Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics
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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 n^1/2-regular graph is n^2-o(1)-hard under the 3-SUM conjecture even when the number of short cycles is small; namely, when the number of k-cycles is O(n^k/2+γ) for γ<1/2. Abboud et al. achieve γ≥ 1/4 by applying structure vs. 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. Consequently, we achieve the best possible γ=0 and the following lower bounds 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 n^1/k) time. For the same stretch, and assuming the query time is n^o(1) Abboud et al. proved an Ω(m^1+1/12.7552 · k) lower bound on the preprocessing time; we improve it to Ω(m^1+1/2k) which is only a factor 2 away from the upper bound. We also obtain tight bounds for stretch 2+o(1) and 3-ϵ 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 (m^1.1927+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(m^4/3,n^2) +t) upper bound is tight up to n^o(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 subquadratic algorithm for 3-SUM if one of the sets has small doubling.
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