AI 中文总结
本文针对预处理3SUM问题，提出基于Fiat–Naor数据结构的统一构造，优化空间开销，实现次二次空间并支持自适应查询，保持查询时间不变。
AI 中文摘要
3SUM问题是指，给定整数集合A、B、C，判断是否存在a∈A和b∈B，使得它们的和属于C。在目标集C未知的预处理变体中，需预处理大小均为n的集合A和B，随后通过求解3SUM实例(A',B',C')来回答由子集A'⊆A、B'⊆B和大小为O(n)的目标集C'指定的查询。Kirkpatrick、Kuszmaul、Mathialagan与Vassilevska Williams[ICALP 2026]首次提出该问题的次二次空间算法，对任意ε∈[0,1/2]，其查询时间为Õ(n^(3/2+ε))，空间为Õ(n^(2-2ε/3))。该算法对重目标和轻目标采用不同机制，对每个重目标显式存储求和对(a,b)的列表，这些列表主导空间开销。本文提出统一构造，对所有查询采用单一机制，不再存储此类对列表，而是利用Fiat–Naor数据结构[SICOMP 1999]对函数(a,b)↦(a+b mod p)求逆，按需恢复这些对，将空间开销优化为Õ(n^max(2−ε,11/6−ε/3))，同时保持相同查询时间，且是首个支持自适应选择查询的次二次空间构造。
英文摘要
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.
Comments16 pages, 5 figures
