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It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c > 0 is a constant. This improves earlier results of Erd˝ os

We revisit a classical load balancing problem in the modern context of decentralized systems and self-interested clients. In particular, there is a set of clients, each of whom must choose a server from a permissible set. Each client selfishly wants to minimize its own latency (job completion time). A server's latency is inversely proportional to its speed,… (More)

We propose new algorithms and improved bounds for interference-aware routing in wireless networks. First, we prove that n arbitrarily matched source-destinations pairs with average distance d, for any 1 ≤ d ≤ √ n, in an O(n) size grid network achieve throughput capacity Ω(n/d). By a simple packing argument, this is also an upper bound in the worst-case. We… (More)

We propose a low-overhead scheme for detecting a network partition or cut in a sensor network. Consider a network <i>S</i> of <i>n</i> sensors, modeled as points in a two-dimensional plane. An ϵ-<i>cut</i>, for any 0 < ϵ < 1, is a linear separation of ϵ<i>n</i> nodes in <i>S</i> from a distinguished node, the <i>base station</i>. Our… (More)

We consider the problem of approximate range counting over streams of <i>d</i>-dimensional points. In the <i>data stream model</i>, the algorithm makes a single scan of the data, which is presented in an arbitrary order, and computes a compact summary (called a <i>sketch</i>). The sketch, whose size depends on the approximation parameter <i>ε</i>, can… (More)

We propose a space-efficient scheme for summarizing multidimensional data streams. Our sketch can be used to solve spatial versions of several classical data stream queries efficiently. For instance, we can track ε-hotspots, which are congruent boxes containing at least an ε fraction of the stream, and maintain hierarchical heavy hitters in d dimensions.… (More)

We generalize the notions of flippable and simultaneously flippable edges in a tri-angulation of a set S of points in the plane to so-called pseudo-simultaneously flippable edges. Such edges are related to the notion of convex decompositions spanned by S. We prove a worst-case tight lower bound for the number of pseudo-simultaneously flippable edges in a… (More)

Heavy hitters, which are items occurring with frequency above a given threshold, are an important aggregation and summary tool when processing data streams or data warehouses. Hierarchical heavy hitters (HHHs) have been introduced as a natural generalization for hierarchical data domains, including multi-dimensional data. An item <i>x</i> in a hierarchy is… (More)

One of the earliest and most well known problems in computational geometry is the so-called art gallery problem. The goal is to compute the minimum possible number guards placed on the vertices of a simple polygon in such a way that they cover the interior of the polygon. In this paper we consider the problem of guarding an art gallery which is modeled as a… (More)

Given n red and n blue points in the plane and a planar straight line matching between the red and the blue points, the matching can be extended into a bipartite planar straight line spanning tree. That is, any red-blue planar matching can be completed into a crossing-free red-blue spanning tree. Such a tree can be constructed in O(n log n) time.