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Competitive Local Routing with Constraints
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This paper shows how to route between any two visible vertices using only 1-local information, while guaranteeing that the returned path has length at most 2 times the Euclidean distance between the source and destination.Abstract:
Let $P$ be a set of $n$ vertices in the plane and $S$ a set of non-crossing line segments between vertices in $P$, called constraints. Two vertices are visible if the straight line segment connecting them does not properly intersect any constraints. The constrained $\Theta_m$-graph is constructed by partitioning the plane around each vertex into $m$ disjoint cones, each with aperture $\theta = 2 \pi/m$, and adding an edge to the `closest' visible vertex in each cone. We consider how to route on the constrained $\Theta_6$-graph. We first show that no deterministic 1-local routing algorithm is $o(\sqrt{n})$-competitive on all pairs of vertices of the constrained $\Theta_6$-graph. After that, we show how to route between any two visible vertices of the constrained $\Theta_6$-graph using only 1-local information. Our routing algorithm guarantees that the returned path is 2-competitive. Additionally, we provide a 1-local 18-competitive routing algorithm for visible vertices in the constrained half-$\Theta_6$-graph, a subgraph of the constrained $\Theta_6$-graph that is equivalent to the Delaunay graph where the empty region is an equilateral triangle. To the best of our knowledge, these are the first local routing algorithms in the constrained setting with guarantees on the length of the returned path.read more
Citations
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Journal ArticleDOI
On the performance of greedy forwarding on Yao and Theta graphs
TL;DR: The worst case shows that the stretches of Greedy Forwarding on both Yao and Theta graphs do not have a constant upper bound in hop metric, and the average-case stretches are measured, and several interesting findings are revealed.
Proceedings ArticleDOI
Routing in Polygonal Domains
Bahareh Banyassady,Man-Kwun Chiu,Matias Korman,Wolfgang Mulzer,André van Renssen,Marcel Roeloffzen,Paul Seiferth,Yannik Stein,Birgit Vogtenhuber,Max Willert +9 more
TL;DR: In this paper, the authors consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes, and propose a routing scheme that always achieves a routing path that exceeds the shortest path by a factor of at most 1 + eps.
Journal ArticleDOI
Routing in polygonal domains
Bahareh Banyassady,Man-Kwun Chiu,Matias Korman,Wolfgang Mulzer,André van Renssen,Marcel Roeloffzen,Paul Seiferth,Yannik Stein,Birgit Vogtenhuber,Max Willert +9 more
TL;DR: In this article, the authors consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes, and present a routing scheme that always achieves a routing path whose length exceeds the shortest path by a factor of at most 1 + e.
Book ChapterDOI
Routing in histograms
Man-Kwun Chiu,Jonas Cleve,Katharina Klost,Matias Korman,Wolfgang Mulzer,André van Renssen,Marcel Roeloffzen,Max Willert +7 more
TL;DR: A routing scheme for double histograms that sends any data packet along a path whose length is at most twice the (unweighted) shortest path distance between the endpoints.
Book ChapterDOI
Bounded-Degree Spanners in the Presence of Polygonal Obstacles.
André van Renssen,Gladys Wong +1 more
TL;DR: This work shows how to construct a plane 2-spanner of the visibility graph of V with respect to S, which can have unbounded degree and is modified in three easy-to-follow steps to bound the degree to 7 at the cost of slightly increasing the spanning ratio to 6.
References
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Book ChapterDOI
Online Routing in Convex Subdivisions
Prosenjit Bose,Pat Morin,Andrej Brodnik,Svante Carlsson,Erik D. Demaine,Rudolf Fleischer,J. Ian Munro,Alejandro López-Ortiz +7 more
TL;DR: There exists a routing algorithm for arbitrary triangulations that has no memory and uses no randomization, and there is no competitive online routing algorithm under the Euclidean distance metric in arbitraryTriangulations.