Theory and Algorithms for Hop-Count-Based Localization with Random Geometric Graph Models of Dense Sensor Networks

Wireless sensor networks can often be viewed in terms of a uniform deployment of a large number of nodes in a region of Euclidean space. Following deployment, the nodes self-organize into a mesh topology with a key aspect being self-localization. Having obtained a mesh topology in a dense, homogeneous deployment, a frequently used approximation is to take the hop distance between nodes to be proportional to the Euclidean distance between them. In this work, we analyze this approximation through two complementary analyses. We assume that the mesh topology is a random geometric graph on the nodes; and that some nodes are designated as anchors with known locations. First, we obtain high probability bounds on the Euclidean distances of all nodes that are h hops away from a fixed anchor node. In the second analysis, we provide a heuristic argument that leads to a direct approximation for the density function of the Euclidean distance between two nodes that are separated by a hop distance h. This approximation is shown, through simulation, to very closely match the true density function. Localization algorithms that draw upon the preceding analyses are then proposed and shown to perform better than some of the well-known algorithms present in the literature. Belief-propagation-based message-passing is then used to further enhance the performance of the proposed localization algorithms. To our knowledge, this is the first usage of message-passing for hop-count-based self-localization.

Factor graph based joint detection/decoding for LDPC coded large-MIMO systems

In this paper, we employ message passing algorithms over graphical models to jointly detect and decode symbols transmitted over large multiple-input multiple-output (MIMO) channels with low density parity check (LDPC) coded bits. We adopt a factor graph based technique to integrate the detection and decoding operations. A Gaussian approximation of spatial interference is used for detection. This serves as a low complexity joint detection/decoding approach for large dimensional MIMO systems coded with LDPC codes of large block lengths. This joint processing achieves significantly better performance than the individual detection and decoding scheme.

Exploiting the Structure of the Constraint Graph for Efficient Points-to Analysis

Points-to analysis is a key compiler analysis. Several memory related optimizations use points-to information to improve their effectiveness. Points-to analysis is performed by building a constraint graph of pointer variables and dynamically updating it to propagate more and more points-to information across its subset edges. So far, the structure of the constraint graph has been only trivially exploited for efficient propagation of information, e.g., in identifying cyclic components or to propagate information in topological order. We perform a careful study of its structure and propose a new inclusion-based flow-insensitive context-sensitive points-to analysis algorithm based on the notion of dominant pointers. We also propose a new kind of pointer-equivalence based on dominant pointers which provides significantly more opportunities for reducing the number of pointers tracked during the analysis. Based on this hitherto unexplored form of pointer-equivalence, we develop a new context-sensitive flow-insensitive points-to analysis algorithm which uses incremental dominator update to efficiently compute points-to information. Using a large suite of programs consisting of SPEC 2000 benchmarks and five large open source programs we show that our points-to analysis is 88% faster than BDD-based Lazy Cycle Detection and 2x faster than Deep Propagation. We argue that our approach of detecting dominator-based pointer-equivalence is a key to improve points-to analysis efficiency.

Diversity in ranking via resistive graph centers

Users can rarely reveal their information need in full detail to a search engine within 1--2 words, so search engines need to "hedge their bets" and present diverse results within the precious 10 response slots. Diversity in ranking is of much recent interest. Most existing solutions estimate the marginal utility of an item given a set of items already in the response, and then use variants of greedy set cover. Others design graphs with the items as nodes and choose diverse items based on visit rates (PageRank). Here we introduce a radically new and natural formulation of diversity as finding centers in resistive graphs. Unlike in PageRank, we do not specify the edge resistances (equivalently, conductances) and ask for node visit rates. Instead, we look for a sparse set of center nodes so that the effective conductance from the center to the rest of the graph has maximum entropy. We give a cogent semantic justification for turning PageRank thus on its head. In marked deviation from prior work, our edge resistances are learnt from training data. Inference and learning are NP-hard, but we give practical solutions. In extensive experiments with subtopic retrieval, social network search, and document summarization, our approach convincingly surpasses recently-published diversity algorithms like subtopic cover, max-marginal relevance (MMR), Grasshopper, DivRank, and SVMdiv.

Representing a cubic graph as the intersection graph of axis-parallel boxes in three dimensions

We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes just touch at their boundaries. Further, this construction can be realized in linear time.

On the secret key capacity of the harary graph PIN model

A pairwise independent network (PIN) model consists of pairwise secret keys (SKs) distributed among m terminals. The goal is to generate, through public communication among the terminals, a group SK that is information-theoretically secure from an eavesdropper. In this paper, we study the Harary graph PIN model, which has useful fault-tolerant properties. We derive the exact SK capacity for a regular Harary graph PIN model. Lower and upper bounds on the fault-tolerant SK capacity of the Harary graph PIN model are also derived.

Information content of molecular graph and prediction of gas phase thermal entropy of organic compounds

Entropy is a fundamental thermodynamic property that has attracted a wide attention across domains, including chemistry. Inference of entropy of chemical compounds using various approaches has been a widely studied topic. However, many aspects of entropy in chemical compounds remain unexplained. In the present work, we propose two new information-theoretical molecular descriptors for the prediction of gas phase thermal entropy of organic compounds. The descriptors reflect the bulk and size of the compounds as well as the gross topological symmetry in their structures, all of which are believed to determine entropy. A high correlation () between the entropy values and our information-theoretical indices have been found and the predicted entropy values, obtained from the corresponding statistically significant regression model, have been found to be within acceptable approximation. We provide additional mathematical result in the form of a theorem and proof that might further help in assessing changes in gas phase thermal entropy values with the changes in molecular structures. The proposed information-theoretical molecular descriptors, regression model and the mathematical result are expected to augment predictions of gas phase thermal entropy for a large number of chemical compounds.

Parallel Flow-Sensitive Pointer Analysis by Graph-Rewriting

Precise pointer analysis is a problem of interest to both the compiler and the program verification community. Flow-sensitivity is an important dimension of pointer analysis that affects the precision of the final result computed. Scaling flow-sensitive pointer analysis to millions of lines of code is a major challenge. Recently, staged flow-sensitive pointer analysis has been proposed, which exploits a sparse representation of program code created by staged analysis. In this paper we formulate the staged flow-sensitive pointer analysis as a graph-rewriting problem. Graph-rewriting has already been used for flow-insensitive analysis. However, formulating flow-sensitive pointer analysis as a graph-rewriting problem adds additional challenges due to the nature of flow-sensitivity. We implement our parallel algorithm using Intel Threading Building Blocks and demonstrate considerable scaling (upto 2.6x) for 8 threads on a set of 10 benchmarks. Compared to the sequential implementation of staged flow-sensitive analysis, a single threaded execution of our implementation performs better in 8 of the benchmarks.

REPRESENTING A CUBIC GRAPH AS THE INTERSECTION GRAPH OF AXIS-PARALLEL BOXES IN THREE DIMENSIONS

We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes touch just at their boundaries.

BELIEF PROPAGATION FOR OPTIMAL EDGE COVER IN THE RANDOM COMPLETE GRAPH

We apply the objective method of Aldous to the problem of finding the minimum-cost edge cover of the complete graph with random independent and identically distributed edge costs. The limit, as the number of vertices goes to infinity, of the expected minimum cost for this problem is known via a combinatorial approach of Hessler and Wastlund. We provide a proof of this result using the machinery of the objective method and local weak convergence, which was used to prove the (2) limit of the random assignment problem. A proof via the objective method is useful because it provides us with more information on the nature of the edge's incident on a typical root in the minimum-cost edge cover. We further show that a belief propagation algorithm converges asymptotically to the optimal solution. This can be applied in a computational linguistics problem of semantic projection. The belief propagation algorithm yields a near optimal solution with lesser complexity than the known best algorithms designed for optimality in worst-case settings.

Rainbow Connection Number of Graph Power and Graph Products

Rainbow connection number, rc(G), of a connected graph G is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same (note that the coloring need not be proper). In this paper we study the rainbow connection number with respect to three important graph product operations (namely the Cartesian product, the lexicographic product and the strong product) and the operation of taking the power of a graph. In this direction, we show that if G is a graph obtained by applying any of the operations mentioned above on non-trivial graphs, then rc(G) a parts per thousand currency sign 2r(G) + c, where r(G) denotes the radius of G and . In general the rainbow connection number of a bridgeless graph can be as high as the square of its radius 1]. This is an attempt to identify some graph classes which have rainbow connection number very close to the obvious lower bound of diameter (and thus the radius). The bounds reported are tight up to additive constants. The proofs are constructive and hence yield polynomial time -factor approximation algorithms.

Belief propagation for minimum weight many-to-one matchings in the random complete graph

In a complete bipartite graph with vertex sets of cardinalities n and n', assign random weights from exponential distribution with mean 1, independently to each edge. We show that, as n -> infinity, with n' = n/alpha] for any fixed alpha > 1, the minimum weight of many-to-one matchings converges to a constant (depending on alpha). Many-to-one matching arises as an optimization step in an algorithm for genome sequencing and as a measure of distance between finite sets. We prove that a belief propagation (BP) algorithm converges asymptotically to the optimal solution. We use the objective method of Aldous to prove our results. We build on previous works on minimum weight matching and minimum weight edge cover problems to extend the objective method and to further the applicability of belief propagation to random combinatorial optimization problems.

PERCOLATION ON THE INFORMATION-THEORETICALLY SECURE SIGNAL TO INTERFERENCE RATIO GRAPH

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R-2 of intensities lambda and lambda(E), respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter gamma. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity lambda(E) of eavesdropper nodes, there exists a critical intensity lambda(c) < infinity such that for all lambda > lambda(c) the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a lambda(0) such that for all lambda < lambda(0) <= lambda(c) a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

Divide and Conquer: Efficient Large-Scale Structure from Motion Using Graph Partitioning

Despite significant advances in recent years, structure-from-motion (SfM) pipelines suffer from two important drawbacks. Apart from requiring significant computational power to solve the large-scale computations involved, such pipelines sometimes fail to correctly reconstruct when the accumulated error in incremental reconstruction is large or when the number of 3D to 2D correspondences are insufficient. In this paper we present a novel approach to mitigate the above-mentioned drawbacks. Using an image match graph based on matching features we partition the image data set into smaller sets or components which are reconstructed independently. Following such reconstructions we utilise the available epipolar relationships that connect images across components to correctly align the individual reconstructions in a global frame of reference. This results in both a significant speed up of at least one order of magnitude and also mitigates the problems of reconstruction failures with a marginal loss in accuracy. The effectiveness of our approach is demonstrated on some large-scale real world data sets.

Graph Modification Problems: A Modern Perspective

We give an overview of recent results and techniques in parameterized algorithms for graph modification problems.