Linear Time Algorithms Based on Multilevel Prefix Tree for Finding Shortest Path with Positive Weights and Minimum Spanning Tree in a Networks

TL;DR

This paper introduces linear time algorithms for shortest path and minimum spanning tree problems in networks, utilizing a novel multilevel prefix tree data structure called PTrie, which considers path weights and vertex count.

Contribution

The paper presents new linear worst-case time algorithms for graph problems using the innovative PTrie data structure, combining prefix trees with efficient operations.

Findings

Algorithms operate in linear worst-case time.

They consider both arc weights and vertex count for path selection.

Implementation in C++ demonstrates practical applicability.

Abstract

In this paper I present general outlook on questions relevant to the basic graph algorithms; Finding the Shortest Path with Positive Weights and Minimum Spanning Tree. I will show so far known solution set of basic graph problems and present my own. My solutions to graph problems are characterized by their linear worst-case time complexity. It should be noticed that the algorithms which compute the Shortest Path and Minimum Spanning Tree problems not only analyze the weight of arcs (which is the main and often the only criterion of solution hitherto known algorithms) but also in case of identical path weights they select this path which walks through as few vertices as possible. I have presented algorithms which use priority queue based on multilevel prefix tree -- PTrie. PTrie is a clever combination of the idea of prefix tree -- Trie, the structure of logarithmic time complexity for insert and remove operations, doubly linked list and queues. In C++ I will implement linear worst-case time algorithm computing the Single-Destination Shortest-Paths problem and I will explain its usage.