minimum_mean_cycle.hpp¶
求有向带权图中平均边权最小的环；先拆强连通分量，再做 Karp DP，单环和等权分量直接处理。返回浮点平均值，无环时为空。
Complexity: Time: \(O(V + E + \sum (\mathrm{Vc} \cdot \mathrm{Ec}))\) over nontrivial cyclic SCCs. Space: \(O(V + E + \max (\mathrm{Vc}^{2}))\), including \(O(V)\) recursive SCC depth.
Implementation¶
当前头文件，省略 include guard；依赖见 #include。
/// @complexity Time: O(V+E+sum(Vc*Ec)) over nontrivial cyclic SCCs.
/// Space: O(V+E+max(Vc^2)), including O(V) recursive SCC depth.
#include "atcoder/internal_scc.hpp"
#include <algorithm>
#include <cassert>
#include <cmath>
#include <limits>
#include <optional>
#include <tuple>
#include <utility>
#include <vector>
namespace noya {
/// @brief Minimum mean weight of a directed cycle; nullopt for an acyclic graph.
/// Karp's dynamic program runs separately inside each strongly connected
/// component. Weights must convert to finite long double; intermediate sums
/// and differences must remain finite. Precision follows that type.
/// Loops and parallel edges are allowed.
template <class Weight>
std::optional<long double>
minimum_mean_cycle(int n, const std::vector<std::tuple<int, int, Weight>> &es) {
assert(n >= 0);
const long double inf = std::numeric_limits<long double>::infinity();
long double answer = inf;
struct edge { int from, to; long double cost; };
std::vector<edge> edges;
edges.reserve(es.size());
for (auto [u, v, w] : es) {
assert(0 <= u && u < n && 0 <= v && v < n);
long double cost = static_cast<long double>(w);
assert(std::isfinite(cost));
if (u == v) answer = std::min(answer, cost);
else edges.push_back({u, v, cost});
}
auto result = [&]() -> std::optional<long double> {
if (answer == inf) return std::nullopt;
return answer;
};
if (edges.empty()) return result();
auto [count, id] = [&] {
atcoder::internal::scc_graph graph(n);
for (auto e : edges) graph.add_edge(e.from, e.to);
return graph.scc_ids();
}();
std::vector<int> size(count), pos(n);
for (int v = 0; v < n; ++v) pos[v] = size[id[v]]++;
std::vector<std::vector<edge>> groups(count);
if (count == 1) {
groups[0] = std::move(edges);
} else {
for (auto e : edges) if (id[e.from] == id[e.to])
groups[id[e.from]].push_back({pos[e.from], pos[e.to], e.cost});
std::vector<edge>().swap(edges);
}
for (int c = 0; c < count; ++c) {
const auto &arcs = groups[c];
if (arcs.empty()) continue;
int s = size[c];
long double low = inf, high = -inf;
for (auto e : arcs) {
low = std::min(low, e.cost);
high = std::max(high, e.cost);
}
if (low >= answer) continue;
// A strongly connected graph with s arcs is one directed cycle.
if (arcs.size() == std::size_t(s) || low == high) {
long double mean = low;
if (low != high) {
long double sum = 0;
for (auto e : arcs) sum += e.cost;
mean = sum / s;
}
answer = std::min(answer, mean);
continue;
}
std::vector dis(s + 1, std::vector<long double>(s, inf));
std::fill(dis[0].begin(), dis[0].end(), 0);
for (int len = 1; len <= s; ++len)
for (auto e : arcs)
dis[len][e.to] = std::min(dis[len][e.to],
dis[len-1][e.from] + e.cost);
std::vector<long double> maximum(s, -inf);
// All vertices are reachable at every length from the implicit zero-cost
// source. Scan complete rows, avoiding a strided read down each column.
for (int len = 0; len < s; ++len)
for (int v = 0; v < s; ++v)
maximum[v] = std::max(maximum[v],
(dis[s][v] - dis[len][v]) / (s - len));
answer = std::min(answer, *std::min_element(maximum.begin(), maximum.end()));
}
return result();
}
} // namespace noya
// NOYA_MINIMUM_MEAN_CYCLE_HPP
#ifndef NOYA_MINIMUM_MEAN_CYCLE_HPP
#define NOYA_MINIMUM_MEAN_CYCLE_HPP 1
/// @complexity Time: O(V+E+sum(Vc*Ec)) over nontrivial cyclic SCCs.
/// Space: O(V+E+max(Vc^2)), including O(V) recursive SCC depth.
#include "atcoder/internal_scc.hpp"
#include <algorithm>
#include <cassert>
#include <cmath>
#include <limits>
#include <optional>
#include <tuple>
#include <utility>
#include <vector>
namespace noya {
/// @brief Minimum mean weight of a directed cycle; nullopt for an acyclic graph.
/// Karp's dynamic program runs separately inside each strongly connected
/// component. Weights must convert to finite long double; intermediate sums
/// and differences must remain finite. Precision follows that type.
/// Loops and parallel edges are allowed.
template <class Weight>
std::optional<long double>
minimum_mean_cycle(int n, const std::vector<std::tuple<int, int, Weight>> &es) {
assert(n >= 0);
const long double inf = std::numeric_limits<long double>::infinity();
long double answer = inf;
struct edge { int from, to; long double cost; };
std::vector<edge> edges;
edges.reserve(es.size());
for (auto [u, v, w] : es) {
assert(0 <= u && u < n && 0 <= v && v < n);
long double cost = static_cast<long double>(w);
assert(std::isfinite(cost));
if (u == v) answer = std::min(answer, cost);
else edges.push_back({u, v, cost});
}
auto result = [&]() -> std::optional<long double> {
if (answer == inf) return std::nullopt;
return answer;
};
if (edges.empty()) return result();
auto [count, id] = [&] {
atcoder::internal::scc_graph graph(n);
for (auto e : edges) graph.add_edge(e.from, e.to);
return graph.scc_ids();
}();
std::vector<int> size(count), pos(n);
for (int v = 0; v < n; ++v) pos[v] = size[id[v]]++;
std::vector<std::vector<edge>> groups(count);
if (count == 1) {
groups[0] = std::move(edges);
} else {
for (auto e : edges) if (id[e.from] == id[e.to])
groups[id[e.from]].push_back({pos[e.from], pos[e.to], e.cost});
std::vector<edge>().swap(edges);
}
for (int c = 0; c < count; ++c) {
const auto &arcs = groups[c];
if (arcs.empty()) continue;
int s = size[c];
long double low = inf, high = -inf;
for (auto e : arcs) {
low = std::min(low, e.cost);
high = std::max(high, e.cost);
}
if (low >= answer) continue;
// A strongly connected graph with s arcs is one directed cycle.
if (arcs.size() == std::size_t(s) || low == high) {
long double mean = low;
if (low != high) {
long double sum = 0;
for (auto e : arcs) sum += e.cost;
mean = sum / s;
}
answer = std::min(answer, mean);
continue;
}
std::vector dis(s + 1, std::vector<long double>(s, inf));
std::fill(dis[0].begin(), dis[0].end(), 0);
for (int len = 1; len <= s; ++len)
for (auto e : arcs)
dis[len][e.to] = std::min(dis[len][e.to],
dis[len-1][e.from] + e.cost);
std::vector<long double> maximum(s, -inf);
// All vertices are reachable at every length from the implicit zero-cost
// source. Scan complete rows, avoiding a strided read down each column.
for (int len = 0; len < s; ++len)
for (int v = 0; v < s; ++v)
maximum[v] = std::max(maximum[v],
(dis[s][v] - dis[len][v]) / (s - len));
answer = std::min(answer, *std::min_element(maximum.begin(), maximum.end()));
}
return result();
}
} // namespace noya
#endif // NOYA_MINIMUM_MEAN_CYCLE_HPP
/// @complexity Time: O(V+E+sum(Vc*Ec)) over nontrivial cyclic SCCs.
/// Space: O(V+E+max(Vc^2)), including O(V) recursive SCC depth.
#include <algorithm>
#include <utility>
#include <vector>
#include <algorithm>
#include <utility>
#include <vector>
namespace atcoder {
namespace internal {
template <class E> struct csr {
std::vector<int> start;
std::vector<E> elist;
explicit csr(int n, const std::vector<std::pair<int, E>>& edges)
: start(n + 1), elist(edges.size()) {
for (auto e : edges) {
start[e.first + 1]++;
}
for (int i = 1; i <= n; i++) {
start[i] += start[i - 1];
}
auto counter = start;
for (auto e : edges) {
elist[counter[e.first]++] = e.second;
}
}
};
} // namespace internal
} // namespace atcoder
namespace atcoder {
namespace internal {
// Reference:
// R. Tarjan,
// Depth-First Search and Linear Graph Algorithms
struct scc_graph {
public:
explicit scc_graph(int n) : _n(n) {}
int num_vertices() { return _n; }
void add_edge(int from, int to) { edges.push_back({from, {to}}); }
// @return pair of (# of scc, scc id)
std::pair<int, std::vector<int>> scc_ids() {
auto g = csr<edge>(_n, edges);
int now_ord = 0, group_num = 0;
std::vector<int> visited, low(_n), ord(_n, -1), ids(_n);
visited.reserve(_n);
auto dfs = [&](auto self, int v) -> void {
low[v] = ord[v] = now_ord++;
visited.push_back(v);
for (int i = g.start[v]; i < g.start[v + 1]; i++) {
auto to = g.elist[i].to;
if (ord[to] == -1) {
self(self, to);
low[v] = std::min(low[v], low[to]);
} else {
low[v] = std::min(low[v], ord[to]);
}
}
if (low[v] == ord[v]) {
while (true) {
int u = visited.back();
visited.pop_back();
ord[u] = _n;
ids[u] = group_num;
if (u == v) break;
}
group_num++;
}
};
for (int i = 0; i < _n; i++) {
if (ord[i] == -1) dfs(dfs, i);
}
for (auto& x : ids) {
x = group_num - 1 - x;
}
return {group_num, ids};
}
std::vector<std::vector<int>> scc() {
auto ids = scc_ids();
int group_num = ids.first;
std::vector<int> counts(group_num);
for (auto x : ids.second) counts[x]++;
std::vector<std::vector<int>> groups(ids.first);
for (int i = 0; i < group_num; i++) {
groups[i].reserve(counts[i]);
}
for (int i = 0; i < _n; i++) {
groups[ids.second[i]].push_back(i);
}
return groups;
}
private:
int _n;
struct edge {
int to;
};
std::vector<std::pair<int, edge>> edges;
};
} // namespace internal
} // namespace atcoder
#include <algorithm>
#include <cassert>
#include <cmath>
#include <limits>
#include <optional>
#include <tuple>
#include <utility>
#include <vector>
namespace noya {
/// @brief Minimum mean weight of a directed cycle; nullopt for an acyclic graph.
/// Karp's dynamic program runs separately inside each strongly connected
/// component. Weights must convert to finite long double; intermediate sums
/// and differences must remain finite. Precision follows that type.
/// Loops and parallel edges are allowed.
template <class Weight>
std::optional<long double>
minimum_mean_cycle(int n, const std::vector<std::tuple<int, int, Weight>> &es) {
assert(n >= 0);
const long double inf = std::numeric_limits<long double>::infinity();
long double answer = inf;
struct edge { int from, to; long double cost; };
std::vector<edge> edges;
edges.reserve(es.size());
for (auto [u, v, w] : es) {
assert(0 <= u && u < n && 0 <= v && v < n);
long double cost = static_cast<long double>(w);
assert(std::isfinite(cost));
if (u == v) answer = std::min(answer, cost);
else edges.push_back({u, v, cost});
}
auto result = [&]() -> std::optional<long double> {
if (answer == inf) return std::nullopt;
return answer;
};
if (edges.empty()) return result();
auto [count, id] = [&] {
atcoder::internal::scc_graph graph(n);
for (auto e : edges) graph.add_edge(e.from, e.to);
return graph.scc_ids();
}();
std::vector<int> size(count), pos(n);
for (int v = 0; v < n; ++v) pos[v] = size[id[v]]++;
std::vector<std::vector<edge>> groups(count);
if (count == 1) {
groups[0] = std::move(edges);
} else {
for (auto e : edges) if (id[e.from] == id[e.to])
groups[id[e.from]].push_back({pos[e.from], pos[e.to], e.cost});
std::vector<edge>().swap(edges);
}
for (int c = 0; c < count; ++c) {
const auto &arcs = groups[c];
if (arcs.empty()) continue;
int s = size[c];
long double low = inf, high = -inf;
for (auto e : arcs) {
low = std::min(low, e.cost);
high = std::max(high, e.cost);
}
if (low >= answer) continue;
// A strongly connected graph with s arcs is one directed cycle.
if (arcs.size() == std::size_t(s) || low == high) {
long double mean = low;
if (low != high) {
long double sum = 0;
for (auto e : arcs) sum += e.cost;
mean = sum / s;
}
answer = std::min(answer, mean);
continue;
}
std::vector dis(s + 1, std::vector<long double>(s, inf));
std::fill(dis[0].begin(), dis[0].end(), 0);
for (int len = 1; len <= s; ++len)
for (auto e : arcs)
dis[len][e.to] = std::min(dis[len][e.to],
dis[len-1][e.from] + e.cost);
std::vector<long double> maximum(s, -inf);
// All vertices are reachable at every length from the implicit zero-cost
// source. Scan complete rows, avoiding a strided read down each column.
for (int len = 0; len < s; ++len)
for (int v = 0; v < s; ++v)
maximum[v] = std::max(maximum[v],
(dis[s][v] - dis[len][v]) / (s - len));
answer = std::min(answer, *std::min_element(maximum.begin(), maximum.end()));
}
return result();
}
} // namespace noya
// NOYA_MINIMUM_MEAN_CYCLE_HPP