Near-Optimal Heaps and Dijkstra on Pointer Machines

TL;DR

This paper introduces a near-optimal working set heap implementation on pointer machines, enabling efficient Dijkstra's algorithm with minimal overhead compared to the optimal running time.

Contribution

It demonstrates that pointer machines can support working set heaps with constant amortized Push and inverse-Ackermann DecreaseKey, leading to near-universal optimality of Dijkstra's algorithm.

Findings

Pointer machines support working set heaps with constant amortized Push.

DecreaseKey operations run in inverse-Ackermann time on pointer machines.

Dijkstra's algorithm is near-universally optimal with only an additive $O(m \, \alpha(m))$ overhead.

Abstract

A heap is a dynamic data structure that stores a set of labeled values under the following operations: pop returns the minimum value of the heap, Push($x_i$) pushes a new value $x_i$ onto the heap, and DecreaseKey($i$, $v$) decreases the value $x_i$ to $v$. A working-set heap is a heap that supports the $x_i \gets$ pop$()$ operation in $O(\log \Gamma(x_i) )$ time where $\Gamma(x_i)$ is the size of the \emph{working set}: the number of elements that were pushed onto the heap while $x_i$ was in the heap. The goal of working set heap design is to maintain the working set property while minimizing the overhead of the Push and DecreaseKey operations. On a word RAM, there exist working set heaps that support Push and DecreaseKey in amortized constant time. In this paper, we show via a simple construction that pointer machines, one of the most general and least-assuming computational models, support working set heaps that support Push in amortized constant time and DecreaseKey in inverse-Ackermann time. A by-product of this analysis is that Dijkstra's shortest path algorithm can be near-universally optimal on a pointer machine -- incurring only an additive $O(m \, \alpha(m))$ overhead compared to the optimal running time for distance ordering, where $m$ denotes the number of edges in the graph.