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#include "graph/flow/dinic.hpp"
最大流を求めるアルゴリズム.
残余グラフ上の辺数最小のパスを BFS により探して, DFS により増加パスがとれなくなるまでフローを流すことを繰り返し, 流せなくなったら終了する.
Dinic(V)
: 頂点数 V
で初期化する.add_edge(from, to, cap, idx = -1)
: 頂点 from
から to
に容量 cap
の辺を追加する.max_flow(s, t)
: 頂点 s
から t
に最大流を流し, その流量を返す.$O(EV^2)$
実測では高速に動作することが多いが, ネットワークが直接入力で与えられるなど最悪ケースを構成できる場合は遅いのでその場合は Push-Relabel などの別の最大流アルゴリズムを検討する必要がある.
$2$ 部グラフの最大マッチングのとき $O(E \sqrt V)$, 辺容量が全て同じ場合 $O(\min (V^{\frac {2} {3}}, E^{\frac {1} {2}}) E)$ となる.
/**
* @brief Dinic(最大流)
*
*/
template <typename flow_t>
struct Dinic {
const flow_t INF;
struct edge {
int to;
flow_t cap;
int rev;
bool isrev;
int idx;
};
vector<vector<edge> > graph;
vector<int> min_cost, iter;
explicit Dinic(int V) : INF(numeric_limits<flow_t>::max()), graph(V) {}
void add_edge(int from, int to, flow_t cap, int idx = -1) {
graph[from].emplace_back(
(edge){to, cap, (int)graph[to].size(), false, idx});
graph[to].emplace_back(
(edge){from, 0, (int)graph[from].size() - 1, true, idx});
}
bool build_augment_path(int s, int t) {
min_cost.assign(graph.size(), -1);
queue<int> que;
min_cost[s] = 0;
que.push(s);
while (!que.empty() && min_cost[t] == -1) {
int p = que.front();
que.pop();
for (auto &e : graph[p]) {
if (e.cap > 0 && min_cost[e.to] == -1) {
min_cost[e.to] = min_cost[p] + 1;
que.push(e.to);
}
}
}
return min_cost[t] != -1;
}
flow_t find_min_dist_augment_path(int idx, const int t, flow_t flow) {
if (idx == t) return flow;
for (int &i = iter[idx]; i < (int)graph[idx].size(); i++) {
edge &e = graph[idx][i];
if (e.cap > 0 && min_cost[idx] < min_cost[e.to]) {
flow_t d = find_min_dist_augment_path(e.to, t, min(flow, e.cap));
if (d > 0) {
e.cap -= d;
graph[e.to][e.rev].cap += d;
return d;
}
}
}
return 0;
}
flow_t max_flow(int s, int t) {
flow_t flow = 0;
while (build_augment_path(s, t)) {
iter.assign(graph.size(), 0);
flow_t f;
while ((f = find_min_dist_augment_path(s, t, INF)) > 0) flow += f;
}
return flow;
}
void output() {
for (int i = 0; i < graph.size(); i++) {
for (auto &e : graph[i]) {
if (e.isrev) continue;
auto &rev_e = graph[e.to][e.rev];
cout << i << "->" << e.to << " (flow: " << rev_e.cap << "/"
<< e.cap + rev_e.cap << ")" << endl;
}
}
}
vector<bool> min_cut(int s) {
vector<bool> used(graph.size());
queue<int> que;
que.emplace(s);
used[s] = true;
while (not que.empty()) {
int p = que.front();
que.pop();
for (auto &e : graph[p]) {
if (e.cap > 0 and not used[e.to]) {
used[e.to] = true;
que.emplace(e.to);
}
}
}
return used;
}
};
#line 1 "graph/flow/dinic.hpp"
/**
* @brief Dinic(最大流)
*
*/
template <typename flow_t>
struct Dinic {
const flow_t INF;
struct edge {
int to;
flow_t cap;
int rev;
bool isrev;
int idx;
};
vector<vector<edge> > graph;
vector<int> min_cost, iter;
explicit Dinic(int V) : INF(numeric_limits<flow_t>::max()), graph(V) {}
void add_edge(int from, int to, flow_t cap, int idx = -1) {
graph[from].emplace_back(
(edge){to, cap, (int)graph[to].size(), false, idx});
graph[to].emplace_back(
(edge){from, 0, (int)graph[from].size() - 1, true, idx});
}
bool build_augment_path(int s, int t) {
min_cost.assign(graph.size(), -1);
queue<int> que;
min_cost[s] = 0;
que.push(s);
while (!que.empty() && min_cost[t] == -1) {
int p = que.front();
que.pop();
for (auto &e : graph[p]) {
if (e.cap > 0 && min_cost[e.to] == -1) {
min_cost[e.to] = min_cost[p] + 1;
que.push(e.to);
}
}
}
return min_cost[t] != -1;
}
flow_t find_min_dist_augment_path(int idx, const int t, flow_t flow) {
if (idx == t) return flow;
for (int &i = iter[idx]; i < (int)graph[idx].size(); i++) {
edge &e = graph[idx][i];
if (e.cap > 0 && min_cost[idx] < min_cost[e.to]) {
flow_t d = find_min_dist_augment_path(e.to, t, min(flow, e.cap));
if (d > 0) {
e.cap -= d;
graph[e.to][e.rev].cap += d;
return d;
}
}
}
return 0;
}
flow_t max_flow(int s, int t) {
flow_t flow = 0;
while (build_augment_path(s, t)) {
iter.assign(graph.size(), 0);
flow_t f;
while ((f = find_min_dist_augment_path(s, t, INF)) > 0) flow += f;
}
return flow;
}
void output() {
for (int i = 0; i < graph.size(); i++) {
for (auto &e : graph[i]) {
if (e.isrev) continue;
auto &rev_e = graph[e.to][e.rev];
cout << i << "->" << e.to << " (flow: " << rev_e.cap << "/"
<< e.cap + rev_e.cap << ")" << endl;
}
}
}
vector<bool> min_cut(int s) {
vector<bool> used(graph.size());
queue<int> que;
que.emplace(s);
used[s] = true;
while (not que.empty()) {
int p = que.front();
que.pop();
for (auto &e : graph[p]) {
if (e.cap > 0 and not used[e.to]) {
used[e.to] = true;
que.emplace(e.to);
}
}
}
return used;
}
};