Universal Optimality of Dijkstra via Beyond-Worst-Case Heaps

TL;DR

This paper proves that Dijkstra's algorithm, when combined with a new heap data structure, is universally optimal across all graph instances, achieving optimal performance in comparison counts and runtime by exploiting locality in heap operations.

Contribution

The paper introduces a new heap with a working-set bound that ensures universal optimality of Dijkstra's algorithm in both runtime and comparison count, a novel application of beyond-worst-case analysis.

Findings

The new heap achieves Fibonacci-heap-like bounds with additional locality advantages.

Dijkstra's algorithm with this heap is universally optimal for the distance order problem.

Heap operations benefit from locality, improving efficiency for recent insertions.

Abstract

In this paper we prove that Dijkstra's shortest-path algorithm, if implemented with a sufficiently efficient heap, is universally optimal in its running time, and with suitable small additions is also universally optimal in its number of comparisons. Universal optimality is a powerful beyond-worst-case performance guarantee for graph algorithms that informally states that a single algorithm on a problem involving graphs with arc and/or vertex weights performs as well as possible on every graph, assuming a worst-case choice of weights. We give the first application of this notion to any sequential algorithm. We design a new heap data structure with a working-set bound, which guarantees that the heap takes advantage of a certain kind of locality in the heap operations. Our heap has the optimal (amortized) bounds of Fibonacci heaps but also has the beyond-worst-case guarantee that the cost of deleting the minimum item is logarithmic in the number of items inserted after it but before it is deleted, instead of logarithmic in the size of the heap when the item is deleted. That is, deletion of recently inserted items is especially efficient. We prove that our working-set bound guarantees universal optimality for the problem of ordering vertices by their distance from the source vertex, which we call the distance order problem. Our result relies on the observation that the sequence of heap operations generated by any run of Dijkstra's algorithm on a fixed graph possesses enough locality that one can couple the number of comparisons performed by any heap with our working-set bound to the minimum number of comparisons required to solve the distance order problem on this graph for a worst-case choice of arc lengths.