On the Depth of a Multiplexer Function with a Small Number of Select Lines

Abstract

This paper continues the research on the circuit synthesis problem for a multiplexer function of logic algebra, which is a component of many integrated circuits and is also used in theoretical study. The exact value of the depth of a multiplexer with \(n\) select lines in the standard basis is found under the assumption that the conjunction and disjunction gates are of depth 1 and the negation gate is of depth 0; the depth equals \(n+2\) if \(10 \le n \le 19\). Thus, it follows from previous results that the exact depth value equals \(n+2\) for all positive integers \(n\) such that either \(2 \le n \le 5\) or \(n \ge 10\). Moreover, for \(n=1\), this value equals 2, and for \(6 \le n \le 9\), it equals either \(n+2\) or \(n+3\). Similar results are also obtained for a basis consisting of all elementary conjunctions and elementary disjunctions of two variables.