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#include "graph/flow/dinic-capacity-scaling.hpp"
最大流を求めるアルゴリズム.
すべての辺の容量が整数の場合, スケーリングを用いて Dinic の計算量を $O(EV \log U)$ に落とすことが出来る($U$ は辺の容量の最大値).
具体的には, フローを残余グラフ上で $k$ が大きい方から $2^k$ 単位で流すようにする.
DinicCapacityScaling(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.$O(EV \log U)$
$U$ は辺の容量の最大値
/**
* @brief Dinic Capacity Scaling(最大流)
*
*/
template< typename flow_t >
struct DinicCapacityScaling {
static_assert(is_integral< flow_t >::value, "template parameter flow_t must be integral type");
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;
flow_t max_cap;
explicit DinicCapacityScaling(int V) : INF(numeric_limits< flow_t >::max()), graph(V), max_cap(0) {}
void add_edge(int from, int to, flow_t cap, int idx = -1) {
max_cap = max(max_cap, cap);
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, const flow_t &base) {
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 >= base && 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_augment_path(int idx, const int t, flow_t base, flow_t flow) {
if(idx == t) return flow;
flow_t sum = 0;
for(int &i = iter[idx]; i < (int)graph[idx].size(); i++) {
edge &e = graph[idx][i];
if(e.cap >= base && min_cost[idx] < min_cost[e.to]) {
flow_t d = find_augment_path(e.to, t, base, min(flow - sum, e.cap));
if(d > 0) {
e.cap -= d;
graph[e.to][e.rev].cap += d;
sum += d;
if(flow - sum < base) break;
}
}
}
return sum;
}
flow_t max_flow(int s, int t) {
if(max_cap == flow_t(0)) return flow_t(0);
flow_t flow = 0;
for(int i = 63 - __builtin_clzll(max_cap); i >= 0; i--) {
flow_t now = flow_t(1) << i;
while(build_augment_path(s, t, now)) {
iter.assign(graph.size(), 0);
flow += find_augment_path(s, t, now, INF);
}
}
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/dinic-capacity-scaling.hpp"
/**
* @brief Dinic Capacity Scaling(最大流)
*
*/
template< typename flow_t >
struct DinicCapacityScaling {
static_assert(is_integral< flow_t >::value, "template parameter flow_t must be integral type");
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;
flow_t max_cap;
explicit DinicCapacityScaling(int V) : INF(numeric_limits< flow_t >::max()), graph(V), max_cap(0) {}
void add_edge(int from, int to, flow_t cap, int idx = -1) {
max_cap = max(max_cap, cap);
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, const flow_t &base) {
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 >= base && 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_augment_path(int idx, const int t, flow_t base, flow_t flow) {
if(idx == t) return flow;
flow_t sum = 0;
for(int &i = iter[idx]; i < (int)graph[idx].size(); i++) {
edge &e = graph[idx][i];
if(e.cap >= base && min_cost[idx] < min_cost[e.to]) {
flow_t d = find_augment_path(e.to, t, base, min(flow - sum, e.cap));
if(d > 0) {
e.cap -= d;
graph[e.to][e.rev].cap += d;
sum += d;
if(flow - sum < base) break;
}
}
}
return sum;
}
flow_t max_flow(int s, int t) {
if(max_cap == flow_t(0)) return flow_t(0);
flow_t flow = 0;
for(int i = 63 - __builtin_clzll(max_cap); i >= 0; i--) {
flow_t now = flow_t(1) << i;
while(build_augment_path(s, t, now)) {
iter.assign(graph.size(), 0);
flow += find_augment_path(s, t, now, INF);
}
}
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;
}
}
}
};