The (maximum receiver-centric) interference of a geometric graph (von (2005)) is studied. It is shown that, with high probability, the following results hold for a set, V, of n points independently and uniformly distributed in the unit d-cube, for constant dimension d: (1) there exists a connected graph with vertex set V that has interference O((log n)^ 1 3 ); (2) no connected graph with vertex set V has interference o((log n)^ 1 4 ); and (3) the minimum spanning tree of @math has interference Theta(( n)^ 1 2 ).
log suggests that point sets with very high interference are unlikely to occur in practice. This intuition is born out by the results of Kranakis al @cite_7 , who show that high interference is unlikely to occur in random point sets in one dimension: Note that, in this one-dimensional case, the minimum spanning tree, @math , is simply a path that connects the points of @math in order, from left to right. Taken together, Part 1 of Theorems and generalize sqrtlogn to arbitrary constant dimensions @math .
@cite_7
1942343061
(MATH) Let P be a set of n points in @math k (P) denote the minimum over all k-flats @math of max peP Dist(p, ). We present an algorithm that computes, for any 0 k (P) from each point of P. The running time of the algorithm is dnO(k e5log(1 e)). The crucial step in obtaining this algorithm is a structural result that says that there is a near-optimal flat that lies in an affine subspace spanned by a small subset of points in P. The size of this "core-set" depends on k and e but is independent of the dimension.This approach also extends to the case where we want to find a k-flat that is close to a prescribed fraction of the entire point set, and to the case where we want to find j flats, each of dimension k, that are close to the point set. No efficient approximation schemes were known for these problems in high-dimensions, when k>1 or j>1.