"Study on Stabbing Rectangles and Circles Induced by Point Sets." by Savindra Singh

Study on Stabbing Rectangles and Circles Induced by Point Sets.

Date of Submission

December 2011

Date of Award

Winter 12-12-2012

Institute Name (Publisher)

Indian Statistical Institute

Document Type

Master's Dissertation

Degree Name

Master of Technology

Subject Name

Computer Science

Department

Advance Computing and Microelectronics Unit (ACMU-Kolkata)

Supervisor

Bishnu, Arijit (ACMU-Kolkata; ISI)

Abstract (Summary of the Work)

Prorimity graph [JT92; Lio; Tou91] is a graph where the edges between the vertices of the graph depends on the neighborliness of vertices. Prozimity graph can be intuitively defined as follows: given a point set P in the plane, the vertices of the graphs, there is an edge between a pair of vertices p, q P if they satisfy some particular notion of neighborliness.Proximity graphs can be used in shape analysis and in data mining |JT92; Toul. In graph drawing, a problem related to proximity graphs is to find the classes of graphs that admit a proximity drawing for some notion of proximity, and whenever possible to efficiently decide, for a given graph, whether such a drawing exists [BETT99; Lio).In the case of Gabriel graphs, GG(P), the notion of neighborliness of a pair of vertices a, b is the closed disk Dab with diameter ab‾. An edge ab is in the Gabriel graph of a point set P if and only if P∩ Dab = {a, b} (see Figure %3D 1.1(Left) [ADH10]. Gabriel graphs were introduced by Gabriel and Sokal (GS69] in the context of geographic variation analysis.In the case of Delaunay graphs, DG(P) [ADH11], the region of influence of a pair of vertices a, b is the set of closed disks Dab with chord ab‾. An edge ab is in the Delaunay graph of a point set P if and only if there exists a disk dab ∈ Dab such that P ∩ das = {a, b}In this thesis, we consider the problems related to the Witness graphs (gen- Figure 1.1: Gabriel graph. Left: The vertices defining the shaded disk are adjacent because their disk doesn't contain any other vertex, in contrast to the other verfices defining the unshaded disk. Right: Witness Gabriel graph. Black points are the vertices of the graph, white points are the witnesses. Each pair of vertices defining a shaded disk are adjacent and the pairs defining the unshaded disks are not. eralization of proximity graphs).1.1 The Witness Gabriel GraphsThe witness Gabriel graph (ADH10] GG─(P,W) is defined by two sets of points P and W; P is the set of vertices of the graph and W is the set of witnesses. There is an edge ab in GG─(P,W) if and only if there is no point of W in Dab\\{a, b} (see Figure I.1(Right)). The witness Gabriel graphs were introduced by Aronov et al. │ADH10│ in 2010.1.2 The Witness Delaunay GraphsThe witress Delaunay groph │JADHI1│ of a point set Pof vertices in the plaar, with respect to a puint set W of witaes, denoted DG─(P.W), is the graph with verte set Pis which two points , y ∈ P are adjacent if and only if there is an open disk that dos ot contain say winess w ∈ W whose bounding cricle passes through x and y.Ia graph drawing, a peobkem that is attracting salstantial revoech is to fiad the mamber of witneo poitta to remove all the edges of a witna graph.

Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843286

Control Number

ISI-DISS-2011-380

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/6913

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