Abstract
We prove a Ω(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 a 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 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.
Highlights
While the problem of computing the shortest paths in general graphs with nonnegative edge weights seems to have been well understood already decades ago, the last 10–15 years have seen tremendous progress when it comes to the specific problem of efficiently computing shortest paths in real-world road networks
One might classify most of the employed techniques into two classes: ones that are based on pruned graph search and others that are based on distance lookups
Besides a very small test graph for sanity checking, we extracted real-world networks graphs based on Open Street Map (OSM) data for our main experiment
Summary
While the problem of computing the shortest paths in general graphs with nonnegative edge weights seems to have been well understood already decades ago, the last 10–15 years have seen tremendous progress when it comes to the specific problem of efficiently computing shortest paths in real-world road networks. This means that a query on a country-sized network like that of Germany (~20 million nodes) can be answered in less than a millisecond compared to few seconds of Dijkstra’s algorithm While these methods directly yield the actual shortest path, the latter class is primarily concerned with the computation of the (exact) distance between given source and target—recovering the actual path often requires some additional effort. As grid-like substructures are quite common in real-world road networks (see Figure 1), one might ask whether better upper bounds for such networks are impossible in general or whether a polylogarithmic upper bound could be shown via more refined proof or CH construction techniques Our work settles this question for contraction hierarchies as well as hub labels up to a logarithmic factor. Note that such an instance-specific lower bound is typically much stronger than an analytical lower bound
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