We introduce a natural variant of the (metric uncapacitated) k-median problem that we call the online median problem. Whereas the k-median 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 worst-case ratio of the cost of an online placement to that of an optimal offline placement. Our main result is a linear-time constant-competitive algorithm for the online median problem. In addition, we present a related, though substantially simpler, linear-time constant-factor approximation algorithm for the (metric uncapacitated) facility location problem. The latter algorithm is similar in spirit to the recent primal-dual-based facility location algorithm of Jain and Vazirani, but our approach is more elementary and yields an improved running time.

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 k--Median 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.

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 so-called width of the problem. 1. Background Packing and covering problems are problems that can be formulated as linear programs using only non-negative coefficients and non-negative variables. Special cases include pure packing problems, which are of the form Ñ�Ü � ¡ Ü � �Ü � � � and pure covering problems, which are of the form Ñ�Ò � ¡ Ü � �Ü � ��. Lagrangian-relaxation algorithms are based on the following basic idea. Given an optimization problem specified as a collection of constraints, modify the problem by selecting some of the constraints and replacing them by a continuous “penalty ” function that, given a partial solution Ü, measures how close Ü is to violating the removed constraints. Construct a solution iteratively in small steps, making each choice to maintain the remaining constraints while minimizing the increase in the penalty function. While Lagrangian-relaxation algorithms have the disadvantage of producing only approximately optimal (or approximately feasible) solutions, the algorithms have the fol-

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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 “push-pull” 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 n-node 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

We then consider the competitive ratio with respect to size. An algorithm is s-size-competitive 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 "bids " 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 cost-competitive online medians extend to approximately metric distance functions, online fractional medians, and online bicriteria approximation.

Peer-to-peer (P2P) architectures support collaboration to accomplish tasks such as downloading data, VOIP telephone, and cooperative backups. However, end-user systems can be extremely heterogeneous, with heavy-tailed distributions of attributes such as storage space and bandwidth. In systems that ignore heterogeneity, performance suffers, hence most of the popular peer-to-peer applications are forced to classify nodes according to capacity, distinguishing super-peers (which play more active roles) from regular ones (which have limited roles). Similar issues arise in large data centers, where nodes may have widely variable configurations and performance. Our paper solves a generalized classification problem called slicing, which involves partitioning the nodes into k subsets using a one-dimensional attribute. Here, we start by arguing that slicing is the most appropriate generalization of existing classification mechanisms. We review prior work on the problem, and introduce our new Sliver protocol. Theoretical and experimental evaluations show that Sliver converges more rapidly than alternatives, and its low cost makes it appealing in a wide range of practical settings.