(Solved): (10 points) In the vertex-disjoint paths problem, we are given as input a dir ...

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(10 points) In the vertex-disjoint paths problem, we are given as input a directed graph \( G= \) \( (V, E) \) and a set of pairs of vertices \( \left(s_{1}, t_{1}\right), \ldots,\left(s_{k}, t_{k}\right) \). We would like to find a collection of \( k \) paths, \( p_{1}, \ldots, p_{k} \) such that - For each \( i, p_{i} \) is a directed path from \( s_{i} \) to \( t_{i} \), and - For every \( i \neq j \), the paths \( p_{i} \) and \( p_{j} \) do not share a vertex. Design an \( O(k \cdot(|V|+|E|) \) time algorithm for this problem.