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Communication Dans Un Congrès Année : 2011

On Parsimonious Explanations For 2-D Tree- and Linearly-Ordered Data

Résumé

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-Hard 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.
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hal-00573639 , version 1 (05-03-2011)

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  • HAL Id : hal-00573639 , version 1

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Howard Karloff, Flip Korn, Konstantin Makarychev, Yuval Rabani. On Parsimonious Explanations For 2-D Tree- and Linearly-Ordered Data. Symposium on Theoretical Aspects of Computer Science (STACS2011), Mar 2011, Dortmund, Germany. pp.332-343. ⟨hal-00573639⟩

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