Problem:
MathJax.Hub.Config({ tex2jax: { inlineMath: [["$","$"], ["\\(","\\)"]], processEscapes: true }});

Maximum Flow

A flow network is a directed graph which has a $source$ and a $sink$. In a flow network, each edge $(u, v)$ has a capacity $c(u, v)$. Each edge receives a flow, but the amount of flow on the edge can not exceed the corresponding capacity. Find the maximum flow from the $source$ to the $sink$.

Input

A flow network is given in the following format.

$|V|\;|E|$

$u_0\;v_0\;c_0$

$u_1\;v_1\;c_1$

:

$u_{|E|-1}\;v_{|E|-1}\;c_{|E|-1}$

$|V|$, $|E|$ is the number of vertices and edges of the flow network respectively. The vertices in $G$ are named with the numbers 0, 1,..., $|V|-1$. The source is 0 and the sink is $|V|-1$.

$u_i$, $v_i$, $c_i$ represent $i$-th edge of the flow network.
A pair of $u_i$ and $v_i$ denotes that there is an edge from $u_i$ to $v_i$ and $c_i$ is the capacity of $i$-th edge.

Output

Print the maximum flow.

Constraints

$2 \leq |V| \leq 100$

$1 \leq |E| \leq 1000$

$0 \leq c_i \leq 10000$

Sample Input 1

4 5
0 1 2
0 2 1
1 2 1
1 3 1
2 3 2

Sample Output 1

3

Solution in JAVA (judge verdict: Wrong Answer)


import java.util.LinkedList;
import java.util.Scanner;

public class Main 
{
	static int[][] res;
	static Scanner sc = new Scanner(System.in);
	static int vn;
	static int en;
	static int[] parent; //parent of each node in the path, for backtracking
	
	static boolean BFS() //find an augmenting path if existing
	{
		System.out.println("BFS starts");
		boolean[] visited = new boolean[vn];
		
		int pos = 0;
		
		LinkedList<Integer> q = new LinkedList<Integer>();
		q.add(0);
		
		while(!q.isEmpty())
		{
			pos = q.pop();
			
			//reached sink?
			if(pos == vn-1)return true;
			
			for(int i = 0; i < vn; i++)
			{
				if(res[pos][i] > 0 && !visited[i])
				{
					visited[i] = true;
					parent[i] = pos;
					q.push(i);
				}
			}
		}
		
		System.out.println("BFS FALSE");
		return false;
	}
	
	public static void main(String[] args) 
	{
		int ans = 0;
		vn = sc.nextInt();
		en = sc.nextInt();

		res = new int[vn][vn];
		parent = new int[vn];
		
		for(int i = 0; i < en; i++)
		{
			int v1 = sc.nextInt();
			int v2 = sc.nextInt();
			int cap = sc.nextInt();
			
			res[v1][v2] = cap;
			res[v2][v1] = -cap;
		}
		
		while(BFS() == true)
		{
			//update residuals on each edge in the path
			int minflow = 999999999;
			int backtrackpos = vn-1;
			while(backtrackpos != 0)
			{
				minflow = Math.min(minflow, res[parent[backtrackpos]][backtrackpos]);
				backtrackpos = parent[backtrackpos];
			}
			
			ans += minflow;
			System.out.println("minflow " + minflow);
			//update flows
			backtrackpos = vn-1;
			while(backtrackpos != 0)
			{
				res[parent[backtrackpos]][backtrackpos] -= minflow;
				res[backtrackpos][parent[backtrackpos]] += minflow;
				backtrackpos = parent[backtrackpos];
			}
		}
		
		System.out.println(ans);
	}

}