Challenge: Shortest Paths

Task

We just discovered our best friend from elementary school on Twitbook. We both want to meet as soon as possible, but we live in two different cites that are far apart. To minimize travel time, we agree to meet at an intermediate city, and then we simultaneously hop in our cars and start driving toward each other. But where exactly should we meet? We are given a weighted graph $G = (V, E)$, where the vertices $V$ represent cities and the edges $E$ represent roads that directly connect cities. Each edge $e$ has a weight $w(e)$ equal to the time required to travel between the two cities. We’re also given a vertex $p$, representing our starting location, and a vertex $q$, representing our friend’s starting location. To find the optimal meeting point for two people traveling along a weighted graph, we can use Dijkstra’s algorithm. We start by computing the shortest paths from both the starting vertices $p$ and $q$ ...