A Coalgebraic Dijkstra Algorithm

TL;DR

This paper introduces a coalgebraic framework for shortest path problems, unifying various algorithms and presenting a Dijkstra-style method with proven correctness and efficiency.

Contribution

It develops a coalgebraic Dijkstra algorithm for a broad class of problems, providing necessary and sufficient conditions for its correctness and efficiency.

Findings

The coalgebraic Dijkstra algorithm correctly solves the CSPP under specific conditions.

The proposed algorithm has asymptotic complexity comparable to classical Dijkstra.

The framework unifies multiple shortest path variants and introduces new problems like the shortest binary tree.

Abstract

The Dijkstra algorithm is a classical method for solving the shortest path problem on weighted graphs. There are several variations of the Dijkstra algorithm, including algorithms for the widest path problem and for two-player games. In this paper, we introduce the coalgebraic shortest path problem (CSPP), a unifying framework for a broad class of optimization problems on state-transition systems. This framework encompasses not only the aforementioned problems but also new ones such as the shortest binary tree problem. We further present a coalgebraic Dijkstra algorithm for solving the CSPP efficiently under a suitable condition. Our condition is necessary and sufficient for the algorithm to return correct solutions, thereby providing a precise criterion for when Dijkstra-style acceleration is possible. We also show that the proposed algorithm achieves asymptotic complexity comparable to that of the classical Dijkstra algorithm.