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We consider the problem of dynamically maintaining the convex hull of a set S of points in the plane under the following special sequence of insertions and deletions (called window-sliding updates): insert a point to the right of all points of S and delete the leftmost point of S. We propose an O(|S|)-space data structure that can handle each update in O(1) amortized time, such that all standard binary-search-based queries on the convex hull of S can be answered in 𝑂(log |S|) time, and the convex hull itself can be output in time linear in its size.
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Dynamic Convex Hulls for Simple Paths
We consider two restricted cases of the planar dynamic convex hull problem with point insertions and deletions. We assume all updates are performed on a deque (double-ended queue) of points. The first case considers the monotonic path case, where all points are sorted in a given direction, say horizontally left-to-right, and only the leftmost and rightmost points can be inserted and deleted. The second case, which is more general, assumes that the points in the deque constitute a simple path. For both cases, we present solutions supporting deque insertions and deletions in worst-case constant time and standard queries on the convex hull of the points in O(log n) time, where n is the number of points in the current point set. The convex hull of the current point set can be reported in O(h+log n) time, where h is the number of edges of the convex hull. For the 1-sided monotone path case, where updates are only allowed on one side, the reporting time can be reduced to O(h), and queries on the convex hull are supported in O(log h) time. All our time bounds are worst case. In addition, we prove lower bounds that match these time bounds, and thus our results are optimal.
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- Award ID(s):
- 2300356
- PAR ID:
- 10546384
- Editor(s):
- Mulzer, Wolfgang; Phillips, Jeff M
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 293
- ISSN:
- 1868-8969
- ISBN:
- 978-3-95977-316-4
- Page Range / eLocation ID:
- 293-293
- Subject(s) / Keyword(s):
- Dynamic convex hull convex hull queries simple paths path updates deque Theory of computation → Computational geometry Theory of computation → Design and analysis of algorithms
- Format(s):
- Medium: X Size: 15 pages; 967012 bytes Other: application/pdf
- Size(s):
- 15 pages 967012 bytes
- Right(s):
- Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.SoCG.2024.24
