On Plane Constrained Bounded-Degree Spanners

Bose, Prosenjit; Fagerberg, Rolf; Renssen, André van; Verdonschot, Sander

doi:10.1007/s00453-018-0476-8

Cited by 26 publications

(29 citation statements)

References 11 publications

(14 reference statements)

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“…Our aim is to route on the constrained half-Θ 6 -graph. This graph was shown to be a plane 2-spanner of Vis(P, S) [2]. The authors also showed a partial routing result (only between visible vertices) on this graph [3].…”

Section: Local Routing On the Visibility Graphmentioning

confidence: 82%

“…Lemma 1 (Lemma 1 of [2]) Let u, v, and w be three arbitrary points in the plane such that uw and vw are visibility edges and w is not the endpoint of a constraint intersecting the interior of triangle uvw. Then there exists a convex chain of visibility edges from u to v in triangle uvw, such that the polygon defined by uw, wv and the convex chain does not contain any constraint or vertex of P .…”

Section: Local Routing On the Visibility Graphmentioning

confidence: 99%

“…Lemma 2 allows us to compute 1-locally which of the edges of Vis(P, S) incident on v are also edges of the constrained half-Θ 6 -graph. Recall that this graph is plane [2], thus we can apply FACE-1 or FACE-2 to route on Vis(P, S).…”

Section: Local Routing On the Visibility Graphmentioning

confidence: 99%

“…Bose and Keil [4] showed that the Constrained Delaunay Triangulation is a 2.42-spanner of Vis(P, S). Recently, the constrained half-Θ 6 -graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) was shown to be a plane 2-spanner of Vis(P, S) [2] and all constrained Θ-graphs with at least 6 cones were shown to be spanners as well [9].…”

Section: Introductionmentioning

confidence: 99%

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Constrained Routing Between Non-Visible Vertices

Bose

Korman

Renssen

et al. 2017

Lecture Notes in Computer Science

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In this paper we study local routing strategies on geometric graphs. Such strategies use geometric properties of the graph like the coordinates of the current and target nodes to route. Specifically, we study routing strategies in the presence of constraints which are obstacles that edges of the graph are not allowed to cross. Let P be a set of n points in the plane and let S be a set of line segments whose endpoints are in P , with no two line segments intersecting properly. We present the first deterministic 1-local O(1)-memory routing algorithm that is guaranteed to find a path between two vertices in the visibility graph of P with respect to a set of constraints S. The strategy never looks beyond the direct neighbors of the current node and does not store more than O(1)-information to reach the target.We then turn our attention to finding competitive routing strategies. We show that when routing on any triangulation T of P such that S ⊆ T , no o(n)-competitive routing algorithm exists when the routing strategy restricts its attention to the triangles intersected by the line segment from the source to the target (a technique commonly used in the unconstrained setting). Finally, we provide an O(n)-competitive deterministic 1-local O(1)-memory routing algorithm on any such T , which is optimal in the worst case, given the lower bound.

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Section: Local Routing On the Visibility Graphmentioning

confidence: 82%

Section: Local Routing On the Visibility Graphmentioning

confidence: 99%

Section: Local Routing On the Visibility Graphmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constrained Routing Between Non-Visible Vertices

Bose

Korman

Renssen

et al. 2017

Lecture Notes in Computer Science

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“…Bose and Keil [7] showed that the Constrained Delaunay Triangulation is a 4π √ 3/9 ≈ 2.42-spanner of Vis(P, S). The constrained Delaunay graph where the empty convex shape is an equilateral triangle was shown to be a 2-spanner of Vis(P, S) [6]. We look at the constrained generalized Delaunay graph, where the empty convex shape can be any convex shape.…”

Section: Introductionmentioning

confidence: 99%

Constrained generalized Delaunay graphs are plane spanners

Bose

Carufel

Renssen

2018

Computational Geometry

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We look at generalized Delaunay graphs in the constrained setting by introducing line segments which the edges of the graph are not allowed to cross. Given an arbitrary convex shape C, a constrained Delaunay graph is constructed by adding an edge between two vertices p and q if and only if there exists a homothet of C with p and q on its boundary that does not contain any other vertices visible to p and q. We show that, regardless of the convex shape C used to construct the constrained Delaunay graph, there exists a constant t (that depends on C) such that it is a plane t-spanner of the visibility graph. Furthermore, we reduce the upper bound on the spanning ratio for the special case where the empty convex shape is an arbitrary rectangle to √ 2 · (2l/s + 1), where l and s are the length of the long and short side of the rectangle.

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Fault-Tolerant Additive Weighted Geometric Spanners

Bhattacharjee

Inkulu

2019

Algorithms and Discrete Applied Mathematics

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Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance dw(p, q) between two points p, q ∈ S is defined as w(p) + d(p, q) + w(q) if p = q and it is zero if p = q. Here, d(p, q) is the geodesic Euclidean distance between p and q. For a real number t > 1, a graph G(S, E) is called a t-spanner for the weighted set S of points if for any two points p and q in S the distance between p and q in graph G is at most t.dw(p, q) for a real number t > 1. For some integer k ≥ 1, a t-spanner G for the set S is a (k, t)-vertex fault-tolerant additive weighted spanner, denoted with (k, t)-VFTAWS, if for any set S ′ ⊂ S with cardinality at most k, the graph G \ S ′ is a t-spanner for the points in S \ S ′ . For any given real number ǫ > 0, we present algorithms to compute a (k, 4 + ǫ)-VFTAWS for the metric space (S, dw) resulting from the points in S belonging to any of the following: R d , simple polygon, polygonal domain, and terrain. Note that d(p, q) is the geodesic Euclidean distance between p and q in the case of simple polygons and terrains whereas in the case of R d it is the Euclidean distance along the line segment joining p and q.

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On Plane Constrained Bounded-Degree Spanners

Cited by 26 publications

References 11 publications

Constrained Routing Between Non-Visible Vertices

Constrained Routing Between Non-Visible Vertices

Constrained generalized Delaunay graphs are plane spanners

Fault-Tolerant Additive Weighted Geometric Spanners

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