1,328 research outputs found
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
Towards a Holistic Integration of Spreadsheets with Databases: A Scalable Storage Engine for Presentational Data Management
Spreadsheet software is the tool of choice for interactive ad-hoc data
management, with adoption by billions of users. However, spreadsheets are not
scalable, unlike database systems. On the other hand, database systems, while
highly scalable, do not support interactivity as a first-class primitive. We
are developing DataSpread, to holistically integrate spreadsheets as a
front-end interface with databases as a back-end datastore, providing
scalability to spreadsheets, and interactivity to databases, an integration we
term presentational data management (PDM). In this paper, we make a first step
towards this vision: developing a storage engine for PDM, studying how to
flexibly represent spreadsheet data within a database and how to support and
maintain access by position. We first conduct an extensive survey of
spreadsheet use to motivate our functional requirements for a storage engine
for PDM. We develop a natural set of mechanisms for flexibly representing
spreadsheet data and demonstrate that identifying the optimal representation is
NP-Hard; however, we develop an efficient approach to identify the optimal
representation from an important and intuitive subclass of representations. We
extend our mechanisms with positional access mechanisms that don't suffer from
cascading update issues, leading to constant time access and modification
performance. We evaluate these representations on a workload of typical
spreadsheets and spreadsheet operations, providing up to 20% reduction in
storage, and up to 50% reduction in formula evaluation time
Space Partitioning Schemes and Algorithms for Generating Regular and Spiral Treemaps
Treemaps have been widely applied to the visualization of hierarchical data.
A treemap takes a weighted tree and visualizes its leaves in a nested planar
geometric shape, with sub-regions partitioned such that each sub-region has an
area proportional to the weight of its associated leaf nodes. Efficiently
generating visually appealing treemaps that also satisfy other quality criteria
is an interesting problem that has been tackled from many directions. We
present an optimization model and five new algorithms for this problem,
including two divide and conquer approaches and three spiral treemap
algorithms. Our optimization model is able to generate superior treemaps that
could serve as a benchmark for comparing the quality of more computationally
efficient algorithms. Our divide and conquer and spiral algorithms either
improve the performance of their existing counterparts with respect to aspect
ratio and stability or perform competitively. Our spiral algorithms also expand
their applicability to a wider range of input scenarios. Four of these
algorithms are computationally efficient as well with quasilinear running times
and the last algorithm achieves a cubic running time. A full version of this
paper with all appendices, data, and source codes is available at
\anonymizeOSF{\OSFSupplementText}
A Divide and Conquer Approximation Algorithm for Partitioning Rectangles
Given a rectangle with area and a set of areas
with , we consider the problem of partitioning into
sub-regions with areas in a way that the total
perimeter of all sub-regions is minimized. The goal is to create square-like
sub-regions, which are often more desired. We propose an efficient
--approximation algorithm for this problem based on a divide and conquer
scheme that runs in time. For the special case when the
aspect ratios of all rectangles are bounded from above by 3, the approximation
factor is . We also present a modified version of out
algorithm as a heuristic that achieves better average and best run times
Stencils and problem partitionings: Their influence on the performance of multiple processor systems
Given a discretization stencil, partitioning the problem domain is an important first step for the efficient solution of partial differential equations on multiple processor systems. Partitions are derived that minimize interprocessor communication when the number of processors is known a priori and each domain partition is assigned to a different processor. This partitioning technique uses the stencil structure to select appropriate partition shapes. For square problem domains, it is shown that non-standard partitions (e.g., hexagons) are frequently preferable to the standard square partitions for a variety of commonly used stencils. This investigation is concluded with a formalization of the relationship between partition shape, stencil structure, and architecture, allowing selection of optimal partitions for a variety of parallel systems
Covering Points by Disjoint Boxes with Outliers
For a set of n points in the plane, we consider the axis--aligned (p,k)-Box
Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together
contain n-k points. In this paper, we consider the boxes to be either squares
or rectangles, and we want to minimize the area of the largest box. For general
p we show that the problem is NP-hard for both squares and rectangles. For a
small, fixed number p, we give algorithms that find the solution in the
following running times:
For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time
for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p}
log^{p-1} k) time for p = 2,3.
In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to
'cover at least n-k points' to avoid having non-feasible solutions. Results
are unchanged. - added Proof to Lemma 11, clarified some sections - corrected
typos and small errors - updated affiliations of two author
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