Nearly Optimal List Labeling
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917655516020736 |
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| author | Bender, Michael A. Conway, Alex Farach-Colton, Martín Komlós, Hanna Koucký, Michal Kuszmaul, William Saks, Michael |
| author_facet | Bender, Michael A. Conway, Alex Farach-Colton, Martín Komlós, Hanna Koucký, Michal Kuszmaul, William Saks, Michael |
| contents | The list-labeling problem captures the basic task of storing a dynamically changing set of up to $n$ elements in sorted order in an array of size $m = (1 + Θ(1))n$. The goal is to support insertions and deletions while moving around elements within the array as little as possible.
Until recently, the best known upper bound stood at $O(\log^2 n)$ amortized cost. This bound, which was first established in 1981, was finally improved two years ago, when a randomized $O(\log^{3/2} n)$ expected-cost algorithm was discovered. The best randomized lower bound for this problem remains $Ω(\log n)$, and closing this gap is considered to be a major open problem in data structures.
In this paper, we present the See-Saw Algorithm, a randomized list-labeling solution that achieves a nearly optimal bound of $O(\log n \operatorname{polyloglog} n)$ amortized expected cost. This bound is achieved despite at least three lower bounds showing that this type of result is impossible for large classes of solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00807 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nearly Optimal List Labeling Bender, Michael A. Conway, Alex Farach-Colton, Martín Komlós, Hanna Koucký, Michal Kuszmaul, William Saks, Michael Data Structures and Algorithms The list-labeling problem captures the basic task of storing a dynamically changing set of up to $n$ elements in sorted order in an array of size $m = (1 + Θ(1))n$. The goal is to support insertions and deletions while moving around elements within the array as little as possible. Until recently, the best known upper bound stood at $O(\log^2 n)$ amortized cost. This bound, which was first established in 1981, was finally improved two years ago, when a randomized $O(\log^{3/2} n)$ expected-cost algorithm was discovered. The best randomized lower bound for this problem remains $Ω(\log n)$, and closing this gap is considered to be a major open problem in data structures. In this paper, we present the See-Saw Algorithm, a randomized list-labeling solution that achieves a nearly optimal bound of $O(\log n \operatorname{polyloglog} n)$ amortized expected cost. This bound is achieved despite at least three lower bounds showing that this type of result is impossible for large classes of solutions. |
| title | Nearly Optimal List Labeling |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2405.00807 |