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# A Divide-and-Conquer Algorithm for Two-Point L_1 Shortest Path Queries in Polygonal Domains

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LIPIcs.SoCG.2019.59.pdf
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## Cite As

Haitao Wang. A Divide-and-Conquer Algorithm for Two-Point L_1 Shortest Path Queries in Polygonal Domains. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.59

## Abstract

Let P be a polygonal domain of h holes and n vertices. We study the problem of constructing a data structure that can compute a shortest path between s and t in P under the L_1 metric for any two query points s and t. To do so, a standard approach is to first find a set of n_s "gateways" for s and a set of n_t "gateways" for t such that there exist a shortest s-t path containing a gateway of s and a gateway of t, and then compute a shortest s-t path using these gateways. Previous algorithms all take quadratic O(n_s * n_t) time to solve this problem. In this paper, we propose a divide-and-conquer technique that solves the problem in O(n_s + n_t log n_s) time. As a consequence, we construct a data structure of O(n+(h^2 log^3 h/log log h)) size in O(n+(h^2 log^4 h/log log h)) time such that each query can be answered in O(log n) time.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
• Theory of computation → Computational geometry
##### Keywords
• shortest paths
• two-point queries
• L_1 metric
• polygonal domains

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