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\documentclass{acmabs}

\begin{document}

\Journal{Journal of the ACM}

\refkey{MiltersenPT96}

\title{The Asymptotic Complexity of Merging Networks}

\author{Peter~Bro Miltersen \and Mike Paterson \and Jun Tarui}

\Pages{147--165}

\Month{January}

\Year{1996}

\Volume{43}

\Number{1}

\maketitle

\begin{abstract}

Let $M(m, n)$ be the minimum number of comparators needed in a comparator
  network that merges $m$ elements $x_1 \le x_2 \le \cdots \le x_m$ and $n$
  elements $y_1 \le y_2 \le \cdots \le y_n$, where $n \ge m$. Batcher's
  odd-even merge yields the following upper bound: \[ M(m, n) \le \frac{1}{2}
  (m + n) \log_2 m + O(n); \] in particular, \[ M(m, n) \le n \log_2 n + O(n).
  \] We prove the following lower bound that matches the upper bound above
  asymptotically as $n \ge m \to \infty$: \[ M(m, n) \ge \frac{1}{2} (m + n)
  \log_2 m - O(n); \] in particular, \[ M(m, n) \ge n \log_2 n - O(n). \] 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.

\end{abstract}

\begin{categories}

F.2.2 [\textbf{Analysis of Algorithms and Problem Complexity}]:
Nonnumerical Algorithms and Problems---\emph{sorting and searching}

\end{categories}

\begin{terms}

Algorithms, Theory

\end{terms}

\begin{keywords}

Comparator network, merging, sorting

\end{keywords}

\end{document}
