Time Bounds for Selection

In this paper we present a new selection algorithm, PICK, and derive by an analysis of its efficiency the (surprising) result that the cost of selection is at most a linear function of the number of input items. In addition, we prove a new lower bound for the cost of selection. The selection problem is perhaps best exemplified by the computation of medians. In general, we may wish to select the i-th smallest of a set of n distinct numbers, or the element ranking closest to a given percentile level. Interest in this problem may be traced to the realm of sports and the design of (traditionally, tennis) tournaments to select the firstand second-best players. In 1883, Lewis Carroll published an article [1] denouncing the unfair method by which the second-best player is usually determined in a "knockout tournament" -the loser of the final match is often not the second-best! (Any of the players who lost only to the best player may be second-best.) Around 1930, Hugo Steinhaus brought the problem into the realm of algorithmic complexity by asking for the minimum number of matches required to (correctly) select both the firstand second-best players from a field of n contestants. In 1932, J. Schreier [8] showed that no more than n + [logg(n)]2 matches are required, and in 1964, S. S. Kislitsin [6] proved this number to be necessary as well. Schreier's method uses a knockout tournament to determine the winner, followed by a second knockout tournament among the