We give the first efficient (1 − ε)-approximation algorithm for the following problem: Given an axis-parallel d-dimensional box R in R d containing n points, compute a maximum-volume empty axis-parallel d-dimensional box contained in R. The minimum of this quantity over all such point sets is of the order ( 1 n ). Our algorithm finds an empty axis-aligned box whose volume is at least (1 − ε) of the maximum in O((8edε −2 ) d · n log d n) time. No previous efficient exact or approximation algorithms were known for this problem for d ≥ 4. As the problem has been recently shown to be NP-hard in arbitrarily high dimensions (i.e., when d is part of the input), the existence of an efficient exact algorithm is unlikely. We also present a (1 − ε)-approximation algorithm that, given an axis-parallel d-dimensional cube R in R d containing n points, computes a maximum-volume empty axis-parallel hypercube contained in R. The minimum of this quantity over all such point sets is also shown to be of the order ( 1 n ). A faster (1 − ε)-approximation algorithm, with a milder dependence on d in the running time, is obtained in this case.
In REF Dumitrescu et al. presented the first efficient 1 − approximation algorithm for the problem of finding the size of the largest MEHR.
On the Largest Empty Axis-Parallel Box Amidst n Points
Algorithmica
Algorithmica
Computer Science Mathematics