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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

We present several near-linear algorithms for problems involving visibility over a 1.5-dimensional terrain. Concretely, we have a 1.5-dimensional terrain T, i.e., a bounded x-monotone polygonal path in the plane, with n vertices, and a set P of m points that lie on or above T. The visibility graph VG(P,T) is the graph with P as its vertex set and {(p,q) | p and q are visible to each other} as its edge set. We present algorithms that perform BFS and DFS on VG(P,T), which run in O(nlog n + mlog³(m+n)) time.
We also consider three optimization problems, in which P is a set of points on T, and we erect a vertical tower of height h at each p ∈ P. In the first problem, called the reverse shortest path problem, we are given two points s, t ∈ P, and an integer k, and wish to find the smallest height h^* for which VG(P(h^*),T) contains a path from s to t of at most k edges, where P(h^*) is the set of the tips of the towers of height h^* erected at the points of P. In the second problem we wish to find the smallest height h^* for which VG(P(h^*),T) contains a cycle, and in the third problem we wish to find the smallest height h^* for which VG(P(h^*),T) is nonempty; we refer to that problem as "Seeing the most without being seen". We present algorithms for the first two problems that run in O^*((m+n)^{6/5}) time, where the O^*(⋅) notation hides subpolynomial factors. The third problem can be solved by a faster algorithm, which runs in O((n+m)log³ (m+n)) time.

Matthew J. Katz, Rachel Saban, and Micha Sharir. Near-Linear Algorithms for Visibility Graphs over a 1.5-Dimensional Terrain. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 77:1-77:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{katz_et_al:LIPIcs.ESA.2024.77, author = {Katz, Matthew J. and Saban, Rachel and Sharir, Micha}, title = {{Near-Linear Algorithms for Visibility Graphs over a 1.5-Dimensional Terrain}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {77:1--77:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.77}, URN = {urn:nbn:de:0030-drops-211482}, doi = {10.4230/LIPIcs.ESA.2024.77}, annote = {Keywords: 1.5-dimensional terrain, visibility, visibility graph, reverse shortest path, parametric search, shrink-and-bifurcate, range searching} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We study the reverse shortest path problem on disk graphs in the plane. In this problem we consider the proximity graph of a set of n disks in the plane of arbitrary radii: In this graph two disks are connected if the distance between them is at most some threshold parameter r. The case of intersection graphs is a special case with r = 0. We give an algorithm that, given a target length k, computes the smallest value of r for which there is a path of length at most k between some given pair of disks in the proximity graph. Our algorithm runs in O^*(n^{5/4}) randomized expected time, which improves to O^*(n^{6/5}) for unit disk graphs, where all the disks have the same radius. Our technique is robust and can be applied to many variants of the problem. One significant variant is the case of weighted proximity graphs, where edges are assigned real weights equal to the distance between the disks or between their centers, and k is replaced by a target weight w. In other variants, we want to optimize a parameter different from r, such as a scale factor of the radii of the disks.
The main technique for the decision version of the problem (determining whether the graph with a given r has the desired property) is based on efficient implementations of BFS (for the unweighted case) and of Dijkstra’s algorithm (for the weighted case), using efficient data structures for maintaining the bichromatic closest pair for certain bicliques and several distance functions. The optimization problem is then solved by combining the resulting decision procedure with enhanced variants of the interval shrinking and bifurcation technique of [R. Ben Avraham et al., 2015].

Haim Kaplan, Matthew J. Katz, Rachel Saban, and Micha Sharir. The Unweighted and Weighted Reverse Shortest Path Problem for Disk Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 67:1-67:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kaplan_et_al:LIPIcs.ESA.2023.67, author = {Kaplan, Haim and Katz, Matthew J. and Saban, Rachel and Sharir, Micha}, title = {{The Unweighted and Weighted Reverse Shortest Path Problem for Disk Graphs}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {67:1--67:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.67}, URN = {urn:nbn:de:0030-drops-187208}, doi = {10.4230/LIPIcs.ESA.2023.67}, annote = {Keywords: Computational geometry, geometric optimization, disk graphs, BFS, Dijkstra’s algorithm, reverse shortest path} }

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