Improved Bounds for 3SUM, K-SUM, and Linear Degeneracy

Given a set of $n$ real numbers, the 3SUM problem is to decide whether there are three of them that sum to zero. Until a recent breakthrough by Gr{\o}nlund and Pettie [FOCS'14], a simple $\Theta(n^2)$-time deterministic algorithm for this problem was conjectured to be optimal. Over the years many algorithmic problems have been shown to be reducible from the 3SUM problem or its variants, including the more generalized forms of the problem, such as k-SUM and k-variate linear degeneracy testing. The conjectured hardness of these problems have become extremely popular for basing conditional lower bounds for numerous algorithmic problems in P. Thus, a better understanding of the complexity of the 3SUM problem and its variants, might shed more light on the complexity of a wide class of problems in P. In this paper we show the following: 1. A deterministic algorithm for 3SUM that runs in $O(n^2 \log\log n / \log n)$ time. 2. The randomized decision tree complexity of 3SUM is $O(n^{3/2})$. 3. The randomized decision tree complexity of $k$-variate linear degeneracy testing (k-LDT) is $O(n^{k/2})$, for any odd $k\ge 3$. These bounds improve the ones obtained by Gr{\o}nlund and Pettie, giving a faster deterministic algorithm and new randomized decision tree bounds for this archetypal algorithmic problems.