Succinct Dynamic Rank/Select: Bypassing the Tree-Structure Bottleneck
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911226194296832 |
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| author | Kuszmaul, William Liang, Jingxun Zhou, Renfei |
| author_facet | Kuszmaul, William Liang, Jingxun Zhou, Renfei |
| contents | We show how to construct a dynamic ordered dictionary, supporting insert/delete/rank/select on a set of $n$ elements from a universe of size $U$, that achieves the optimal amortized expected time complexity of $O(1 + \log n / \log \log U)$, while achieving a nearly optimal space consumption of $\log \binom{U}{n} + n / 2^{(\log n)^{Ω(1)}} + \text{polylog}\, U$ bits in the regime where $U = \text{poly}(n)$. This resolves an open question by Pibiri and Venturini as to whether a redundancy (a.k.a. space overhead) of $o(n)$ bits is possible, and is the first dynamic solution to bypass the so-called tree-structure bottleneck, in which the bits needed to encode some dynamic tree structure are themselves enough to force a redundancy of $\widetildeΩ(n)$ bits. Our main technical building block is a dynamic balanced binary search tree, which we call the compressed tabulation-weighted treap, that itself achieves a surprising time/space tradeoff. The tree supports $\text{polylog}\, n$-time operations and requires a static lookup table of size $\text{poly}(n) + \text{polylog}\, U$ -- but, in exchange for these, the tree is able to achieve a remarkable space guarantee. Its total space redundancy is $O(\log U)$ bits. In fact, if the tree is given $n$ and $U$ for free, then the redundancy further drops to $O(1)$ bits. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_19175 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Succinct Dynamic Rank/Select: Bypassing the Tree-Structure Bottleneck Kuszmaul, William Liang, Jingxun Zhou, Renfei Data Structures and Algorithms We show how to construct a dynamic ordered dictionary, supporting insert/delete/rank/select on a set of $n$ elements from a universe of size $U$, that achieves the optimal amortized expected time complexity of $O(1 + \log n / \log \log U)$, while achieving a nearly optimal space consumption of $\log \binom{U}{n} + n / 2^{(\log n)^{Ω(1)}} + \text{polylog}\, U$ bits in the regime where $U = \text{poly}(n)$. This resolves an open question by Pibiri and Venturini as to whether a redundancy (a.k.a. space overhead) of $o(n)$ bits is possible, and is the first dynamic solution to bypass the so-called tree-structure bottleneck, in which the bits needed to encode some dynamic tree structure are themselves enough to force a redundancy of $\widetildeΩ(n)$ bits. Our main technical building block is a dynamic balanced binary search tree, which we call the compressed tabulation-weighted treap, that itself achieves a surprising time/space tradeoff. The tree supports $\text{polylog}\, n$-time operations and requires a static lookup table of size $\text{poly}(n) + \text{polylog}\, U$ -- but, in exchange for these, the tree is able to achieve a remarkable space guarantee. Its total space redundancy is $O(\log U)$ bits. In fact, if the tree is given $n$ and $U$ for free, then the redundancy further drops to $O(1)$ bits. |
| title | Succinct Dynamic Rank/Select: Bypassing the Tree-Structure Bottleneck |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2510.19175 |