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 relative positions of the nodes within the diamond determine the possible intersections between their representative regions. The new data structure adds the capability to detect intersections between rectangles in a segment tree. We use this to solve the “Rectangle Intersection Problem” with a more straightforward algorithm than has been used previously.