A clustered back-bone for routing in ad-hoc networks

In the recent years, a lot of research work has been undertaken in the area of ad-hoc networks due to the increasing potential of putting them to commercial use in various types of mobile computing devices. Topology control in ad-hoc networks is a widely researched topic; with a number of algorithms being proposed for the construction of a power-efficient topology that optimizes the battery usage of the mobile nodes. This research proposes a novel technique of partitioning the ad-hoc network into virtually-disjoint clusters. The ultimate aim of forming a routing graph over which power-efficient routing can be implemented in a simple and effective manner is realized by partitioning the network into disjoint clusters and thereafter joining them through gateways to form a connected, planar back-bone which is also a t-spanner of the original Unit Disk Graph (UDG). Some of the previously proposed algorithms require the nodes to construct local variations of the Delaunay Triangulation and undertake several complicated steps for ensuring the planarity of the back-bone graph. The construction of the Delaunay Triangulation is very complex and time-consuming. This work achieves the objective of constructing a routing graph which is a planar spanner, without requiring the expensive construction of the Delaunay Triangulation, thus saving the node power, an important resource in the ad-hoc network. Moreover, the algorithm guarantees that the total number of messages required to be sent by each node is O(n). This makes the topology easily reconfigurable in case of node motion

Spanning Properties of Theta-Theta Graphs

We study the spanning properties of Theta-Theta graphs. Similar in spirit with the Yao-Yao graphs, Theta-Theta graphs partition the space around each vertex into a set of k cones, for some fixed integer k > 1, and select at most one edge per cone. The difference is in the way edges are selected. Yao-Yao graphs select an edge of minimum length, whereas Theta-Theta graphs select an edge of minimum orthogonal projection onto the cone bisector. It has been established that the Yao-Yao graphs with parameter k = 6k' have spanning ratio 11.67, for k' >= 6. In this paper we establish a first spanning ratio of $7.82$ for Theta-Theta graphs, for the same values of $k$. We also extend the class of Theta-Theta spanners with parameter 6k', and establish a spanning ratio of $16.76$ for k' >= 5. We surmise that these stronger results are mainly due to a tighter analysis in this paper, rather than Theta-Theta being superior to Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta graphs decreases to 4.64 as k' increases to 8. These are the first results on the spanning properties of Theta-Theta graphs.Comment: 20 pages, 6 figures, 3 table

Relaxed spanners for directed disk graphs

Let $(V,\delta)$ be a finite metric space, where $V$ is a set of $n$ points and $\delta$ is a distance function defined for these points. Assume that $(V,\delta)$ has a constant doubling dimension $d$ and assume that each point $p\in V$ has a disk of radius $r(p)$ around it. The disk graph that corresponds to $V$ and $r(\cdot)$ is a \emph{directed} graph $I(V,E,r)$, whose vertices are the points of $V$ and whose edge set includes a directed edge from $p$ to $q$ if $\delta(p,q)\leq r(p)$. In \cite{PeRo08} we presented an algorithm for constructing a (1+\eps)-spanner of size O(n/\eps^d \log M), where $M$ is the maximal radius $r(p)$. The current paper presents two results. The first shows that the spanner of \cite{PeRo08} is essentially optimal, i.e., for metrics of constant doubling dimension it is not possible to guarantee a spanner whose size is independent of $M$. The second result shows that by slightly relaxing the requirements and allowing a small perturbation of the radius assignment, considerably better spanners can be constructed. In particular, we show that if it is allowed to use edges of the disk graph I(V,E,r_{1+\eps}), where r_{1+\eps}(p) = (1+\eps)\cdot r(p) for every $p\in V$, then it is possible to get a (1+\eps)-spanner of size O(n/\eps^d) for $I(V,E,r)$. Our algorithm is simple and can be implemented efficiently

Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph

Data-sensitive metrics adapt distances locally based the density of data points with the goal of aligning distances and some notion of similarity. In this paper, we give the first exact algorithm for computing a data-sensitive metric called the nearest neighbor metric. In fact, we prove the surprising result that a previously published $3$-approximation is an exact algorithm. The nearest neighbor metric can be viewed as a special case of a density-based distance used in machine learning, or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric to compute sparse spanners and persistent homology. We also explore the behavior of the metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest.Comment: 15 page

Topology design for time-varying networks

Traditional wireless networks seek to support end-to-end communication through either a single-hop wireless link to infrastructure or multi-hop wireless path to some destination. However, in some wireless networks (such as delay tolerant networks, or mobile social networks), due to sparse node distribution, node mobility, and time-varying network topology, end-to-end paths between the source and destination are not always available. In such networks, the lack of continuous connectivity, network partitioning, and long delays make design of network protocols very challenging. Previous DTN or time-varying network research mainly focuses on routing and information propagation. However, with large number of wireless devices' participation, and a lot of network functionality depends on the topology, how to maintain efficient and dynamic topology of a time-varying network becomes crucial. In this dissertation, I model a time-evolving network as a directed time-space graph which includes both spacial and temporal information of the network, then I study various topology control problems with such time-space graphs. First, I study the basic topology design problem where the links of the network are reliable. It aims to build a sparse structure from the original time-space graph such that (1) the network is still connected over time and/or supports efficient routing between any two nodes; (2) the total cost of the structure is minimized. I first prove that this problem is NP-hard, and then propose several greedy-based methods as solutions. Second, I further study a cost-efficient topology design problem, which not only requires the above two objective, but also guarantees that the spanning ratio of the topology is bounded by a given threshold. This problem is also NP-hard, and I give several greedy algorithms to solve it. Last, I consider a new topology design problem by relaxing the assumption of reliable links. Notice that in wireless networks the topologies are not quit predictable and the links are often unreliable. In this new model, each link has a probability to reflect its reliability. The new reliable topology design problem aims to build a sparse structure from the original space-time graph such that (1) for any pair of devices, there is a space-time path connecting them with the reliability larger than a required threshold; (2) the total cost of the structure is minimized. Several heuristics are proposed, which can significantly reduce the total cost of the topology while maintain the connectivity or reliability over time. Extensive simulations on both random networks and real-life tracing data have been conducted, and results demonstrate the efficiency of the proposed methods

The localized Delaunay triangulation and ad-hoc routing in heterogeneous environments

Ad-Hoc Wireless routing has become an important area of research in the last few years due to the massive increase in wireless devices. Computational Geometry is relevant in attempts to build stable, low power routing schemes. It is only recently, however, that models have been expanded to consider devices with a non-uniform broadcast range, and few properties are known. In particular, we find, via both theoretical and experimental methods, extremal properties for the Localized Delaunay Triangulation over the Mutual Inclusion Graph. We also provide a distributed, sub-quadratic algorithm for the generation of the structure

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