5 research outputs found
Doctor of Philosophy
dissertationThe contributions of this dissertation are centered around designing new algorithms in the general area of sublinear algorithms such as streaming, core sets and sublinear verification, with a special interest in problems arising from data analysis including data summarization, clustering, matrix problems and massive graphs. In the first part, we focus on summaries and coresets, which are among the main techniques for designing sublinear algorithms for massive data sets. We initiate the study of coresets for uncertain data and study coresets for various types of range counting queries on uncertain data. We focus mainly on the indecisive model of locational uncertainty since it comes up frequently in real-world applications when multiple readings of the same object are made. In this model, each uncertain point has a probability density describing its location, defined as distinct locations. Our goal is to construct a subset of the uncertain points, including their locational uncertainty, so that range counting queries can be answered by examining only this subset. For each type of query we provide coreset constructions with approximation-size trade-offs. We show that random sampling can be used to construct each type of coreset, and we also provide significantly improved bounds using discrepancy-based techniques on axis-aligned range queries. In the second part, we focus on designing sublinear-space algorithms for approximate computations on massive graphs. In particular, we consider graph MAXCUT and correlation clustering problems and develop sampling based approaches to construct truly sublinear () sized coresets for graphs that have polynomial (i.e., for any ) average degree. Our technique is based on analyzing properties of random induced subprograms of the linear program formulations of the problems. We demonstrate this technique with two examples. Firstly, we present a sublinear sized core set to approximate the value of the MAX CUT in a graph to a factor. To the best of our knowledge, all the known methods in this regime rely crucially on near-regularity assumptions. Secondly, we apply the same framework to construct a sublinear-sized coreset for correlation clustering. Our coreset construction also suggests 2-pass streaming algorithms for computing the MAX CUT and correlation clustering objective values which are left as future work at the time of writing this dissertation. Finally, we focus on streaming verification algorithms as another model for designing sublinear algorithms. We give the first polylog space and sublinear (in number of edges) communication protocols for any streaming verification problems in graphs. We present efficient streaming interactive proofs that can verify maximum matching exactly. Our results cover all flavors of matchings (bipartite/ nonbipartite and weighted). In addition, we also present streaming verifiers for approximate metric TSP and exact triangle counting, as well as for graph primitives such as the number of connected components, bipartiteness, minimum spanning tree and connectivity. In particular, these are the first results for weighted matchings and for metric TSP in any streaming verification model. Our streaming verifiers use only polylogarithmic space while exchanging only polylogarithmic communication with the prover in addition to the output size of the relevant solution. We also initiate a study of streaming interactive proofs (SIPs) for problems in data analysis and present efficient SIPs for some fundamental problems. We present protocols for clustering and shape fitting including minimum enclosing ball (MEB), width of a point set, -centers and -slab problem. We also present protocols for fundamental matrix analysis problems: We provide an improved protocol for rectangular matrix problems, which in turn can be used to verify (approximate) eigenvectors of an integer matrix . In general our solutions use polylogarithmic rounds of communication and polylogarithmic total communication and verifier space
Range Queries on Uncertain Data
Given a set of uncertain points on the real line, each represented by
its one-dimensional probability density function, we consider the problem of
building data structures on to answer range queries of the following three
types for any query interval : (1) top- query: find the point in that
lies in with the highest probability, (2) top- query: given any integer
as part of the query, return the points in that lie in
with the highest probabilities, and (3) threshold query: given any threshold
as part of the query, return all points of that lie in with
probabilities at least . We present data structures for these range
queries with linear or nearly linear space and efficient query time.Comment: 26 pages. A preliminary version of this paper appeared in ISAAC 2014.
In this full version, we also present solutions to the most general case of
the problem (i.e., the histogram bounded case), which were left as open
problems in the preliminary versio
ε-Kernel Coresets for Stochastic Points
With the dramatic growth in the number of application domains that generate probabilistic, noisy and uncertain data, there has been an increasing interest in designing algorithms for geometric or combinatorial optimization problems over such data. In this paper, we initiate the study of constructing epsilon-kernel coresets for uncertain points. We consider uncertainty in the existential model where each point\u27s location is fixed but only occurs with a certain probability, and the locational model where each point has a probability distribution describing its location. An epsilon-kernel coreset approximates the width of a point set in any direction. We consider approximating the expected width (an ε-EXP-KERNEL), as well as the probability distribution on the width (an (ε, tau)-QUANT-KERNEL) for any direction. We show that there exists a set of O(ε^{-(d-1)/2}) deterministic points which approximate the expected width under the existential and locational models, and we provide efficient algorithms for constructing such coresets. We show, however, it is not always possible to find a subset of the original uncertain points which provides such an approximation. However, if the existential probability of each point is lower bounded by a constant, an ε-EXP-KERNEL is still possible. We also provide efficient algorithms for construct an (ε, τ)-QUANT-KERNEL coreset in nearly linear time. Our techniques utilize or connect to several important notions in probability and geometry, such as Kolmogorov distances, VC uniform convergence and Tukey depth, and may be useful in other geometric optimization problem in stochastic settings. Finally, combining with known techniques, we show a few applications to approximating the extent of uncertain functions, maintaining extent measures for stochastic moving points and some shape fitting problems under uncertainty
Range Counting Coresets for Uncertain Data
We study coresets for various types of range counting queries on uncertain data. In our model each uncertain point has a probability density describing its location, sometimes defined as k distinct locations. Our goal is to construct a subset of the uncertain points, including their locational uncertainty, so that range counting queries can be answered by just examining this subset. We study three distinct types of queries. RE queries return the expected number of points in a query range. RC queries return the number of points in the range with probability at least a threshold. RQ queries returns the probability that fewer than some threshold fraction of the points are in the range. In both RC and RQ coresets the threshold is provided as part of the query. And for each type of query we provide coreset constructions with approximation-size tradeoffs. We show that random sampling can be used to construct each type of coreset, and we also provide significantly improved bounds using discrepancy-based approaches on axis-aligned range queries