Multiparty Selection

Given a sequence A of n numbers and an integer (target) parameter 1 ? i ? n, the (exact) selection problem is that of finding the i-th smallest element in A. An element is said to be (i,j)-mediocre if it is neither among the top i nor among the bottom j elements of S. The approximate selection problem is that of finding an (i,j)-mediocre element for some given i,j; as such, this variant allows the algorithm to return any element in a prescribed range. In the first part, we revisit the selection problem in the two-party model introduced by Andrew Yao (1979) and then extend our study of exact selection to the multiparty model. In the second part, we deduce some communication complexity benefits that arise in approximate selection. In particular, we present a deterministic protocol for finding an approximate median among k players

Algorithmic and Combinatorial Results in Selection and Computational Geometry

This dissertation investigates two sets of algorithmic and combinatorial problems. Thefirst part focuses on the selection problem under the pairwise comparison model. For the classic “median of medians” scheme, contrary to the popular belief that smaller group sizes cause superlinear behavior, several new linear time algorithms that utilize small groups are introduced. Then the exact number of comparisons needed for an optimal selection algorithm is studied. In particular, the implications of a long standing conjecture known as Yao’s hypothesis are explored. For the multiparty model, we designed low communication complexity protocols for selecting an exact or an approximate median of data that is distributed among multiple players. In the second part, three computational geometry problems are studied. For the longestspanning tree with neighborhoods, approximation algorithms are provided. For the stretch factor of polygonal chains, upper bounds are proved and almost matching lower bound constructions in \mathbb{R}^2 and higher dimensions are developed. For the piercing number τ and independence number ν of a family of axis-parallel rectangles in the plane, a lower bound construction for ν = 4 that matches Wegner’s conjecture is analyzed. The previous matching construction for ν = 3, due to Wegner himself, dates back to 1968