Given a graph which represents a flow network where every edge has a capacity. Also given two vertices *source *‘s’ and *sink* ‘t’ in the graph, find the maximum possible flow from s to t with following constraints:**a)** Flow on an edge doesn’t exceed the given capacity of the edge.**b)** Incoming flow is equal to outgoing flow for every vertex except s and t.

For example, consider the following graph from CLRS book.

The maximum possible flow in the above graph is 23.

Prerequisite : **Max Flow Problem Introduction**

Ford-Fulkerson AlgorithmThe following is simple idea of Ford-Fulkerson algorithm:1)Start with initial flow as 0.2)While there is a augmenting path from source to sink. Add this path-flow to flow.3)Return flow.

**Time Complexity:** Time complexity of the above algorithm is O(max_flow * E). We run a loop while there is an augmenting path. In worst case, we may add 1 unit flow in every iteration. Therefore the time complexity becomes O(max_flow * E).**How to implement the above simple algorithm? **

Let us first define the concept of Residual Graph which is needed for understanding the implementation. * Residual Graph* of a flow network is a graph which indicates additional possible flow. If there is a path from source to sink in residual graph, then it is possible to add flow. Every edge of a residual graph has a value called

*which is equal to original capacity of the edge minus current flow. Residual capacity is basically the current capacity of the edge.*

**residual capacity**Let us now talk about implementation details. Residual capacity is 0 if there is no edge between two vertices of residual graph. We can initialize the residual graph as original graph as there is no initial flow and initially residual capacity is equal to original capacity. To find an augmenting path, we can either do a BFS or DFS of the residual graph. We have used BFS in below implementation. Using BFS, we can find out if there is a path from source to sink. BFS also builds parent[] array. Using the parent[] array, we traverse through the found path and find possible flow through this path by finding minimum residual capacity along the path. We later add the found path flow to overall flow.

The important thing is, we need to update residual capacities in the residual graph. We subtract path flow from all edges along the path and we add path flow along the reverse edges We need to add path flow along reverse edges because may later need to send flow in reverse direction (See following link for example).

https://www.geeksforgeeks.org/max-flow-problem-introduction/amp/

Below is the implementation of Ford-Fulkerson algorithm. To keep things simple, graph is represented as a 2D matrix.

`// C++ program for implementation of Ford Fulkerson` `// algorithm` `#include <iostream>` `#include <limits.h>` `#include <queue>` `#include <string.h>` `using` `namespace` `std;`
`// Number of vertices in given graph` `#define V 6` `/* Returns true if there is a path from source 's' to sink` ` ` `'t' in residual graph. Also fills parent[] to store the`
` ` `path */`
`bool` `bfs(` `int` `rGraph[V][V], ` `int` `s, ` `int` `t, ` `int` `parent[])`
`{` ` ` `// Create a visited array and mark all vertices as not`
` ` `// visited`
` ` `bool` `visited[V];`
` ` `memset` `(visited, 0, ` `sizeof` `(visited));`
` ` `// Create a queue, enqueue source vertex and mark source`
` ` `// vertex as visited`
` ` `queue<` `int` `> q;`
` ` `q.push(s);`
` ` `visited[s] = ` `true` `;`
` ` `parent[s] = -1;`
` ` `// Standard BFS Loop`
` ` `while` `(!q.empty()) {`
` ` `int` `u = q.front();`
` ` `q.pop();`
` ` `for` `(` `int` `v = 0; v < V; v++) {`
` ` `if` `(visited[v] == ` `false` `&& rGraph[u][v] > 0) {`
` ` `// If we find a connection to the sink node,`
` ` `// then there is no point in BFS anymore We`
` ` `// just have to set its parent and can return`
` ` `// true`
` ` `if` `(v == t) {`
` ` `parent[v] = u;`
` ` `return` `true` `;`
` ` `}`
` ` `q.push(v);`
` ` `parent[v] = u;`
` ` `visited[v] = ` `true` `;`
` ` `}`
` ` `}`
` ` `}`
` ` `// We didn't reach sink in BFS starting from source, so`
` ` `// return false`
` ` `return` `false` `;`
`}` `// Returns the maximum flow from s to t in the given graph` `int` `fordFulkerson(` `int` `graph[V][V], ` `int` `s, ` `int` `t)`
`{` ` ` `int` `u, v;`
` ` `// Create a residual graph and fill the residual graph`
` ` `// with given capacities in the original graph as`
` ` `// residual capacities in residual graph`
` ` `int` `rGraph[V]`
` ` `[V]; ` `// Residual graph where rGraph[i][j]`
` ` `// indicates residual capacity of edge`
` ` `// from i to j (if there is an edge. If`
` ` `// rGraph[i][j] is 0, then there is not)`
` ` `for` `(u = 0; u < V; u++)`
` ` `for` `(v = 0; v < V; v++)`
` ` `rGraph[u][v] = graph[u][v];`
` ` `int` `parent[V]; ` `// This array is filled by BFS and to`
` ` `// store path`
` ` `int` `max_flow = 0; ` `// There is no flow initially`
` ` `// Augment the flow while tere is path from source to`
` ` `// sink`
` ` `while` `(bfs(rGraph, s, t, parent)) {`
` ` `// Find minimum residual capacity of the edges along`
` ` `// the path filled by BFS. Or we can say find the`
` ` `// maximum flow through the path found.`
` ` `int` `path_flow = INT_MAX;`
` ` `for` `(v = t; v != s; v = parent[v]) {`
` ` `u = parent[v];`
` ` `path_flow = min(path_flow, rGraph[u][v]);`
` ` `}`
` ` `// update residual capacities of the edges and`
` ` `// reverse edges along the path`
` ` `for` `(v = t; v != s; v = parent[v]) {`
` ` `u = parent[v];`
` ` `rGraph[u][v] -= path_flow;`
` ` `rGraph[v][u] += path_flow;`
` ` `}`
` ` `// Add path flow to overall flow`
` ` `max_flow += path_flow;`
` ` `}`
` ` `// Return the overall flow`
` ` `return` `max_flow;`
`}` `// Driver program to test above functions` `int` `main()`
`{` ` ` `// Let us create a graph shown in the above example`
` ` `int` `graph[V][V]`
` ` `= { { 0, 16, 13, 0, 0, 0 }, { 0, 0, 10, 12, 0, 0 },`
` ` `{ 0, 4, 0, 0, 14, 0 }, { 0, 0, 9, 0, 0, 20 },`
` ` `{ 0, 0, 0, 7, 0, 4 }, { 0, 0, 0, 0, 0, 0 } };`
` ` `cout << ` `"The maximum possible flow is "`
` ` `<< fordFulkerson(graph, 0, 5);`
` ` `return` `0;`
`}` |

`// Java program for implementation of Ford Fulkerson` `// algorithm` `import` `java.io.*;`
`import` `java.lang.*;`
`import` `java.util.*;`
`import` `java.util.LinkedList;`
`class` `MaxFlow {`
` ` `static` `final` `int` `V = ` `6` `; ` `// Number of vertices in graph`
` ` `/* Returns true if there is a path from source 's' to`
` ` `sink 't' in residual graph. Also fills parent[] to`
` ` `store the path */`
` ` `boolean` `bfs(` `int` `rGraph[][], ` `int` `s, ` `int` `t, ` `int` `parent[])`
` ` `{`
` ` `// Create a visited array and mark all vertices as`
` ` `// not visited`
` ` `boolean` `visited[] = ` `new` `boolean` `[V];`
` ` `for` `(` `int` `i = ` `0` `; i < V; ++i)`
` ` `visited[i] = ` `false` `;`
` ` `// Create a queue, enqueue source vertex and mark`
` ` `// source vertex as visited`
` ` `LinkedList<Integer> queue`
` ` `= ` `new` `LinkedList<Integer>();`
` ` `queue.add(s);`
` ` `visited[s] = ` `true` `;`
` ` `parent[s] = -` `1` `;`
` ` `// Standard BFS Loop`
` ` `while` `(queue.size() != ` `0` `) {`
` ` `int` `u = queue.poll();`
` ` `for` `(` `int` `v = ` `0` `; v < V; v++) {`
` ` `if` `(visited[v] == ` `false`
` ` `&& rGraph[u][v] > ` `0` `) {`
` ` `// If we find a connection to the sink`
` ` `// node, then there is no point in BFS`
` ` `// anymore We just have to set its parent`
` ` `// and can return true`
` ` `if` `(v == t) {`
` ` `parent[v] = u;`
` ` `return` `true` `;`
` ` `}`
` ` `queue.add(v);`
` ` `parent[v] = u;`
` ` `visited[v] = ` `true` `;`
` ` `}`
` ` `}`
` ` `}`
` ` `// We didn't reach sink in BFS starting from source,`
` ` `// so return false`
` ` `return` `false` `;`
` ` `}`
` ` `// Returns tne maximum flow from s to t in the given`
` ` `// graph`
` ` `int` `fordFulkerson(` `int` `graph[][], ` `int` `s, ` `int` `t)`
` ` `{`
` ` `int` `u, v;`
` ` `// Create a residual graph and fill the residual`
` ` `// graph with given capacities in the original graph`
` ` `// as residual capacities in residual graph`
` ` `// Residual graph where rGraph[i][j] indicates`
` ` `// residual capacity of edge from i to j (if there`
` ` `// is an edge. If rGraph[i][j] is 0, then there is`
` ` `// not)`
` ` `int` `rGraph[][] = ` `new` `int` `[V][V];`
` ` `for` `(u = ` `0` `; u < V; u++)`
` ` `for` `(v = ` `0` `; v < V; v++)`
` ` `rGraph[u][v] = graph[u][v];`
` ` `// This array is filled by BFS and to store path`
` ` `int` `parent[] = ` `new` `int` `[V];`
` ` `int` `max_flow = ` `0` `; ` `// There is no flow initially`
` ` `// Augment the flow while tere is path from source`
` ` `// to sink`
` ` `while` `(bfs(rGraph, s, t, parent)) {`
` ` `// Find minimum residual capacity of the edhes`
` ` `// along the path filled by BFS. Or we can say`
` ` `// find the maximum flow through the path found.`
` ` `int` `path_flow = Integer.MAX_VALUE;`
` ` `for` `(v = t; v != s; v = parent[v]) {`
` ` `u = parent[v];`
` ` `path_flow`
` ` `= Math.min(path_flow, rGraph[u][v]);`
` ` `}`
` ` `// update residual capacities of the edges and`
` ` `// reverse edges along the path`
` ` `for` `(v = t; v != s; v = parent[v]) {`
` ` `u = parent[v];`
` ` `rGraph[u][v] -= path_flow;`
` ` `rGraph[v][u] += path_flow;`
` ` `}`
` ` `// Add path flow to overall flow`
` ` `max_flow += path_flow;`
` ` `}`
` ` `// Return the overall flow`
` ` `return` `max_flow;`
` ` `}`
` ` `// Driver program to test above functions`
` ` `public` `static` `void` `main(String[] args)`
` ` `throws` `java.lang.Exception`
` ` `{`
` ` `// Let us create a graph shown in the above example`
` ` `int` `graph[][] = ` `new` `int` `[][] {`
` ` `{ ` `0` `, ` `16` `, ` `13` `, ` `0` `, ` `0` `, ` `0` `}, { ` `0` `, ` `0` `, ` `10` `, ` `12` `, ` `0` `, ` `0` `},`
` ` `{ ` `0` `, ` `4` `, ` `0` `, ` `0` `, ` `14` `, ` `0` `}, { ` `0` `, ` `0` `, ` `9` `, ` `0` `, ` `0` `, ` `20` `},`
` ` `{ ` `0` `, ` `0` `, ` `0` `, ` `7` `, ` `0` `, ` `4` `}, { ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `}`
` ` `};`
` ` `MaxFlow m = ` `new` `MaxFlow();`
` ` `System.out.println(` `"The maximum possible flow is "`
` ` `+ m.fordFulkerson(graph, ` `0` `, ` `5` `));`
` ` `}`
`}` |

`# Python program for implementation ` `# of Ford Fulkerson algorithm` `from` `collections ` `import` `defaultdict`
`# This class represents a directed graph ` `# using adjacency matrix representation` `class` `Graph:`
` ` `def` `__init__(` `self` `, graph):`
` ` `self` `.graph ` `=` `graph ` `# residual graph`
` ` `self` `. ROW ` `=` `len` `(graph)`
` ` `# self.COL = len(gr[0])`
` ` `'''Returns true if there is a path from source 's' to sink 't' in`
` ` `residual graph. Also fills parent[] to store the path '''`
` ` `def` `BFS(` `self` `, s, t, parent):`
` ` `# Mark all the vertices as not visited`
` ` `visited ` `=` `[` `False` `]` `*` `(` `self` `.ROW)`
` ` `# Create a queue for BFS`
` ` `queue ` `=` `[]`
` ` `# Mark the source node as visited and enqueue it`
` ` `queue.append(s)`
` ` `visited[s] ` `=` `True`
` ` `# Standard BFS Loop`
` ` `while` `queue:`
` ` `# Dequeue a vertex from queue and print it`
` ` `u ` `=` `queue.pop(` `0` `)`
` ` `# Get all adjacent vertices of the dequeued vertex u`
` ` `# If a adjacent has not been visited, then mark it`
` ` `# visited and enqueue it`
` ` `for` `ind, val ` `in` `enumerate` `(` `self` `.graph[u]):`
` ` `if` `visited[ind] ` `=` `=` `False` `and` `val > ` `0` `:`
` ` `# If we find a connection to the sink node, `
` ` `# then there is no point in BFS anymore`
` ` `# We just have to set its parent and can return true`
` ` `if` `ind ` `=` `=` `t:`
` ` `visited[ind] ` `=` `True`
` ` `return` `True`
` ` `queue.append(ind)`
` ` `visited[ind] ` `=` `True`
` ` `parent[ind] ` `=` `u`
` ` `# We didn't reach sink in BFS starting `
` ` `# from source, so return false`
` ` `return` `False`
` ` ` ` ` ` `# Returns tne maximum flow from s to t in the given graph`
` ` `def` `FordFulkerson(` `self` `, source, sink):`
` ` `# This array is filled by BFS and to store path`
` ` `parent ` `=` `[` `-` `1` `]` `*` `(` `self` `.ROW)`
` ` `max_flow ` `=` `0` `# There is no flow initially`
` ` `# Augment the flow while there is path from source to sink`
` ` `while` `self` `.BFS(source, sink, parent) :`
` ` `# Find minimum residual capacity of the edges along the`
` ` `# path filled by BFS. Or we can say find the maximum flow`
` ` `# through the path found.`
` ` `path_flow ` `=` `float` `(` `"Inf"` `)`
` ` `s ` `=` `sink`
` ` `while` `(s !` `=` `source):`
` ` `path_flow ` `=` `min` `(path_flow, ` `self` `.graph[parent[s]][s])`
` ` `s ` `=` `parent[s]`
` ` `# Add path flow to overall flow`
` ` `max_flow ` `+` `=` `path_flow`
` ` `# update residual capacities of the edges and reverse edges`
` ` `# along the path`
` ` `v ` `=` `sink`
` ` `while` `(v !` `=` `source):`
` ` `u ` `=` `parent[v]`
` ` `self` `.graph[u][v] ` `-` `=` `path_flow`
` ` `self` `.graph[v][u] ` `+` `=` `path_flow`
` ` `v ` `=` `parent[v]`
` ` `return` `max_flow`
` ` `# Create a graph given in the above diagram` `graph ` `=` `[[` `0` `, ` `16` `, ` `13` `, ` `0` `, ` `0` `, ` `0` `],`
` ` `[` `0` `, ` `0` `, ` `10` `, ` `12` `, ` `0` `, ` `0` `],`
` ` `[` `0` `, ` `4` `, ` `0` `, ` `0` `, ` `14` `, ` `0` `],`
` ` `[` `0` `, ` `0` `, ` `9` `, ` `0` `, ` `0` `, ` `20` `],`
` ` `[` `0` `, ` `0` `, ` `0` `, ` `7` `, ` `0` `, ` `4` `],`
` ` `[` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `]]`
`g ` `=` `Graph(graph)`
`source ` `=` `0` `; sink ` `=` `5`
` ` `print` `(` `"The maximum possible flow is %d "` `%` `g.FordFulkerson(source, sink))`
`# This code is contributed by Neelam Yadav` |

`// C# program for implementation` `// of Ford Fulkerson algorithm` `using` `System;`
`using` `System.Collections.Generic;`
`public` `class` `MaxFlow {`
` ` `static` `readonly` `int` `V = 6; ` `// Number of vertices in`
` ` `// graph`
` ` `/* Returns true if there is a path`
` ` `from source 's' to sink 't' in residual`
` ` `graph. Also fills parent[] to store the`
` ` `path */`
` ` `bool` `bfs(` `int` `[, ] rGraph, ` `int` `s, ` `int` `t, ` `int` `[] parent)`
` ` `{`
` ` `// Create a visited array and mark`
` ` `// all vertices as not visited`
` ` `bool` `[] visited = ` `new` `bool` `[V];`
` ` `for` `(` `int` `i = 0; i < V; ++i)`
` ` `visited[i] = ` `false` `;`
` ` `// Create a queue, enqueue source vertex and mark`
` ` `// source vertex as visited`
` ` `List<` `int` `> queue = ` `new` `List<` `int` `>();`
` ` `queue.Add(s);`
` ` `visited[s] = ` `true` `;`
` ` `parent[s] = -1;`
` ` `// Standard BFS Loop`
` ` `while` `(queue.Count != 0) {`
` ` `int` `u = queue[0];`
` ` `queue.RemoveAt(0);`
` ` `for` `(` `int` `v = 0; v < V; v++) {`
` ` `if` `(visited[v] == ` `false`
` ` `&& rGraph[u, v] > 0) {`
` ` `// If we find a connection to the sink`
` ` `// node, then there is no point in BFS`
` ` `// anymore We just have to set its parent`
` ` `// and can return true`
` ` `if` `(v == t) {`
` ` `parent[v] = u;`
` ` `return` `true` `;`
` ` `}`
` ` `queue.Add(v);`
` ` `parent[v] = u;`
` ` `visited[v] = ` `true` `;`
` ` `}`
` ` `}`
` ` `}`
` ` `// We didn't reach sink in BFS starting from source,`
` ` `// so return false`
` ` `return` `false` `;`
` ` `}`
` ` `// Returns tne maximum flow`
` ` `// from s to t in the given graph`
` ` `int` `fordFulkerson(` `int` `[, ] graph, ` `int` `s, ` `int` `t)`
` ` `{`
` ` `int` `u, v;`
` ` `// Create a residual graph and fill`
` ` `// the residual graph with given`
` ` `// capacities in the original graph as`
` ` `// residual capacities in residual graph`
` ` `// Residual graph where rGraph[i,j]`
` ` `// indicates residual capacity of`
` ` `// edge from i to j (if there is an`
` ` `// edge. If rGraph[i,j] is 0, then`
` ` `// there is not)`
` ` `int` `[, ] rGraph = ` `new` `int` `[V, V];`
` ` `for` `(u = 0; u < V; u++)`
` ` `for` `(v = 0; v < V; v++)`
` ` `rGraph[u, v] = graph[u, v];`
` ` `// This array is filled by BFS and to store path`
` ` `int` `[] parent = ` `new` `int` `[V];`
` ` `int` `max_flow = 0; ` `// There is no flow initially`
` ` `// Augment the flow while tere is path from source`
` ` `// to sink`
` ` `while` `(bfs(rGraph, s, t, parent)) {`
` ` `// Find minimum residual capacity of the edhes`
` ` `// along the path filled by BFS. Or we can say`
` ` `// find the maximum flow through the path found.`
` ` `int` `path_flow = ` `int` `.MaxValue;`
` ` `for` `(v = t; v != s; v = parent[v]) {`
` ` `u = parent[v];`
` ` `path_flow`
` ` `= Math.Min(path_flow, rGraph[u, v]);`
` ` `}`
` ` `// update residual capacities of the edges and`
` ` `// reverse edges along the path`
` ` `for` `(v = t; v != s; v = parent[v]) {`
` ` `u = parent[v];`
` ` `rGraph[u, v] -= path_flow;`
` ` `rGraph[v, u] += path_flow;`
` ` `}`
` ` `// Add path flow to overall flow`
` ` `max_flow += path_flow;`
` ` `}`
` ` `// Return the overall flow`
` ` `return` `max_flow;`
` ` `}`
` ` `// Driver code`
` ` `public` `static` `void` `Main()`
` ` `{`
` ` `// Let us create a graph shown in the above example`
` ` `int` `[, ] graph = ` `new` `int` `[, ] {`
` ` `{ 0, 16, 13, 0, 0, 0 }, { 0, 0, 10, 12, 0, 0 },`
` ` `{ 0, 4, 0, 0, 14, 0 }, { 0, 0, 9, 0, 0, 20 },`
` ` `{ 0, 0, 0, 7, 0, 4 }, { 0, 0, 0, 0, 0, 0 }`
` ` `};`
` ` `MaxFlow m = ` `new` `MaxFlow();`
` ` `Console.WriteLine(` `"The maximum possible flow is "`
` ` `+ m.fordFulkerson(graph, 0, 5));`
` ` `}`
`}` `/* This code contributed by PrinciRaj1992 */` |

**Output:**

The maximum possible flow is 23

The above implementation of Ford Fulkerson Algorithm is called **Edmonds-Karp Algorithm**. The idea of Edmonds-Karp is to use BFS in Ford Fulkerson implementation as BFS always picks a path with minimum number of edges. When BFS is used, the worst case time complexity can be reduced to O(VE^{2}). The above implementation uses adjacency matrix representation though where BFS takes O(V^{2}) time, the time complexity of the above implementation is O(EV^{3}) (Refer CLRS book for proof of time complexity)

This is an important problem as it arises in many practical situations. Examples include, maximizing the transportation with given traffic limits, maximizing packet flow in computer networks.

Dinc’s Algorithm for Max-Flow.**Exercise:**

Modify the above implementation so that it that runs in O(VE^{2}) time.**References:**

http://www.stanford.edu/class/cs97si/08-network-flow-problems.pdf

Introduction to Algorithms 3rd Edition by Clifford Stein, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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