Dynamic Data Structures for Fat Objects and Their Applications

Alon Efrat, Matthew J. Katz, Frank Nielsen, Micha Sharir

Abstract: We present several efficient dynamic data structures for point-enclosure queries involving convex fat objects in ℝ2 or ℝ3. These structures are more efficient than alternative known structures because they exploit the fatness of the objects. We then apply these structures to obtain efficient 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 off 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 $$\overrightarrow r$$ whose length is relatively short, find the first object hit by $$\overrightarrow r$$ .