The Mono- and Bichromatic Empty Rectangle and Square Problems in All Dimensions

Maximal Empty Boxes Amidst Random Points

“…Around the same time, Backer and Keil [5] also obtained the tight Θ(n d ) bound. In retrospect, we observe that the three constructions for the Ω(n d ) lower bound in [17,13,5] are based on essentially the same simple idea. For the O(n d ) upper bound, Kaplan et al [17] use an elegant shifting technique and cite Boissonnat, Sharir, Tagansky, and Yvinec [8] for a similar analysis, and Backer and Keil [5] use essentially the same technique (which they call deflation-inflation) in their proof and cite (the Hence the maximum number of maximal empty boxes is Θ(n d ) for each fixed d. This means that any algorithm that computes a maximum-volume empty box by enumerating all maximal empty boxes is bound to be inefficient in the worst case.…”

“…In retrospect, we observe that the three constructions for the Ω(n d ) lower bound in [17,13,5] are based on essentially the same simple idea. For the O(n d ) upper bound, Kaplan et al [17] use an elegant shifting technique and cite Boissonnat, Sharir, Tagansky, and Yvinec [8] for a similar analysis, and Backer and Keil [5] use essentially the same technique (which they call deflation-inflation) in their proof and cite (the Hence the maximum number of maximal empty boxes is Θ(n d ) for each fixed d. This means that any algorithm that computes a maximum-volume empty box by enumerating all maximal empty boxes is bound to be inefficient in the worst case. Indeed, by a standard result in parameterized complexity theory [21,Section 6.3], the aforementioned W[1]-hardness result of Giannopoulos, Knauer, Wahlström, and Werner [15,Theorem 3] implies that the existence of an exact algorithm running in n o(d) time is unlikely, i.e., unless the so-called Exponential Time Hypothesis (ETH) fails, i.e., unless 3-SAT can be solved in 2 o(n) time.…”

“…Shortly after this, following the earlier work [12,23] on the Maximum Empty Box problem, Dumitrescu and Jiang [13] obtained the lower bound Ω(n d ) independently, unaware of the recent work by Kaplan et al [17]. Around the same time, Backer and Keil [5] also obtained the tight Θ(n d ) bound. In retrospect, we observe that the three constructions for the Ω(n d ) lower bound in [17,13,5] are based on essentially the same simple idea.…”

“…Augustine et al [4] and Kaplan, Mozes, Nussbaum, and Sharir [16] studied a related problem, that of finding the largest empty rectangle containing a query point. For the Maximum Empty Box problem in higher dimensions, Backer and Keil [5,6] proved that the problem is NP-hard when the dimension d is part of the input, and Giannopoulos, Knauer, Wahlström, and Werner [15] further showed that the problem is W [1]-hard with the dimension d as the parameter. From the positive direction, Dumitrescu and Jiang [13] presented an approximation algorithm that finds an empty box whose volume is at least 1−ε of the optimal in O (…”

“…Kaplan, Rubin, Sharir, and Verbin [17] presented an output-sensitive algorithm running in O((n + k) log d−1 n) time, where k is the number of maximal empty boxes. Backer and Keil [5,6] also reported an output-sensitive algorithm running in O(k log d−2 n) time; in particular, the worst-case running time of their algorithm for d = 3 is O(n 3 log n). Previously, Datta and Soundaralakshmi [12] had reported an O(n 3 ) time exact algorithm for the d = 3 case, but their analysis for the running time seems incomplete.…”