We introduce hierarchical neighbor graphs, a new architecture for connecting ad hoc wireless nodes distributed in a plane. The structure has the flavor of hierarchical clustering and requires only local knowledge and minimal computation at each node to be formed and repaired. Hence, it is a suitable interconnection model for an ad hoc wireless sensor network. The structure is able to use energy efficiently by reorganizing dynamically when the battery power of heavily utilized nodes degrades and is able to achieve throughput, energy efficiency and network lifetimes that compare favorably with the leading proposals for data collation in sensor networks such as LEACH (Heinzelman et. al., 2002). Additionally, hierarchical neighbor graphs have low power stretch i.e. the power required to connect nodes through the network is a small factor higher than the power required to connect them directly. Our structure also compares favorably to mathematical structures proposed for connecting points in a plane e.g. nearest-neighbor graphs (Ballister et. al., 2005), @math -graphs (Ruppert and Seidel, 1991), in that it has expected constant degree and does not require any significant computation or global information to be formed.
Finally, we end by mentioning that hierarchical neighbor graphs bear more than a passing resemblance to the skip list data structure @cite_19 and its biased version @cite_2 .
@cite_19 @cite_2
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We consider the problem of representing graphs compactly while supporting queries efficiently. In particular we describe a data structure for representing n-vertex unlabeled graphs that satisfy an O(nc)-separator theorem, c < 1. The structure uses O(n) bits, and supports adjacency and degree queries in constant time, and neighbor listing in constant time per neighbor. This generalizes previous results for graphs with constant genus, such as planar graphs.We present experimental results using many "real world" graphs including 3-dimensional finite element meshes, link graphs of the web, internet router graphs, VLSI circuits, and street map graphs. Compared to adjacency lists, our approach reduces space usage by almost an order of magnitude, while supporting depthfirst traversal in about the same running time. Nearest neighbor (k-NN) graphs are widely used in machine learning and data mining applications, and our aim is to better understand what they reveal about the cluster structure of the unknown underlying distribution of points. Moreover, is it possible to identify spurious structures that might arise due to sampling variability? Our first contribution is a statistical analysis that reveals how certain subgraphs of a k-NN graph form a consistent estimator of the cluster tree of the underlying distribution of points. Our second and perhaps most important contribution is the following finite sample guarantee. We carefully work out the tradeoff between aggressive and conservative pruning and are able to guarantee the removal of all spurious cluster structures at all levels of the tree while at the same time guaranteeing the recovery of salient clusters. This is the first such finite sample result in the context of clustering. The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50). We design a variation of skip lists that performs well for generally biased access sequences. Given n items, each with a positive weight wi, 1 ≤ i ≤ n, the time to access item i is O(1 + log (W wi)), where W=∑i=1nwi; the data structure is dynamic. We present two instantiations of biased skip lists, one of which achieves this bound in the worst case, the other in the expected case. The structures are nearly identical; the deterministic one simply ensures the balance condition that the randomized one achieves probabilistically. We use the same method to analyze both.