All Pairs Shortest Paths Algorithms

Given a communication network or a road network one of the most natural algorithmic question is how to determine the shortest path from one point to another. In this paper we deal with one of the most fundamental problems of Graph Theory, the All Pairs Shortest Path (APSP) problem. We study three algorithms namely The FloydWarshall algorithm, APSP via Matrix Multiplication and the Johnson’s algorithm for this problem. We also give a slight modification to the FloydWarshall Algorithm which decreases the number of computations but the asymptotic order remains the same. Index Terms — All Pairs Shortest Path (APSP), Floyd Warshall algorithm (F-W), APSP via Matrix Multiplication, Johnson’s algorithm. INTRODUCTION Shortest paths computation is one of the most fundamental problems in graph theory. The huge interest in the problem is mainly due to the wide spectrum of its applications, ranging from routing in communication networks to robot motion planning, scheduling, sequence alignment in molecular biology and length-limited Huffman coding, to name only a very few. The problem divides into two related categories: single-source shortest-paths problems and all-pairs shortest-paths problems. The single-source shortest-path problem in a directed graph consists of determining the shortest path from a fixed source vertex to all other vertices. The all-pairs shortest-distance problem is that of finding the shortest paths between all pairs of vertices of a graph.