Maximum Flow (ADU Feb 21)
a WimpyPoint presentation owned by Mark Dettinger

    * Maximum Flow in a Network
    * The Ford-Fulkerson Algorithm in C 

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];
    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;
    while (head!=tail) {
	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 () {
    read_input_file();
    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.

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Last modified 2001-02-21
mdettinger@arsdigita.com