article

Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces

Authors:

Lyudmil Aleksandrov,

Hristo N. Djidjev,

Anil Maheshwari,

Doron Nussbaum,

Jörg-Rüdiger SackAuthors Info & Claims

Discrete & Computational Geometry, Volume 44, Issue 4

Pages 762 - 801

Published: 01 December 2010 Publication History

Abstract

We consider the well-known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the sub-paths within each face of P.

We present query algorithms that compute approximate distances and/or approximate shortest paths on P. Our all-pairs query algorithms take as input an approximation parameter źź(0,1) and a query time parameter $\mathfrak{q}$, in a certain range, and build a data structure $\mathrm{APQ}(P,\varepsilon;\mathfrak{q})$, which is then used for answering ź-approximate distance queries in $O(\mathfrak{q})$ time. As a building block of the $\mathrm{APQ}(P,\varepsilon;\mathfrak{q})$ data structure, we develop a single-source query data structure SSQ(a;P,ź) that can answer ź-approximate distance queries from a fixed point a to any query point on P in logarithmic time. Our algorithms answer shortest path queries in weighted surfaces, which is an important extension, both theoretically and practically, to the extensively studied Euclidean distance case. In addition, our algorithms improve upon previously known query algorithms for shortest paths on surfaces. The algorithms are based on a novel graph separator algorithm introduced and analyzed here, which extends and generalizes previously known separator algorithms.

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Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces
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cover image Discrete & Computational Geometry

Discrete & Computational Geometry Volume 44, Issue 4

December 2010

248 pages

ISSN:0179-5376

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Copyright © Copyright © 2010 Springer Science+Business Media, LLC.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 December 2010

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Cited By

Wei VWong RLong CMount DSamet H(2024)On Efficient Shortest Path Computation on Terrain Surface: A Direction-Oriented ApproachIEEE Transactions on Knowledge and Data Engineering10.1109/TKDE.2024.336314736:8(4129-4143)Online publication date: 1-Aug-2024
https://dl.acm.org/doi/10.1109/TKDE.2024.3363147
Yuan NWang PMeng WChen SXu JXin SHe YWang W(2023)A Variational Framework for Curve Shortening in Various Geometric DomainsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2021.313502129:4(1951-1963)Online publication date: 1-Apr-2023
https://dl.acm.org/doi/10.1109/TVCG.2021.3135021
Ma XKeates SJiang YKosinka J(2019)Subdivision surface fitting to a dense mesh using ridges and umbilicsComputer Aided Geometric Design10.1016/j.cagd.2014.10.00132:C(5-21)Online publication date: 1-Jan-2019
https://dl.acm.org/doi/10.1016/j.cagd.2014.10.001
Jaklin NTibboel MGeraerts RShapiro AAmato NHodgins J(2014)Computing high-quality paths in weighted regionsProceedings of the Seventh International Conference on Motion in Games10.1145/2668064.2668097(77-86)Online publication date: 6-Nov-2014
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Aleksandrov LDjidjev HMaheshwari ASack J(2013)An Approximation Algorithm for Computing Shortest Paths in Weighted 3-d DomainsDiscrete & Computational Geometry10.1007/s00454-013-9486-050:1(124-184)Online publication date: 1-Jul-2013
https://dl.acm.org/doi/10.1007/s00454-013-9486-0
Djidjev HSommer C(2011)Approximate distance queries for weighted polyhedral surfacesProceedings of the 19th European conference on Algorithms10.5555/2040572.2040636(579-590)Online publication date: 5-Sep-2011
https://dl.acm.org/doi/10.5555/2040572.2040636

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