Problem
Research questions and friction points this paper is trying to address.
3SUM
preprocessing
space efficiency
query
Recovery Beats Storage: Improved Space for Preprocessed 3SUM
🤖 AI Summary
该研究通过使用Fiat-Naor数据结构来恢复求和对，而非存储它们，从而改进了预处理3SUM问题的空间复杂度至亚二次级别。
📝 Abstract
The 3SUM problem asks, given sets $A,B,C$ of integers, whether there exist $a\in A$ and $b\in B$ whose sum belongs to $C$. In the preprocessed variant with unknown $C$, one preprocesses sets $A$ and $B$, each of size $n$, and subsequently answers a query specified by subsets $A'\subseteq A$, $B'\subseteq B$ and a target set $C'$ of size $O(n)$, by solving the 3SUM instance $(A',B',C')$. Kirkpatrick, Kuszmaul, Mathialagan, and Vassilevska Williams [ICALP 2026] gave the first algorithm with subquadratic space for this problem, achieving $\tilde{O}(n^{3/2+ε})$ query time using $\tilde{O}(n^{2-2ε/3})$ space, for every $ε\in[0,1/2]$. Their algorithm employs separate mechanisms for heavy and light targets, and for each heavy target it stores explicitly the list of pairs $(a,b)$ summing to it; these lists dominate the space bound. We present a unified construction that uses a single mechanism for all queries. Instead of storing these lists of pairs, we recover them on demand by leveraging the Fiat--Naor data structure [SICOMP 1999] to invert the function $(a,b)\mapsto (a+b\bmod p)$. This simplification improves the space bound to $\tilde{O}(n^{\max(2-ε, 11/6-ε/3)})$, while maintaining the same query time. Moreover, our construction is the first to achieve subquadratic space while supporting adaptively chosen queries.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Unified Construction
Fiat-Naor Data Structure
On-demand Recovery
Subquadratic Space
Adaptively Chosen Queries
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