On the I/O Complexity of the CYK Algorithm and of a Family of Related DP Algorithms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929561407586304 |
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| author | De Stefani, Lorenzo Gupta, Vedant |
| author_facet | De Stefani, Lorenzo Gupta, Vedant |
| contents | Asymptotically tight lower bounds are derived for the Input/Output (I/O) complexity of a class of dynamic programming algorithms including matrix chain multiplication, optimal polygon triangulation, and the construction of optimal binary search trees. Assuming no recomputation of intermediate values, we establish an $Ω\left(\frac{n^3}{\sqrt{M}B}\right)$ I/O lower bound, where $n$ denotes the size of the input and $M$ denotes the size of the available fast memory (cache). When recomputation is allowed, we show the same bound holds for $M < cn$, where $c$ is a positive constant. In the case where $M \ge 2n$, we show an $Ω\left(n/B\right)$ I/O lower bound. We also discuss algorithms for which the number of executed I/O operations matches asymptotically each of the presented lower bounds, which are thus asymptotically tight.
Additionally, we refine our general method to obtain a lower bound for the I/O complexity of the Cocke-Younger-Kasami algorithm, where the size of the grammar impacts the I/O complexity. An upper bound with asymptotically matching performance in many cases is also provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the I/O Complexity of the CYK Algorithm and of a Family of Related DP Algorithms De Stefani, Lorenzo Gupta, Vedant Data Structures and Algorithms F.2.0 Asymptotically tight lower bounds are derived for the Input/Output (I/O) complexity of a class of dynamic programming algorithms including matrix chain multiplication, optimal polygon triangulation, and the construction of optimal binary search trees. Assuming no recomputation of intermediate values, we establish an $Ω\left(\frac{n^3}{\sqrt{M}B}\right)$ I/O lower bound, where $n$ denotes the size of the input and $M$ denotes the size of the available fast memory (cache). When recomputation is allowed, we show the same bound holds for $M < cn$, where $c$ is a positive constant. In the case where $M \ge 2n$, we show an $Ω\left(n/B\right)$ I/O lower bound. We also discuss algorithms for which the number of executed I/O operations matches asymptotically each of the presented lower bounds, which are thus asymptotically tight. Additionally, we refine our general method to obtain a lower bound for the I/O complexity of the Cocke-Younger-Kasami algorithm, where the size of the grammar impacts the I/O complexity. An upper bound with asymptotically matching performance in many cases is also provided. |
| title | On the I/O Complexity of the CYK Algorithm and of a Family of Related DP Algorithms |
| topic | Data Structures and Algorithms F.2.0 |
| url | https://arxiv.org/abs/2410.20337 |