This book is primarily a textbook introduction to various areas of discrete geometry, in which several key results and methods are explained, in an accessible and concrete manner, in each area.Expand

A simple randomized algorithm which solves linear programs withn constraints andd variables in expected time, and computes the lexicographically smallest nonnegative point satisfyingn given linear inequalities ind variables.Expand

The halfspace itrange itreporting problem, given a finite set P of points in R d, can be solved substantially more efficiently that the more general simplex range searching problem.Expand

1. Introduction 1.1 Discrepancy for Rectangles and Uniform Distribution 1.2 Geometric Discrepancy in a More General Setting 1.3 Combinatorial Discrepancy 1.4 On Applications and Connections 2.… Expand

It is shown that multilevel range searching data structures can be built with only a polylogarithmic overhead in space and query time per level (the previous solutions require at least a small fixed power ofn) and Hopcroft's problem can be solved in time.Expand

It is shown that with recently developed derandomization techniques, one can convert Clarkson’s randomized algorithm for linear programming in fixed dimension into a linear-time deterministic algorithm, which works in a fairly general abstract setting, which allows us to solve various other problems.Expand

The complexity of these problems when G is restricted to be a partial k -tree is discussed, and a polynomial time algorithm is given for the n disjoint connecting paths problem restricted topartial k -trees (with n part of input).Expand

A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not… Expand

We prove a theorem on partitioning point sets inEd (d fixed) and give an efficient construction of partition trees based on it. This yields a simplex range searching structure with linear space,O(n… Expand