Selection Algorithms

Given a set X containing n distinct elements, and given a number 1 i n, we would like to nd the element whose rank in X is exactly i, i.e., the element of X larger than exactly i 1 elements of X and smaller than the other n i elements of X. In particular we are interested cases where i = n for some 0 < < 1. The median of X is the dn=2e-th element. Improving a long standing result of Sch onhage, Paterson and Pippenger we show that the median (as well as every other element) of a set containing n elements can always be found using at most 2:9423n comparisons. We also describe an algorithm for selecting the n-th largest element, using at most (1 + (1 + o(1))H( )) n comparisons, where H( ) is the binary entropy function and the o(1) stands for a function that tends to 0 as tends to 0. For small values of this is almost the best possible as there is a lower bound of about (1 + H( )) n comparisons. We use some ideas from the median selection algorithm and we obtain a better selection algorithm for some intermediate values of (e.g., the dn=4e-th element can be found using only 2:69n comparisons). The underlying technique for our upper bounds is mass production. Mass production is carried out in factories which are originally used by Schonhage et al. in their 3n+ o(n) median algorithm. We introduce green factories and perform an amortised analysis of their production costs. On the other hand, we show a new lower bound for selection algorithms. We obtain a lower bound of (1 + H( ) + ( 4)) n o(n) comparisons for restricted selection algorithms. The known selection algorithms, including the ones in [BFP+73] and [SPP76] as well as the selection algorithms presented in this thesis, comply with this restriction. This bound gives in particular, a lower bound of 2:00247 n comparisons on median selection. We then extend this lower bound and obtain a lower bound of (2 + ) n comparisons for any median selection algorithm (for some > 0). Unfortunately, the improvement obtained for the general case is very small. We extend another result of Schonhage, Paterson and Pippenger [SPP76] and show that a conjecture of Yao implies that the n-th element can be found using at most (1 + log 1 +O( )) n vii comparisons (but does not give an explicit algorithm). This is even closer, but still above, the (1+H( )) n lower bound. We show, in particular, that if Yao's conjecture is true then the n-th element, where 1=3 1=2 can be found using at most (2 + ) n + o(n) comparisons. Yao conjectured that extra elements can not be used to obtain better selection algorithms. We describe an algorithm that selects the n-th element given (1 + )n elements (i.e., given n extra elements), whose complexity is better than the worst case complexity of any other known algorithm when 0:4711 < < 0:1724. This confronts Yao's conjecture with a concrete challenge.