In this report, we initiate study on understanding a theoretical model for distributed computing called Congested Clique. This report presents constant-time and near-constant-time distributed algorithms for a variety of problems in the Congested Clique model. We start by showing how to compute a 3-ruling set in expected O(log log log n) rounds and using this, we obtain a constant-approximation to metric facility location, also in expected O(log log log n) rounds. In addition, assuming an input metric space of constant doubling dimension, we obtain constant-round algorithms to compute maximal independent set on distance-threshold graphs and constant-factor approximation to the metric facility location problem. These results significantly improve on the running time of the fastest known algorithms for these problems in the Congested Clique setting. Then, we study two fundamental graph problems, Graph Connectivity (GC) and Minimum Spanning Tree (MST), in the Congested Clique model, and present several new bounds on the time and message complexities of randomized algorithms for these problems. No non-trivial (i.e., super-constant) time lower bounds are known for either of the aforementioned problems; in particular, an important open question is whether or not constant-round algorithms exist for these problems. We make progress toward answering this question by presenting randomized Monte Carlo algorithms for both problems that run in O(log log log n) rounds (where n is the size of the clique). In addition, assuming an input metric space of constant doubling dimension, we obtain constant-round algorithm the MST problem. Our results improve by an exponential factor on the long-standing (deterministic) time bound of O(log log n) rounds for these problems due to Lotker et al. (SICOMP 2005). Our algorithms make use of several algorithmic tools including graph sketching, random sampling, and fast sorting. Thus far there has been little work on understanding the message complexity of problems in the Congested Clique. In this report, we initiate a study on the message complexity of Congested Clique algorithms. We study two graph problems, Graph Connectivity (GC) and Minimum Spanning Tree (MST), in the Congested Clique model, focusing on the design of fast algorithms with low message complexity. Our motivation comes from recently established connections between the Congested Clique model and models of large-scale distributed computing such as MapReduce (Hegeman et al., SIROCCO 2014) and the “big data” model (Klauck et al., SODA 2015). For these connections to be fruitful, Congested Clique algorithms not only need to be fast, they also need to have low message complexity. While the aforementioned algorithms are fast, they have an Ω(n2) message complexity, which makes them impractical in the context of the MapReduce and “big data” models. This motivates our goal of achieving low message complexity, without sacrificing the speed of the algorithm. We start with the simpler GC problem and show that it can be solved in O(log log log n) rounds using only O(n poly log n) messages. Then we derive subroutines to aid our earlier MST algorithm to run in O(log log log n) rounds using O(m poly log n) messages on an m-edge input graph. Then, we present an algorithm running in O(log* n) rounds, with message complexity O (sqrt(m · n)) and then build on this algorithm to derive a family of algorithms, containing for any ε, 0 < ε ≤ 1, an algorithm running in O(log*n/ε) rounds, using O(n^(1+ε/ε)) messages. Setting ε = log log n/ log n leads to the first sub-logarithmic round Congested Clique MST algorithm that uses only O (n) messages. Our results are a step toward understanding the power of randomization in the Congested Clique with respect to both time and message complexity.

In a search game, a seeker searches for a hider in some space. The seeker wishes to find the hider as quickly as possible, and the hider wishes to avoid capture as long as possible. In this paper, I will focus on the case where the search space is a star, and the only information the seeker has is the speed of the hider. I will provide algorithms for some cases where the seeker is guaranteed to find the hider and prove optimality for some of these cases. Also, I will look at some cases where the hider can avoid capture indefinitely. I will also present some results for searching on trees.

We examine the problem of counting interval graphs. We answer the question posed by Hanlon, of whether the formal power series generating function of the number of interval graphs on n vertices has a positive radius of convergence. We have found that it is zero. We have obtained a lower bound and an upper bound on the number of interval graphs on n vertices. We also study the application of interval graphs to the dynamic storage allocation problem. Dynamic storage allocation has been shown to be NP-complete by Stockmeyer. Coloring interval graphs on-line has applications to dynamic storage allocation. The most colors used by Kierstead's algorithm is 3 ω -2, where ω is the size of the largest clique in the graph. We determine a lower bound on the colors used. One such lower bound is 2 ω -1.

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Community, a notion transversal to all areas of Social Network Analysis, has drawn tremendous amount of attention across the sciences in the past decades. Numerous attempts to characterize both the sociological embodiment of the concept as well as its observable structural manifestation in the social network have to this date only converged in spirit. No formal consensus has been reached on the quantifiable aspects of community, despite it being deeply linked to topological and dynamic aspects of the underlying social network. Presenting a fresh approach to the evaluation of communities, this thesis introduces and builds upon the cohesion, a novel metric which captures the intrinsic quality, as a community, of a set of nodes in a network. The cohesion, defined in terms of social triads, was found to be highly correlated to the subjective perception of communitiness through the use of a large-scale online experiment in which users were able to compute and rate the quality of their social groups on Facebook. Adequately reflecting the complexity of social interactions, the problem of finding a maximally cohesive group inside a given social network is shown to be NP-hard. Using a heuristic approximation algorithm, applications of the cohesion to broadly different use cases are highlighted, ranging from its application to network visualization, to the study of the evolution of agreement groups in the United States Senate, to the understanding of the intertwinement between subjects' psychological traits and the cohesive structures in their social neighborhood. The use of the cohesion proves invaluable in that it offers non-trivial insights on the network structure and its relation to the associated semantic.

The work in this thesis can be divided into two different parts. In the first part, we suggest an approximate edge 3-coloring polynomial time algorithm for cubic graphs. For any cubic graph with n vertices, using this coloring algorithm, we get an edge 3-coloring with at most n/3 error vertices. In the second part, we study Jim Propp's Rotor-Router model on some non-bipartite graph. We find the difference between the number of chips at vertices after performing a walk on this graph using Propp model and the expected number of chips after a random walk. It is known that for line of integers and d-dimenional grid, this deviation is constant. However, it is also proved that for k-ary infinite trees, for some initial configuration the deviation is no longer a constant and say it is D. We present a similar study on some non-bipartite graph constructed from k-ary infinite trees and conclude that for this graph with the same initial configuration, the deviation is almost (k²)D.

Thesis (M.S.)--University of Texas at El Paso, 2007. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.

Thesis (Ph.D.)--Computing, Georgia Institute of Technology, 2009. / Committee Chair: Bader, David; Committee Member: Berry, Jonathan; Committee Member: Fujimoto, Richard; Committee Member: Saini, Subhash; Committee Member: Vuduc, Richard

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (Ph.D.) -- Mississippi State University. Department of Computer Science and Engineering. / Title from title screen. Includes bibliographical references.