In as Few Comparisons as Possible

Abstract

We review a variety of data ordering problems with the goal of solving them in as few comparisons as possible. En route we highlight a number of open problems, some new, some a couple of decades old, and others open for up to a half century. The first is that of sorting and the Ford-Johnson Merge-Insertion algorithm [8] of 1959, which remains the “best”, at least for the “best and worst” values of n. Is it optimal, or are its extra .028.. n or so comparisons beyond the information theoretic lower bound necessary?

Moving to selection problems we first examine a special case. The problem of finding the second largest member of a set is fairly straightforward in the worst case. The best expected case method remains the $n+ \Theta(\lg \lg n)$ method of Matula from 1973 [10]. It begs the question as to whether the $\lg \lg n$ term is necessary. The status of median finding has remained unchanged for a couple of decades, since the work of Dor and Zwick [4,5]. (3 − δ)n comparisons are sufficient, while (2 + ε)n are necessary. So the constant isn’t an integer, but is it log_4/3 2 as conjectured by Paterson [11]? This worst case behavior is in sharp contrast with the expected case of median finding where the answer has been known since the mid-’80’s [3,6].

Finally we look at the problem of partial sorting (arranging elements according to a given partial order) and completing a sort given partially ordered data. The latter problem was posed and solved within n or so comparisons of optimal by Fredman in 1975 [7]. The method, though, could use exponential time to determine which comparisons to perform. The more recent approaches of Cardinal et al [2,1] to these problems are based on graph entropy arguments and require only polynomial time to determine the comparisons to be made. Indeed the solution to the partial ordering problem involves a reduction to multiple selection [9]. In both cases the number of comparisons used differs from the information theoretic lower bound by only a lower order term plus a linear term.