Lower Bounds for Relaxation-Based Shortest Path Algorithms Public

Abstract

Computing shortest paths in directed, weighted graphs is a classical algorithms problem with applications to transportation, social networking, and many other fields. Although recent years have seen the development of fast shortest path procedures, the traditional Bellman-Ford algorithm and others like it remain the only shortest path algorithms that can operate on a general graph and therefore still see use in fields like packet routing. This paper examines existing lower bounds for the performance of Bellman-Ford-like shortest path algorithms. It moves beyond the non-adaptive setting that has received frequent attention to examine adaptive algorithms, which are attentive to information beyond the mere topology of the input graph. Specifically, I will expand some existing results about non-adaptive algorithms to a more general set of non-adaptive and weakly adaptive approaches. Additionally, I will produce new lower bounds for several Bellman-Ford-like adaptive algorithms that show the inclusion of adaptive heuristics does not improve the minimum number of relaxation operations beyond n-cubed on a weighted graph with n vertices.

Table of Contents

1 Introduction

2 Definitions

3 Related Work

3.1 The Supersequence Problem

3.2 Existing Upper Bounds

3.2.1 Yen's Algorithm

3.2.2 Randomized Approaches

3.2.3 Related Problems

3.3 Existing Lower Bounds

3.3.1 Non-Adaptive Deterministic Algorithms

3.3.2 Non-Adaptive Randomized Algorithms

3.3.3 Stochastic Algorithms

3.3.4 Hardness of Bellman-Ford

4 Our Results

4.1 Improving Existing Results

4.1.1 Meyer e al.

4.1.2 Eppstein

4.2 New Results

4.2.1 Problem Statement

4.2.2 Deterministic Weakly Adaptive Algorithms

4.2.3 Removing Assumptions

4.2.4 Randomized Weakly Adaptive Algorithms

5 Conclusion

Bibliography

About this Honors Thesis

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School	Emory College
Department	Computer Science
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Computer Science Mathematics
Mot-clé	Bellman-Ford Computer Science Mathematics Shortest Path Asymptotic Algorithms
Committee Chair / Thesis Advisor	Grigni, Michelangelo, Emory University
Committee Members	Weissman, Daniel, Emory University Ettinger, Bree, Emory University

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