We present an optimal parallel algorithm to construct the well-separated pair decomposition of a point set P in < d. We show how this leads to a deterministic optimal O(logn) time parallel algorithm for nding the k-nearest-neighbors of each point in P, where k is a constant. We discuss several additional applications of the well-separated pair decomposition for which we can derive faster parall...

Let S be a set of points in R. Given a geometric spanner graph, G = (S, E), with constant dilation t, and a positive constant ε, we show how to construct a (1 + ε)-spanner of G with O(|S|) edges in time O(|E|+ |S| log |S|). Previous algorithms require a preliminary step in which the edges are sorted in non-decreasing order of their lengths and, thus, have running time Ω(|E| log |S|). We obtain ...

We discuss the structure of the equilibrium states of a regularized Keller-Segel model describing the chemotaxis of bacterial populations. We consider the limit of high degradation of the secreted chemical where analytical results can be obtained. Below a critical effective temperature, the system experiences a second order phase transition from a homogeneous phase to an inhomogeneous phase for...

This paper studies colour filtering with particular reference to colour bleeding for median and mode filters. While confirming that multiple single-channel implementation gives especially serious colour bleeding, the paper goes on to argue that the conventional solution — of using the vector median — is over-strenuous for many types of application. In fact, instead of insisting that the output ...

Let 〈X,Γ〉 be a uniform space with its uniformity generated by a set of pseudo-metrics Γ . Let the symbol “ ' " denote the usual infinitesimal relation on ∗X , and define a new infinitesimal relation “ ≈ " by writing x ≈ y whenever ∗ρ(x, p) ' ∗ρ(y, p) for each ρ ∈ Γ and each p ∈ X . Let us call a uniform space 〈X,Γ〉 an S-space if the relations ' and ≈ coincide on fin(∗X). In [14], we showed that...

In this paper, we consider sampling and reconstruction of signals in a reproducing kernel subspace of L(R), 1 ≤ p ≤ ∞, associated with an idempotent integral operator whose kernel has certain off-diagonal decay and regularity. The space of p-integrable non-uniform splines and the shift-invariant spaces generated by finitely many localized functions are our model examples of such reproducing ker...

The world of mathematics is not confined to the three dimensions of the space that we inhabit. Mathematicians study sphere-packing problems in spaces of arbitrary dimension. Geometrical puzzles can be posed and solved in such spaces. Some practical challenges end up in such a form. With this chapter we present a short excursion into the outer space of high dimensions. The topic is explored in a...