6,316 research outputs found

    Anchored Rectangle and Square Packings

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    For points p_1,...,p_n in the unit square [0,1]^2, an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles r_1,...,r_n in [0,1]^2 such that point p_i is a corner of the rectangle r_i (that is, r_i is anchored at p_i) for i=1,...,n. We show that for every set of n points in [0,1]^2, there is an anchored rectangle packing of area at least 7/12-O(1/n), and for every n, there are point sets for which the area of every anchored rectangle packing is at most 2/3. The maximum area of an anchored square packing is always at least 5/32 and sometimes at most 7/27. The above constructive lower bounds immediately yield constant-factor approximations, of 7/12 -epsilon for rectangles and 5/32 for squares, for computing anchored packings of maximum area in O(n log n) time. We prove that a simple greedy strategy achieves a 9/47-approximation for anchored square packings, and 1/3 for lower-left anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS and a PTAS for anchored rectangle and square packings in n^{O(1/epsilon)} and exp(poly(log (n/epsilon))) time, respectively

    On Guillotine Separability of Squares and Rectangles

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    Guillotine separability of rectangles has recently gained prominence in combinatorial optimization, computational geometry, and combinatorics. Consider a given large stock unit (say glass or wood) and we need to cut out a set of required rectangles from it. Many cutting technologies allow only end-to-end cuts called guillotine cuts. Guillotine cuts occur in stages. Each stage consists of either only vertical cuts or only horizontal cuts. In k-stage packing, the number of cuts to obtain each rectangle from the initial packing is at most k (plus an additional trimming step to separate the rectangle itself from a waste area). Pach and Tardos [Pach and Tardos, 2000] studied the following question: Given a set of n axis-parallel rectangles (in the weighted case, each rectangle has an associated weight), cut out as many rectangles (resp. weight) as possible using a sequence of guillotine cuts. They provide a guillotine cutting sequence that recovers 1/(2 log n)-fraction of rectangles (resp. weights). Abed et al. [Fidaa Abed et al., 2015] claimed that a guillotine cutting sequence can recover a constant fraction for axis-parallel squares. They also conjectured that for any set of rectangles, there exists a sequence of axis-parallel guillotine cuts that recovers a constant fraction of rectangles. This conjecture, if true, would yield a combinatorial O(1)-approximation for Maximum Independent Set of Rectangles (MISR), a long-standing open problem. We show the conjecture is not true, if we only allow o(log log n) stages (resp. o(log n/log log n)-stages for the weighted case). On the positive side, we show a simple O(n log n)-time 2-stage cut sequence that recovers 1/(1+log n)-fraction of rectangles. We improve the extraction of squares by showing that 1/40-fraction (resp. 1/160 in the weighted case) of squares can be recovered using guillotine cuts. We also show O(1)-fraction of rectangles, even in the weighted case, can be recovered for many special cases of rectangles, e.g. fat (bounded width/height), ?-large (large in one of the dimensions), etc. We show that this implies O(1)-factor approximation for Maximum Weighted Independent Set of Rectangles, the weighted version of MISR, for these classes of rectangles

    Packing-limited growth of irregular objects

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    We study growth limited by packing for irregular objects in two dimensions. We generate packings by seeding objects randomly in time and space and allowing each object to grow until it collides with another object. The objects we consider allow us to investigate the separate effects of anisotropy and non-unit aspect ratio. By means of a connection to the decay of pore-space volume, we measure power law exponents for the object size distribution. We carry out a scaling analysis, showing that it provides an upper bound for the size distribution exponent. We find that while the details of the growth mechanism are irrelevant, the exponent is strongly shape dependent. Potential applications lie in ecological and biological environments where sessile organisms compete for limited space as they grow.Comment: 6 pages, 4 figures, 1 table, revtex
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