TY - GEN

T1 - Computing majority by constant depth majority circuits with low fan-in gates

AU - Kulikov, Alexander S.

AU - Podolskii, Vladimir V.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - We study the following computational problem: for which values of k, the majority of n bits MAJn can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJk o MAJk. We observe that the minimum value of k for which there exists a MAJk o MAJk circuit that has high correlation with the majority of n bits is equal to Θ(n1/2). We then show that for a randomized MAJk o MAJk circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n2/3+o(1). We show a worst case lower bound: if a MAJk o MAJk circuit computes the majority of n bits correctly on all inputs, then k ≥ n13/19+o(1). This lower bound exceeds the optimal value for randomized circuits and thus is unreachable for pure randomized techniques. For depth 3 circuits we show that a circuit with k = O(n2/3) can compute MAJn correctly on all inputs.

AB - We study the following computational problem: for which values of k, the majority of n bits MAJn can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJk o MAJk. We observe that the minimum value of k for which there exists a MAJk o MAJk circuit that has high correlation with the majority of n bits is equal to Θ(n1/2). We then show that for a randomized MAJk o MAJk circuit computing the majority of n input bits with high probability for every input, the minimum value of k is equal to n2/3+o(1). We show a worst case lower bound: if a MAJk o MAJk circuit computes the majority of n bits correctly on all inputs, then k ≥ n13/19+o(1). This lower bound exceeds the optimal value for randomized circuits and thus is unreachable for pure randomized techniques. For depth 3 circuits we show that a circuit with k = O(n2/3) can compute MAJn correctly on all inputs.

KW - Circuit complexity

KW - Computational complexity

KW - Lower bound

KW - Majority

KW - Threshold

KW - Upper bound

UR - http://www.scopus.com/inward/record.url?scp=85016194461&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.STACS.2017.49

DO - 10.4230/LIPIcs.STACS.2017.49

M3 - Conference contribution

AN - SCOPUS:85016194461

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017

A2 - Vallee, Brigitte

A2 - Vollmer, Heribert

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

T2 - 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017

Y2 - 8 March 2017 through 11 March 2017

ER -