The Recursive Structure of Some Ordering Problems

Some classical ordering problems (sorting, finding the maximum, finding the maximum and the minimum, finding the largest and the next largest, merging, and finding the median) are considered from a recursive viewpoint. If X(n) denotes an instance of size n of any one of these problems then X(n) can be solved by finding the solution to a number ~(n, k) of problems X(k) for some fixed k; ~(n, k) is called the relative complexity. Upper and lower bounds on the relative complexity are found. For the problem of finding the maximum, finding the maximum and the minimum, and finding the largest and the next largest these bounds are optimal. Divide and conquer is one of the most useful paradigms in algorithm design. One of its key aspects is the technique of solving a problem X(n), of size n, by making use of solutions to problems X(k) with k < n; thus divide and conquer algorithms are applicable for those problems whose solution can be expressed recursively. We shall investigate, for several classical ordering problems X(n), the number of solutions to problems X(r) with r bounded by some fixed k, needed to solve X(n). In all the problems that we consider X(2) will be the problem of comparing two numbers. Our results reduce to known ones on the number of comparisons in the case k = 2. In various degrees of detail we shall consider the following problems: 1. A(n): sort n elements into ascending order, 2. B(n): find the maximum in a set of n elements, 3. C(n): find the maximum and the minimum in a set of n elements, 4. D(n): find the largest and the next largest in a set of n elements, 5. E(n): merge two sorted lists of length n/2, 6. F(n): find the medians in a set of n elements. All of these problems have traditionally been studied for their worst case comparison complexity and substantial progress has been made on all of them. In fact, apart