Exploiting Hopsets: Improved Distance Oracles for Graphs of Constant Highway Dimension and Beyond

TL;DR

This paper introduces advanced 3-hopset based distance oracles that significantly reduce the number of shortcuts needed for graphs with constant highway or skeleton dimensions, improving efficiency over previous methods.

Contribution

It demonstrates the advantages of 3-hopsets in practical scenarios, including exponential reduction in shortcuts for certain graph classes and approximation algorithms for minimum-size hopsets.

Findings

3-hopsets require exponentially fewer shortcuts in certain graphs.

3-hopset oracles offer speedup over 2-hopset based oracles.

Polylogarithmic approximation algorithms for minimum-size hopsets.

Abstract

For fixed $h \geq 2$, we consider the task of adding to a graph $G$ a set of weighted shortcut edges on the same vertex set, such that the length of a shortest $h$-hop path between any pair of vertices in the augmented graph is exactly the same as the original distance between these vertices in $G$. A set of shortcut edges with this property is called an exact $h$-hopset and may be applied in processing distance queries on graph $G$. In particular, a $2$-hopset directly corresponds to a distributed distance oracle known as a hub labeling. In this work, we explore centralized distance oracles based on $3$-hopsets and display their advantages in several practical scenarios. In particular, for graphs of constant highway dimension, and more generally for graphs of constant skeleton dimension, we show that $3$-hopsets require exponentially fewer shortcuts per node than any previously described distance oracle while incurring only a quadratic increase in the query decoding time, and actually offer a speedup when compared to simple oracles based on a direct application of $2$-hopsets. Finally, we consider the problem of computing minimum-size $h$-hopset (for any $h \geq 2$) for a given graph $G$, showing a polylogarithmic-factor approximation for the case of unique shortest path graphs. When $h=3$, for a given bound on the space used by the distance oracle, we provide a construction of hopsets achieving polylog approximation both for space and query time compared to the optimal $3$-hopset oracle given the space bound.