Bounds on the size of merging networks

Abstract

Let M( m, n) be the minimum number of comparators needed in an ( m, n)-merging network. Batcher's odd-even merge provides upper bounds, whereas the best general lower bounds were established by Yao and Yao (1976) and Miltersen et al. (to appear). In this paper, we concentrate on small fixed m and arbitrary n. M(1, n) = n has long been known. In their 1976 paper, Yao and Yao showed M(2,n) = ⌈ 3n 2 ⌉ and asked for the exact value of M(3, n). We prove M(3,n) = ⌈ (7n + 3) 4 ⌉ for all n. Furthermore, M(4,n)> 11 6 n, M(5,n) > 2n are shown, improving previous bounds. Some related questions are discussed.