Solving problems on concurrent processors. Vol. 1: General techniques and regular problems
Solving problems on concurrent processors. Vol. 1: General techniques and regular problems
Partitioning Problems in Parallel, Pipeline, and Distributed Computing
IEEE Transactions on Computers
Equi-depth multidimensional histograms
SIGMOD '88 Proceedings of the 1988 ACM SIGMOD international conference on Management of data
Partition trees for triangle counting and other range searching problems
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Structured partitioning problems
Operations Research
On the optimal binary plane partition for sets of isothetic rectangles
Information Processing Letters
Almost optimal set covers in finite VC-dimension: (preliminary version)
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Balancing histogram optimality and practicality for query result size estimation
SIGMOD '95 Proceedings of the 1995 ACM SIGMOD international conference on Management of data
Improved histograms for selectivity estimation of range predicates
SIGMOD '96 Proceedings of the 1996 ACM SIGMOD international conference on Management of data
Histogram-based estimation techniques in database systems
Histogram-based estimation techniques in database systems
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
On approximating rectangle tiling and packing
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
The Grid File: An Adaptable, Symmetric Multikey File Structure
ACM Transactions on Database Systems (TODS)
Accurate estimation of the number of tuples satisfying a condition
SIGMOD '84 Proceedings of the 1984 ACM SIGMOD international conference on Management of data
On Rectangular Partitionings in Two Dimensions: Algorithms, Complexity, and Applications
ICDT '99 Proceedings of the 7th International Conference on Database Theory
Partitioning an Array onto a Mesh of Processors
PARA '96 Proceedings of the Third International Workshop on Applied Parallel Computing, Industrial Computation and Optimization
Optimal Histograms with Quality Guarantees
VLDB '98 Proceedings of the 24rd International Conference on Very Large Data Bases
Universality of Serial Histograms
VLDB '93 Proceedings of the 19th International Conference on Very Large Data Bases
Selectivity Estimation Without the Attribute Value Independence Assumption
VLDB '97 Proceedings of the 23rd International Conference on Very Large Data Bases
On the Complexity of the Generalized Block Distribution
IRREGULAR '96 Proceedings of the Third International Workshop on Parallel Algorithms for Irregularly Structured Problems
ICALP '97 Proceedings of the 24th International Colloquium on Automata, Languages and Programming
Polynomial time approximation schemes for Euclidean TSP and other geometric problems
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
The optimization of queries in relational databases
The optimization of queries in relational databases
Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
A Las Vegas algorithm for linear programming when the dimension is small
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
Dynamic multidimensional histograms
Proceedings of the 2002 ACM SIGMOD international conference on Management of data
On Rectangular Partitionings in Two Dimensions: Algorithms, Complexity, and Applications
ICDT '99 Proceedings of the 7th International Conference on Database Theory
Fast Optimal Genome Tiling with Applications to Microarray Design and Homology Search
WABI '02 Proceedings of the Second International Workshop on Algorithms in Bioinformatics
Vmhist: Efficient Multidimensional Histograms with Improved Accuracy
DaWaK 2000 Proceedings of the Second International Conference on Data Warehousing and Knowledge Discovery
APPROX '00 Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization
Tiling Multi-dimensional Arrays
FCT '99 Proceedings of the 12th International Symposium on Fundamentals of Computation Theory
A Probabilistic Framework for Building Privacy-Preserving Synopses of Multi-dimensional Data
SSDBM '08 Proceedings of the 20th international conference on Scientific and Statistical Database Management
Multiple-Objective Compression of Data Cubes in Cooperative OLAP Environments
ADBIS '08 Proceedings of the 12th East European conference on Advances in Databases and Information Systems
Geometric algorithms for optimal airspace design and air traffic controller workload balancing
Journal of Experimental Algorithmics (JEA)
Approximation algorithms for min-max generalization problems
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Differentially private data release through multidimensional partitioning
SDM'10 Proceedings of the 7th VLDB conference on Secure data management
Location-aware type ahead search on spatial databases: semantics and efficiency
Proceedings of the 2011 ACM SIGMOD International Conference on Management of data
A quad-tree based multiresolution approach for two-dimensional summary data
Information Systems
Efficient construction of histograms for multidimensional data using quad-trees
Decision Support Systems
Information Sciences: an International Journal
k-ARQ: k-anonymous ranking queries
DASFAA'10 Proceedings of the 15th international conference on Database Systems for Advanced Applications - Volume Part I
Relations between two common types of rectangular tilings
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
A new approximation algorithm for multidimensional rectangle tiling
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
Improving the accuracy of histograms for geographic data objects
DASFAA'12 Proceedings of the 17th international conference on Database Systems for Advanced Applications - Volume Part I
Seal: spatio-textual similarity search
Proceedings of the VLDB Endowment
Secure multidimensional range queries over outsourced data
The VLDB Journal — The International Journal on Very Large Data Bases
Synopses for Massive Data: Samples, Histograms, Wavelets, Sketches
Foundations and Trends in Databases
Sensitivity of self-tuning histograms: query order affecting accuracy and robustness
SSDBM'12 Proceedings of the 24th international conference on Scientific and Statistical Database Management
A class of R-tree histograms for spatial databases
Proceedings of the 20th International Conference on Advances in Geographic Information Systems
Efficiently adapting graphical models for selectivity estimation
The VLDB Journal — The International Journal on Very Large Data Bases
Compiling mockups to flexible UIs
Proceedings of the 2013 9th Joint Meeting on Foundations of Software Engineering
Proceedings of the VLDB Endowment
Data & Knowledge Engineering
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Partitioning a multi-dimensional data set into rectangular partitions subject to certain constraints is an important problem that arises in many database applications, including histogram-based selectivity estimation, load-balancing, and construction of index structures. While provably optimal and efficient algorithms exist for partitioning one-dimensional data, the multi-dimensional problem has received less attention, except for a few special cases. As a result, the heuristic partitioning techniques that are used in practice are not well understood, and come with no guarantees on the quality of the solution. In this paper, we present algorithmic and complexity-theoretic results for the fundamental problem of partitioning a two-dimensional array into rectangular tiles of arbitrary size in a way that minimizes the number of tiles required to satisfy a given constraint. Our main results are approximation algorithms for several partitioning problems that provably approximate the optimal solutions within small constant factors, and that run in linear or close to linear time. We also establish the NP-hardness of several partitioning problems, therefore it is unlikely that there are efficient, i.e., polynomial time, algorithms for solving these problems exactly. We also discuss a few applications in which partitioning problems arise. One of the applications is the problem of constructing multi-dimensional histograms. Our results, for example, give an efficient algorithm to construct the V-Optimal histograms which are known to be the most accurate histograms in several selectivity estimation problems. Our algorithms are the first to provide guaranteed bounds on the quality of the solution.