Fast Distributed Algorithms for LP-Type Problems of Bounded Dimension

In this paper we present various distributed algorithms for LP-type problems in the well-known gossip model. LP-type problems include many important classes of problems such as (integer) linear programming, geometric problems like smallest enclosing ball and polytope distance, and set problems like hitting set and set cover. In the gossip model, a node can only push information to or pull information from nodes chosen uniformly at random. Protocols for the gossip model are usually very practical due to their fast convergence, their simplicity, and their stability under stress and disruptions. Our algorithms are very efficient (logarithmic rounds or better with just polylogarithmic communication work per node per round) whenever the combinatorial dimension of the given LP-type problem is constant, even if the size of the given LP-type problem is polynomially large in the number of nodes

Covering many points with a small-area box

Let $P$ be a set of $n$ points in the plane. We show how to find, for a given integer $k>0$, the smallest-area axis-parallel rectangle that covers $k$ points of $P$ in $O(nk^2 \log n+ n\log^2 n)$ time. We also consider the problem of, given a value $\alpha>0$, covering as many points of $P$ as possible with an axis-parallel rectangle of area at most $\alpha$. For this problem we give a probabilistic $(1-\varepsilon)$-approximation that works in near-linear time: In $O((n/\varepsilon^4)\log^3 n \log (1/\varepsilon))$ time we find an axis-parallel rectangle of area at most $\alpha$ that, with high probability, covers at least $(1-\varepsilon)\mathrm{\kappa^*}$ points, where $\mathrm{\kappa^*}$ is the maximum possible number of points that could be covered

Net and Prune: A Linear Time Algorithm for Euclidean Distance Problems

We provide a general framework for getting expected linear time constant factor approximations (and in many cases FPTAS's) to several well known problems in Computational Geometry, such as $k$-center clustering and farthest nearest neighbor. The new approach is robust to variations in the input problem, and yet it is simple, elegant and practical. In particular, many of these well studied problems which fit easily into our framework, either previously had no linear time approximation algorithm, or required rather involved algorithms and analysis. A short list of the problems we consider include farthest nearest neighbor, $k$-center clustering, smallest disk enclosing $k$ points, $k$th largest distance, $k$th smallest $m$-nearest neighbor distance, $k$th heaviest edge in the MST and other spanning forest type problems, problems involving upward closed set systems, and more. Finally, we show how to extend our framework such that the linear running time bound holds with high probability

On Approximating the Riemannian 1-Center

International audienceIn this paper, we generalize the simple Euclidean 1-center approximation algorithm of Badoiu and Clarkson (2003) to Riemannian geometries and study accordingly the convergence rate. We then show how to instantiate this generic algorithm to two particular cases: (1) hyperbolic geometry, and (2) Riemannian manifold of symmetric positive definite matrices

Triangles and Girth in Disk Graphs and Transmission Graphs

Let S subset R^2 be a set of n sites, where each s in S has an associated radius r_s > 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t in S if and only if |st| <= r_s + r_t, i.e., if the disks with centers s and t and respective radii r_s and r_t intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if |st| <= r_s, i.e., if t lies in the disk with center s and radius r_s. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest