Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics

The "short cycle removal"technique was recently introduced by Abboud, Bringmann, Khoury and Zamir (STOC '22) to prove fine-grained hardness of approximation. Its main technical result is that listing all triangles in an n1/2-regular graph is n2-o(1)-hard even when the number of short cycles is small; namely, when the number of k-cycles is O(nk/2+-3) for-3<1/2. Its corollaries are based on the 3-SUM conjecture and their strength depends on-3, i.e. on how effectively the short cycles are removed. Abboud et al. achieve-3≥ 1/4 by applying structure versus randomness arguments on graphs. In this paper, we take a step back and apply conceptually similar arguments on the numbers of the 3-SUM problem, from which the hardness of triangle listing is derived. Consequently, we achieve the best possible., -3=0 and the following lower bound corollaries under the 3-SUM conjecture: ∗Approximate distance oracles: The seminal Thorup-Zwick distance oracles achieve stretch 2k± O(1) after preprocessing a graph in O(m n1/k) time. For the same stretch, and assuming the query time isno(1) Abboud et al. proved an ω(m1+1/12.7552 · k) lower bound on the preprocessing time; we improve it to ω(m1+1/2k) which is only a factor2 away from the upper bound. Additionally, we obtain tight bounds for stretch 2+o(1) and 3-"and higher lower bounds for dynamic shortest paths. ∗Listing 4-cycles: Abboud et al. proved the first super-linear lower bound for listing all 4-cycles in a graph, ruling out (m1.1927+t)1+o(1) time algorithms where t is the number of 4-cycles. We settle the complexity of this basic problem by showing that the O(min(m4/3,n2) +t) upper bound is tight up to no(1) factors. Our results exploit a rich tool set from additive combinatorics, most notably the Balog-Szemerédi-Gowers theorem and Rusza's covering lemma. A key ingredient that may be of independent interest is a truly subquadratic algorithm for 3-SUM if one of the sets has small doubling.