How to Compose Shortest Paths

The composition problem for shortest paths asks the following: given shortest paths on weighted graphs M and N which share a common boundary, find the shortest paths on their union. This problem is a crucial step in any algorithm which uses the divide and conquer method to find shortest paths. This extended abstract details how this problem may be understood categorically. Finding shortest paths is represented by a functor and the composition problem asks to find the value of this functor on a pushout using the values of the functor on the components. Furthermore, we present an algorithm which solves the composition problem for shortest paths. When implemented in Python, this algorithm reduces the computation time for finding shortest paths by relying on precompilation.