The asymptotic complexity of merging networks

Abstract

Let M(m, n) be the minimum number of comparators in a comparator network that merges two ordered chains x1 = m. Batcher's odd-even merge yields the following upper bound: M(m, n) = n/2. log2 (m + 1); M (n, n) >= n/2. log2 n + O (n). We prove a new lower bound that matches the upper bound asymptotically: M (m, n) >= (m + n)/2. log2 (m + 1) - O (m), e.g., M (n, n) >= n log2 n - O (n). Our proof technique extends to give similarly tight lower bounds for the size of monotone Boolean circuits for merging, and for the size of switching networks capable of realizing the set of permutations that arise from merging