### Abstract

The Floyd-Warshall algorithm [Flo62, Roy59, War62] is a classic
dynamic programming algorithm to compute the length of all shortest
paths between any two vertices in a graph (i.e. to solve the all-pairs
shortest path problem, or APSP for short). Given a representation of
the graph as a matrix of weights M, it computes another matrix M'
which represents a graph with the same path lengths and contains the
length of the shortest path between any two vertices i and j. This is
only possible if the graph does not contain any negative cycles.
However, in this case the Floyd-Warshall algorithm will detect the
situation by calculating a negative diagonal entry. This entry
includes a formalization of the algorithm and of these key properties.
The algorithm is refined to an efficient imperative version using the
Imperative Refinement Framework.