Revisiting the Sparse Matrix Compression Problem
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arXiv
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| Autores principales: | , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915801508872192 |
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| author | Jugé, Vincent Köppl, Dominik Limouzy, Vincent Marino, Andrea Olblich, Jannik Punzi, Giulia Uno, Takeaki |
| author_facet | Jugé, Vincent Köppl, Dominik Limouzy, Vincent Marino, Andrea Olblich, Jannik Punzi, Giulia Uno, Takeaki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15314 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Revisiting the Sparse Matrix Compression Problem Jugé, Vincent Köppl, Dominik Limouzy, Vincent Marino, Andrea Olblich, Jannik Punzi, Giulia Uno, Takeaki Data Structures and Algorithms 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. |
| title | Revisiting the Sparse Matrix Compression Problem |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2602.15314 |