From d0767d3d181401b902ff0e5fb72cbe2bbf70e5ef Mon Sep 17 00:00:00 2001 From: hitonanode <32937551+hitonanode@users.noreply.github.com> Date: Sun, 6 Oct 2024 15:03:01 +0900 Subject: [PATCH 1/2] Add Monge shortest path --- other_algorithms/monge_shortest_path.hpp | 84 +++++++++++++++++++ other_algorithms/monge_shortest_path.md | 47 +++++++++++ .../test/monge_shortest_path.yuki705.test.cpp | 28 +++++++ 3 files changed, 159 insertions(+) create mode 100644 other_algorithms/monge_shortest_path.hpp create mode 100644 other_algorithms/monge_shortest_path.md create mode 100644 other_algorithms/test/monge_shortest_path.yuki705.test.cpp diff --git a/other_algorithms/monge_shortest_path.hpp b/other_algorithms/monge_shortest_path.hpp new file mode 100644 index 00000000..e242198e --- /dev/null +++ b/other_algorithms/monge_shortest_path.hpp @@ -0,0 +1,84 @@ +#pragma once +#include +#include + +// Shortest path of Monge-weighted graph +// Variant of LARSCH Algorithm: https://noshi91.hatenablog.com/entry/2023/02/18/005856 +// Complexity: O(n log n) +// +// Given a directed graph with n vertices and weighted edges +// (w(i, j) = cost_callback(i, j) (i < j)), +// this class calculates the shortest path from vertex 0 to all other vertices. +template struct monge_shortest_path { + std::vector dist; // dist[i] = shortest distance from 0 to i + std::vector amin; // amin[i] = previous vertex of i in the shortest path + + template void _check(int i, int k, F cost_callback) { + if (i <= k) return; + if (Cost c = dist[k] + cost_callback(k, i); c < dist[i]) dist[i] = c, amin[i] = k; + } + + template void _rec_solve(int l, int r, F cost_callback) { + if (r - l == 1) return; + + const int m = (l + r) / 2; + for (int k = amin[l]; k <= amin[r]; ++k) _check(m, k, cost_callback); + + _rec_solve(l, m, cost_callback); + for (int k = l + 1; k <= m; ++k) _check(r, k, cost_callback); + _rec_solve(m, r, cost_callback); + } + + template Cost solve(int n, F cost_callback) { + assert(n > 0); + dist.resize(n); + amin.assign(n, 0); + + dist[0] = Cost(); + for (int i = 1; i < n; ++i) dist[i] = cost_callback(0, i); + + _rec_solve(0, n - 1, cost_callback); + + return dist.back(); + } + + template int num_edges() const { + int ret = 0; + for (int c = (int)amin.size() - 1; c >= 0; c = amin[c]) ++ret; + return ret; + } +}; + +// Find shortest path length from 0 to n - 1 with k edges, min_edges <= k <= max_edges +// https://noshi91.hatenablog.com/entry/2022/01/13/001217 +template +Cost monge_shortest_path_with_specified_edges(int n, int min_edges, int max_edges, + Cost max_abs_cost, F cost_callback) { + + assert(1 <= n); + assert(0 <= min_edges); + assert(min_edges <= max_edges); + assert(max_edges <= n - 1); + + monge_shortest_path msp; + + auto eval = [&](Cost p) -> Cost { + msp.solve(n, [&](int i, int j) { return cost_callback(i, j) - p; }); + return -msp.dist.back() - p * (p < 0 ? max_edges : min_edges); + }; + + Cost lo = -max_abs_cost * 3, hi = max_abs_cost * 3; + + while (lo + 1 < hi) { + Cost p = (lo + hi) / 2, f = eval(p), df = eval(p + 1) - f; + if (df == Cost()) { + return -f; + } else { + (df < Cost() ? lo : hi) = p; + } + } + + Cost flo = eval(lo), fhi = eval(hi); + + return flo < fhi ? -flo : -fhi; +} diff --git a/other_algorithms/monge_shortest_path.md b/other_algorithms/monge_shortest_path.md new file mode 100644 index 00000000..b39e34b8 --- /dev/null +++ b/other_algorithms/monge_shortest_path.md @@ -0,0 +1,47 @@ +--- +title: Shortest path of DAG with Monge weights +documentation_of: ./monge_shortest_path.hpp +--- + +$n$ 頂点の DAG で辺重みが Monge となるようなものに対して最短路長を高速に求める． [1] で紹介されている簡易版 LARSCH Algorithm が実装されていて，計算量は $O(n \log n)$ ． + +また，辺数が `min_edges` 以上 `max_edges` 以下であるようなものに限った最短路長を高速に求めることも可能（計算量にさらに重み二分探索の $\log$ がつく）． + +## 使用方法 + +### 最短路長の計算 + +```cpp +auto f = [&](int s, int t) -> Cost { + // +}; + +monge_shortest_path msp; +Cost ret = msp.solve(n, f); +``` + +### 辺の本数の下限・上限を指定した最短路長の計算 + +```cpp +auto f = [&](int s, int t) -> Cost { + // +}; + +int n; // 頂点数 +int l, r; // 辺の本数が [l, r] の範囲に収まる最短路を見つけたい +Cost max_weight; // f() が返す値の絶対値の上界 + +Cost ret = monge_shortest_path_with_specified_edges(n, l, r, max_weight, f); +``` + +## 問題例 + +- [No.705 ゴミ拾い Hard - yukicoder](https://yukicoder.me/problems/no/705) +- [AtCoder Beginner Contest 218 H - Red and Blue Lamps](https://atcoder.jp/contests/abc218/tasks/abc218_h) +- [東京海上日動プログラミングコンテスト2024（AtCoder Beginner Contest 355） G - Baseball](https://atcoder.jp/contests/abc355/tasks/abc355_g) +- [東北大学プログラミングコンテスト 2022 K - Lebesgue Integral](https://atcoder.jp/contests/tupc2022/tasks/tupc2022_k) + +## Links + +- [1] [簡易版 LARSCH Algorithm - noshi91のメモ](https://noshi91.hatenablog.com/entry/2023/02/18/005856) +- [2] [Aliens DP における二分探索の色々 - noshi91のメモ](https://noshi91.hatenablog.com/entry/2023/11/20/052227#fn-c9578a2a) diff --git a/other_algorithms/test/monge_shortest_path.yuki705.test.cpp b/other_algorithms/test/monge_shortest_path.yuki705.test.cpp new file mode 100644 index 00000000..671f812f --- /dev/null +++ b/other_algorithms/test/monge_shortest_path.yuki705.test.cpp @@ -0,0 +1,28 @@ +#define PROBLEM "https://yukicoder.me/problems/no/705" +#include "../monge_shortest_path.hpp" + +#include +#include +#include +using namespace std; + +int main() { + cin.tie(nullptr), ios::sync_with_stdio(false); + + int N; + cin >> N; + vector A(N), X(N), Y(N); + for (auto &a : A) cin >> a; + for (auto &x : X) cin >> x; + for (auto &y : Y) cin >> y; + + auto weight = [&](int j, int i) { + assert(j < i); + --i; + const long long dx = abs(A.at(i) - X.at(j)), dy = Y.at(j); + return dx * dx * dx + dy * dy * dy; + }; + + monge_shortest_path msp; + cout << msp.solve(N + 1, weight) << '\n'; +} From 2003db0857b9236543e6fac89424f8d89f51edbc Mon Sep 17 00:00:00 2001 From: hitonanode <32937551+hitonanode@users.noreply.github.com> Date: Sun, 6 Oct 2024 15:03:32 +0900 Subject: [PATCH 2/2] add concave max plus convolution test --- .../concave_max_plus_convolution.test.cpp | 22 +++++++++++++++++++ 1 file changed, 22 insertions(+) create mode 100644 other_algorithms/test/concave_max_plus_convolution.test.cpp diff --git a/other_algorithms/test/concave_max_plus_convolution.test.cpp b/other_algorithms/test/concave_max_plus_convolution.test.cpp new file mode 100644 index 00000000..08a1a519 --- /dev/null +++ b/other_algorithms/test/concave_max_plus_convolution.test.cpp @@ -0,0 +1,22 @@ +#define PROBLEM "https://judge.yosupo.jp/problem/min_plus_convolution_convex_arbitrary" + +#include "../smawk.hpp" + +#include +#include +using namespace std; + +int main() { + cin.tie(nullptr); + ios::sync_with_stdio(false); + + int N, M; + cin >> N >> M; + vector A(N), B(M); + for (auto &a : A) cin >> a, a = -a; + for (auto &b : B) cin >> b, b = -b; + + auto ret = concave_max_plus_convolution(B, A); + + for (int i = 0; i < N + M - 1; ++i) cout << -ret[i] << " \n"[i + 1 == N + M - 1]; +}