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Amitava Datta

Bio: Amitava Datta is an academic researcher from Indian Institute of Technology Madras. The author has contributed to research in topics: Largest empty rectangle & Parallel algorithm. The author has an hindex of 1, co-authored 3 publications receiving 17 citations.

Papers
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Journal ArticleDOI
TL;DR: Some efficient algorithms for the largest rectangle problem are presented that run in O ( nlogn + K ) time for all three problems and the worst-case running time is O (n 2 ) time.

16 citations

Journal ArticleDOI
TL;DR: This paper presents efficient parallel algorithms for the maximum empty rectangle problem on crew pram and anO(logn) time algorithm on a mesh-of-trees architecture.
Abstract: We present efficient parallel algorithms for the maximum empty rectangle problem in this paper. On crew pram, we solve the area version of this problem inO(log 2 n) time usingO(nlogn) processors. The perimeter version of this problem is solved inO(logn) time usingO(nlog 2 n) processors. On erew pram, we solve both the problems inO(logn) time usingO(n 2/logn) processors. We also present anO(logn) time algorithm on a mesh-of-trees architecture.

3 citations

Journal ArticleDOI
TL;DR: This work considers two limiting cases of this problem when the cardinalities of set A is much greater than that of set B, and presents efficient sequential and parallel algorithms for these two problems.
Abstract: Given a bounding isothetic rectangle BR and two sets of points A and B with cardinalities n and m inside it, we have to find an isothetic rectangle containing maximum number of points from set A and no point from set B. We consider two limiting cases of this problem when the cardinalities of set A (resp. set B) is much greater than that of set B (resp. set ,A). We present efficient sequential and parallel algorithms for these two problems. Our sequential algorithms run in O((n + m)log m + m 2) and O((m+ n) log n + n 2) time respectively. The parallel algorithms in CREW PRAM run in o(log n) ando(log m 2) time using O(max(n,m 2/logm)) and O(max(m,n 2/logn)) processors respectively. Our sequential algorithms are faster than a previous algorithm under these constraints on cardinality. No previous parallel algorithm was known for this problem. We also present an optimal systolic algorithm for the original problem.

1 citations


Cited by
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Journal ArticleDOI
TL;DR: In this article, a (1−e)-approximation algorithm for the problem of finding an empty axis-aligned box whose volume is at least (1 − e) of the maximum was given.
Abstract: We give the first efficient (1−e)-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 $\Theta (\frac {1}{n} )$ . Our algorithm finds an empty axis-aligned box whose volume is at least (1−e) of the maximum in O((8ede −2) d ⋅nlog 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−e)-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 $\Theta (\frac{1}{n} )$ . A faster (1−e)-approximation algorithm, with a milder dependence on d in the running time, is obtained in this case.

68 citations

Posted Content
TL;DR: The first efficient (1−ε)-approximation algorithm is given for the following problem: Given an axis-parallel d-dimensional box R in ℝd containing n points, compute a maximum-volume empty axis-paralleld-dimensionalbox contained in R.
Abstract: We give the first nontrivial upper and lower bounds on the maximum volume of an empty axis-parallel box inside an axis-parallel unit hypercube in $\RR^d$ containing $n$ points. For a fixed $d$, we show that the maximum volume is of the order $\Theta(\frac{1}{n})$. We then use the fact that the maximum volume is $\Omega(\frac{1}{n})$ in our design of the first efficient $(1-\eps)$-approximation algorithm for the following problem: Given an axis-parallel $d$-dimensional box $R$ in $\RR^d$ containing $n$ points, compute a maximum-volume empty axis-parallel $d$-dimensional box contained in $R$. The running time of our algorithm is nearly linear in $n$, for small $d$, and increases only by an $O(\log{n})$ factor when one goes up one dimension. No previous efficient exact or approximation algorithms were known for this problem for $d \geq 4$. As the problem has been recently shown to be NP-hard in arbitrary high dimensions (i.e., when $d$ is part of the input), the existence of efficient exact algorithms is unlikely. We also obtain tight estimates on the maximum volume of an empty axis-parallel hypercube inside an axis-parallel unit hypercube in $\RR^d$ containing $n$ points. For a fixed $d$, this maximum volume is of the same order order $\Theta(\frac{1}{n})$. A faster $(1-\eps)$-approximation algorithm, with a milder dependence on $d$ in the running time, is obtained in this case.

41 citations

Journal ArticleDOI
TL;DR: This paper considers the problem of placing efficiently a rectangle in a two-dimensional layout that may not have the bottom-left placement property and presents a @Q(nlogn)-time algorithm for solving this problem.

35 citations

Journal ArticleDOI
TL;DR: This work presents an efficient algorithm for computing the maximum empty hyper rectangle (MEHR), a problem to find a maximum volume or surface area hyper rectangle R within BHR such that R does not contain any point from the set P.

22 citations

Journal ArticleDOI
TL;DR: It is shown that the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube [0,1]d in R is (1 ± o(1)) $\frac{(2d-2)!}{(d-1)!}$n lnd−1n, if d is fixed.
Abstract: We show that the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube [0,1] d in ℝ d is \((1 \pm o(1))\allowbreak \frac{(2d-2)!}{(d-1)!} \, n \ln^{d-1} n\), if d is fixed. This estimate is relevant for analyzing the performance of any exact algorithm for computing the largest empty axis-parallel box amidst n points in a given axis-parallel box R, that proceeds by examining all maximal empty boxes. While the Θ(n log d − 1 n) bound has been claimed for d = 3 for more than ten years by now, and has been recently used for all d ≥ 3 in the analysis of algorithms for computing the largest empty box, it did not rely on a valid proof. Here we present the first valid proof for the Θ(n log d − 1 n) bound; only an O(n log d − 1 n) bound was previously proved.

13 citations