Table-driven quadtree traversal algorithms

Abstract

Two quadtree algorithms are presented that use table driven traversals to reduce the time complexity required to achieve their respective goals. The first algorithm is a two step process that converts a boundary representation of a polygon into a corresponding region representation of the same image. The first step orders the border pixels of the polygon. The second step fills in the polygon in O(B) time where B is the number of border pixels for the polygon of interest. A table propagates the correct values of upcoming nodes in a simulated traversal of the final region quadtree. This is unique because the pointer representation of the tree being traversed does not exist. A linear quadtree representation is constructed as this traversal proceeds. The second algorithm is an update algorithm for a quadtree (or octtree) of moving particles. Particle simulations have had the long-standing problem of calculating the interactions among n particles. It takes O(n2) time for direct computation of all the interactions between n particles. Greengard [Gree87, Carr87] has devised a way to approximate these calculations in linear time using a tree data structure. However, the particle simulation must still rebuild the particle tree after every iteration, which requires O(n log n) time. Our algorithm updates the existing tree of particles, rather than building a new tree. It operates in near linear time in the number of particles being simulated. The update algorithm uses a table to store particles as they move between nodes of the tree.