关于二维树序和线性序数据的简约表示
On Parsimonious Explanations for 2-D Tree- and Linearly-Ordered Data
摘要：
本文研究了树和线性有序数组数据的“解释问题”，这个问题是由数据库应用程序激发的，最近解决了一维树有序的情况。本文给出了一个矩阵A，它的行和列都有语义：行的特殊子集和列的特殊子集有意义，其他子集没有意义。A中的一个子矩阵被称为有意义的，当且仅当它是一个有意义的行子集和一个有意义的列子集的叉积，在这种情况下，我们称它为“允许矩形”。我们的目标是将A“解释”为加权允许矩形的稀疏和。具体地，我们希望找到尽可能少的加权允许矩形，使得对于所有i，j，a_{ij}等于包括单元（i，j）的所有矩形的权重之和。在本文中，我们考虑的自然情况下，矩阵的维数是树有序或线性有序。在树序的情况下，我们给出一棵有根树T1，它的叶子是A的行，另一棵树T2，它的叶子是列。树的节点以一种明显的方式对应于它们的叶子后代的集合。在线性排序的情况下，一组行或列当且仅当它是连续的时才有意义。对于树型数据，证明了解释问题的NP-Hard性，并给出了一个随机2-近似算法;对于线性数据，证明了解释问题的NP-Hard性，并给出了一个2.56-近似算法。据我们所知，这些是第一个结果的问题稀疏和准确表示矩阵的加权矩形。
英文摘要：
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 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 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.
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