Stuart J. Sidney

A distributed selection algorithm and its expected communication complexity

Abstract We consider the distributed K -selection problem defined as follows: a set S of n elements is distributed among d processors and an originator processor wants to know the value of the K th smallest element in S . The goal is to obtain an algorithm that minimizes the communication activities among the processors. We propose an algorithm whose expected communication complexity is O(log log n ) whereas the worst-case complexity is O(log n ); in the point-to-point model, this yields an O( d log log n ) upper bound on the expected number of messages for distributed selection.

Circle orders, N-gon orders and the crossing number

Let Φ={P1,...,Pm } be a family of sets. A partial order P(Φ, <) on Φ is naturally defined by the condition Pi <Pj iff Pi is contained in Pj. When the elements of Φ are disks (i.e. circles together with their interiors), P(Φ, <) is called a circle order; if the elements of Φ are n-polygons, P(Φ, <) is called an n-gon order. In this paper we study circle orders and n-gon orders. The crossing number of a partial order introduced in [5] is studied here. We show that for every n, there are partial orders with crossing number n. We prove next that the crossing number of circle orders is at most 2 and that the crossing number of n-gon orders is at most 2n. We then produce for every n≥4 partial orders of dimension n which are not circle orders. Also for every n>3, we prove that there are partial orders of dimension 2n+2 which are not n-gon orders. Finally, we prove that every partial order of dimension ≤2n is an n-gon order.

A combinatorial result about points and balls in Euclidean space

For eachn≥1 there iscn>0 such that for any finite sexX ⊆ ℝ″ there isA ⊆X, |A|≤1/2(n+3), having the following property: ifB ⊇A is ann-ball, then |B ∩X|≥cn|X|. This generalizes a theorem of Neumann-Lara and Urrutia which states thatc2≥1/60.

Geometric containment and vector dominance

In this paper the following problem is addressed: given a family of geometric figures in the plane, how many real variables are required to parameterize the family so that one figure from the fammily is contained in another (perhaps after translation, rotation and reflection) if and only if the parameter values for the first figure are not greater than those for the second figure? It is shown that for the family o rectangles no such finite parametrization exists; thuus, geometric containment of polygons with k ? 4 sides is not reducible to vector dominance, regardless of the choice and the finite number of parameters. It is also shown that, for plane rectangles, an infinite but countable number of parameters suffice for the reduction.

Geometric containment and partial orders

Given two geometric sets A and B, it is said that A is containable in B provided A is isometric to a subset of B. Containability induces a partial order on any set of geometric figures, such as rectangles in the plane. A recent result states that for the set of rectangles in the plane, the containability partial order is of countably infinite dimension. In this paper the rectangle result is extended to other families of geometric figures and to a partial order obtained from quadratic polynomials.

Average properties of two-dimensional partial orders

The complexity of many combinatorial algorithms is directly related to the number of upsets (filters) in an underlying partial order or, equivalently, to the number of downsets (ideals) that are the complements of upsets. In this paper, we investigate the expected number of upsets in randomly generated two-dimensional partial orders and present a number of new results, together with simple combinatorial proofs of some known results. As an application of our results, we show that some well-known algorithms for precedence constrained scheduling have nonpolynomial expected time and storage complexity.

Geometric Containment, Common Roots of Polynomials and Partial Orders

The reduction to vector dominance of two unrelated problems is studied; the two problems are 1. Geometric containment: given a class of geometric figures, determine for any two figures in the class whether one is contained in the other (perhaps after translation, rotation and even reflection). 2. Common roots of polynomials: given a class of polynomials, determine for any two polynomials in the class whether they have positive common roots.