A result of Johnson and Lindenstrauss  shows that a set of n points in high dimensional Euclidean space can be mapped into an O(log n/⑀ 2)-dimensional Euclidean space such that the distance between any two points changes by only a factor of (1 Ϯ ⑀). In this note, we prove this theorem using elementary probabilistic techniques.
The doubling constant of a metric space (X, d) is the smallest value λ such that every ball in X can be covered by λ balls of half the radius. The doubling dimension of X is then defined as dim(X) = log 2 λ. A metric (or sequence of metrics) is called doubling precisely when its doubling dimension is bounded. This is a robust class of metric spaces which… (More)
Consider a setting in which a group of nodes, situated in a large underlying network, wishes to reserve bandwidth on which to support communication. <italic>Virtual private networks</italic> (VPNs) are services that support such a construct; rather than building a new physical network on the group of nodes that must be connected, bandwidth in the underlying… (More)
Several combinatorial optimization problems choose elements to minimize the total cost of constructing a feasible solution that satisfies requirements of clients. In the S<sc>teiner</sc> T<sc>ree</sc> problem, for example, edges must be chosen to connect terminals (clients); in V<sc>ertex</sc> C<sc>over</sc>, vertices must be chosen to cover edges… (More)
Approximation algorithms for clustering points in metric spaces is a flourishing area of research, with much research effort spent on getting a better understanding of the approximation guarantees possible for many objective functions such as k-median, k-means, and min-sum clustering. This quest for better approximation algorithms is further fueled by the… (More)
The Johnson-Lindenstrauss lemma shows that a set of n points in high dimensional Euclidean space can be mapped down into an O(log n== 2) dimensional Euclidean space such that the distance between any two points changes by only a factor of (1). In this note, we prove this lemma using elementary probabilistic techniques.
In many applications, one has to actively select among a set of expensive observations before making an informed decision. For example, in environmental monitoring, we want to select locations to measure in order to most effectively predict spatial phenomena. Often, we want to select observations which are robust against a number of possible objective… (More)
We give simple and easy-to-analyze randomized approximation algorithms for several well-studied NP-hard network design problems. Our algorithms improve over the previously best known approximation ratios. Our main results are the following.<ul><li>We give a randomized 3.55-approximation algorithm for the <i> connected facility location</i> problem. The… (More)
In this paper we study the problem of approximately releasing the cut function of a graph while preserving differential privacy, and give new algorithms (and new analyses of existing algorithms) in both the interactive and non-interactive settings. Our algorithms in the interactive setting are achieved by revisiting the problem of releasing differentially… (More)