P13942 [EC Final 2019] Flow

One of $\textit{Pang}$'s research interests is the maximum flow problem. A directed graph $G$ with $n$ vertices is $\textit{universe}$ if the following condition is satisfied: - $G$ is the union of $k$ vertex-independent simple paths from vertex $1$ to vertex $n$ of the same length. A set of paths is vertex-independent if they do not have any internal vertex in common. A vertex in a path is called internal if it is not an endpoint of that path. A path is simple if its vertices are distinct. Let $G$ be a $\textit{universe}$ graph with $n$ vertices and $m$ edges. Each edge has a non-negative integral capacity. You are allowed to perform the following operation any (including $0$) times to make the maximum flow from vertex $1$ to vertex $n$ as large as possible: Let $e$ be an edge with positive capacity. Reduce the capacity of $e$ by $1$ and increase the capacity of another edge by $1$. $\textit{Pang}$ wants to know what is the minimum number of operations to achieve it?

The first line contains two integers $n$ and $m$ ($2\leq n\leq 100000, 1\leq m \leq 200000$). Each of the next $m$ lines contains three integers $x, y$ and $z$, denoting an edge from $x$ to $y$ with capacity $z$ ($1 \leq x, y \leq n$, $0\le z\le 1000000000$). It's guaranteed that the input is a $universe$ graph without multiple edges and self-loops.

Output a single integer --- the minimum number of operations.