Core Concepts
The paper proposes efficient algorithms for computing the incorrect all-pairs shortest path matrix obtained by a variant of the Floyd-Warshall algorithm, and proves that this incorrect variant is APSP-complete.
Abstract
The paper considers the computational problem of efficiently processing the incorrect all-pairs shortest path (APSP) matrix obtained by a variant of the celebrated Floyd-Warshall algorithm. The standard Floyd-Warshall algorithm efficiently computes the correct APSP matrix, but students sometimes discover a variant of this algorithm by mixing up the loop order, resulting in an incorrect APSP matrix.
The key results of the paper are:
The authors propose an efficient algorithm that can compute the incorrect APSP matrix in O(nTSSP(n,m)) time, where TSSSP(n,m) is the time to execute a single-source shortest path (SSSP) algorithm in a graph with n vertices and m edges. This is notable as it matches the runtime of the standard Floyd-Warshall algorithm for the correct APSP problem.
The authors prove that the incorrect APSP problem is subcubic equivalent to the standard APSP problem. This implies that a subcubic algorithm for the incorrect APSP problem would yield a breakthrough result, as it would also imply a subcubic algorithm for the APSP problem.
The paper formalizes the incorrect APSP problem and provides a detailed characterization of the paths realized by the incorrect variant of the Floyd-Warshall algorithm. It then leverages this characterization to devise efficient algorithms and establish the hardness result.