Improved Time-Space Tradeoffs for 3SUM-Indexing

2025-12-03

AI Summary

This paper investigates the time–space trade-off for the 3SUM-Indexing problem. To overcome query-efficiency bottlenecks in the preprocessing phase, we propose a novel construction based on function decomposition and the Fiat–Naor inversion algorithm: we decompose the target function into structured subfunctions, thereby extending the applicability of the Fiat–Naor framework to 3SUM-Indexing. Our approach achieves the first tight trade-off (TS = n^{2.5}) between query time (T) and space (S), significantly improving upon the prior best bound (TS^3 = n^6) in the regime (n^{3/2} ll S ll n^{7/4}). Moreover, our method naturally generalizes to related indexing problems—including (k)-SUM-Indexing, (k)-XOR-Indexing, and string indexing—thereby strengthening lower bounds for multiple fundamental data structures.

Technology Category

• Search and Optimization: Distributed Search
• Data Mining & Knowledge Management: Data Compression
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web crawling and indexing
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

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. Additionally, we improve the best known time-space tradeoffs for the Gapped String Indexing and Jumbled Indexing problems, which are well-known data structure problems 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.

Problem

• Improves time-space tradeoffs for 3SUM-Indexing preprocessing problem
• Extends improvements to kSUM-Indexing and kXOR-Indexing generalizations
• Enhances related data structure problems like Gapped String Indexing

Innovation

• Decomposes function into sub-functions for inversion.
• Applies improved Fiat-Naor algorithm to 3SUM-Indexing.
• Extends technique to kSUM-Indexing and kXOR-Indexing problems.