Maximum Flow (ADU Feb 21)

## Maximum Flow (ADU Feb 21)

a WimpyPoint presentation owned by Mark Dettinger

## Maximum Flow in a Network

Given a directed graph with a source and a sink and capacities assigned to the edges, determine the maximum flow from the source to the sink.
• For each edge, the flow must not exceed the edge's capacity.
• For each node, the incoming flow must be equal to the outgoing flow.
```

```

## The Ford-Fulkerson Algorithm in C

```#include <stdio.h>
```

#### Basic Definitions

```#define WHITE 0
#define GRAY 1
#define BLACK 2
#define MAX_NODES 1000
#define oo 1000000000
```

#### Declarations

```int n;  // number of nodes
int e;  // number of edges
int capacity[MAX_NODES][MAX_NODES]; // capacity matrix
int flow[MAX_NODES][MAX_NODES];     // flow matrix
int color[MAX_NODES]; // needed for breadth-first search
int pred[MAX_NODES];  // array to store augmenting path

int min (int x, int y) {
return x<y ? x : y;  // returns minimum of x and y
}
```

#### A Queue for Breadth-First Search

```int head,tail;
int q[MAX_NODES+2];

void enqueue (int x) {
q[tail] = x;
tail++;
color[x] = GRAY;
}

int dequeue () {
int x = q[head];
color[x] = BLACK;
return x;
}
```

#### Breadth-First Search for an augmenting path

```int bfs (int start, int target) {
int u,v;
for (u=0; u<n; u++) {
color[u] = WHITE;
}
head = tail = 0;
enqueue(start);
pred[start] = -1;
u = dequeue();
// Search all adjacent white nodes v. If the capacity
// from u to v in the residual network is positive,
// enqueue v.
for (v=0; v<n; v++) {
if (color[v]==WHITE && capacity[u][v]-flow[u][v]>0) {
enqueue(v);
pred[v] = u;
}
}
}
// If the color of the target node is black now,
// it means that we reached it.
return color[target]==BLACK;
}
```

#### Ford-Fulkerson Algorithm

```int max_flow (int source, int sink) {
int i,j,u;
// Initialize empty flow.
int max_flow = 0;
for (i=0; i<n; i++) {
for (j=0; j<n; j++) {
flow[i][j] = 0;
}
}
// While there exists an augmenting path,
// increment the flow along this path.
while (bfs(source,sink)) {
// Determine the amount by which we can increment the flow.
int increment = oo;
for (u=n-1; pred[u]>=0; u=pred[u]) {
increment = min(increment,capacity[pred[u]][u]-flow[pred[u]][u]);
}
// Now increment the flow.
for (u=n-1; pred[u]>=0; u=pred[u]) {
flow[pred[u]][u] += increment;
flow[u][pred[u]] -= increment;
}
max_flow += increment;
}
// No augmenting path anymore. We are done.
return max_flow;
}
```

#### Reading the input file and the main program

```void read_input_file() {
int a,b,c,i,j;
FILE* input = fopen("mf.in","r");
// read number of nodes and edges
fscanf(input,"%d %d",&n,&e);
// initialize empty capacity matrix
for (i=0; i<n; i++) {
for (j=0; j<n; j++) {
capacity[i][j] = 0;
}
}
// read edge capacities
for (i=0; i<e; i++) {
fscanf(input,"%d %d %d",&a,&b,&c);
capacity[a][b] = c;
}
fclose(input);
}

int main () {
printf("%d\n",max_flow(0,n-1));
return 0;
}
```

#### The Input File

```6 10    // 6 nodes, 10 edges
0 1 16  // capacity from 0 to 1 is 16
0 2 13  // capacity from 0 to 2 is 13
1 2 10  // capacity from 1 to 2 is 10
2 1 4   // capacity from 2 to 1 is 4
3 2 9   // capacity from 3 to 2 is 9
1 3 12  // capacity from 1 to 3 is 12
2 4 14  // capacity from 2 to 4 is 14
4 3 7   // capacity from 4 to 3 is 7
3 5 20  // capacity from 3 to 5 is 20
4 5 4   // capacity from 4 to 5 is 4
```

#### Output of the Program

The program computes the maximum flow from 0 to 5.
```23
```

```

```