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- W2494755989 abstract "We revisit the following problem (along with its higher dimensional variant): Given a set S of n points inside an axis-parallel rectangle U in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in U but contains no points of S. 1. We prove that the number of maximum-area empty rectangles amidst n points in the plane is O(n log n 2^alpha(n)), where alpha(n) is the extremely slowly growing inverse of Ackermann's function. The previous best bound, O(n^2), is due to Naamad, Lee, and Hsu (1984).2. For any d at least 3, we prove that the number of maximum-volume empty boxes amidst n points in R^d is always O(n^d) and sometimes Omega(n^floor(d/2)). This is the first superlinear lower bound derived for this problem. 3. We discuss some algorithmic aspects regarding the search for a maximum empty box in R^3. In particular, we present an algorithm that finds a (1-epsilon)-approximation of the maximum empty box amidst n points in O(epsilon^{-2} n^{5/3} log^2{n}) time." @default.
- W2494755989 created "2016-08-23" @default.
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- W2494755989 date "2016-01-01" @default.
- W2494755989 modified "2023-09-24" @default.
- W2494755989 title "On the Number of Maximum Empty Boxes Amidst n Points" @default.
- W2494755989 doi "https://doi.org/10.4230/lipics.socg.2016.36" @default.
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