Template -「网络流 & 二分图」
EK.
很少用到，知道思想即可。
懒得写封装的屑。
queue<int> q;
int Cap[MAXN][MAXN], Flow[MAXN][MAXN], Aug[MAXN], fa[MAXN], n;
void Add_Cap(int u, int v, int C) { Cap[u][v] += C; }
int bfs(int s, int t) {
for(int i = 1; i <= n; i++) {
Aug[i] = 0;
fa[i] = 0;
}
while(!q.empty())
q.pop();
Aug[s] = INF;
q.push(s);
while(!q.empty()) {
int u = q.front();
q.pop();
for(int v = 1; v <= n; v++)
if(Cap[u][v] - Flow[u][v] > 0 && !Aug[v]) {
Aug[v] = Min(Aug[u], Cap[u][v] - Flow[u][v]);
fa[v] = u;
if(v == t)
return Aug[v];
q.push(v);
}
}
return 0;
}
int EK(int s, int t) {
int Delta = bfs(s, t), res = 0, u;
while(Delta) {
res += Delta;
for(int v = t; v != s; v = u) {
u = fa[v];
Flow[u][v] += Delta;
Flow[v][u] -= Delta;
}
Delta = bfs(s, t);
}
return res;
}
Dinic.
时间复杂度玄学的常用最大流板子。
struct Maximum_Flow {
#define Type int
struct Edge {
int v, Nxt;
Edge () {}
Edge (int V, int N) {
v = V, Nxt = N;
}
} e[Maxm << 1];
int n, Cnt, s, t;
Type Cap[Maxm << 1], Flow[Maxm << 1];
int Lab[Maxn], Cur[Maxn], Head[Maxn], Col[Maxn];
queue <int> q;
void Init (int N, int S, int T) {
for (int i = 0; i <= Cnt; i++)
Flow[i] = 0, Cap[i] = 0;
n = N, Cnt = 0, s = S, t = T;
for (int i = 1; i <= n; i++)
Head[i] = -1, Col[i] = 0;
}
void Add_Edge (int u, int v, Type w) {
Cap[Cnt] += w;
e[Cnt] = Edge (v, Head[u]), Head[u] = Cnt++;
e[Cnt] = Edge (u, Head[v]), Head[v] = Cnt++;
}
bool Lab_Vertex () {
for (int i = 1; i <= n; i++)
Lab[i] = 0;
Lab[t] = 1;
while (!q.empty ())
q.pop ();
q.push (t);
while (!q.empty ()) {
int v = q.front (); q.pop ();
for (int i = Head[v], u; ~i; i = e[i].Nxt) {
u = e[i].v;
if (!Lab[u] && Cap[i ^ 1] - Flow[i ^ 1]) {
Lab[u] = Lab[v] + 1, q.push (u);
if (u == s)
return Lab[s];
}
}
}
return Lab[s];
}
Type Widen (int u, Type Limit) {
if (u == t)
return Limit;
Type Used = 0, Delta;
for (int i = Cur[u], v; ~i; i = e[i].Nxt) {
v = e[i].v, Cur[u] = i;
if (Lab[v] + 1 != Lab[u] || Cap[i] - Flow[i] <= 0)
continue;
Delta = Widen (v, Min (Limit - Used, Cap[i] - Flow[i]));
Used += Delta, Flow[i] += Delta, Flow[i ^ 1] -= Delta;
if (Used == Limit)
return Used;
}
return Used;
}
Type Dinic () {
Type Res = 0;
while (Lab_Vertex ()) {
for (int i = 1; i <= n; i++)
Cur[i] = Head[i];
Res += Widen (s, Inf);
if (Res < 0)
return Inf;
}
return Res;
}
void Color (int u) {
Col[u] = 1;
for (int i = Head[u]; ~i; i = e[i].Nxt)
if (Cap[i] - Flow[i] > 0 && !Col[e[i].v])
Color (e[i].v);
}
#undef Type
} Flow_Graph;
spfa + Dinic.
普通最小费用最大流。（小心被卡？
struct Minimum_Cost_Maximum_Flow {
#define Type int
struct edge {
int v, nxt;
Type Wei, Cap, Flow;
edge() {}
edge(int V, int Nxt, Type C, Type W, Type F) {
v = V, nxt = Nxt, Cap = C, Wei = W, Flow = F;
}
} e[MAXM << 1];
int head[MAXM << 1], cnt = 0;
void Add_Edge(int u, int v, Type c, Type w) {
e[cnt] = edge(v, head[u], c, w, 0);
head[u] = cnt++;
e[cnt] = edge(u, head[v], 0, -w, 0);
head[v] = cnt++;
}
int Cur[MAXN], n, s, t;
bool vis[MAXN], Flag[MAXN];
Type Dist[MAXN], Aug[MAXN], Flow, Cost;
void init(int N, int S, int T) {
n = N, s = S, t = T, cnt = 0;
for(int i = 1; i <= n; i++)
head[i] = -1;
}
bool spfa() {
for(int i = 1; i <= n; i++)
Dist[i] = INF, vis[i] = false, Aug[i] = INF;
Dist[s] = 0, vis[s] = true, Aug[s] = INF;
queue<int> q;
q.push(s);
while(!q.empty()) {
int u = q.front(); q.pop();
vis[u] = false;
for(int i = head[u], v; ~i; i = e[i].nxt) {
v = e[i].v;
if(e[i].Cap - e[i].Flow > 0 && Dist[v] > Dist[u] + e[i].Wei) {
Dist[v] = Dist[u] + e[i].Wei;
Aug[v] = Min(Aug[u], e[i].Cap - e[i].Flow);
if(!vis[v])
vis[v] = true, q.push(v);
}
}
}
return Dist[t] != INF;
}
Type Widen(int u, Type Limit) {
if(u == t)
return Limit;
Flag[u] = true;
Type Used = 0, Delta;
for(int i = Cur[u], v; ~i; i = e[i].nxt) {
v = e[i].v;
if(Flag[v] || Dist[u] + e[i].Wei != Dist[v] || e[i].Cap - e[i].Flow <= 0)
continue;
Cur[u] = i;
Delta = Widen(v, Min(Limit - Used, e[i].Cap - e[i].Flow));
Used += Delta, e[i].Flow += Delta, e[i ^ 1].Flow -= Delta;
if(Used == Limit)
break;
}
Flag[u] = false;
return Used;
}
void Dinic() {
Flow = 0, Cost = 0;
Type Delta;
while(spfa()) {
for(int i = 1; i <= n; i++)
Cur[i] = head[i];
Delta = Widen(s, INF);
Flow += Delta, Cost += Delta * Dist[t];
}
}
#undef Type
} Flow_Graph;
spfa + Dijkstra + EK.
时间复杂度稳定，不会被卡。呃不会被卡吧。
struct Minimum_Cost_Maximum_Flow {
#define Type int
struct edge {
int v, nxt;
Type Wei, Cap, Flow;
edge() {}
edge(int V, int Nxt, Type C, Type W, Type F) {
v = V, nxt = Nxt, Cap = C, Wei = W, Flow = F;
}
} e[MAXM << 1];
int head[MAXM << 1], cnt = 0;
void Add_Edge(int u, int v, Type c, Type w) {
e[cnt] = edge(v, head[u], c, w, 0);
head[u] = cnt++;
e[cnt] = edge(u, head[v], 0, -w, 0);
head[v] = cnt++;
}
struct node {
int x;
Type dis;
node() {}
node(int X, Type Dis) {
x = X, dis = Dis;
}
friend bool operator < (node One, node TheOther) {
return One.dis > TheOther.dis;
}
};
struct Back {
int Pre, id;
Back() {}
Back(int P, int Id) {
Pre = P, id = Id;
}
} Last[MAXN];
int Cur[MAXN], n, s, t;
bool vis[MAXN], Flag[MAXN];
Type Dist[MAXN], Aug[MAXN], h[MAXN], Flow, Cost;
void init(int N, int S, int T) {
n = N, s = S, t = T, cnt = 0;
for(int i = 1; i <= n; i++)
head[i] = -1;
}
void spfa(int s, int t) {
for(int i = 1; i <= n; i++)
h[i] = INF, vis[i] = false;
h[s] = 0, vis[s] = true;
queue<int> q;
q.push(s);
while(!q.empty()) {
int u = q.front(); q.pop();
vis[u] = false;
for(int i = head[u], v; ~i; i = e[i].nxt) {
v = e[i].v;
if(e[i].Cap - e[i].Flow > 0 && h[v] > h[u] + e[i].Wei) {
h[v] = h[u] + e[i].Wei;
if(!vis[v])
vis[v] = true, q.push(v);
}
}
}
}
bool Dijkstra(int s, int t) {
for(int i = 1; i <= n; i++)
Dist[i] = INF, Last[i] = Back(-1, -1), vis[i] = false, Aug[i] = INF;
priority_queue<node> q;
Dist[s] = 0;
q.push(node(s, Dist[s]));
while(!q.empty()) {
int u = q.top().x; q.pop();
if(vis[u])
continue;
vis[u] = true;
for(int i = head[u], v; ~i; i = e[i].nxt) {
v = e[i].v;
if(e[i].Cap - e[i].Flow > 0 && Dist[v] > Dist[u] + e[i].Wei + h[u] - h[v]) {
Last[v] = Back(u, i);
Dist[v] = Dist[u] + e[i].Wei + h[u] - h[v];
Aug[v] = Min(Aug[u], e[i].Cap - e[i].Flow);
q.push(node(v, Dist[v]));
}
}
}
return Dist[t] != INF;
}
void EK() {
Flow = 0, Cost = 0;
spfa(s, t);
while(Dijkstra(s, t)) {
Flow += Aug[t];
Cost += Aug[t] * (Dist[t] + h[t]);
int pos = t;
while(pos != s) {
e[Last[pos].id].Flow += Aug[t];
e[Last[pos].id ^ 1].Flow -= Aug[t];
pos = Last[pos].Pre;
}
for(int i = 1; i <= n; i++)
h[i] += Dist[i];
}
}
#undef Type
} Flow_Graph;
Kuhn-Munkres.
最大匹配板子。当然你也可以直接 Dinic。
struct Bipartite_Graph {
struct edge {
int v, nxt;
edge() {}
edge(int V, int Nxt) {
v = V, nxt = Nxt;
}
} e[MAXM << 1];
int head[MAXN], n, m, cnt;
void Add_Edge(int u, int v) {
e[cnt] = edge(v, head[u]);
head[u] = cnt++;
}
int Mat[MAXN][2], Tim[MAXN], tot;
void init(int N, int M) {
n = N, m = M;
for(int i = 1; i <= n; i++)
head[i] = -1, Tim[i] = 0, Mat[i][0] = 0, Mat[i][1] = 0;
cnt = 0, tot = 0;
}
bool dfs(int u) {
if (Tim[u] == tot)
return false;
Tim[u] = tot;
for (int i = head[u], v; ~i; i = e[i].nxt) {
v = e[i].v;
if (!Mat[v][1] || dfs(Mat[v][1])) {
Mat[v][1] = u, Mat[u][0] = v;
return true;
}
}
return false;
}
int calc() {
int ans = 0;
for (int i = 1; i <= m; i++)
Mat[i][1] = 0;
for (int i = 1; i <= n; i++)
Tim[i] = 0, Mat[i][0] = 0;
for (int i = n; i >= 1; i--) {
tot++;
ans += dfs(i);
}
return ans;
}
} Graph;
KM.
最小权最大匹配。
int n, m;
bool Vis[MAXN];
int Mat[MAXN], fa[MAXN];
LL w[MAXN][MAXN], Slack[MAXN], Val[MAXN][2];
void bfs(int S) {
for(int i = 0; i <= n; i++)
Slack[i] = INF, Vis[i] = false;
LL d;
int pos = 0, p, u;
for(Mat[pos] = S; Mat[pos]; pos = p) {
Vis[pos] = true, u = Mat[pos], d = INF;
for(int v = 1; v <= n; v++) {
if(Vis[v])
continue;
if(Val[u][0] + Val[v][1] - w[u][v] < Slack[v]) {
Slack[v] = Val[u][0] + Val[v][1] - w[u][v];
fa[v] = pos;
}
if(Slack[v] < d) {
d = Slack[v];
p = v;
}
}
for(int v = 0; v <= n; v++)
if(Vis[v])
Val[Mat[v]][0] -= d, Val[v][1] += d;
else
Slack[v] -= d;
}
for(; pos; pos = fa[pos])
Mat[pos] = Mat[fa[pos]];
}
LL KM() {
LL res = 0;
for(int i = 1; i <= n; i++)
bfs(i);
for(int i = 1; i <= n; i++)
res += w[Mat[i]][i];
return res;
}
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