On a family of strong geometric spanners that admit local routing strategies

We introduce a family of directed geometric graphs, denoted \paz, that depend on two parameters $\lambda$ and $\theta$. For $0\leq \theta<\frac{\pi}{2}$ and ${1/2} < \lambda < 1$, the \paz graph is a strong $t$-spanner, with $t=\frac{1}{(1-\lambda)\cos\theta}$. The out-degree of a node in the \paz graph is at most $\lfloor2\pi/\min(\theta, \arccos\frac{1}{2\lambda})\rfloor$. Moreover, we show that routing can be achieved locally on \paz. Next, we show that all strong $t$-spanners are also $t$-spanners of the unit disk graph. Simulations for various values of the parameters $\lambda$ and $\theta$ indicate that for random point sets, the spanning ratio of \paz is better than the proven theoretical bounds

Undirected Connectivity of Sparse Yao Graphs

Given a finite set S of points in the plane and a real value d > 0, the d-radius disk graph G^d contains all edges connecting pairs of points in S that are within distance d of each other. For a given graph G with vertex set S, the Yao subgraph Y_k[G] with integer parameter k > 0 contains, for each point p in S, a shortest edge pq from G (if any) in each of the k sectors defined by k equally-spaced rays with origin p. Motivated by communication issues in mobile networks with directional antennas, we study the connectivity properties of Y_k[G^d], for small values of k and d. In particular, we derive lower and upper bounds on the minimum radius d that renders Y_k[G^d] connected, relative to the unit radius assumed to render G^d connected. We show that d=sqrt(2) is necessary and sufficient for the connectivity of Y_4[G^d]. We also show that, for d = 2/sqrt(3), Y_3[G^d] is always connected. Finally, we show that Y_2[G^d] can be disconnected, for any d >= 1.Comment: 7 pages, 11 figure

An Efficient Construction of Yao-Graph in Data-Distributed Settings

A sparse graph that preserves an approximation of the shortest paths between all pairs of points in a plane is called a geometric spanner. Using range trees of sublinear size, we design an algorithm in massively parallel computation (MPC) model for constructing a geometric spanner known as Yao-graph. This improves the total time and the total memory of existing algorithms for geometric spanners from subquadratic to near-linear

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

Pi/2-Angle Yao Graphs are Spanners

We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor 8(29+23sqrt(2)). Enroute to this, we also show that the Yao graph Y4 in the Linf metric is a planar spanner with stretch factor 8.Comment: 20 pages, 9 figure