4,768 research outputs found
Minimum Perimeter Rectangles That Enclose Congruent Non-Overlapping Circles
We use computational experiments to find the rectangles of minimum perimeter
into which a given number n of non-overlapping congruent circles can be packed.
No assumption is made on the shape of the rectangles. In many of the packings
found, the circles form the usual regular square-grid or hexagonal patterns or
their hybrids. However, for most values of n in the tested range n =< 5000,
e.g., for n = 7, 13, 17, 21, 22, 26, 31, 37, 38, 41, 43...,4997, 4998, 4999,
5000, we prove that the optimum cannot possibly be achieved by such regular
arrangements. Usually, the irregularities in the best packings found for such n
are small, localized modifications to regular patterns; those irregularities
are usually easy to predict. Yet for some such irregular n, the best packings
found show substantial, extended irregularities which we did not anticipate. In
the range we explored carefully, the optimal packings were substantially
irregular only for n of the form n = k(k+1)+1, k = 3, 4, 5, 6, 7, i.e., for n =
13, 21, 31, 43, and 57. Also, we prove that the height-to-width ratio of
rectangles of minimum perimeter containing packings of n congruent circles
tends to 1 as n tends to infinity.Comment: existence of irregular minimum perimeter packings for n not of the
form (10) is conjectured; smallest such n is n=66; existence of irregular
minimum area packings is conjectured, e.g. for n=453; locally optimal
packings for the two minimization criteria are conjecturally the same (p.22,
line 5); 27 pages, 12 figure
On three soft rectangle packing problems with guillotine constraints
We investigate how to partition a rectangular region of length and
height into rectangles of given areas using
two-stage guillotine cuts, so as to minimize either (i) the sum of the
perimeters, (ii) the largest perimeter, or (iii) the maximum aspect ratio of
the rectangles. These problems play an important role in the ongoing Vietnamese
land-allocation reform, as well as in the optimization of matrix multiplication
algorithms. We show that the first problem can be solved to optimality in
, while the two others are NP-hard. We propose mixed
integer programming (MIP) formulations and a binary search-based approach for
solving the NP-hard problems. Experimental analyses are conducted to compare
the solution approaches in terms of computational efficiency and solution
quality, for different objectives
Defragmenting the Module Layout of a Partially Reconfigurable Device
Modern generations of field-programmable gate arrays (FPGAs) allow for
partial reconfiguration. In an online context, where the sequence of modules to
be loaded on the FPGA is unknown beforehand, repeated insertion and deletion of
modules leads to progressive fragmentation of the available space, making
defragmentation an important issue. We address this problem by propose an
online and an offline component for the defragmentation of the available space.
We consider defragmenting the module layout on a reconfigurable device. This
corresponds to solving a two-dimensional strip packing problem. Problems of
this type are NP-hard in the strong sense, and previous algorithmic results are
rather limited. Based on a graph-theoretic characterization of feasible
packings, we develop a method that can solve two-dimensional defragmentation
instances of practical size to optimality. Our approach is validated for a set
of benchmark instances.Comment: 10 pages, 11 figures, 1 table, Latex, to appear in "Engineering of
Reconfigurable Systems and Algorithms" as a "Distinguished Paper
Hard and Easy Instances of L-Tromino Tilings
We study tilings of regions in the square lattice with L-shaped trominoes.
Deciding the existence of a tiling with L-trominoes for an arbitrary region in
general is NP-complete, nonetheless, we identify restrictions to the problem
where it either remains NP-complete or has a polynomial time algorithm. First,
we characterize the possibility of when an Aztec rectangle and an Aztec diamond
has an L-tromino tiling. Then, we study tilings of arbitrary regions where only
rotations of L-trominoes are available. For this particular case we
show that deciding the existence of a tiling remains NP-complete; yet, if a
region does not contains certain so-called "forbidden polyominoes" as
sub-regions, then there exists a polynomial time algorithm for deciding a
tiling.Comment: Full extended version of LNCS 11355:82-95 (WALCOM 2019
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