Shortest Path Algorithms between Theory and Practice

TL;DR

This paper reviews the SSSP problem, introduces a new algorithm that outperforms existing methods in practice by selecting the most-effective vertices to scan, and provides extensive empirical evaluation.

Contribution

It proposes a novel SSSP algorithm that improves practical performance while maintaining theoretical bounds, with detailed analysis and implementation strategies.

Findings

The new algorithm outperforms state-of-the-art algorithms on several graph families.

It achieves similar theoretical bounds as Bellman-Ford-Moore.

Extensive computational study validates the algorithm's effectiveness.

Abstract

Utilizing graph algorithms is a common activity in computer science. Algorithms that perform computations on large graphs are not always efficient. This work investigates the Single-Source Shortest Path (SSSP) problem, which is considered to be one of the most important and most studied graph problems. This thesis contains a review of the SSSP problem in both theory and practice. In addition, it discusses a new single-source shortest-path algorithm that achieves the same $O(n \cdot m)$ time bound as the traditional Bellman-Ford-Moore algorithm but outperforms it and other state-of-the-art algorithms in practice. The work is comprised of three parts. The first discusses some basic shortest-path and negative-cycle-detection algorithms in literature from the theoretical and practical point of view. The second contains a discussion of a new algorithm for the single-source shortest-path problem that outperforms most state-of-the-art algorithms for several well-known families of graphs. The main idea behind the proposed algorithm is to select the fewest most-effective vertices to scan. We also propose a discussion of correctness, termination, and the proof of the worst-case time bound of the proposed algorithm. This section also suggests two different implementations for the proposed algorithm, the first runs faster while the second performs a fewer number of operations. Finally, an extensive computational study of the different shortest paths algorithms is conducted. The results are proposed using a new evaluation metric for shortest-path algorithms. A discussion of outcomes, strengths, and weaknesses of the various shortest path algorithms are also included in this work.

Full text

This thesis contains a review about the SSSP problem in both theory and practice. In addition, it discusses a new single-source shortest-path algorithm that achieves the same $O(n\cdot m)$ time bound as the traditional Bellman-Ford-Moore algorithm but outperforms it and other state-of-the-art algorithms.

The work is comprised of three parts. The first, it discusses some basic shortest-path and negative-cycle-detection algorithms in literature from the theoretical and practical point of view. The second, it contains a discussion of a new algorithm for the single-source shortest-path problem that outperforms most state-of-the-art algorithms for several well-known families of graphs. The main idea behind the proposed algorithm is to select the fewest most-effective vertices to scan. We also propose a discussion of correctness, termination, and the proof of the worst-case time bound of the proposed algorithm. This section also suggests two different implementations for the proposed algorithm, the first runs faster while the second performs a fewer number of operations. Finally, an extensive computational study of the different shortest paths algorithms is conducted. The results are proposed using a new evaluation metric for shortest-path algorithms. A discussion of the outcomes and the strengths and the weaknesses of various shortest path algorithms are also included in this work.

Dedication : This thesis is dedicated to my friend Safia Salem may Allah have mercy on her.

Foremost, I would like to express my sincere gratitude to my supervisors : Prof.Amr Elmasry and Dr.Ayman Khalafalla for guiding me through my MSc. study, introducing me to many exciting research problems, and working by my side over the past couple of years.

I am most indebted to my friends who supported me in many ways. I have to thank Christine Basta, Riham Salah, Rania Sherif, Sarah Ahmed, Doaa Nasser, Ahmed Elamrawy, Ahmed Ezzet, Ahmed Azaz, Ahmed Zaitoon, Ziad Ragab, Mohamed Youssef, Mostafa Gad, Walaa Wanis and Eman Ahmed for their good company along this journey.

Last but not least; I would like to thank my family..… \Declaration I declare that no part of the work referred to in this thesis has been submitted in support of an application for another degree or qualification from this or any other University or Institution.

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\listofsymbols

ll SSSP & Single Source Shortest Path.

ZDO Zero Degree Only algorithm.

ZDO-Bits Zero Degree Only Bits algorithm.

GoR Goldberg and Radzik algorithm.

Tar Tarjan algorithm.

Pal Pallotino algorithm.

S-grids Square grids family.

W-grids Wide grids family.

L-grids Long grids family.

PH-grids Positive Hard grids family.

NH-grids Negative Hard grids family.

S-rand Sparse random family.

D-rand Dense random family.

P-rand Potential random family.

PD2S-rand Potential Dense to Sparse random family.

PS-rand Potential with artificial Source random family.

PC-rand Potential with Hamiltonian Cycle random family.

SPACYC Shortest Path acyclic generator.

FP-acyc Fully Positive acyclic family.

FN-acyc Fully Negative acyclic family.

P2N-acyc Positive to Negative acyclic family.

SQNC Square grids with Negative Cycles family.