The 'Cross" Rectangle Intersection Problem

Intersection of Rectangle and Ellipse

The Unified Segment Tree and its Application to the Rectangle Intersection Problem

In this paper we introduce a variation on the multidimensional segment tree, formed by unifying different interpretations of the dimensionalities of the levels within the tree. Nodes in the resulting d-dimensional structure can have up to d parents and 2d children. In order to better visualize these relationships we introduce a diamond representation of the data structure. We show how the relat...

On a restricted cross-intersection problem

Suppose A and B are families of subsets of an n-element set and L is a set of s numbers. We say that the pair (A,B) is L-cross-intersecting if |A∩B| ∈ L for every A ∈A and B ∈ B. Among such pairs (A,B) we write PL(n) for the maximum possible value of |A||B|. In this paper we find an exact bound for PL(n) when n is sufficiently large, improving earlier work of Sgall. We also determine P{2}(n) an...

The Runaway Rectangle Escape Problem

Motivated by the applications of routing in PCB buses, the Rectangle Escape Problem was recently introduced and studied. In this problem, we are given a set of rectangles S in a rectangular region R, and we would like to extend these rectangles to one of the four sides of R. Define the density of a point p in R as the number of extended rectangles that contain p. The question is then to find an...

On the Rectangle Escape Problem

Motivated by a bus routing application, we study the following rectangle escape problem: Given a set S of n rectangles inside a rectangular region R, extend each rectangle in S toward one of the four borders of R so that the maximum density over the region R is minimized, where the density of each point p ∈ R is defined as the number of extended rectangles containing p. We show that the problem...