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#include "graph/flow/ford-fulkerson.hpp"
最大流を求めるアルゴリズム.
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.
FordFulkerson(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.$O(FE)$
$F$ は最大流
/**
* @brief Ford Fulkerson(最大流)
* @docs docs/ford-fulkerson.md
*/
template< typename flow_t >
struct FordFulkerson {
struct edge {
int to;
flow_t cap;
int rev;
bool isrev;
int idx;
};
const flow_t INF;
vector< vector< edge > > graph;
vector< int > used;
int timestamp;
explicit FordFulkerson(int V) : INF(numeric_limits< flow_t >::max()), graph(V), used(V, -1), timestamp(0) {}
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});
}
flow_t find_augment_path(int idx, const int t, flow_t flow) {
if(idx == t) return flow;
used[idx] = timestamp;
for(auto &e : graph[idx]) {
if(e.cap > 0 && used[e.to] != timestamp) {
flow_t d = find_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;
for(flow_t f; (f = find_augment_path(s, t, INF)) > 0; timestamp++) {
flow += f;
}
timestamp++;
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;
}
}
}
};
#line 1 "graph/flow/ford-fulkerson.hpp"
/**
* @brief Ford Fulkerson(最大流)
* @docs docs/ford-fulkerson.md
*/
template< typename flow_t >
struct FordFulkerson {
struct edge {
int to;
flow_t cap;
int rev;
bool isrev;
int idx;
};
const flow_t INF;
vector< vector< edge > > graph;
vector< int > used;
int timestamp;
explicit FordFulkerson(int V) : INF(numeric_limits< flow_t >::max()), graph(V), used(V, -1), timestamp(0) {}
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});
}
flow_t find_augment_path(int idx, const int t, flow_t flow) {
if(idx == t) return flow;
used[idx] = timestamp;
for(auto &e : graph[idx]) {
if(e.cap > 0 && used[e.to] != timestamp) {
flow_t d = find_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;
for(flow_t f; (f = find_augment_path(s, t, INF)) > 0; timestamp++) {
flow += f;
}
timestamp++;
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;
}
}
}
};