Improved bounds for 3SUM, k-SUM, and linear degeneracy

Omer Gold, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 Grønlund and Pettie [FOCS'14], a simple ϵ(n2)-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 (k-LDT). The conjectured hardness of these problems have become extremely popular for basing conditional lower bounds for numerous algorithmic problems in P. In this paper, we show that the randomized 4-linear decision tree complexity1 of 3SUM is O(n3/2), and that the randomized (2k - 2)-linear decision tree complexity of k-SUM and k- LDT is O(nk/2), for any odd k ≥ 3. These bounds improve (albeit being randomized) the corresponding O(n3/2√log n) and O(nk/2√log n) bounds obtained by Grønlund and Pettie. Our technique includes a specialized randomized variant of the fractional cascading data structure. Additionally, we give another deterministic algorithm for 3SUM that runs in O(n2 log log n/ log n) time. The latter bound matches a recent independent bound by Freund [Algorithmica 2017], but our algorithm is somewhat simpler, due to a better use of the word-RAM model.

Original languageEnglish
Title of host publication25th European Symposium on Algorithms, ESA 2017
EditorsChristian Sohler, Christian Sohler, Kirk Pruhs
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770491
StatePublished - 1 Sep 2017
Event25th European Symposium on Algorithms, ESA 2017 - Vienna, Austria
Duration: 4 Sep 20176 Sep 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference25th European Symposium on Algorithms, ESA 2017


  • 3SUM
  • Fractional cascading
  • K-SUM
  • Linear decision trees
  • Linear degeneracy


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