Static Retrieval Revisited: To Optimality and Beyond
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909860902207488 |
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| author | Hu, Yang Kuszmaul, William Liang, Jingxun Yu, Huacheng Zhang, Junkai Zhou, Renfei |
| author_facet | Hu, Yang Kuszmaul, William Liang, Jingxun Yu, Huacheng Zhang, Junkai Zhou, Renfei |
| contents | In the static retrieval problem, a data structure must answer retrieval queries mapping a set of $n$ keys in a universe $[U]$ to $v$-bit values. Information-theoretically, retrieval data structures can use as little as $nv$ bits of space. For small value sizes $v$, it is possible to achieve $O(1)$ query time while using space $nv + o(n)$ bits -- whether or not such a result is possible for larger values of $v$ (e.g., $v = Θ(\log n)$) has remained open.
In this paper, we obtain a tight lower bound (as well as matching upper bounds) for the static retrieval problem. In the case where values are large, we show that there is actually a significant tension between time and space. It is not possible, for example, to get $O(1)$ query time using $nv + o(n)$ bits of space, when $v = Θ(\log n)$ (and assuming the word RAM model with $O(\log n)$-bit words).
At first glance, our lower bound would seem to render retrieval unusable in many settings that aim to achieve very low redundancy. However, our second result offers a way around this: We show that, whenever a retrieval data structure $D_1$ is stored along with another data structure $D_2$ (whose size is similar to or larger than the size of $D_1$), it is possible to implement the combined data structure $D_1 \cup D_2$ so that queries to $D_1$ take $O(1)$ time, operations on $D_2$ take the same asymptotic time as if $D_2$ were stored on its own, and the total space is $nv + \mathrm{Space}(D_2) + n^{0.67}$ bits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18237 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Static Retrieval Revisited: To Optimality and Beyond Hu, Yang Kuszmaul, William Liang, Jingxun Yu, Huacheng Zhang, Junkai Zhou, Renfei Data Structures and Algorithms In the static retrieval problem, a data structure must answer retrieval queries mapping a set of $n$ keys in a universe $[U]$ to $v$-bit values. Information-theoretically, retrieval data structures can use as little as $nv$ bits of space. For small value sizes $v$, it is possible to achieve $O(1)$ query time while using space $nv + o(n)$ bits -- whether or not such a result is possible for larger values of $v$ (e.g., $v = Θ(\log n)$) has remained open. In this paper, we obtain a tight lower bound (as well as matching upper bounds) for the static retrieval problem. In the case where values are large, we show that there is actually a significant tension between time and space. It is not possible, for example, to get $O(1)$ query time using $nv + o(n)$ bits of space, when $v = Θ(\log n)$ (and assuming the word RAM model with $O(\log n)$-bit words). At first glance, our lower bound would seem to render retrieval unusable in many settings that aim to achieve very low redundancy. However, our second result offers a way around this: We show that, whenever a retrieval data structure $D_1$ is stored along with another data structure $D_2$ (whose size is similar to or larger than the size of $D_1$), it is possible to implement the combined data structure $D_1 \cup D_2$ so that queries to $D_1$ take $O(1)$ time, operations on $D_2$ take the same asymptotic time as if $D_2$ were stored on its own, and the total space is $nv + \mathrm{Space}(D_2) + n^{0.67}$ bits. |
| title | Static Retrieval Revisited: To Optimality and Beyond |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2510.18237 |