On Geometric Spanners of Euclidean and Unit Disk Graphs

We consider the problem of constructing bounded-degree planar geometric spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor C_{del\approx 2.42; however, its degree may not be bounded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor \rho =1+2\pi(k\cos{\frac{\pi{k)^{-1 and degree bounded by $k$, for any integer parameter $k\geq 14$. This result immediately implies an algorithm for constructing a planar geometric spanner of a Euclidean graph with stretch factor \rho \cdot C_{del and degree bounded by $k$, for any integer parameter $k\geq 14$. Moreover, the resulting spanner contains a Euclidean Minimum Spanning Tree (EMST) as a subgraph. Our second contribution lies in developing the structural results necessary to transfer our analysis and algorithm from Euclidean graphs to unit disk graphs, the usual model for wireless ad-hoc networks. We obtain a very simple distributed, {\em strictly-localized algorithm that, given a unit disk graph embedded in the plane, constructs a geometric spanner with the above stretch factor and degree bound, and also containing an EMST as a subgraph. The obtained results dramatically improve the previous results in all aspects, as shown in the paper

Computing Lightweight Spanning Subgraphs Locally

We consider the problem of computing bounded-degree lightweight plane spanning subgraphs of unit disk graphs in the local distributed model of computation. We are motivated by the hypothesis that such subgraphs can provide the underlying network topology for efficient unicasting and/or multicasting in wireless distributed systems. We start by showing that, for any integer $k \geq 2$, there exists a $k$-local distributed algorithm that, given a unit disk graph $U$ embedded in the plane, constructs a plane subgraph of $U$ containing a Euclidean Minimum Spanning Tree (EMST) of $V(U)$, whose degree is at most 6, and whose total weight is at most $(1 + \frac{2}{k-1})$ times the weight of an EMST of $V(U)$. We show that this bound is tight by proving that, for any $\epsilon \u3e 0$, there exists a unit disk graph $U$ such that no $k$-local distributed algorithm can construct a spanning subgraph of $U$ whose total weight is at most $(1 + \frac{2}{k-1} -\epsilon)$ times the weight of an EMST of $V(U)$. We then go further and present the first $k$-local distributed algorithm, where $k$ is a constant, that computes a bounded-degree plane lightweight {\em spanner} of a given unit disk graph. The upper bounds on the number of communication rounds of the algorithm, the degree, the stretch factor, and the weight of the spanner, are very small. For example, our results imply an $18$-local distributed algorithm that computes a plane spanner of a given unit disk graph $U$, whose degree is at most 14, stretch factor at most 8.81, and weight at most $8.81$ times the weight of an EMST of $V(U)$. All the obtained results rely on an elegant structural result that we develop for weighted planar graphs. We show a wider application of this result by giving an $O(n\lg{n})$ time centralized algorithm that constructs bounded-degree plane lightweight spanners of unit disk graphs (which include Euclidean graphs), with the best upper bounds on the spanner degree, stretch factor, and weight

Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition

We provide efficient constant factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves partitioning the plane minus the input points into subsets with exactly three boundary components, with approximately minimum total length. In the Euclidean case, these curves are squares; in the hyperbolic case, they combine our Euclidean square pants decomposition with our tree clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now Lemma 5.2, as the previous proof was erroneou

Algorithms for Fast Aggregated Convergecast in Sensor Networks

Fast and periodic collection of aggregated data is of considerable interest for mission-critical and continuous monitoring applications in sensor networks. In the many-to-one communication paradigm, referred to as convergecast, we focus on applications wherein data packets are aggregated at each hop en-route to the sink along a tree-based routing topology, and address the problem of minimizing the convergecast schedule length by utilizing multiple frequency channels. The primary hindrance in minimizing the schedule length is the presence of interfering links. We prove that it is NP-complete to determine whether all the interfering links in an arbitrary network can be removed using at most a constant number of frequencies. We give a sufficient condition on the number of frequencies for which all the interfering links can be removed, and propose a polynomial time algorithm that minimizes the schedule length in this case. We also prove that minimizing the schedule length for a given number of frequencies on an arbitrary network is NP-complete, and describe a greedy scheme that gives a constant factor approximation on unit disk graphs. When the routing tree is not given as an input to the problem, we prove that a constant factor approximation is still achievable for degree-bounded trees. Finally, we evaluate our algorithms through simulations and compare their performance under different network parameters

Topology Control in Heterogeneous Wireless Networks: Problems and Solutions

Previous work on topology control usually assumes homogeneous wireless nodes with uniform transmission ranges. In this paper, we propose two localized topology control algorithms for heterogeneous wireless multi-hop networks with nonuniform transmission ranges: Directed Relative Neighborhood Graph (DRNG) and Directed Local Spanning Subgraph (DLSS). In both algorithms, each node selects a set of neighbors based on the locally collected information. We prove that (1) the topologies derived under DRNG and DLSS preserve the network connectivity; (2) the out degree of any node in the resulting topology by DLSS is bounded, while the out degree cannot be bounded in DRNG; and (3) the topologies generated by DRNG and DLSS preserve the network bi-directionality