken1215/algorithm

:heavy_check_mark: monge_shortest_path_by_cht
(Optimization/monge_shortest_path_by_cht.hpp)

概要

General Li Chao Tree を用いた Monge 単一始点最短路アルゴリズム。

仕様

制約

問題例

参考文献

Depends on

Verified with

Code

#pragma once
#include <bits/stdc++.h>
#include <cassert>
using namespace std;

#include "general_li_chao_tree.hpp"

// find shortest path on DAG with Pairwise Single Crossing cost, by Li Chao Tree, O(N log N)
// vertex: 0, 1, 2, ..., N
// A(j, i) = dp[i] + f(i, j) must be Pairwise Single Crossing (must be in all 0 <= i, j <= N, including i >= j)
// Attention: Even if f is Pairwise Single Crossing, A is not necessarily Pairwise Single Crossing
// edges: (i, j), j >= start[i] (i + 1 <= start[i] <= N + 1), start is non-decreasing
template<class VAL, class FUNC> vector<pair<VAL, int>> MongeShortestPathByCHT(int N, const FUNC &f, const vector<int> &start) {
    assert((int)start.size() == N);
    for (int i = 0; i < N; i++) {
        assert(i + 1 <= start[i] && start[i] <= N + 1);
        if (i > 0) assert(start[i - 1] <= start[i]);
    }
    VAL INF = numeric_limits<VAL>::max() / 2;
    vector<pair<VAL, int>> dp(N + 1, make_pair(numeric_limits<VAL>::max() / 2, -1));
    auto make_func = [&](int i) {
        return [&, i](long long j) -> VAL { return dp[i].first + f(i, j); };
    };
    DynamicGeneralLiChaoTree<VAL, decltype(make_func(0))> cht(1, N + 1, INF);
    dp[0].first = 0;
    int i = 0;
    for (int j = 1; j <= N; j++) {
        while (i < N && j >= start[i]) {
            cht.insert(make_func(i));
            i++;
        }
        dp[j] = cht.query(j);
    };
    return dp;
}
template<class VAL, class FUNC> vector<pair<VAL, int>> MongeShortestPathByCHT(int N, const FUNC &f) {
    vector<int> start(N);
    for (int i = 0; i < N; i++) start[i] = i + 1;
    return MongeShortestPathByCHT<VAL>(N, f, start);
}
#line 2 "Optimization/monge_shortest_path_by_cht.hpp"
#include <bits/stdc++.h>
#line 4 "Optimization/monge_shortest_path_by_cht.hpp"
using namespace std;

#line 4 "Optimization/general_li_chao_tree.hpp"
using namespace std;


// Dynamic Li Chao Tree for Pairwise Single Crossing Functions
/*
    Func: Pairwise Single Crossing 性を仮定
    注意点：i < j なる領域だけでなく、全領域での Pairwise Single Crossing 性が必要！！！

    x 座標のとりうる値の範囲: [XL, XR)
*/
template<class VAL, class FUNC, class X = long long> struct DynamicGeneralLiChaoTree {
    using Func = function<VAL(X)>;
    struct IndexedFunc {
        Func f;
        int id;
        VAL eval(X x) const { return f(x); };
    };
    struct Node {
        optional<IndexedFunc> func;
        unique_ptr<Node> left, right;
    };
    
    // inner values
    X XL, XR;
    VAL INF = numeric_limits<VAL>::max() / 2;
    int next_id = 0;
    unique_ptr<Node> root;
    
    // constructors
    DynamicGeneralLiChaoTree() {}
    DynamicGeneralLiChaoTree(X xl, X xr, VAL inf = numeric_limits<VAL>::max() / 2) : XL(xl), XR(xr), INF(inf) {
        assert(xl < xr);
    }
    static bool better(const IndexedFunc &a, const IndexedFunc &b, X x) {
        return make_pair(a.eval(x), a.id) < make_pair(b.eval(x), b.id);
    }
    void clear() {
        root.reset();
        next_id = 0;
    }

    // insert(f), in O(log X)
    void insert(unique_ptr<Node> &p, X left, X right, IndexedFunc f) {
        if (!p) p = make_unique<Node>();
        if (!p->func) {
            p->func.emplace(f);
            return;
        }
        X mid = (left + right) / 2;
        bool left_better = better(f, *p->func, left);
        bool right_better = better(f, *p->func, right - 1);

        if (!left_better && !right_better) return;
        if (left_better && right_better) {
            p->func.emplace(f);
            return;
        }
        bool mid_better = better(f, *p->func, mid);
        if (mid_better) {
            IndexedFunc old = std::move(*p->func);
            p->func.emplace(f);
            if (left_better != mid_better) insert(p->left, left, mid, old);
            else insert(p->right, mid, right, old);
        } else {
            if (left_better != mid_better) insert(p->left, left, mid, f);
            else insert(p->right, mid, right, f);
        }
    }
    int insert(Func f) {
        int id = next_id++;
        insert(root, XL, XR, IndexedFunc{std::move(f), id});
        return id;
    }

    // insert_segment(l, r, f), in O(log^2 X)
    void insert_segment(unique_ptr<Node> &p, X left, X right, X l, X r, const IndexedFunc &f) {
        if (r <= left || right <= l) return;
        if (l <= left && right <= r) {
            insert(p, left, right, f);
            return;
        }
        if (!p) p = make_unique<Node>();
        X mid = (left + right) / 2;
        insert_segment(p->left, left, mid, l, r, f);
        insert_segment(p->right, mid, right, l, r, f);
    }
    int insert_segment(X l, X r, Func f) {
        assert(XL <= l && l <= r && r <= XR);
        int id = next_id++;
        if (l != r) insert_segment(root, XL, XR, l, r, IndexedFunc{std::move(f), id});
        return id;
    }

    // query(x), in O(log X)
    pair<VAL, int> query(X x) const {
        assert(XL <= x && x < XR);
        pair<VAL, int> res{INF, -1};
        const Node *p = root.get();
        X left = XL, right = XR;
        while (p) {
            if (p->func) {
                auto cur = make_pair(p->func->eval(x), p->func->id);
                if (res.second == -1 || cur < res) res = cur;
            }
            if (right - left == 1) break;
            X mid = (left + right) / 2;
            if (x < mid) {
                p = p->left.get();
                right = mid;
            } else {
                p = p->right.get();
                left = mid;
            }
        }
        return res;
    }
};

#line 7 "Optimization/monge_shortest_path_by_cht.hpp"

// find shortest path on DAG with Pairwise Single Crossing cost, by Li Chao Tree, O(N log N)
// vertex: 0, 1, 2, ..., N
// A(j, i) = dp[i] + f(i, j) must be Pairwise Single Crossing (must be in all 0 <= i, j <= N, including i >= j)
// Attention: Even if f is Pairwise Single Crossing, A is not necessarily Pairwise Single Crossing
// edges: (i, j), j >= start[i] (i + 1 <= start[i] <= N + 1), start is non-decreasing
template<class VAL, class FUNC> vector<pair<VAL, int>> MongeShortestPathByCHT(int N, const FUNC &f, const vector<int> &start) {
    assert((int)start.size() == N);
    for (int i = 0; i < N; i++) {
        assert(i + 1 <= start[i] && start[i] <= N + 1);
        if (i > 0) assert(start[i - 1] <= start[i]);
    }
    VAL INF = numeric_limits<VAL>::max() / 2;
    vector<pair<VAL, int>> dp(N + 1, make_pair(numeric_limits<VAL>::max() / 2, -1));
    auto make_func = [&](int i) {
        return [&, i](long long j) -> VAL { return dp[i].first + f(i, j); };
    };
    DynamicGeneralLiChaoTree<VAL, decltype(make_func(0))> cht(1, N + 1, INF);
    dp[0].first = 0;
    int i = 0;
    for (int j = 1; j <= N; j++) {
        while (i < N && j >= start[i]) {
            cht.insert(make_func(i));
            i++;
        }
        dp[j] = cht.query(j);
    };
    return dp;
}
template<class VAL, class FUNC> vector<pair<VAL, int>> MongeShortestPathByCHT(int N, const FUNC &f) {
    vector<int> start(N);
    for (int i = 0; i < N; i++) start[i] = i + 1;
    return MongeShortestPathByCHT<VAL>(N, f, start);
}
Back to top page