The asymptotic complexity of merging networks

Let M(m,n) be the minimum number of comparators needed in a comparator network that merges m elements x₁⩽x₂⩽. . .⩽x_m and n elements y₁ ⩽y₂. . .⩽y_n, where n⩾m. Batcher´s odd-even merge yields the following upper bound: M(m,n)⩽¹/₂ (m+n)log₂(m+1)+O( n); in particular, M(n,n)⩽n log₂n+O(n). The authors prove the following lower bound that matches the upper bound above asymptotically as n⩾m→∞:M(m,n )⩾¹/₂(m+n)log₂ (m+1)-O(m); in particular, M( n,n)⩾nlog₂n-O (n). The authors´ 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