13 、最大流
13.1 SAP邻接 矩阵形式
/*
* SAP算法（矩阵形式）
* 结点编号从0开始
*/
const int MAXN=1100;
int maze[MAXN][MAXN];
int gap[MAXN],dis[MAXN],pre[MAXN],cur[MAXN];
int sap(int start,int end,int nodenum)
{
memset(cur,0,sizeof(cur));
memset(dis,0,sizeof(dis));
memset(gap,0,sizeof(gap));
int u=pre[start]=start,maxflow=0,aug=-1;
gap[0]=nodenum;
while(dis[start]<nodenum)
{
loop:
for(int v=cur[u];v<nodenum;v++)
if(maze[u][v] && dis[u]==dis[v]+1)
{
if(aug==-1 || aug>maze[u][v])aug=maze[u][v];
pre[v]=u;
u=cur[u]=v;
if(v==end)
{
maxflow+=aug;
for(u=pre[u];v!=start;v=u,u=pre[u])
{
maze[u][v]-=aug;
maze[v][u]+=aug;
}
aug=-1;
}
goto loop;
}
int mindis=nodenum-1;
for(int v=0;v<nodenum;v++)
if(maze[u][v]&&mindis>dis[v])
{
cur[u]=v;
mindis=dis[v];
}
if((--gap[dis[u]])==0)break;
gap[dis[u]=mindis+1]++;
u=pre[u];
}
return maxflow;
}
13.2 SAP邻接矩阵形式 2
保留原矩阵，可用于多次使最大流
/*
* SAP邻接矩阵形式
* 点的编号从0开始
* 增加个flow数组，保留原矩阵maze,可用于多次使用最大流
*/
const int MAXN=1100;
int maze[MAXN][MAXN];
int gap[MAXN],dis[MAXN],pre[MAXN],cur[MAXN];
int flow[MAXN][MAXN];//存最大流的容量
int sap(int start,int end,int nodenum)
{
memset(cur,0,sizeof(cur));
memset(dis,0,sizeof(dis));
memset(gap,0,sizeof(gap));
98 / 152 ACM模板 kuangbin
memset(flow,0,sizeof(flow));
int u=pre[start]=start,maxflow=0,aug=-1;
gap[0]=nodenum;
while(dis[start]<nodenum)
{
loop:
for(int v=cur[u];v<nodenum;v++)
if(maze[u][v]-flow[u][v] && dis[u]==dis[v]+1)
{
if(aug==-1 || aug>maze[u][v]-flow[u][v])aug=maze[u][v]-flow[u][v];
pre[v]=u;
u=cur[u]=v;
if(v==end)
{
maxflow+=aug;
for(u=pre[u];v!=start;v=u,u=pre[u])
{
flow[u][v]+=aug;
flow[v][u]-=aug;
}
aug=-1;
}
goto loop;
}
int mindis=nodenum-1;
for(int v=0;v<nodenum;v++)
if(maze[u][v]-flow[u][v]&&mindis>dis[v])
{
cur[u]=v;
mindis=dis[v];
}
if((--gap[dis[u]])==0)break;
gap[dis[u]=mindis+1]++;
u=pre[u];
}
return maxflow;
}
13.3 ISAPSAPSAP邻接表形式
const int MAXN = 100010;//点数的最大值
const int MAXM = 400010;//边数的最大值
const int INF = 0x3f3f3f3f;
struct Edge
{
int to,next,cap,flow;
}edge[MAXM];//注意是MAXM
int tol;
int head[MAXN];
int gap[MAXN],dep[MAXN],pre[MAXN],cur[MAXN];
void init()
{
tol = 0;
memset(head,-1,sizeof(head));
}
//加边，单向图三个参数，双向图四个参数
void addedge(int u,int v,int w,int rw=0)
{
edge[tol].to = v;edge[tol].cap = w;edge[tol].next = head[u];
edge[tol].flow = 0;head[u] = tol++;
edge[tol].to = u;edge[tol].cap = rw;edge[tol].next = head[v];
edge[tol].flow = 0;head[v]=tol++;
}
//输入参数：起点、终点、点的总数
//点的编号没有影响，只要输入点的总数
int sap(int start,int end,int N)
{
memset(gap,0,sizeof(gap));
memset(dep,0,sizeof(dep));
memcpy(cur,head,sizeof(head));
int u = start;
pre[u] = -1;
gap[0] = N;
int ans = 0;
while(dep[start] < N)
{
if(u == end)
{
int Min = INF;
for(int i = pre[u];i != -1; i = pre[edge[i^1].to])
if(Min > edge[i].cap - edge[i].flow)
Min = edge[i].cap - edge[i].flow;
for(int i = pre[u];i != -1; i = pre[edge[i^1].to])
{
edge[i].flow += Min;
edge[i^1].flow -= Min;
}
u = start;
ans += Min;
continue;
}
bool flag = false;
int v;
for(int i = cur[u]; i != -1;i = edge[i].next)
{
v = edge[i].to;
if(edge[i].cap - edge[i].flow && dep[v]+1 == dep[u])
{
flag = true;
cur[u] = pre[v] = i;
break;
}
}
if(flag)
{
u = v;
continue;
}
int Min = N;
for(int i = head[u]; i != -1;i = edge[i].next)
if(edge[i].cap - edge[i].flow && dep[edge[i].to] < Min)
{
Min = dep[edge[i].to];
cur[u] = i;
}
gap[dep[u]]--;
if(!gap[dep[u]])return ans;
dep[u] = Min+1;
gap[dep[u]]++;
if(u != start) u = edge[pre[u]^1].to;
}
return ans;
}
ISAP13.4 ISAP+bfs +bfs+bfs初始化 +栈优化
const int MAXN = 100010;//点数的最大值
const int MAXM = 400010;//边数的最大值
const int INF = 0x3f3f3f3f;
struct Edge
{
int to,next,cap,flow;
}edge[MAXM];//注意是MAXM
int tol;
int head[MAXN];
int gap[MAXN],dep[MAXN],cur[MAXN];
void init()
{
tol = 0;
memset(head,-1,sizeof(head));
}
void addedge(int u,int v,int w,int rw = 0)
{
edge[tol].to = v; edge[tol].cap = w; edge[tol].flow = 0;
edge[tol].next = head[u]; head[u] = tol++;
edge[tol].to = u; edge[tol].cap = rw; edge[tol].flow = 0;
edge[tol].next = head[v]; head[v] = tol++;
}
int Q[MAXN];
void BFS(int start,int end)
{
memset(dep,-1,sizeof(dep));
memset(gap,0,sizeof(gap));
gap[0] = 1;
int front = 0, rear = 0;
dep[end] = 0;
Q[rear++] = end;
while(front != rear)
{
int u = Q[front++];
for(int i = head[u]; i != -1; i = edge[i].next)
{
int v = edge[i].to;
if(dep[v] != -1)continue;
Q[rear++] = v;
dep[v] = dep[u] + 1;
gap[dep[v]]++;
}
}
}
int S[MAXN];
int sap(int start,int end,int N)
{
BFS(start,end);
memcpy(cur,head,sizeof(head));
int top = 0;
int u = start;
int ans = 0;
while(dep[start] < N)
{
if(u == end)
{
int Min = INF;
int inser;
for(int i = 0;i < top;i++)
if(Min > edge[S[i]].cap - edge[S[i]].flow)
{
Min = edge[S[i]].cap - edge[S[i]].flow;
inser = i;
}
for(int i = 0;i < top;i++)
{
edge[S[i]].flow += Min;
edge[S[i]^1].flow -= Min;
}
ans += Min;
top = inser;
u = edge[S[top]^1].to;
continue;
}
bool flag = false;
int v;
for(int i = cur[u]; i != -1; i = edge[i].next)
{
v = edge[i].to;
if(edge[i].cap - edge[i].flow && dep[v]+1 == dep[u])
{
flag = true;
cur[u] = i;
break;
}
}
if(flag)
{
S[top++] = cur[u];
u = v;
continue;
}
int Min = N;
for(int i = head[u]; i != -1; i = edge[i].next)
if(edge[i].cap - edge[i].flow && dep[edge[i].to] < Min)
{
Min = dep[edge[i].to];
cur[u] = i;
}
gap[dep[u]]--;
if(!gap[dep[u]])return ans;
dep[u] = Min + 1;
gap[dep[u]]++;
if(u != start)u = edge[S[--top]^1].to;
}
return ans;
}