Tiering in Contraction and Edge Hierarchies for Stochastic Route Planning

Stochastic route planning is a hard problem, since it deals with uncertain edge weights, usually modeled as probability distributions. Stochastic shortest path queries are very expensive, as they must compute convolutions of edge weight distributions, whose representations can have a major impact on query costs. Effective speedup techniques for shortest path queries exist for deterministic edge weights, but their extensions to stochastic settings have had limited success, and real-time stochastic routing queries remain beyond reach. We introduce the tiering technique for Contraction and Edge Hierarchies (CHs and EHs) to address this challenge. We divide the hierarchy into tiers, and represent edge weights in each tier in ways that permit effective tradeoffs between accuracy, convolution costs, and space use. We show how to use Gaussians to approximate histograms, and bound errors using the KL divergence and Hellinger distance measures. We develop Uncertain Contraction Hierarchies (UCHs) and Uncertain Edge Hierarchies (UEHs) using these methods, and show that they improve both CH and EH performance for three different stochastic query types: probabilistic budget routes, non-dominated routes, and routes to minimize the mean-risk objective. We evaluate our methods using real-world data from Mapbox Traffic Data for a section of Los Angeles. Finally, our results show that query times for EHs can be competitive with CHs for stochastic edge weights, contrary to current belief.