Problem Description

Given $n$ vertices and $m$ directed edges, with a capacity on each edge, find the maximum flow from vertex $s$ to vertex $t$.

Input Format

The first line contains four positive integers $n$, $m$, $s$, $t$, separated by spaces, representing the number of vertices, the number of directed edges, the source vertex index, and the sink vertex index.

The next $m$ lines each contain three positive integers $u_i$, $v_i$, $c_i$, separated by spaces, meaning that the $i$-th directed edge starts from $u_i$, ends at $v_i$, and has capacity $c_i$.

Output Format

Output one integer, the maximum flow from $s$ to $t$.

7 14 1 7
1 2 5
1 3 6
1 4 5
2 3 2
2 5 3
3 2 2
3 4 3
3 5 3
3 6 7
4 6 5
5 6 1
6 5 1
5 7 8
6 7 7

14

10 16 1 2
1 3 2
1 4 2
5 2 2
6 2 2
3 5 1
3 6 1
4 5 1
4 6 1
1 7 2147483647
9 2 2147483647
7 8 2147483647
10 9 2147483647
8 5 2
8 6 2
3 10 2
4 10 2

8

Hint

Constraints: $1\leqslant n \leqslant 1200$, $1\leqslant m \leqslant 120000$, $1\leqslant c \leqslant 2^{31}-1$.

It is guaranteed that the answer does not exceed $2^{31}-1$.

The time complexity of common network flow algorithms is $O(n^2 m)$, so please optimize your algorithm as much as possible.

testdata provider: @negiizhao.

(If anyone passes this problem using Dinic’s algorithm, please send a private message to the uploader.)

Translated by ChatGPT 5