TY - GEN

T1 - Improved bounds for 3SUM, k-SUM, and linear degeneracy

AU - Gold, Omer

AU - Sharir, Micha

N1 - Funding Information:
∗ For the full version of this paper see [21]. Work on this paper has been supported by Grant 892/13 from the Israel Science Foundation, by Grant 2012/229 from the U.S.-Israeli Binational Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), by the Blavatnik Research Fund in Computer Science at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. † A full version of the paper is available at http://arxiv.org/abs/1512.05279. 1 An r-linear decision tree is one in which each branching is based on a sign test of a linear expression with at most r terms. The complexity of the tree is its depth.

PY - 2017/9/1

Y1 - 2017/9/1

N2 - 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.

AB - 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.

KW - 3SUM

KW - Fractional cascading

KW - K-SUM

KW - Linear decision trees

KW - Linear degeneracy

UR - http://www.scopus.com/inward/record.url?scp=85030527280&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2017.42

DO - 10.4230/LIPIcs.ESA.2017.42

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AN - SCOPUS:85030527280

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 25th European Symposium on Algorithms, ESA 2017

A2 - Sohler, Christian

A2 - Sohler, Christian

A2 - Pruhs, Kirk

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 25th European Symposium on Algorithms, ESA 2017

Y2 - 4 September 2017 through 6 September 2017

ER -