Yuval Rabani

Efficient search for approximate nearest neighbor in high dimensional spaces

We address the problem ofdesigning data structures that allow efficient search f or approximate nearest neighbors. More specifically, given a database consisting ofa set ofvectors in some high dimensional Euclidean space, we want to construct a space-efficient data structure that would allow us to search, given a query vector, for the closest or nearly closest vector in the database. We also address this problem when distances are measured by the L1 norm and in the Hamming cube. Significantly improving and extending recent results ofKleinberg, we construct data structures whose size is polynomial in the size ofthe database and search algorithms that run in time nearly linear or nearly quadratic in the dimension. (Depending on the case, the extra factors are polylogarithmic in the size ofthe database.)

An O (log k ) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm

It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for k-commodity flow instances with arbitrary capacities and demands. This improves upon the previously best-known bound of O(log2 k) and is existentially tight, up to a constant factor. An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal min-cut ratio, is presented.

On the hardness of approximating MULTICUT and SPARSEST-CUT

We show that the MULTICUT, SPARSEST-CUT, and MIN-2CNF/spl equiv/DELETION problems are NP-hard to approximate within every constant factor, assuming the unique games conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of /spl Omega/(log log n).

Competitive algorithms for distributed data management

We deal with the competitive analysis of algorithms for managing data in a distributed environment. We deal with the file allocation problem, where copies of a file may be be stored in the local storage of some subsets of processors. Copies may be replicated and discarded over time so as to optimize communication costs, but multiple copies must be kept consistent and at least one copy must be stored somewhere in the network at all times. We deal with competitive algorithms for minimizing communication costs, over arbitrary sequences of reads and writes, and arbitrary network topologies. We define the constrained file allocation problem to be the solution of many individual file allocation problems simultaneously, subject to the constraints of local memory size. We give competitive algorithms for this problem on the uniform network topology. We then introduce distributed competitive algorithms for on-line data tracking (a generalization of mobile user tracking) to transform our competitive data management algorithms into distributed algorithms themselves.

Efficient Search for Approximate Nearest Neighbor in High Dimensional Spaces

We address the problem of designing data structures that allow efficient search for approximate nearest neighbors. More specifically, given a database consisting of a set of vectors in some high dimensional Euclidean space, we want to construct a space-efficient data structure that would allow us to search, given a query vector, for the closest or nearly closest vector in the database. We also address this problem when distances are measured by the L1 norm and in the Hamming cube. Significantly improving and extending recent results of Kleinberg, we construct data structures whose size is polynomial in the size of the database and search algorithms that run in time nearly linear or nearly quadratic in the dimension. (Depending on the case, the extra factors are polylogarithmic in the size of the database.)

An improved approximation algorithm for multiway cut

Given an undirected graph with edge costs and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. Multiway Cut is the problem of finding a multiway cut of minimum cost. Previously, a very simple combinatorial algorithm due to Dahlhaus, Johnson, Papadimitriou, Seymour, and Yannakakis gave a performance guarantee of 2(1?1k). In this paper, we present a new linear programming relaxation for Multiway Cut and a new approximation algorithm based on it. The algorithm breaks the threshold of 2 for approximating Multiway Cut, achieving a performance ratio of at most 1.5?1k. This improves the previous result for every value of k. In particular, for k=3 we get a ratio of 76<43.

The Effectiveness of Lloyd-Type Methods for the k-Means Problem

We investigate variants of Lloyds heuristic for clustering high dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a clusterability criterion for data sets. We present variants of Lloyds heuristic that quickly lead to provably near-optimal clustering solutions when applied to well-clusterable instances. This is the first performance guarantee for a variant of Lloyds heuristic. The provision of a guarantee on output quality does not come at the expense of speed: some of our algorithms are candidates for being faster in practice than currently used variants of Lloyds method. In addition, our other algorithms are faster on well-clusterable instances than recently proposed approximation algorithms, while maintaining similar guarantees on clustering quality. Our main algorithmic contribution is a novel probabilistic seeding process for the starting configuration of a Lloyd-type iteration.

Approximation schemes for clustering problems

Let <i>k</i> be a fixed integer. We consider the problem ofpartitioning an input set of points endowed with a distancefunction into <i>k</i> clusters. We give polynomial timeapproximation schemes for the following three clustering problems:Metric <i>k</i>-Clustering, l ²₂<i>k</i>-Clustering, and l²₂ <i>k</i>-Median.In the <i>k</i>-Clustering problem, the objective is to minimizethe sum of all intra-cluster distances. In the <i>k</i>-Medianproblem, the goal is to minimize the sum of distances from pointsin a cluster to the (best choice of) cluster center. In metricinstances, the input distance function is a metric. In l²₂ instances, the points are in R^<i>d</i> and the distance between two points <i>x,y</i>is measured by <i>x−y</i> ²₂ (noticethat (R ^<i>d</i>, ṡ ²₂ is nota metric space). For the first two problems, our results are thefirst polynomial time approximation schemes. For the third problem,the running time of our algorithms is a vast improvement overprevious work.

Local divergence of Markov chains and the analysis of iterative load-balancing schemes

We develop a general technique for the quantitative analysis of iterative distributed load balancing schemes. We illustrate the technique by studying two simple, intuitively appealing models that are prevalent in the literature: the diffusive paradigm, and periodic balancing circuits (or the dimension exchange paradigm). It is well known that such load balancing schemes can be roughly modeled by Markov chains, but also that this approximation can be quite inaccurate. Our main contribution is an effective way of characterizing the deviation between the actual loads and the distribution generated by a related Markov chain, in terms of a natural quantity which we call the local divergence. We apply this technique to obtain bounds on the number of rounds required to achieve coarse balancing in general networks, cycles and meshes in these models. For balancing circuits, we also present bounds for the stronger requirement of perfect balancing, or counting.

Competitive k-server algorithms

Deterministic competitive k-server algorithms are given for all k and all metric spaces. This settles the k-server conjecture of M.S. Manasse et al. (1988) up to the competitive ratio. The best previous result for general metric spaces was a three-server randomized competitive algorithm and a nonconstructive proof that a deterministic three-server competitive algorithm exists. The competitive ratio the present authors can prove is exponential in the number of servers. Thus, the question of the minimal competitive ratio for arbitrary metric spaces is still open. The methods set forth here also give competitive algorithms for a natural generalization of the k-server problem, called the k-taxicab problem. >