A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels

Abstract

We prove a \(\varOmega (\sqrt{n})\) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular it is planar and has bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 83% of an upper bound given by a constructed CH.