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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2011.332
URN: urn:nbn:de:0030-drops-30246
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2011/3024/
Karloff, Howard ;
Korn, Flip ;
Makarychev, Konstantin ;
Rabani, Yuval
On Parsimonious Explanations For 2-D Tree- and Linearly-Ordered Data
Abstract
This paper studies the ``explanation problem'' for tree- and linearly-ordered array data, a problem motivated by database applications and recently solved for the one-dimensional tree-ordered case. In this paper, one is given a matrix A=(a_{ij}) whose rows and columns have semantics: special subsets of the rows and special subsets of the columns are meaningful, others are not. A submatrix in A is said to be meaningful if and only if it is the cross product of a meaningful row subset and a meaningful column subset, in which case we call it an ``allowed rectangle.'' The goal is to ``explain'' A as a sparse sum of weighted allowed rectangles. Specifically, we wish to find as few weighted allowed rectangles as possible such that, for all i,j, a_ij equals the sum of the weights of all rectangles which include cell (i,j).
In this paper we consider the natural cases in which the matrix dimensions are tree-ordered or linearly-ordered. In the tree-ordered case, we are given a rooted tree $T_1$ whose leaves are the rows of $A$ and another, $T_2$, whose leaves are the columns. Nodes of the trees correspond in an obvious way to the sets of their leaf descendants. In the linearly-ordered case, a set of rows or columns is meaningful if and only if it is contiguous.
For tree-ordered data, we prove the explanation problem NP-Hard and give a randomized $2$-approximation algorithm for it. For linearly-ordered data, we prove the explanation problem NP-Har and give a $2.56$-approximation algorithm. To our knowledge, these are the first results for the problem of sparsely and exactly representing matrices by weighted rectangles.
BibTeX - Entry
@InProceedings{karloff_et_al:LIPIcs:2011:3024,
author = {Howard Karloff and Flip Korn and Konstantin Makarychev and Yuval Rabani},
title = {{On Parsimonious Explanations For 2-D Tree- and Linearly-Ordered Data}},
booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
pages = {332--343},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-25-5},
ISSN = {1868-8969},
year = {2011},
volume = {9},
editor = {Thomas Schwentick and Christoph D{\"u}rr},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2011/3024},
URN = {urn:nbn:de:0030-drops-30246},
doi = {10.4230/LIPIcs.STACS.2011.332},
annote = {Keywords: ordered data, explanation problem}
}
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Keywords: |
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ordered data, explanation problem |
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Collection: |
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28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) |
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Issue Date: |
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2011 |
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Date of publication: |
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11.03.2011 |
