We provide a formal mathematical definition of the Shortest Paths for All Flows (SP-AF) problem and provide many efficient algorithms. The SP-AF problem combines the well known Shortest Paths (SP) and Bottleneck Paths (BP) problems, and can be solved by utilising matrix multiplication. Thus in our research of the SP-AF problem, we also make a series of contributions to the underlying topics of the SP problem, the BP problem, and matrix multiplication.

For the topic of matrix multiplication we show that on an n-by-n two dimensional (2D) square mesh array, two n-by-n matrices can be multiplied in exactly 1.5n ‒ 1 communication steps. This halves the number of communication steps required by the well known Cannon’s algorithm that runs on the same sized mesh array.

We provide two contributions for the SP problem. Firstly, we enhance the breakthrough algorithm by Alon, Galil and Margalit (AGM), which was the first algorithm to achieve a deeply sub-cubic time bound for solving the All Pairs Shortest Paths (APSP) problem on dense directed graphs. Our enhancement allows the algorithm by AGM to remain sub-cubic for larger upper bounds on integer edge costs. Secondly, we show that for graphs with n vertices, the APSP problem can be solved in exactly 3n ‒ 2 communication steps on an n-by-n 2D square mesh array. This improves on the previous result of 3.5n communication steps achieved by Takaoka and Umehara.

For the BP problem, we show that we can compute the bottleneck of the entire graph without solving the All Pairs Bottleneck Paths (APBP) problem, resulting in a much more efficient time bound.

Finally we define an algebraic structure called the distance/flow semi-ring to formally introduce the SP-AF problem, and we provide many algorithms for solving the Single Source SP-AF (SSSP-AF) problem and the All Pairs SP-AF (APSP-AF) problem. For the APSP-AF problem, algebraic algorithms are given that utilise faster matrix multiplication over a ring.