TY - GEN

T1 - On parsimonious explanations for 2-D tree- and linearly-ordered data

AU - Karloff, Howard

AU - Korn, Flip

AU - Makarychev, Konstantin

AU - Rabani, Yuval

PY - 2011

Y1 - 2011

N2 - 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 = (aij) 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, aij 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 T1 whose leaves are the rows of A and another, T2, 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.

AB - 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 = (aij) 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, aij 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 T1 whose leaves are the rows of A and another, T2, 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.

UR - http://www.scopus.com/inward/record.url?scp=84880273640&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.STACS.2011.332

DO - 10.4230/LIPIcs.STACS.2011.332

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AN - SCOPUS:84880273640

SN - 9783939897255

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 332

EP - 343

BT - 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011

T2 - 28th International Symposium on Theoretical Aspects of Computer Science, STACS 2011

Y2 - 10 March 2011 through 12 March 2011

ER -