(1 + ε)-Approximation for Facility Location in Data Streams

Abstract

We consider the Euclidean facility location problem with uniform opening cost. In this problem, we are given a set of n points P ⊆ R 2 and an opening cost f ∈ R + , and we want to find a set of facilities F ⊆ R 2 that minimizeswhere d(p, q) is the Euclidean distance between p and q. We obtain two main results:• A (1 + ε)-approximation algorithm with running time O(n log n(log n log log n + (log log n)which is O(n log 2 n log log n) for any constant ε.• The first (1 + ε)-approximation algorithm for the cost of the facility location problem for dynamic geometric data streams, i.e., when the stream consists of insert and delete operations of points from a discrete space {1, . . . , ∆} 2 . The streaming algorithm usesOur PTAS is significantly faster than any previously known (1 + ε)-approximation algorithm for the problem, and is also relatively simple. Our algorithm for dynamic geometric data streams is the first (1 + ε)-approximation algorithm for the cost of the facility location problem with polylogarithmic space, and it resolves an open problem in the streaming area.Both algorithms are based on a novel and simple decomposition of an input point set P into small subsets Pi, such that:• the cost of solving the facility location problem for each Pi is small (which means that for each Pi one needs to open only a small, polylogarithmic number of facilities),, where for a point set P , OPT(P ) denotes the cost of an optimal solution for P . The decomposition can be used directly to obtain the PTAS by splitting the point set in the subsets and efficiently solve the problem for each subset independently. By combining our partitioning with techniques to process dynamic data streams of sampling from the cells of the partition and estimating the cost from the sample, we obtain our data streaming algorithm.

Clustering Problems on Sliding Windows

“…In this model, the goal is to maintain a computation on only the most recent W elements of the stream, rather than on the stream in its entirety.We consider the problem of clustering in the sliding window model. Algorithms have been developed for a number of streaming clustering problems, including k-median [26,12,29,25], k-means [13, 23] and facility location [19]. However, while the sliding window model has received renewed attention recently [18,6], no major clustering results in this model have been published since Babcock, Datar, Motwani and O'Callaghan [5] presented a solution to the k-median problem.…”

Quantizing Multiple Sources to a Common Cluster Center: An Asymptotic Analysis

“…, y 2,m , respectively. The cost of conveying a package from one location to another can be modeled to be proportional to the rth power of the distance between the two locations [26][27][28][29]. Thus, the total cost of conveying the ith package through the facility at u k is given by…”

Online Facility Location with Deletions

“…For example, a number of graph optimization problems have been considered in fully dynamic models in the setting of data streams, in the so-called turnstile model. This model has been investigated in two scenarios: in the context of geometric graph optimization problems (see, e.g., [15,28,35]), and only very recently, in the context of standard graph optimization problems (see, e.g., a recent survey [38] and the references therein). The main focus of these studies is to design algorithms that process a stream of data (in this case, edge or vertex insertions and deletions) and using very limited space, to maintain some basic graph features.…”

“…the optimal cost of the Facility Location problem within a factor of O(log 2 ∆). The best currently known streaming algorithm using poly(log ∆)-space gives an (1+ε)-approximation for this problem [15]. The research in data streaming for standard graph optimization problems has been traditionally focusing on the insertion-only model, where one was aiming to design streaming algorithms with O(n poly log n) space (cf.…”