Dynamic Data Structures for Fat Objects and Their Applications

We present several eecient dynamic data structures for point-enclosure queries involving convex fat objects in R 2 or R 3. These structures are more eecient than alternative known structures because they exploit the fatness of the objects. We then apply these structures to obtain eecient solutions to three problems: (i) Finding a perfect matching between a set of points and a set of convex fat objects. (ii) Finding a piercing set for a collection of convex fat objects, whose size is within a constant factor oo the optimum size. (iii) Constructing a data structure for answering bounded-length segment-shooting queries: Given a set of fat objects in the plane as above, and a query oriented segment ! r , whose length is relatively short, nd the rst object hit by ! r .