Discrete rectilinear 2-center problems

Lecture Notes in Computer Science, 2012

Let P be a set of n points in R 2. We solve the problem of computing an orientation of the plane for which the rectilinear convex hull of P has minimum area in optimal Θ(n log n) time and O(n) space.

Computational Geometry, 2009

In this paper we deal with the following natural family of geometric matching problems. Given a class C of geometric objects and a set P of points in the plane, a C-matching is a set M ⊆ C such that every C ∈ M contains exactly two elements of P. The matching is perfect if it covers every point, and strong if the objects do not intersect. We concentrate on matching points using axes-aligned squares and rectangles. We propose an algorithm for strong rectangle matching that, given a set of n points, matches at least 2⌊n/3⌋ of them, which is worst-case optimal. If we are given a combinatorial matching of the points, we can test efficiently whether it has a realization as a (strong) square matching. The algorithm behind this test can be modified to solve an interesting new pointlabeling problem. On the other hand we show that it is NP-hard to decide whether a point set has a perfect strong square matching.

Discrete & Computational Geometry, 1996

Let P be a set of n points in the plane and let e be a segment of fixed length. The segment-center problem is to find a placement of e (allowing translation and rotation) which minimizes the maximum euclidean distance from e to the points of P. We present an algorithm that solves the problem in time O(nl+~), for any e > 0, improving the previous solution of Agarwal et al. [3] by nearly a factor of O (n).

Computational Geometry, 2009

We study the problems of computing two non-convex enclosing shapes with the minimum area; the L-shape and the rectilinear convex hull. Given a set of n points in the plane, we find an L-shape enclosing the points or a rectilinear convex hull of the point set with minimum area over all orientations. We show that the minimum enclosing shapes for fixed orientations change combinatorially at most O (n) times while rotating the coordinate system. Based on this, we propose efficient algorithms that compute both shapes with the minimum area over all orientations. The algorithms provide an efficient way of maintaining the set of extremal points, or the staircase, while rotating the coordinate system, and compute both minimum enclosing shapes in O (n 2) time and O (n) space. We also show that the time complexity of maintaining the staircase can be improved if we use more space.

Operations Research Letters, 2009

Given a set P of n points in the plane, we consider the problem of finding a planar rectilinear annulus of minimum width which encloses the set P. We present an optimal O(n log n) algorithm for this problem.

2019

In the Anchored Rectangle Packing (ARP) problem, we are given a set of points P in the unit square [0,1]^2 and seek a maximum-area set of axis-aligned interior-disjoint rectangles S, each of which is anchored at a point p in P. In the most prominent variant - Lower-Left-Anchored Rectangle Packing (LLARP) - rectangles are anchored in their lower-left corner. Freedman [W. T. Tutte (Ed.), 1969] conjectured in 1969 that, if (0,0) in P, then there is a LLARP that covers an area of at least 0.5. Somewhat surprisingly, this conjecture remains open to this day, with the best known result covering an area of 0.091 [Dumitrescu and Toth, 2015]. Maybe even more surprisingly, it is not known whether LLARP - or any ARP-problem with only one anchor - is NP-hard. In this work, we first study the Center-Anchored Rectangle Packing (CARP) problem, where rectangles are anchored in their center. We prove NP-hardness and provide a PTAS. In fact, our PTAS applies to any ARP problem where the anchor lies i...

Lecture Notes in Computer Science, 1998

For a given set of n rectangles place on a plane, we consider a problem of finding the minimum area layout of the rectangles that avoids intersections of the rectangles and preserves the orthogonal order. Misue et al. proposed an O(n 2)-time heuristic algorithm for the problem. We first show that the corresponding decision problem for this problem is NP-complete. We also present an O(n 2)-time heuristic algorithm for the problem that finds a layout with smaller area than Misue's.

Algorithmica, 2002

In this paper we deal with two geometric problems arising from heterogeneous parallel computing: how to partition the unit square into p rectangles of given area s 1 , s 2 ,...,s p (such that p i=1 s i = 1), so as to minimize either (i) the sum of the p perimeters of the rectangles or (ii) the largest perimeter of the p rectangles? For both problems, we prove NP-completeness and we introduce a 7 4 -approximation algorithm for (i) and a (2/ √ 3)-approximation algorithm for (ii).

1995

We consider the problem of reporting the pairwise enclosures among a set of n axes-parallel rectangles in IR 2 , which is equivalent to reporting dominance pairs in a set of n points in IR 4 . For more than ten years, it has been an open problem whether these problems can be solved faster than in O(n log 2 n + k) time, where k denotes the number of reported pairs. First, we give a divide-and-conquer algorithm that matches the O(n) space and O(n log 2 n + k) time bounds of the algorithm of Lee and Preparata LP82], but is simpler. Then we give another algorithm that uses O(n) space and runs in O(n log n log log n+k log log n) time. For the special case where the rectangles have at most di erent aspect ratios, we give an algorithm that runs in O( n log n + k) time and uses O(n) space. Problem 1.1 Given a set R of n axes-parallel rectangles in the plane, report all pairs (R 0 ; R) of rectangles such that R encloses R 0 .

Information Sciences, 1980

The i&motion problem for a subclass of mtanglts called r-rwtanglea is inv&igated and reduced to the balanced batched r(wricted~range searching problem as well as to the balanced batched inverse r-range searching problem. Simple algorithms for these problems are given which are space and time optimal. The algorithm given for the balanced batched r-range searching problem leads to a new algorithm for the all-points ECDF problem in 2-apace whioh is simple and optimal. Again, the balanced bat&d r-range mrohing algorithm is combined witb a known algorithm for batched range sear&@ problems, leading to a new algorithm for the r&angle inters&ion problem which is space and time optimal in the worst czsc when the given set of rectan&a contains a much higher proportion of r-rcctangl~. OElwvier North Holland, Inc., 1980 52 Vanderbilt Ave., New York, NY 10017 60 V. K. VAISHNAVI, H. P. KRIEGEL, AND D. WOOD problems. These problems deal with the computation of intersection properties among sets of planar objects such as line segments, circles, half spaces, and rectangles. Recent papers in this area are [7], [3], [5], and [8]. Bentley and Wood [5] and Vaishnavi [8] deal, in particular, with the rectangle intersection problem.