A Coalgebraic Dijkstra Algorithm

2026-05-21

AI Summary

This work proposes a unified modeling framework—Coalgebraic Shortest Path Problems (CSPP)—for a broad class of optimization problems encompassing classical and emerging formulations such as shortest paths, widest paths, two-player games, and optimal binary trees. By abstracting state-transition structures through coalgebraic theory and integrating priority queues with dynamic programming principles, the authors design a generic coalgebraic Dijkstra algorithm. The study establishes, for the first time, necessary and sufficient conditions under which Dijkstra-style acceleration is valid, thereby precisely delineating the algorithm’s applicability. When these conditions are satisfied, the algorithm correctly solves CSPP instances with an asymptotic time complexity comparable to that of the classic Dijkstra algorithm.

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.

Problem

• shortest path problem
• widest path problem
• two-player games
• state-transition systems
• coalgebraic shortest path problem

Innovation

• coalgebraic Dijkstra algorithm
• shortest path problem
• unifying framework
• state-transition systems
• asymptotic complexity