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In this paper we study two geometric data structure problems in the special case when input objects or queries are fat rectangles. We show that in this case a significant improvement compared to the general case can be achieved. We describe data structures that answer two- and three-dimensional orthogonal range reporting queries in the case when the query range is a fat rectangle. Our two-dimensional data structure uses O(n) words and supports queries in O(loglog U + k) time, where n is the number of points in the data structure, U is the size of the universe and k is the number of points in the query range. Our three-dimensional data structure needs O(n log^ε U) words of space and answers queries in O(loglog U + k) time. We also consider the rectangle stabbing problem on a set of three-dimensional fat rectangles. Our data structure uses O(n) space and answers stabbing queries in O(log U loglog U + k) time.
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The Maximum Exposure Problem
Given a set of points P and axis-aligned rectangles R in the plane, a point p ∈ P is called exposed if it lies outside all rectangles in R. In the max-exposure problem, given an integer parameter k, we want to delete k rectangles from R so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible complexity conjectures is also hard to approximate even when rectangles in R are translates of two fixed rectangles. However, if R only consists of translates of a single rectangle, we present a polynomial-time approximation scheme. For general rectangle range space, we present a simple O(k) bicriteria approximation algorithm; that is by deleting O(k2) rectangles, we can expose at least Ω(1/k) of the optimal number of points.
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- Award ID(s):
- 1814172
- PAR ID:
- 10168737
- Date Published:
- Journal Name:
- Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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