Improved Time-Space Tradeoffs for 3SUM-Indexing
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910160768729088 |
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| author | Dinur, Itai Golovnev, Alexander |
| author_facet | Dinur, Itai Golovnev, Alexander |
| contents | 3SUM-Indexing is a preprocessing variant of the 3SUM problem that has recently received a lot of attention. The best known time-space tradeoff for the problem is $T S^3 = n^{6}$ (up to logarithmic factors), where $n$ is the number of input integers, $S$ is the length of the preprocessed data structure, and $T$ is the running time of the query algorithm. This tradeoff was achieved in [KP19, GGHPV20] using the Fiat-Naor generic algorithm for Function Inversion. Consequently, [GGHPV20] asked whether this algorithm can be improved by leveraging the structure of 3SUM-Indexing.
In this paper, we exploit the structure of 3SUM-Indexing to give a time-space tradeoff of $T S = n^{2.5}$, which is better than the best known one in the range $n^{3/2} \ll S \ll n^{7/4}$. We further extend this improvement to the $k$SUM-Indexing problem-a generalization of 3SUM-Indexing-and to the related $k$XOR-Indexing problem, where addition is replaced with XOR. we improve the known time-space tradeoff for the Jumbled Indexing problem, which is a well-known data structure problem related to 3SUM-Indexing.
Our improvement comes from an alternative way to apply the Fiat-Naor algorithm to 3SUM-Indexing. Specifically, we exploit the structure of the function to be inverted by decomposing it into "sub-functions" with certain properties. This allows us to apply an improvement to the Fiat-Naor algorithm (which is not directly applicable to 3SUM-Indexing), obtained in [GGPS23] in a much larger range of parameters. We believe that our techniques may be useful in additional application-dependent optimizations of the Fiat-Naor algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04258 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved Time-Space Tradeoffs for 3SUM-Indexing Dinur, Itai Golovnev, Alexander Data Structures and Algorithms 3SUM-Indexing is a preprocessing variant of the 3SUM problem that has recently received a lot of attention. The best known time-space tradeoff for the problem is $T S^3 = n^{6}$ (up to logarithmic factors), where $n$ is the number of input integers, $S$ is the length of the preprocessed data structure, and $T$ is the running time of the query algorithm. This tradeoff was achieved in [KP19, GGHPV20] using the Fiat-Naor generic algorithm for Function Inversion. Consequently, [GGHPV20] asked whether this algorithm can be improved by leveraging the structure of 3SUM-Indexing. In this paper, we exploit the structure of 3SUM-Indexing to give a time-space tradeoff of $T S = n^{2.5}$, which is better than the best known one in the range $n^{3/2} \ll S \ll n^{7/4}$. We further extend this improvement to the $k$SUM-Indexing problem-a generalization of 3SUM-Indexing-and to the related $k$XOR-Indexing problem, where addition is replaced with XOR. we improve the known time-space tradeoff for the Jumbled Indexing problem, which is a well-known data structure problem related to 3SUM-Indexing. Our improvement comes from an alternative way to apply the Fiat-Naor algorithm to 3SUM-Indexing. Specifically, we exploit the structure of the function to be inverted by decomposing it into "sub-functions" with certain properties. This allows us to apply an improvement to the Fiat-Naor algorithm (which is not directly applicable to 3SUM-Indexing), obtained in [GGPS23] in a much larger range of parameters. We believe that our techniques may be useful in additional application-dependent optimizations of the Fiat-Naor algorithm. |
| title | Improved Time-Space Tradeoffs for 3SUM-Indexing |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2512.04258 |