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Maximum Clique Problem of Rectangle Graphs

01 Jan 1983-
About: The article was published on 1983-01-01 and is currently open access. It has received 32 citations till now. The article focuses on the topics: Chordal graph & Clique-sum.
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Journal ArticleDOI
TL;DR: The state of the art of computational geometry is surveyed, a discipline that deals with the complexity of geometric problems within the framework of the analysis of algorithms.
Abstract: We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis of algorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computer-aided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areas—convex hulls, intersections, searching, proximity, and combinatorial optimizations—are discussed. Seven algorithmic techniques—incremental construction, plane-sweep, locus, divide-and-conquer, geometric transformation, prune-and-search, and dynamization—are each illustrated with an example. A collection of problem transformations to establish lower bounds for geo-metric problems in the algebraic computation/decision model is also included.

271 citations

Journal ArticleDOI
TL;DR: A new method for detecting rearrangements, GRIDSS (Genome Rearrangement IDentification Software Suite), is a multithreaded structural variant (SV) caller that performs efficient genome-wide break-end assembly prior to variant calling using a novel positional de Bruijn graph-based assembler.
Abstract: The identification of genomic rearrangements with high sensitivity and specificity using massively parallel sequencing remains a major challenge, particularly in precision medicine and cancer research. Here, we describe a new method for detecting rearrangements, GRIDSS (Genome Rearrangement IDentification Software Suite). GRIDSS is a multithreaded structural variant (SV) caller that performs efficient genome-wide break-end assembly prior to variant calling using a novel positional de Bruijn graph-based assembler. By combining assembly, split read, and read pair evidence using a probabilistic scoring, GRIDSS achieves high sensitivity and specificity on simulated, cell line, and patient tumor data, recently winning SV subchallenge #5 of the ICGC-TCGA DREAM8.5 Somatic Mutation Calling Challenge. On human cell line data, GRIDSS halves the false discovery rate compared to other recent methods while matching or exceeding their sensitivity. GRIDSS identifies nontemplate sequence insertions, microhomologies, and large imperfect homologies, estimates a quality score for each breakpoint, stratifies calls into high or low confidence, and supports multisample analysis.

214 citations

Journal ArticleDOI
TL;DR: AnO ( n2) algorithm is presented for the maximum clique problem and is better than a previously known algorithm which is based on sorting and runs inO (n2 logn) time.
Abstract: We consider the following circle placement problem: given a set of pointsp i ,i=1,2, ...,n, each of weightw i , in the plane, and a fixed disk of radiusr, find a location to place the disk such that the total weight of the points covered by the disk is maximized. The problem is equivalent to the so-called maximum weighted clique problem for circle intersection graphs. That is, given a setS ofn circles,D i ,i=1,2, ...,n, of the same radiusr, each of weightw i , find a subset ofS whose common intersection is nonempty and whose total weight is maximum. AnO (n 2) algorithm is presented for the maximum clique problem. The algorithm is better than a previously known algorithm which is based on sorting and runs inO (n 2 logn) time.

104 citations


Cites background from "Maximum Clique Problem of Rectangle..."

  • ...In [6, 7] the case in which the objects involved are rectangles is studied....

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Journal ArticleDOI
TL;DR: A tutorial survey is presented of hierarchical data structures for representing collections of small rectangles, which are often used as an approximation of shapes for which they serve as the minimum rectilinear enclosing object.
Abstract: A tutorial survey is presented of hierarchical data structures for representing collections of small rectangles. Rectangles are often used as an approximation of shapes for which they serve as the minimum rectilinear enclosing object. They arise in applications in cartography as well as very large-scale integration (VLSI) design rule checking. The different data structures are discussed in terms of how they support the execution of queries involving proximity relations. The focus is on intersection and subset queries. Several types of representations are described. Some are designed for use with the plane-sweep paradigm, which works well for static collections of rectangles. Others are oriented toward dynamic collections. In this case, one representation reduces each rectangle to a point in a higher multidimensional space and treats the problem as one involving point data. The other representation is area based—that is, it depends on the physical extent of each rectangle.

87 citations

Journal ArticleDOI
TL;DR: In this paper, a new algorithm based on interval tree data structure is presented, which runs in O(nlogn) time and consumes O(n) space, which can be tailored for locating the position of the plate to enclose maximum or minimum number of objects with the same time and space complexity.
Abstract: Given a set of n points in R 2 bounded within a rectangular floor F, and a rectangular plate P of specified size, we consider the following two problems: find an isothetic position of P such that it encloses (i) maximum and (ii) minimum number of points, keeping P totally contained within F. For both of these problems, a new algorithm based on interval tree data structure is presented, which runs in O(nlogn) time and consumes O(n) space. If polygonal objects of arbitrary size and shape are distributed in R 2, the proposed algorithm can be tailored for locating the position of the plate to enclose maximum or minimum number of objects with the same time and space complexity. Finally, the algorithm is extended for identifying a cuboid, i.e., a rectangular parallelepiped that encloses maximum number of polyhedral objects in R 3. Thus, the proposed technique serves as a unified paradigm for solving a general class of enclosure problems encountered in computational geometry and pattern recognition.

67 citations


Cites background or methods from "Maximum Clique Problem of Rectangle..."

  • ...In a d-dimensional space (d >_ 3), finding the m a x i m u m clique in the hyper-rectangle intersection graph or equivalently, the max-enclosure problem, can be solved in O(n d-l) t ime [13]....

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  • ...The algorithm for finding maximum clique in a hyper-rectangle intersection graph [13] can be used to solve this problem in O(n 2) time....

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