Recursive Shortest Path Algorithm with Application to Density-integration of Weighted Graphs

TL;DR

This paper introduces a recursive shortest path algorithm that efficiently computes topological metrics across different graph densities by updating edges incrementally, reducing the need for repeated Dijkstra's algorithm applications.

Contribution

A novel recursive shortest path algorithm based on edge updates that improves efficiency in density-integration of weighted graphs compared to traditional methods.

Findings

Algorithm is more efficient than iterative Dijkstra's when using adjacency lists.

Reduces computational complexity in density-integration tasks.

Applicable to weighted graphs in genetics, proteomics, neuroimaging.

Abstract

Graph theory is increasingly commonly utilised in genetics, proteomics and neuroimaging. In such fields, the data of interest generally constitute weighted graphs. Analysis of such weighted graphs often require the integration of topological metrics with respect to the density of the graph. Here, density refers to the proportion of the number of edges present in that graph. When topological metrics based on shortest paths are of interest, such density-integration usually necessitates the iterative application of Dijkstra's algorithm in order to compute the shortest path matrix at each density level. In this short note, we describe a recursive shortest path algorithm based on single edge updating, which replaces the need for the iterative use of Dijkstra's algorithm. Our proposed procedure is based on pairs of breadth-first searches around each of the vertices incident to the edge added at each recursion. An algorithmic analysis of the proposed technique is provided. When the graph of interest is coded as an adjacency list, our algorithm can be shown to be more efficient than an iterative use of Dijkstra's algorithm.