Theta graph

Abstract: We propose a new geometric spanner for static wireless ad hoc networks, which can be constructed efficiently in a localized manner. It integrates the connected dominating set and the local Delaunay graph to form a backbone of the wireless network. Priori arts showed that both structures can be constructed locally with bounded communication costs. This new spanner has these following attractive properties: 1) the backbone is a planar graph, 2) the node degree of the backbone is bounded from above by a positive constant, 3) it is a spanner for both hops and length, 4) it can be constructed locally and is easy to maintain when the nodes move around, and 5) moreover, the communication cost of each node is bounded by a constant. Simulation results are also presented for studying its practical performance.

Abstract: We propose a new geometric spanner, for wireless ad hoc networks, which can be constructed efficiently in a distributed manner. It combines the connected dominating set and the local Delaunay graph to form the backbone of a wireless network. This new spanner has the following attractive properties: (1) the backbone is a planar graph; (2) the node degree of the backbone is bounded from above by a positive constant; (3) it is a spanner both for hops and length; moreover, we show that, given any two nodes u and /spl upsi/, there is a path connecting them in the backbone such that its length is no more than 6 times that of the shortest path and the number of links is no more than 3 times that of the shortest path; (4) it can be constructed locally and is easy to maintain when the nodes move around; and (5) we show that the computation cost of each node is at most O(d log d), where d is its l-hop neighbors in the original unit disk graph, and the communication cost of each node is bounded by a constant. Simulation results are also presented for studying its practical performance.

Abstract: We present a new method for approximate nearest neighbour search on large datasets of high dimensional feature vectors, such as SIFT or GIST descriptors. Our approach constructs a directed graph that can be efficiently explored for nearest neighbour queries. Each vertex in this graph represents a feature vector from the dataset being searched. The directed edges are computed by exploiting the fact that, for these datasets, the intrinsic dimensionality of the local manifold-like structure formed by the elements of the dataset is significantly lower than the embedding space. We also provide an efficient search algorithm that uses this graph to rapidly find the nearest neighbour to a query with high probability. We show how the method can be adapted to give a strong guarantee of 100% recall where the query is within a threshold distance of its nearest neighbour. We demonstrate that our method is significantly more efficient than existing state of the art methods. In particular, our GPU implementation can deliver 90% recall for queries on a data set of 1 million SIFT descriptors at a rate of over 1.2 million queries per second on a Titan X. Finally we also demonstrate how our method scales to datasets of 5M and 20M entries.

Abstract: Spanner of an undirected graph G = (V,E) is a subgraph that is sparse and yet preserves all-pairs distances approximately. More formally, a spanner with stretcht ∈ N is a subgraph (V,ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. Though G is trivially a t-spanner of itself, the research as well as applications of spanners invariably deal with a t-spanner that has as small number of edges as possible.We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner.Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a t-spanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a t-spanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal.