 License: Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license (CC BY-NC-ND 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2011.332
URN: urn:nbn:de:0030-drops-30246
URL: https://drops.dagstuhl.de/opus/volltexte/2011/3024/
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### On Parsimonious Explanations For 2-D Tree- and Linearly-Ordered Data

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### 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},
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