A Concentration Inequality for the Facility Location Problem

We give a concentration inequality for a stochastic version of the facility location problem on the plane. We show the objective $$C_n(X) = \min_{F \subseteq [0,1]^2} \, |F| + \sum_{x\in X} \min_{f \in F} \| x-f\|$$ is concentrated in an interval of length $O(n^{1/6})$ and $\mathbb{E}[C_n] = \Theta(n^{2/3})$ if the input $X$ consists of $n$ i.i.d. uniform points in the unit square. Our main tool is to use a suitable geometric quantity, previously used in the design of approximation algorithms for the facility location problem, to analyze a martingale process.Comment: 6 pages, 1 figur

One-step-ahead kinematic compressive sensing

A large portion of work on compressive sampling and sensing has focused on reconstructions from a given measurement set. When the individual samples are expensive and optional, as is the case with autonomous agents operating in a physical domain and under specific energy limits, the CS problem takes on a new aspect because the projection is column-sparse, and the number of samples is not necessarily large. As a result, random sampling may no longer be the best tactic. The underlying incoherence properties in l0 reconstruction, however, can still motivate the purposeful design of samples in planning for CS with one or more agents; we develop here a greedy and computationally tractable sampling rule that will improve errors relative to random points. Several example cases illustrate that the approach is effective and robust.United States. Office of Naval Research (Grant N00014-09-1-0700

Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability

A limit theorem is established for a class of random processes (called here subadditive Euclidean functionals) which arise in problems of geometric probability. Particular examples include the length of shortest path through a random sample, the length of a rectilinear Steiner tree spanned by a sample, and the length of a minimal matching. Also, a uniform convergence theorem is proved which is needed in Karp\u27s probabilistic algorithm for the traveling salesman problem

Traveling in randomly embedded random graphs

We consider the problem of traveling among random points in Euclidean space, when only a random fraction of the pairs are joined by traversable connections. In particular, we show a threshold for a pair of points to be connected by a geodesic of length arbitrarily close to their Euclidean distance, and analyze the minimum length Traveling Salesperson Tour, extending the Beardwood-Halton-Hammersley theorem to this setting.Comment: 25 pages, 2 figure