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#include "other_algorithms/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).
auto f = [&](int s, int t) -> Cost {
//
};
monge_shortest_path<Cost> msp;
Cost ret = msp.solve(n, f);
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);
#pragma once
#include <cassert>
#include <vector>
// 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 <class Cost> struct monge_shortest_path {
std::vector<Cost> dist; // dist[i] = shortest distance from 0 to i
std::vector<int> amin; // amin[i] = previous vertex of i in the shortest path
template <class F> 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 <class F> 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 <class F> 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 <class F> 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 <class Cost, class F>
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<Cost> 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;
}
#line 2 "other_algorithms/monge_shortest_path.hpp"
#include <cassert>
#include <vector>
// 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 <class Cost> struct monge_shortest_path {
std::vector<Cost> dist; // dist[i] = shortest distance from 0 to i
std::vector<int> amin; // amin[i] = previous vertex of i in the shortest path
template <class F> 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 <class F> 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 <class F> 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 <class F> 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 <class Cost, class F>
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<Cost> 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;
}