Reach for A*: shortest path algorithms with preprocessing

Abstract

We study the point-to-point shortest path problem with preprocessing. Given an input graph, we preprocess it so as to be able to answer a series of source-to-destination queries efficiently. Our work is motivated by an algorithm of Gutman [ALENEX'04], based on the notion of reach, which measures how important each vertex is with respect to shortest paths. We present a simplified version of his algorithm that does not require explicit lower bounds during queries. We also show how the addition of shortcuts to the graph greatly improves the performance of both preprocessing and queries. Finally, we combine a reach-based algorithm with landmark-based A * search to obtain a wide range of space-time trade-offs. For our motivating application, driving directions for road networks, the resulting algorithm is very efficient and practical. The road networks of the USA and Western Europe have roughly 20 million vertices, but on average our algorithm must visit fewer than a thousand to find the distance between two points. Our algorithm also works reasonably well on 2-dimensional grid graphs with random arc weights.

On k-skip shortest paths

“…Immediately by symmetry, the SP problem can also be solved by a backward search from t to s (if G is directed, the backward search implicitly reverses the direction of each edge). The bi-directional algorithm [10,20] achieves better efficiency by performing both searches synchronously, the effect of which is essentially to explore the vertices u ∈ V in ascending order of rs,t(u), where…”

Customizable Point-of-Interest Queries in Road Networks

“…The prototypical hierarchical technique is contraction hierarchies [13], but other methods can be used as well [38], [39], [40]. Conceptually, a typical hierarchical s-t (point-to-point) query is simply a pruned version of bidirectional Dijkstra's algorithm, in which the forward search (from s) and the backward search (from t) only traverse arcs that lead to "more important" vertices.…”

Core Routing on Dynamic Time-Dependent Road Networks

“…In order to compute valid lower bounds to the distances from s or to t, proxy nodes have been introduced in [19] and used for the CALT algorithm (i.e., core-based ALT on a static graph) in [3]. See also [21] for a description. We here report the main idea: on the graph G weighted by λ, let t ′ = arg min v∈VC {d(t, v)} be the core node closest to t. By triangle inequalities it is easy to derive a valid potential function for the forward search which uses landmark distances for t ′ as a proxy for t:…”