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Lower Bounds for Adaptive Relaxation-Based Algorithms for Single-Source Shortest Paths

Sunny Atalig, Alexander Hickerson, Arrdya Srivastav, Tingting Zheng, Marek Chrobak

TL;DR

This model captures as a special case the operations on tentative distances used by Dijkstra's algorithm and extends this lower bound to adaptive algorithms that, in addition to relaxations, can perform queries involving some simple types of linear inequalities between edge weights and tentative distances.

Abstract

We consider the classical single-source shortest path problem in directed weighted graphs. D.~Eppstein proved recently an $Ω(n^3)$ lower bound for oblivious algorithms that use relaxation operations to update the tentative distances from the source vertex. We generalize this result by extending this $Ω(n^3)$ lower bound to \emph{adaptive} algorithms that, in addition to relaxations, can perform queries involving some simple types of linear inequalities between edge weights and tentative distances. Our model captures as a special case the operations on tentative distances used by Dijkstra's algorithm.

Lower Bounds for Adaptive Relaxation-Based Algorithms for Single-Source Shortest Paths

TL;DR

This model captures as a special case the operations on tentative distances used by Dijkstra's algorithm and extends this lower bound to adaptive algorithms that, in addition to relaxations, can perform queries involving some simple types of linear inequalities between edge weights and tentative distances.

Abstract

We consider the classical single-source shortest path problem in directed weighted graphs. D.~Eppstein proved recently an lower bound for oblivious algorithms that use relaxation operations to update the tentative distances from the source vertex. We generalize this result by extending this lower bound to \emph{adaptive} algorithms that, in addition to relaxations, can perform queries involving some simple types of linear inequalities between edge weights and tentative distances. Our model captures as a special case the operations on tentative distances used by Dijkstra's algorithm.

Paper Structure

This paper contains 6 sections, 5 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lower Bound for Deterministic Algorithms with Edge Queries
  4. Lower Bound for Deterministic Algorithms with Three Types of Queries
  5. Lower Bounds for Randomized Algorithms
  6. Final Comments and Open Problems

Key Result

Theorem 1

(a) Let $\mathcal{A}$ be a deterministic query/relaxation-based algorithm for the single-source shortest path problem that uses only edge queries. Then the running time of $\mathcal{A}$ is $\Omega(n^3)$, even if the weights are non-negative and symmetric (that is, the graph is undirected). (b) If $\

Figures (1)

  • Figure 1: The state of the game right after phase $k$ ends. Framed numbers next to vertices represent their D-values.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof : Proof sketch
  • Lemma 5.1
  • Claim 5.2
  • Claim 5.3