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15
Better Streaming Algorithms for Clustering Problems
 In Proc. of 35th ACM Symposium on Theory of Computing (STOC
, 2003
"... We study cluster ng pr blems in the str aming model, wher e the goal is to cluster a set of points by making one pass (or a few passes) over the data using a small amount of storSD space.Our mainr esult is a r ndomized algor ithm for kMedian prE lem which p duces a constant factor a ..."
Abstract

Cited by 91 (1 self)
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We study cluster ng pr blems in the str aming model, wher e the goal is to cluster a set of points by making one pass (or a few passes) over the data using a small amount of storSD space.Our mainr esult is a r ndomized algor ithm for kMedian prE lem which p duces a constant factor appr oximation in one pass using storR4 space O(kpolylog n). This is a significant imp r vement of the prS ious best algor5 hm which yielded a 2 appr ximation using O(n )space.
The Online Median Problem
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... We introduce a natural variant of the (metric uncapacitated) kmedian problem that we call the online median problem. Whereas the kmedian problem involves optimizing the simultaneous placement of k facilities, the online median problem imposes the following additional constraints: the facilities ar ..."
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Cited by 85 (2 self)
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We introduce a natural variant of the (metric uncapacitated) kmedian problem that we call the online median problem. Whereas the kmedian problem involves optimizing the simultaneous placement of k facilities, the online median problem imposes the following additional constraints: the facilities are placed one at a time; a facility cannot be moved once it is placed, and the total number of facilities to be placed, k, is not known in advance. The objective of an online median algorithm is to minimize the competitive ratio, that is, the worstcase ratio of the cost of an online placement to that of an optimal offline placement. Our main result is a lineartime constantcompetitive algorithm for the online median problem. In addition, we present a related, though substantially simpler, lineartime constantfactor approximation algorithm for the (metric uncapacitated) facility location problem. The latter algorithm is similar in spirit to the recent primaldualbased facility location algorithm of Jain and Vazirani, but our approach is more elementary and yields an improved running time.
Sequential and parallel algorithms for mixed packing and covering
 IN 42ND ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2001
"... We describe sequential and parallel algorithms that approximately solve linear programs with no negative coefficients (a.k.a. mixed packing and covering problems). For explicitly given problems, our fastest sequential algorithm returns a solution satisfying all constraints within a ¦ ¯ factor in Ç ..."
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Cited by 69 (6 self)
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We describe sequential and parallel algorithms that approximately solve linear programs with no negative coefficients (a.k.a. mixed packing and covering problems). For explicitly given problems, our fastest sequential algorithm returns a solution satisfying all constraints within a ¦ ¯ factor in Ç Ñ � ÐÓ � Ñ � ¯ time, where Ñ is the number of constraints and � is the maximum number of constraints any variable appears in. Our parallel algorithm runs in time polylogarithmic in the input size times ¯ � and uses a total number of operations comparable to the sequential algorithm. The main contribution is that the algorithms solve mixed packing and covering problems (in contrast to pure packing or pure covering problems, which have only “� ” or only “� ” inequalities, but not both) and run in time independent of the socalled width of the problem.
The reverse greedy algorithm for the metric kmedian problem
 Information Processing Letters
"... The Reverse Greedy algorithm (RGreedy) for the kmedian problem works as follows. It starts by placing facilities on all nodes. At each step, it removes a facility to minimize the total distance to the remaining facilities. It stops when k facilities remain. We prove that, if the distance function i ..."
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Cited by 7 (0 self)
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The Reverse Greedy algorithm (RGreedy) for the kmedian problem works as follows. It starts by placing facilities on all nodes. At each step, it removes a facility to minimize the total distance to the remaining facilities. It stops when k facilities remain. We prove that, if the distance function is metric, then the approximation ratio of RGreedy is between Ω(log n / loglog n) and O(log n).
Playing push vs pull: Models and algorithms for disseminating dynamic data in networks
 In Proceedings of the ACM Symposium on Parallelism in Algorithms and Architectures
, 2006
"... Consider a network in which a collection of source nodes maintain and periodically update data objects for a collection of sink nodes, each of which periodically accesses the data originating from some specified subset of the source nodes. We consider the task of efficiently relaying the dynamically ..."
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Cited by 3 (1 self)
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Consider a network in which a collection of source nodes maintain and periodically update data objects for a collection of sink nodes, each of which periodically accesses the data originating from some specified subset of the source nodes. We consider the task of efficiently relaying the dynamically changing data objects to the sinks from their sources of interest. Our focus is on the following “pushpull” approach for this data dissemination problem. Whenever a data object is updated, its source relays the update to a designated subset of nodes, its push set; similarly, whenever a sink requires an update, it propagates its query to a designated subset of nodes, its pull set. The push and pull sets need to be chosen such that every pull set of a sink intersects the push sets of all its sources of interest. We study the problem of choosing push sets and pull sets to minimize total global communication while satisfying all communication requirements. We formulate and study several variants of the above data dissemination problem, that take into account different paradigms for routing between sources (resp., sinks) and their push sets (resp., pull sets) – multicast, unicast, and controlled broadcast – as well as the aggregability of the data objects. Under the multicast model, we present an optimal polynomial time algorithm for tree networks, which yields a randomized O(log n)approximation algorithm for nnode general networks, for which the problem is hard to approximate within a constant factor. Under the unicast ∗ Chakinala, Kumarasubramanian, and Manokaran were partially supported by a generous gift from Northeastern
Nearly LinearWork Algorithms for Mixed Packing/Covering and FacilityLocation Linear Programs, preprint
, 1407
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Nearly lineartime approximation schemes for mixed packing/covering and facilitylocation linear programs. ArXiv eprints, abs/1407.3015
, 2014
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Online Medians via Online Bribery (Extended Abstract)
"... We then consider the competitive ratio with respect to size. An algorithm is ssizecompetitive if, for each k, the cost of Fk is at most the minimum cost of any set of k facilities, while the size of Fk is at most sk. We present optimally competitive algorithms for this problem. Our proofs reduce o ..."
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We then consider the competitive ratio with respect to size. An algorithm is ssizecompetitive if, for each k, the cost of Fk is at most the minimum cost of any set of k facilities, while the size of Fk is at most sk. We present optimally competitive algorithms for this problem. Our proofs reduce online medians to the following online bribery problem: faced with some unknown threshold T 2 R+, an algorithm must submit &quot;bids &quot; b 2 R+ until it submits a bid as large as T. The algorithm pays the sum of its bids. We describe optimally competitive algorithms for online bribery. Our results on costcompetitive online medians extend to approximately metric distance functions, online fractional medians, and online bicriteria approximation.
unknown title
, 2006
"... Abstract In the kmedian problem we are given sets of facilities and customers, and distances between them. For a given set F of facilities, the cost of serving a customer u is the minimum distance between u and a facility in F. The goal is to find a set F of k facilities that minimizes the sum, ove ..."
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Abstract In the kmedian problem we are given sets of facilities and customers, and distances between them. For a given set F of facilities, the cost of serving a customer u is the minimum distance between u and a facility in F. The goal is to find a set F of k facilities that minimizes the sum, over all customers, of their service costs. Following the work of Mettu and Plaxton, we study the incremental medians problem, where k is not known in advance. An incremental algorithm produces a nested sequence of facility sets F1 ` F2 `::: ` Fn, where jFkj = k for each k. Such an algorithm is called ccostcompetitive if the cost of each Fk is at most c times the optimum kmedian cost. We give improved incremental algorithms for the metric version of this problem: an 8costcompetitive deterministic algorithm, a 2e ss 5:44costcompetitive randomized algorithm, (24 + ffl)costcompetitive, polynomialtime deterministic algorithm, and a 6e + ffl ss 16:31costcompetitive, polynomialtime randomized algorithm. We also consider the competitive ratio with respect to size. An algorithm is ssizecompetitive if the cost of each Fk is at most the minimum cost of any set of k facilities, while the size of Fk is at most sk. We show that the optimal sizecompetitive ratios for this problem, in the deterministic and randomized cases, are 4 and e. For polynomialtime algorithms, we present the first polynomialtime O(log m)sizeapproximation algorithm for the offline problem, as well as a polynomialtime O(log m)sizecompetitive algorithm for the incremental problem. Our upper bound proofs reduce the incremental medians problem to the following online bidding problem: faced with some unknown threshold T 2 R+, an algorithm must submit &quot;bids &quot; b 2 R+ until it submits a bid b * T, paying the sum of all its bids. We present folklore algorithms for online bidding and prove that they are optimally competitive.