重新审视稀疏矩阵压缩问题
Revisiting the Sparse Matrix Compression Problem
AI总结：
本文重新审视稀疏矩阵压缩问题，证明贪心算法的近似比为Θ(√(ℓ+ρ))，并提出动态规划算法解决双对数宽度或对数宽度的限制问题，同时引入修剪非零边框的变种问题。
摘要：
稀疏矩阵压缩问题要求为一个二进制n×ℓ矩阵提供一维表示，该表示由一个整数数组的行索引和每个行的位移函数组成，使得通过此表示可以以常数时间访问矩阵中的任一元素。已证明寻找长度为ℓ+ρ的整数数组或限制位移函数至ρ值的决策问题为NP完全问题（参见Garey和Johnson的教科书）。作为一种实用启发式方法，提出了一种贪心算法，该算法将第i行向右移动，直到与前一行形成解。尽管该贪心算法因在实践中具有良好的近似性而受到青睐，但我们证明其实际上具有Θ(√(ℓ+ρ))的近似比。我们还对诸如不同行数或每行最大非零元素数等参数化问题给出了进一步的难度结果。最后，我们设计了一种动态规划算法，该算法在双对数宽度或对数宽度的进一步限制下解决该问题。我们从新视角研究了所有这些发现，通过引入一个问题变种，即通过修剪非零边框来最小化所得到整数数组的长度，该变种在文献中尚未被研究，但具有实际动机。
英文摘要：
The sparse matrix compression problem asks for a one-dimensional representation of a binary $n \times \ell$ matrix, formed by an integer array of row indices and a shift function for each row, such that accessing a matrix entry is possible in constant time by consulting this representation. It has been shown that the decision problem for finding an integer array of length $\ell+ρ$ or restricting the shift function up to values of $ρ$ is NP-complete (cf. the textbook of Garey and Johnson). As a practical heuristic, a greedy algorithm has been proposed to shift the $i$-th row until it forms a solution with its predecessor rows. Despite that this greedy algorithm is cherished for its good approximation in practice, we show that it actually exhibits an approximation ratio of $Θ(\sqrt{\ell+ρ})$. We give further hardness results for parameterizations such as the number of distinct rows or the maximum number of non-zero entries per row. Finally, we devise a DP-algorithm that solves the problem for double-logarithmic matrix widths or logarithmic widths for further restrictions. We study all these findings also under a new perspective by introducing a variant of the problem, where we wish to minimize the length of the resulting integer array by trimming the non-zero borders, which has not been studied in the literature before but has practical motivations.
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