We revisit a classical graph-theoretic problem, the single-source shortest-path (SSSP) problem, in weighted unit-disk graphs. We first propose an exact (and deterministic) algorithm which solves the problem in O(n log 2 n) time using linear space, where n is the number of the vertices of the graph. This significantly improves the previous deterministic algorithm by Cabello and Jejčič [CGTA'15] which uses O(n 1+δ ) time and O(n 1+δ ) space (for any small constant δ > 0) and the previous randomized algorithm by Kaplan et al. [SODA'17] which uses O(n log 12+o(1) n) expected time and O(n log 3 n) space. More specifically, we show that if the 2D offline insertion-only (additively-)weighted nearest-neighbor problem with k operations (i.e., insertions and queries) can be solved in f (k) time, then the SSSP problem in weighted unit-disk graphs can be solved in O(n log n + f (n)) time. Using the same framework with some new ideas, we also obtain a (1+ε)-approximate algorithm for the problem, using O(n log n + n log 2 (1/ε)) time and linear space. This improves the previous (1 + ε)-approximate algorithm by Chan and Skrepetos [SoCG'18] which uses O((1/ε) 2 n log n) time and O((1/ε) 2 n) space. Because of the Ω(n log n)-time lower bound of the problem (even when approximation is allowed), both of our algorithms are almost optimal.

The range closest-pair (RCP) problem is the range-search version of the classical closest-pair problem, which aims to store a given dataset of points in some data structure such that when a query range X is specified, the closest pair of points contained in X can be reported efficiently. A natural generalization of the RCP problem is the colored range closest-pair (CRCP) problem in which the given data points are colored and the goal is to find the closest bichromatic pair contained in the query range. All the previous work on the RCP problem was restricted to the uncolored version and the Euclidean distance function. In this paper, we make the first progress on the CRCP problem. We investigate the problem under a general distance function induced by a monotone norm; in particular, this covers all the Lp-metrics for p > 0 and the L∞-metric. We design efficient (1 + ε)-approximate CRCP data structures for orthogonal queries in R 2 , where ε > 0 is a pre-specified parameter. The highlights are two data structures for answering rectangle queries, one of which uses O(ε −1 n log 4 n) space and O(log 4 n + ε −1 log 3 n + ε −2 log n) query time while the other uses O(ε −1 n log 3 n) space and O(log 5 n + ε −1 log 4 n + ε −2 log 2 n) query time. In addition, we also apply our techniques to the CRCP problem in higher dimensions, obtaining efficient data structures for slab, 2-box, and 3D dominance queries. Before this paper, almost all the existing results for the RCP problem were achieved in R 2 . arXiv:1807.09977v1 [cs.CG] 26 Jul 2018 points to the x-axis. Again, this projection argument does not apply to the RCP problem as it changes the pairwise distances of the points, and the RCP problem for vertical strip queries is actually nontrivial [21,25].Similarly to the single-shot closest-pair problem, the RCP problem can be naturally generalized to the colored range closest-pair (CRCP) problem in which we want to store a colored dataset and report the closest bichromatic pair of points contained in a query range. Surprisingly, despite of much effort made on the RCP problem, this generalization has never been considered previously. In this paper, we make the first progress on the CRCP problem. Unlike the previous work, we do not restrict ourselves to the Euclidean metric. Instead, we investigate the problem under a general metric satisfying some certain condition. This covers all L pmetrics for p > 0 (including the L ∞ -metric). The CRCP problem is even harder than the (uncolored) RCP problem, especially when considered under such a general metric. As such, we are interested in answering CRCP queries approximately. That is, for a specified query range X, we want to report a bichromatic pair of points in X whose distance is at most (1 + ε) · Opt where Opt is the distance of the closest bichromatic pair of points in X, where ε is a pre-specified parameter. Our main goal is to design efficient (1 + ε)-approximate CRCP data structures in terms of space and query time. Related work. The closest-pair problem and range search are ...

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