On the Average-Case Complexity of Selecting the kth Best

Let v,(n) be the minimum average number of pairwise comparisons needed to find the k-th largest of n numbers (k > 2) , assuming that all n! orderings are equally likely. D. W. Matula proved that, for sme absolute constant c , Vk(n)-n 5 c k log log n as n 3 = . In the -present-paper, we show that there exists an absolute constant c' > 0 such that v,(n)-n 2 c' k log log n as n 3 w , proving a conjecture of Matula. gywords: algorithm, average-case, binary tree, comparison, complexity, decision tree, selection. f* This research was supported in part by National Science Foundation grants MCS-72-03752 A03 and MCS-77-05313. 9 Part of this work was done while this author was visiting Bell Laboratories, Murray Hill, New Jersey 07974.