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Let P be a set of h pairwise-disjoint polygonal obstacles with a total of n vertices in the plane. In this paper, we consider the problem of building a data structure that can quickly compute an L1 shortest obstacle-avoiding path between any two query points s and t. We build a data structure of size O(n + h2 · log h · 4√log h) in O(n + h2 · log2 h · 4√log h) time that answers each query in O(log n + k) time, where k is the number of edges of the output path. Note that n + h2 · log2 h · 4√log h = O(n+h2+ϵ) for any constant ϵ > 0. We also extend our techniques to the weighted rectilinear version in which the "obstacles" of P are rectilinear regions with "weights" and allow L1 paths to travel through them with weighted costs. Our algorithm answers each query in O(log n + k) time with a data structure of size O(n2 · log n · 4√log n) that is built in O(n2 · log2 n · 4√log n) time.

Two-Point L1 Shortest Path Queries in the Plane