Extended Dijkstra algorithm and Moore-Bellman-Ford algorithm

TL;DR

The paper generalizes single-source shortest path problems by introducing GSSSP, defined via a path functional $f$ on a complete path system with source $s$ and cost evaluation time $M(n)$. It formalizes order-related properties (WOP, SOP, OP) and proves structural propositions that enable solvability under conservative SOP, culminating in two extended algorithms, EDA and EMBFA, which output an arborescence with $f(P_T[v])=m_f(v)$ and run in $M(n)O(n^2)$ and $M(n)O(nm)$ time, respectively. The framework recovers classical CSSSP as a special case and extends to anti-risk/path variants like ARP, with applications demonstrated through several examples. Overall, the work deepens the theoretical understanding of shortest-path problems and broadens their applicability to generalized path costs.

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.