The Online Median Problem

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 constant-competitive algorithm for the online median problem running in time that is linear in the input size. In addition, we present a related, though substantially simpler, constant-factor approximation algorithm for the (metric uncapacitated) facility location problem that runs in time linear in the input size. 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. While our primary focus is on problems which ask us to minimize the weighted average service distance to facilities, we also show that our results can be generalized to hold, to within constant factors, for more general objective functions. For example, we show that all of our approximation results hold, to within constant factors, for the k-means objective function

High-Throughput Inference of Protein-Protein Interaction Sites from Unassigned NMR Data by Analyzing Arrangements Induced By Quadratic Forms on 3-Manifolds

We cast the problem of identifying protein-protein interfaces, using only unassigned NMR spectra, into a geometric clustering problem. Identifying protein-protein interfaces is critical to understanding inter- and intra-cellular communication, and NMR allows the study of protein interaction in solution. However it is often the case that NMR studies of a protein complex are very time-consuming, mainly due to the bottleneck in assigning the chemical shifts, even if the apo structures of the constituent proteins are known. We study whether it is possible, in a high-throughput manner, to identify the interface region of a protein complex using only unassigned chemical shift and residual dipolar coupling (RDC) data. We introduce a geometric optimization problem where we must cluster the cells in an arrangement on the boundary of a 3-manifold. The arrangement is induced by a spherical quadratic form, which in turn is parameterized by SO(3)xR^2. We show that this formalism derives directly from the physics of RDCs. We present an optimal algorithm for this problem that runs in O(n^3 log n) time for an n-residue protein. We then use this clustering algorithm as a subroutine in a practical algorithm for identifying the interface region of a protein complex from unassigned NMR data. We present the results of our algorithm on NMR data for 7 proteins from 5 protein complexes and show that our approach is useful for high-throughput applications in which we seek to rapidly identify the interface region of a protein complex

The online median problem

We introduce a natural variant of the (metric uncapacitated) k-median problem that we call the onlinemedian problem. Whereas the k-median problem involves optimizing the simultaneous placement of kfacilities, 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 tobe placed, k, is not known in advance. The objective of an online median algorithm is to minimize thecompetitive ratio, that is, the worst-case ratio of the cost of an online placement to that of an optima

The Online Median Problem

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

Optimal time bounds for approximate clustering

Clustering is a fundamental problem in unsuper-vised learning, and has been studied widely both as a problem of learning mixture models andas an optimization problem. In this paper, we study clustering with respect the k-median ob-jective function, a natural formulation of clustering in which we attempt to minimize the av-erage distance to cluster centers. One of the main contributions of this paper is a simple butpowerful sampling technique that we call successive sampling that could be of independent inter-est. We show that our sampling procedure can rapidly identify a small set of points (of size just O(k log n=k)) that summarize the input pointsfor the purpose of clustering. Using successive sampling, we develop an algorithm for the k-median problem that runs in O(nk) time for awide range of values of k and is guaranteed, withhigh probability, to return a solution with cost at most a constant factor times optimal. We also es-tablish a lower bound of \Omega (nk) on any random-ized constant-factor approximation algorithm for the k-median problem that succeeds with even anegligible (say 1 100) probability. The best pre-vious upper bound for the problem wa