A subquadratic algorithm for 3XOR

Given a set $X$ of $n$ binary words of equal length $w$, the 3XOR problem asks for three elements $a, b, c \in X$ such that $a \oplus b=c$, where $\oplus$ denotes the bitwise XOR operation. The problem can be easily solved on a word RAM with word length $w$ in time $O(n^2 \log{n})$. Using Han's fast integer sorting algorithm (2002/2004) this can be reduced to $O(n^2 \log{\log{n}})$. With randomization or a sophisticated deterministic dictionary construction, creating a hash table for $X$ with constant lookup time leads to an algorithm with (expected) running time $O(n^2)$. At present, seemingly no faster algorithms are known. We present a surprisingly simple deterministic, quadratic time algorithm for 3XOR. Its core is a version of the Patricia trie for $X$, which makes it possible to traverse the set $a \oplus X$ in ascending order for arbitrary $a\in \{0, 1\}^{w}$ in linear time. Furthermore, we describe a randomized algorithm for 3XOR with expected running time $O(n^2\cdot\min\{\log^3{w}/w, (\log\log{n})^2/\log^2 n\})$. The algorithm transfers techniques to our setting that were used by Baran, Demaine, and P\u{a}tra\c{s}cu (2005/2008) for solving the related int3SUM problem (the same problem with integer addition in place of binary XOR) in expected time $o(n^2)$. As suggested by Jafargholi and Viola (2016), linear hash functions are employed. The latter authors also showed that assuming 3XOR needs expected running time $n^{2-o(1)}$ one can prove conditional lower bounds for triangle enumeration just as with 3SUM. We demonstrate that 3XOR can be reduced to other problems as well, treating the examples offline SetDisjointness and offline SetIntersection, which were studied for 3SUM by Kopelowitz, Pettie, and Porat (2016)

On Multidimensional and Monotone k-SUM

The well-known k-SUM conjecture is that integer k-SUM requires time Omega(n^{ceil{k/2}-o(1)}). Recent work has studied multidimensional k-SUM in F_p^d, where the best known algorithm takes time tilde O(n^{ceil{k/2}}). Bhattacharyya et al. [ICS 2011] proved a min(2^{Omega(d)},n^{Omega(k)}) lower bound for k-SUM in F_p^d under the Exponential Time Hypothesis. We give a more refined lower bound under the standard k-SUM conjecture: for sufficiently large p, k-SUM in F_p^d requires time Omega(n^{k/2-o(1)}) if k is even, and Omega(n^{ceil{k/2}-2k(log k)/(log p)-o(1)}) if k is odd. For a special case of the multidimensional problem, bounded monotone d-dimensional 3SUM, Chan and Lewenstein [STOC 2015] gave a surprising tilde O(n^{2-2/(d+13)}) algorithm using additive combinatorics. We show this algorithm is essentially optimal. To be more precise, bounded monotone d-dimensional 3SUM requires time Omega(n^{2-frac{4}{d}-o(1)}) under the standard 3SUM conjecture, and time Omega(n^{2-frac{2}{d}-o(1)}) under the so-called strong 3SUM conjecture. Thus, even though one might hope to further exploit the structural advantage of monotonicity, no substantial improvements beyond those obtained by Chan and Lewenstein are possible for bounded monotone d-dimensional 3SUM

On the Fine-Grained Complexity of Parity Problems

We consider the parity variants of basic problems studied in fine-grained complexity. We show that finding the exact solution is just as hard as finding its parity (i.e. if the solution is even or odd) for a large number of classical problems, including All-Pairs Shortest Paths (APSP), Diameter, Radius, Median, Second Shortest Path, Maximum Consecutive Subsums, Min-Plus Convolution, and 0/1-Knapsack. A direct reduction from a problem to its parity version is often difficult to design. Instead, we revisit the existing hardness reductions and tailor them in a problem-specific way to the parity version. Nearly all reductions from APSP in the literature proceed via the (subcubic-equivalent but simpler) Negative Weight Triangle (NWT) problem. Our new modified reductions also start from NWT or a non-standard parity variant of it. We are not able to establish a subcubic-equivalence with the more natural parity counting variant of NWT, where we ask if the number of negative triangles is even or odd. Perhaps surprisingly, we justify this by designing a reduction from the seemingly-harder Zero Weight Triangle problem, showing that parity is (conditionally) strictly harder than decision for NWT

Faster Algorithms for the Sparse Random 3XOR Problem

We present two new algorithms for a variant of the 3XOR problem with lists consisting of N n-bit 10 vectors whose coefficients are drawn randomly according to a Bernoulli distribution of parameter 11 p 0.13. The analysis of these algorithms reveal a "phase change" for a 16 certain threshold p. 17 2012 ACM Subject Classification Theory of computation → Computational complexity and cryp-18 tography; Theory of computation 1

Threesomes, Degenerates, and Love Triangles

The 3SUM problem is to decide, given a set of $n$ real numbers, whether any three sum to zero. It is widely conjectured that a trivial $O(n^2)$-time algorithm is optimal and over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies $\Omega(n^2)$ lower bounds on numerous problems in computational geometry and a variant of the conjecture implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures. In this paper we refute the 3SUM conjecture. We prove that the decision tree complexity of 3SUM is $O(n^{3/2}\sqrt{\log n})$ and give two subquadratic 3SUM algorithms, a deterministic one running in $O(n^2 / (\log n/\log\log n)^{2/3})$ time and a randomized one running in $O(n^2 (\log\log n)^2 / \log n)$ time with high probability. Our results lead directly to improved bounds for $k$-variate linear degeneracy testing for all odd $k\ge 3$. The problem is to decide, given a linear function $f(x_1,\ldots,x_k) = \alpha_0 + \sum_{1\le i\le k} \alpha_i x_i$ and a set $A \subset \mathbb{R}$, whether $0\in f(A^k)$. We show the decision tree complexity of this problem is $O(n^{k/2}\sqrt{\log n})$. Finally, we give a subcubic algorithm for a generalization of the $(\min,+)$-product over real-valued matrices and apply it to the problem of finding zero-weight triangles in weighted graphs. We give a depth-$O(n^{5/2}\sqrt{\log n})$ decision tree for this problem, as well as an algorithm running in time $O(n^3 (\log\log n)^2/\log n)$

Data Structures Meet Cryptography: 3SUM with Preprocessing

This paper shows several connections between data structure problems and cryptography against preprocessing attacks. Our results span data structure upper bounds, cryptographic applications, and data structure lower bounds, as summarized next. First, we apply Fiat--Naor inversion, a technique with cryptographic origins, to obtain a data structure upper bound. In particular, our technique yields a suite of algorithms with space $S$ and (online) time $T$ for a preprocessing version of the $N$-input 3SUM problem where $S^3\cdot T = \widetilde{O}(N^6)$. This disproves a strong conjecture (Goldstein et al., WADS 2017) that there is no data structure that solves this problem for $S=N^{2-\delta}$ and $T = N^{1-\delta}$ for any constant $\delta>0$. Secondly, we show equivalence between lower bounds for a broad class of (static) data structure problems and one-way functions in the random oracle model that resist a very strong form of preprocessing attack. Concretely, given a random function $F: [N] \to [N]$ (accessed as an oracle) we show how to compile it into a function $G^F: [N^2] \to [N^2]$ which resists $S$-bit preprocessing attacks that run in query time $T$ where $ST=O(N^{2-\varepsilon})$ (assuming a corresponding data structure lower bound on 3SUM). In contrast, a classical result of Hellman tells us that $F$ itself can be more easily inverted, say with $N^{2/3}$-bit preprocessing in $N^{2/3}$ time. We also show that much stronger lower bounds follow from the hardness of kSUM. Our results can be equivalently interpreted as security against adversaries that are very non-uniform, or have large auxiliary input, or as security in the face of a powerfully backdoored random oracle. Thirdly, we give non-adaptive lower bounds for 3SUM and a range of geometric problems which match the best known lower bounds for static data structure problems

Clustered Integer 3SUM via Additive Combinatorics

We present a collection of new results on problems related to 3SUM, including: 1. The first truly subquadratic algorithm for $\ \ \ \ \$ 1a. computing the (min,+) convolution for monotone increasing sequences with integer values bounded by $O(n)$, $\ \ \ \ \$1b. solving 3SUM for monotone sets in 2D with integer coordinates bounded by $O(n)$, and $\ \ \ \ \$1c. preprocessing a binary string for histogram indexing (also called jumbled indexing). The running time is: $O(n^{(9+\sqrt{177})/12}\,\textrm{polylog}\,n)=O(n^{1.859})$ with randomization, or $O(n^{1.864})$ deterministically. This greatly improves the previous $n^2/2^{\Omega(\sqrt{\log n})}$ time bound obtained from Williams' recent result on all-pairs shortest paths [STOC'14], and answers an open question raised by several researchers studying the histogram indexing problem. 2. The first algorithm for histogram indexing for any constant alphabet size that achieves truly subquadratic preprocessing time and truly sublinear query time. 3. A truly subquadratic algorithm for integer 3SUM in the case when the given set can be partitioned into $n^{1-\delta}$ clusters each covered by an interval of length $n$, for any constant $\delta>0$. 4. An algorithm to preprocess any set of $n$ integers so that subsequently 3SUM on any given subset can be solved in $O(n^{13/7}\,\textrm{polylog}\,n)$ time. All these results are obtained by a surprising new technique, based on the Balog--Szemer\'edi--Gowers Theorem from additive combinatorics

All non-trivial variants of 3-LDT are equivalent

The popular 3-SUM conjecture states that there is no strongly subquadratic time algorithm for checking if a given set of integers contains three distinct elements that sum up to zero. A closely related problem is to check if a given set of integers contains distinct $x_1, x_2, x_3$ such that $x_1+x_2=2x_3$. This can be reduced to 3-SUM in almost-linear time, but surprisingly a reverse reduction establishing 3-SUM hardness was not known. We provide such a reduction, thus resolving an open question of Erickson. In fact, we consider a more general problem called 3-LDT parameterized by integer parameters $\alpha_1, \alpha_2, \alpha_3$ and $t$. In this problem, we need to check if a given set of integers contains distinct elements $x_1, x_2, x_3$ such that $\alpha_1 x_1+\alpha_2 x_2 +\alpha_3 x_3 = t$. For some combinations of the parameters, every instance of this problem is a NO-instance or there exists a simple almost-linear time algorithm. We call such variants trivial. We prove that all non-trivial variants of 3-LDT are equivalent under subquadratic reductions. Our main technical contribution is an efficient deterministic procedure based on the famous Behrend's construction that partitions a given set of integers into few subsets that avoid a chosen linear equation

Fine-Grained Completeness for Optimization in P

We initiate the study of fine-grained completeness theorems for exact and approximate optimization in the polynomial-time regime. Inspired by the first completeness results for decision problems in P (Gao, Impagliazzo, Kolokolova, Williams, TALG 2019) as well as the classic class MaxSNP and MaxSNP-completeness for NP optimization problems (Papadimitriou, Yannakakis, JCSS 1991), we define polynomial-time analogues MaxSP and MinSP, which contain a number of natural optimization problems in P, including Maximum Inner Product, general forms of nearest neighbor search and optimization variants of the $k$-XOR problem. Specifically, we define MaxSP as the class of problems definable as $\max_{x_1,\dots,x_k} \#\{ (y_1,\dots,y_\ell) : \phi(x_1,\dots,x_k, y_1,\dots,y_\ell) \}$, where $\phi$ is a quantifier-free first-order property over a given relational structure (with MinSP defined analogously). On $m$-sized structures, we can solve each such problem in time $O(m^{k+\ell-1})$. Our results are: - We determine (a sparse variant of) the Maximum/Minimum Inner Product problem as complete under *deterministic* fine-grained reductions: A strongly subquadratic algorithm for Maximum/Minimum Inner Product would beat the baseline running time of $O(m^{k+\ell-1})$ for *all* problems in MaxSP/MinSP by a polynomial factor. - This completeness transfers to approximation: Maximum/Minimum Inner Product is also complete in the sense that a strongly subquadratic $c$-approximation would give a $(c+\varepsilon)$-approximation for all MaxSP/MinSP problems in time $O(m^{k+\ell-1-\delta})$, where $\varepsilon > 0$ can be chosen arbitrarily small. Combining our completeness with~(Chen, Williams, SODA 2019), we obtain the perhaps surprising consequence that refuting the OV Hypothesis is *equivalent* to giving a $O(1)$-approximation for all MinSP problems in faster-than-$O(m^{k+\ell-1})$ time.Comment: Full version of APPROX'21 paper, abstract shortened to fit ArXiv requirement