An Infinite Class of Sparse-Yao Spanners

We show that, for any integer k > 5, the Sparse-Yao graph YY_{6k} (also known as Yao-Yao) is a spanner with stretch factor 11.67. The stretch factor drops down to 4.75 for k > 7.Comment: 17 pages, 12 figure

Spanners for Geometric Intersection Graphs

Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.Comment: 16 pages, 5 figures, Late

Parabolic Yao-type Geometric Spanners in Wireless Ad Hoc Networks

Geometric graphs are frequently used to model a wireless ad hoc network in order to build efficient routing algorithms. The network topology in mobile wireless networks may often change therefore position-based routing that uses the idea of localized routing has an advantage over other types of routing protocols. Since a wireless network has limited memory and energy resources, topology control has an important role in enhancing certain desirable properties of these networks. To achieve the goal of topology control, spanning subgraphs of the UDG graph such as Relative Neighborhood Graph, Gabriel Graph and Yao Graph are constructed, which are then routed upon rather than the original UDG. In this thesis we introduce a spanning subgraph of the UDG graph that is a variation of the Displaced Apex Adaptive Yao (DAAY) graph. In this subgraph an exclusion zone based on a parabola is defined with respect to each non-excluded nearest neighbor and positioned on each node, instead of a cone as with the DAAY subgraph, such that each nearest neighbor is inside the parabola. It also lets the apex of the parabola to move along the line segment between the node and its neighbor. The subgraph has two adjustable parameters, one each for the position of the apex with respect to the nearest neighbor, and the width of the parabola. Thus a directed or undirected spanning subgraph of a UDG is constructed. We show that this spanning subgraph is connected, has a conditional bounded out-degree, is a t-spanner with bounded stretch factor, and contains the Euclidean minimum spanning tree as a subgraph. Experimental comparisons with related spanning subgraphs are also presented

Almost All Even Yao-Yao Graphs Are Spanners

It is an open problem whether Yao-Yao graphs YY_{k} (also known as sparse-Yao graphs) are all spanners when the integer parameter k is large enough. In this paper we show that, for any integer k >= 42, the Yao-Yao graph YY_{2k} is a t_k-spanner, with stretch factor t_k = 6.03+O(k^{-1}) when k tends to infinity. Our result generalizes the best known result which asserts that all YY_{6k} are spanners for k >= 6 [Bauer and Damian, SODA\u2713]. Our proof is also somewhat simpler

Ramsey-type theorems for lines in 3-space

We prove geometric Ramsey-type statements on collections of lines in 3-space. These statements give guarantees on the size of a clique or an independent set in (hyper)graphs induced by incidence relations between lines, points, and reguli in 3-space. Among other things, we prove that: (1) The intersection graph of n lines in R^3 has a clique or independent set of size Omega(n^{1/3}). (2) Every set of n lines in R^3 has a subset of n^{1/2} lines that are all stabbed by one line, or a subset of Omega((n/log n)^{1/5}) such that no 6-subset is stabbed by one line. (3) Every set of n lines in general position in R^3 has a subset of Omega(n^{2/3}) lines that all lie on a regulus, or a subset of Omega(n^{1/3}) lines such that no 4-subset is contained in a regulus. The proofs of these statements all follow from geometric incidence bounds -- such as the Guth-Katz bound on point-line incidences in R^3 -- combined with Tur\'an-type results on independent sets in sparse graphs and hypergraphs. Although similar Ramsey-type statements can be proved using existing generic algebraic frameworks, the lower bounds we get are much larger than what can be obtained with these methods. The proofs directly yield polynomial-time algorithms for finding subsets of the claimed size.Comment: 18 pages including appendi