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

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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.
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Dates and versions

hal-01737210 , version 1 (19-03-2018)
hal-01737210 , version 2 (23-05-2019)

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Cite

Siddharth Gupta, Adrian Kosowski, Laurent Viennot. Exploiting Hopsets: Improved Distance Oracles for Graphs of Constant Highway Dimension and Beyond. ICALP 2019 - 46th International Colloquium on Automata, Languages, and Programming, Jul 2019, Patras, Greece. ⟨hal-01737210v2⟩
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