Extended Dijkstra algorithm and Moore-Bellman-Ford algorithm

TL;DR

This paper generalizes single-source shortest path problems by defining a path function, then extends Dijkstra's and Moore-Bellman-Ford algorithms to solve this generalized problem efficiently under certain conditions.

Contribution

It introduces a general framework for shortest path problems and develops extended algorithms with proven complexity bounds, expanding the applicability of classical algorithms.

Findings

Extended algorithms solve GSSSP in $O(n^2)M(n)$ and $O(mn)M(n)$ time.

Algorithms are applicable under conditions like order-preserving in last road.

Applications demonstrate the algorithms' effectiveness in various scenarios.

Abstract

Study the general single-source shortest path problem. Firstly, define a path function on a set of some path with same source on a graph, and develop a kind of general single-source shortest path problem (GSSSP) on the defined path function. Secondly, following respectively the approaches of the well known Dijkstra's algorithm and Moore-Bellman-Ford algorithm, design an extended Dijkstra's algorithm (EDA) and an extended Moore-Bellman-Ford algorithm (EMBFA) to solve the problem GSSSP under certain given conditions. Thirdly, introduce a few concepts, such as order-preserving in last road (OPLR) of path function, and so on. And under the assumption that the value of related path function for any path can be obtained in $M(n)$ time, prove respectively the algorithm EDA solving the problem GSSSP in $O(n^2)M(n)$ time and the algorithm EMBFA solving the problem GSSSP in $O(mn)M(n)$ time. Finally, some applications of the designed algorithms are shown with a few examples. What we done can improve both the researchers and the applications of the shortest path theory.