gomory_hu_tree.hpp¶
用至多 \(n-1\) 次最大流构造任意两点最小割树；复用残量网络，跳过孤立点，查询省去全树数组初始化。
Complexity: Time: \(O(V + E + K^{2} + K \cdot E)\) plus \(K - 1\) maximum-flow computations on \(K\) vertices; \(K\) includes vertex 0 and nonisolated vertices. Query \(O(V)\). Space: \(O(V + E)\) plus the maximum-flow workspace; query \(O(V)\).
Implementation¶
当前头文件，省略 include guard；依赖见 #include。
/// @complexity Time: O(V + E + K^2 + K*E) plus K-1 maximum-flow
/// computations on K vertices; K includes vertex 0 and nonisolated vertices.
/// Query O(V). Space: O(V + E) plus the maximum-flow workspace; query O(V).
#include "atcoder/maxflow.hpp"
#include <algorithm>
#include <cassert>
#include <limits>
#include <tuple>
#include <utility>
#include <vector>
namespace noya {
/// @brief Gomory-Hu tree representing every pairwise minimum cut of an
/// undirected nonnegative-capacity graph.
template <class Cap> struct gomory_hu_tree_result {
std::vector<int> fa;
std::vector<Cap> cut;
std::vector<std::vector<std::pair<int, Cap>>> tr;
/// @brief Return the minimum cut value between two vertices in O(n) time.
Cap min_cut(int s, int t) const {
const int n = int(tr.size());
assert(0 <= s && s < n);
assert(0 <= t && t < n);
if (s == t) {
return std::numeric_limits<Cap>::max();
}
Cap best = std::numeric_limits<Cap>::max();
auto dfs = [&](auto &self, int u, int p, Cap value) -> bool {
if (u == t) { best = value; return true; }
for (auto [v, w] : tr[u])
if (v != p && self(self, v, u, std::min(value, w))) return true;
return false;
};
if (dfs(dfs, s, -1, best)) return best;
assert(false);
return Cap{};
}
};
/// @brief Build a Gomory-Hu tree in O(n) maximum-flow computations. Each edge
/// is (u, v, capacity); parallel edges are supported and self-loops ignored.
/// All pairwise minimum cuts must fit in Cap. Uses ACL's recursive Dinic.
template <class Cap>
gomory_hu_tree_result<Cap>
gomory_hu_tree(int n, const std::vector<std::tuple<int, int, Cap>> &es) {
assert(n >= 0);
gomory_hu_tree_result<Cap> res;
res.fa.assign(n, 0);
res.cut.assign(n, Cap{});
res.tr.resize(n);
if (n <= 1) {
return res;
}
std::vector<int> id(n, -1), vs;
id[0] = 0;
for (const auto &[a, b, cap] : es) {
assert(0 <= a && a < n);
assert(0 <= b && b < n);
assert(!(cap < Cap{}));
if (a != b && cap != Cap{}) {
id[a] = id[b] = 0;
}
}
for (int u = 0; u < n; u++) {
if (id[u] == 0) {
id[u] = int(vs.size());
vs.push_back(u);
}
}
const int k = int(vs.size());
atcoder::mf_graph<Cap> g(k);
std::vector<std::pair<Cap, Cap>> init;
for (const auto &[a, b, cap] : es) {
if (a == b || cap == Cap{}) {
continue;
}
if (!(std::numeric_limits<Cap>::max() - cap < cap)) {
// capacity 2*c and initial flow c give residual capacity c both ways.
Cap twice = cap;
twice += cap;
g.add_edge(id[a], id[b], twice);
init.emplace_back(twice, cap);
} else {
// Preserve the full capacity range when 2*c cannot be represented.
g.add_edge(id[a], id[b], cap);
init.emplace_back(cap, Cap{});
g.add_edge(id[b], id[a], cap);
init.emplace_back(cap, Cap{});
}
}
for (int i = 1; i < k; i++) {
int s = vs[i], t = res.fa[s];
for (int e = 0; e < int(init.size()); e++) {
g.change_edge(e, init[e].first, init[e].second);
}
res.cut[s] = g.flow(i, id[t]);
std::vector<bool> sd = g.min_cut(i);
for (int j = i + 1; j < k; j++) {
int u = vs[j];
if (res.fa[u] == t && sd[j]) {
res.fa[u] = s;
}
}
if (sd[id[res.fa[t]]]) {
res.fa[s] = res.fa[t];
res.fa[t] = s;
std::swap(res.cut[s], res.cut[t]);
}
}
for (int u = 1; u < n; u++) {
int fa = res.fa[u];
Cap val = res.cut[u];
res.tr[u].emplace_back(fa, val);
res.tr[fa].emplace_back(u, val);
}
return res;
}
} // namespace noya
// NOYA_GOMORY_HU_TREE_HPP
#ifndef NOYA_GOMORY_HU_TREE_HPP
#define NOYA_GOMORY_HU_TREE_HPP 1
/// @complexity Time: O(V + E + K^2 + K*E) plus K-1 maximum-flow
/// computations on K vertices; K includes vertex 0 and nonisolated vertices.
/// Query O(V). Space: O(V + E) plus the maximum-flow workspace; query O(V).
#include "atcoder/maxflow.hpp"
#include <algorithm>
#include <cassert>
#include <limits>
#include <tuple>
#include <utility>
#include <vector>
namespace noya {
/// @brief Gomory-Hu tree representing every pairwise minimum cut of an
/// undirected nonnegative-capacity graph.
template <class Cap> struct gomory_hu_tree_result {
std::vector<int> fa;
std::vector<Cap> cut;
std::vector<std::vector<std::pair<int, Cap>>> tr;
/// @brief Return the minimum cut value between two vertices in O(n) time.
Cap min_cut(int s, int t) const {
const int n = int(tr.size());
assert(0 <= s && s < n);
assert(0 <= t && t < n);
if (s == t) {
return std::numeric_limits<Cap>::max();
}
Cap best = std::numeric_limits<Cap>::max();
auto dfs = [&](auto &self, int u, int p, Cap value) -> bool {
if (u == t) { best = value; return true; }
for (auto [v, w] : tr[u])
if (v != p && self(self, v, u, std::min(value, w))) return true;
return false;
};
if (dfs(dfs, s, -1, best)) return best;
assert(false);
return Cap{};
}
};
/// @brief Build a Gomory-Hu tree in O(n) maximum-flow computations. Each edge
/// is (u, v, capacity); parallel edges are supported and self-loops ignored.
/// All pairwise minimum cuts must fit in Cap. Uses ACL's recursive Dinic.
template <class Cap>
gomory_hu_tree_result<Cap>
gomory_hu_tree(int n, const std::vector<std::tuple<int, int, Cap>> &es) {
assert(n >= 0);
gomory_hu_tree_result<Cap> res;
res.fa.assign(n, 0);
res.cut.assign(n, Cap{});
res.tr.resize(n);
if (n <= 1) {
return res;
}
std::vector<int> id(n, -1), vs;
id[0] = 0;
for (const auto &[a, b, cap] : es) {
assert(0 <= a && a < n);
assert(0 <= b && b < n);
assert(!(cap < Cap{}));
if (a != b && cap != Cap{}) {
id[a] = id[b] = 0;
}
}
for (int u = 0; u < n; u++) {
if (id[u] == 0) {
id[u] = int(vs.size());
vs.push_back(u);
}
}
const int k = int(vs.size());
atcoder::mf_graph<Cap> g(k);
std::vector<std::pair<Cap, Cap>> init;
for (const auto &[a, b, cap] : es) {
if (a == b || cap == Cap{}) {
continue;
}
if (!(std::numeric_limits<Cap>::max() - cap < cap)) {
// capacity 2*c and initial flow c give residual capacity c both ways.
Cap twice = cap;
twice += cap;
g.add_edge(id[a], id[b], twice);
init.emplace_back(twice, cap);
} else {
// Preserve the full capacity range when 2*c cannot be represented.
g.add_edge(id[a], id[b], cap);
init.emplace_back(cap, Cap{});
g.add_edge(id[b], id[a], cap);
init.emplace_back(cap, Cap{});
}
}
for (int i = 1; i < k; i++) {
int s = vs[i], t = res.fa[s];
for (int e = 0; e < int(init.size()); e++) {
g.change_edge(e, init[e].first, init[e].second);
}
res.cut[s] = g.flow(i, id[t]);
std::vector<bool> sd = g.min_cut(i);
for (int j = i + 1; j < k; j++) {
int u = vs[j];
if (res.fa[u] == t && sd[j]) {
res.fa[u] = s;
}
}
if (sd[id[res.fa[t]]]) {
res.fa[s] = res.fa[t];
res.fa[t] = s;
std::swap(res.cut[s], res.cut[t]);
}
}
for (int u = 1; u < n; u++) {
int fa = res.fa[u];
Cap val = res.cut[u];
res.tr[u].emplace_back(fa, val);
res.tr[fa].emplace_back(u, val);
}
return res;
}
} // namespace noya
#endif // NOYA_GOMORY_HU_TREE_HPP
/// @complexity Time: O(V + E + K^2 + K*E) plus K-1 maximum-flow
/// computations on K vertices; K includes vertex 0 and nonisolated vertices.
/// Query O(V). Space: O(V + E) plus the maximum-flow workspace; query O(V).
#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <vector>
#include <vector>
namespace atcoder {
namespace internal {
template <class T> struct simple_queue {
std::vector<T> payload;
int pos = 0;
void reserve(int n) { payload.reserve(n); }
int size() const { return int(payload.size()) - pos; }
bool empty() const { return pos == int(payload.size()); }
void push(const T& t) { payload.push_back(t); }
T& front() { return payload[pos]; }
void clear() {
payload.clear();
pos = 0;
}
void pop() { pos++; }
};
} // namespace internal
} // namespace atcoder
namespace atcoder {
template <class Cap> struct mf_graph {
public:
mf_graph() : _n(0) {}
explicit mf_graph(int n) : _n(n), g(n) {}
int add_edge(int from, int to, Cap cap) {
assert(0 <= from && from < _n);
assert(0 <= to && to < _n);
assert(0 <= cap);
int m = int(pos.size());
pos.push_back({from, int(g[from].size())});
int from_id = int(g[from].size());
int to_id = int(g[to].size());
if (from == to) to_id++;
g[from].push_back(_edge{to, to_id, cap});
g[to].push_back(_edge{from, from_id, 0});
return m;
}
struct edge {
int from, to;
Cap cap, flow;
};
edge get_edge(int i) {
int m = int(pos.size());
assert(0 <= i && i < m);
auto _e = g[pos[i].first][pos[i].second];
auto _re = g[_e.to][_e.rev];
return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
}
std::vector<edge> edges() {
int m = int(pos.size());
std::vector<edge> result;
for (int i = 0; i < m; i++) {
result.push_back(get_edge(i));
}
return result;
}
void change_edge(int i, Cap new_cap, Cap new_flow) {
int m = int(pos.size());
assert(0 <= i && i < m);
assert(0 <= new_flow && new_flow <= new_cap);
auto& _e = g[pos[i].first][pos[i].second];
auto& _re = g[_e.to][_e.rev];
_e.cap = new_cap - new_flow;
_re.cap = new_flow;
}
Cap flow(int s, int t) {
return flow(s, t, std::numeric_limits<Cap>::max());
}
Cap flow(int s, int t, Cap flow_limit) {
assert(0 <= s && s < _n);
assert(0 <= t && t < _n);
assert(s != t);
std::vector<int> level(_n), iter(_n);
internal::simple_queue<int> que;
auto bfs = [&]() {
std::fill(level.begin(), level.end(), -1);
level[s] = 0;
que.clear();
que.push(s);
while (!que.empty()) {
int v = que.front();
que.pop();
for (auto e : g[v]) {
if (e.cap == 0 || level[e.to] >= 0) continue;
level[e.to] = level[v] + 1;
if (e.to == t) return;
que.push(e.to);
}
}
};
auto dfs = [&](auto self, int v, Cap up) {
if (v == s) return up;
Cap res = 0;
int level_v = level[v];
for (int& i = iter[v]; i < int(g[v].size()); i++) {
_edge& e = g[v][i];
if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;
Cap d =
self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
if (d <= 0) continue;
g[v][i].cap += d;
g[e.to][e.rev].cap -= d;
res += d;
if (res == up) return res;
}
level[v] = _n;
return res;
};
Cap flow = 0;
while (flow < flow_limit) {
bfs();
if (level[t] == -1) break;
std::fill(iter.begin(), iter.end(), 0);
Cap f = dfs(dfs, t, flow_limit - flow);
if (!f) break;
flow += f;
}
return flow;
}
std::vector<bool> min_cut(int s) {
std::vector<bool> visited(_n);
internal::simple_queue<int> que;
que.push(s);
while (!que.empty()) {
int p = que.front();
que.pop();
visited[p] = true;
for (auto e : g[p]) {
if (e.cap && !visited[e.to]) {
visited[e.to] = true;
que.push(e.to);
}
}
}
return visited;
}
private:
int _n;
struct _edge {
int to, rev;
Cap cap;
};
std::vector<std::pair<int, int>> pos;
std::vector<std::vector<_edge>> g;
};
} // namespace atcoder
#include <algorithm>
#include <cassert>
#include <limits>
#include <tuple>
#include <utility>
#include <vector>
namespace noya {
/// @brief Gomory-Hu tree representing every pairwise minimum cut of an
/// undirected nonnegative-capacity graph.
template <class Cap> struct gomory_hu_tree_result {
std::vector<int> fa;
std::vector<Cap> cut;
std::vector<std::vector<std::pair<int, Cap>>> tr;
/// @brief Return the minimum cut value between two vertices in O(n) time.
Cap min_cut(int s, int t) const {
const int n = int(tr.size());
assert(0 <= s && s < n);
assert(0 <= t && t < n);
if (s == t) {
return std::numeric_limits<Cap>::max();
}
Cap best = std::numeric_limits<Cap>::max();
auto dfs = [&](auto &self, int u, int p, Cap value) -> bool {
if (u == t) { best = value; return true; }
for (auto [v, w] : tr[u])
if (v != p && self(self, v, u, std::min(value, w))) return true;
return false;
};
if (dfs(dfs, s, -1, best)) return best;
assert(false);
return Cap{};
}
};
/// @brief Build a Gomory-Hu tree in O(n) maximum-flow computations. Each edge
/// is (u, v, capacity); parallel edges are supported and self-loops ignored.
/// All pairwise minimum cuts must fit in Cap. Uses ACL's recursive Dinic.
template <class Cap>
gomory_hu_tree_result<Cap>
gomory_hu_tree(int n, const std::vector<std::tuple<int, int, Cap>> &es) {
assert(n >= 0);
gomory_hu_tree_result<Cap> res;
res.fa.assign(n, 0);
res.cut.assign(n, Cap{});
res.tr.resize(n);
if (n <= 1) {
return res;
}
std::vector<int> id(n, -1), vs;
id[0] = 0;
for (const auto &[a, b, cap] : es) {
assert(0 <= a && a < n);
assert(0 <= b && b < n);
assert(!(cap < Cap{}));
if (a != b && cap != Cap{}) {
id[a] = id[b] = 0;
}
}
for (int u = 0; u < n; u++) {
if (id[u] == 0) {
id[u] = int(vs.size());
vs.push_back(u);
}
}
const int k = int(vs.size());
atcoder::mf_graph<Cap> g(k);
std::vector<std::pair<Cap, Cap>> init;
for (const auto &[a, b, cap] : es) {
if (a == b || cap == Cap{}) {
continue;
}
if (!(std::numeric_limits<Cap>::max() - cap < cap)) {
// capacity 2*c and initial flow c give residual capacity c both ways.
Cap twice = cap;
twice += cap;
g.add_edge(id[a], id[b], twice);
init.emplace_back(twice, cap);
} else {
// Preserve the full capacity range when 2*c cannot be represented.
g.add_edge(id[a], id[b], cap);
init.emplace_back(cap, Cap{});
g.add_edge(id[b], id[a], cap);
init.emplace_back(cap, Cap{});
}
}
for (int i = 1; i < k; i++) {
int s = vs[i], t = res.fa[s];
for (int e = 0; e < int(init.size()); e++) {
g.change_edge(e, init[e].first, init[e].second);
}
res.cut[s] = g.flow(i, id[t]);
std::vector<bool> sd = g.min_cut(i);
for (int j = i + 1; j < k; j++) {
int u = vs[j];
if (res.fa[u] == t && sd[j]) {
res.fa[u] = s;
}
}
if (sd[id[res.fa[t]]]) {
res.fa[s] = res.fa[t];
res.fa[t] = s;
std::swap(res.cut[s], res.cut[t]);
}
}
for (int u = 1; u < n; u++) {
int fa = res.fa[u];
Cap val = res.cut[u];
res.tr[u].emplace_back(fa, val);
res.tr[fa].emplace_back(u, val);
}
return res;
}
} // namespace noya
// NOYA_GOMORY_HU_TREE_HPP