Problem 59-3Author(s): Paul BrockSource: SIAM Review, Vol. 2, No. 2 (Apr., 1960), pp. 155-156Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2027377 .

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PROBLEMS AND SOLUTIONS 155

where a and b are constants. If

a(T) = cT (T > To) in region 1

a(T) = 0 (T ? To) in region 2,

where c and To are known constants, a solution of equation (1) which includes the determination of the shape of the boundary separating regions 1 and 2 is sought. The boundary conditions to be satisfied are

1. the separating boundary B passes through the point (xo, 0), 2. T and OT/cn are continuous across B, 3. T bounded, 4. limV2+2> T = 0.

The following questions are of particular interest:

(a) Does a stable solution exist? (b) If it does, is the solution unique? (c) Is the separating boundary B closed?

A similar alternative but simpler problem in which

a(T) = c, (T > To) in region 1,

a(T) = 0, (T ? To) in region2,

is also of interest.

SOLUTIONS

Problem 59-3, Optimum Sorting Procedure, by Paul Brock.

A fundamental procedure in all business operations is that of filing informa- tion. Whenever the information in the file is to be updated, the updating items are first sorted in accordance with the key of the file. This is the standard alter- native to a direct use of the file which is a random access procedure. In many cases, particularly in multiple file problems, the sorting procedure must be done by a computer. This is an expensive and time consuming operation. Many different procedures have been suggested, and each takes a certain amount of time. It would be useful to determine a minimum computer time procedure.

A general investigation of this problem is under way. In this investigation the following general problem arises: Given a sequence of positive integers, what is the expected length of its maximal monotonic nondecreasing subsequence.

The solution depends upon the length of the sequence and the number of allow- able integers. The special case for two integers has been solved. The solution depends upon the following lemma which is proposed as a problem: Among sequences of fixed length M consisting the p l's and q 2's, the number of se- quences whose maximal monotonic nondecreasing subsequences are of length n is the same for all p, q such that 0 < p, q < n ? M.

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156 PROBLEMS AND SOLUTIONS

Solution by the proposer.

Let F(p, q, n) be the totality of F sequences of length n for a given p, q choice where an F sequence of length n is defined as a sequence containing a maximum monotone nondecreasing subsequence of length n. The proof is by induction on M. The smallest value of M for which

(1) 0 < p, q < n <

holds is Ml = 2. For this case the theorem is obvious. Now consider the theorem for sequences of length M < M: choose a p, q satisfying p + q = 31I and (1), and fix n < M. Case 1: n = M: For any values of p, q > 0, there is only one F sequence, namely,

11 **. 122**.2. Case 2: n < M: Assume q < n - 1. Since n < M1, it is clear that p, q > 2 for

(1) to hold. Consider those sequences starting with a 2. Since q < n, the 2 cannot be counted towards any F sequence. Thus, the total number of F sequences starting with a 2 is F(p, q - 1, n). Now con- sider those sequences starting with a 1. The 1 must be contained in every F sequence. Hence, the totality with this condition satisfied is F(p - 1, q, n- 1). Consequently, F(p, q, n) = F(p, q - 1, n) + F(p - 1,q,n -1).

Since

p + q - 1 < M,

p, q - 1 <n <211,

p - 1, q < n - 1 <;I,

the conditions on the right satisfy the inductive hypothesis. To remove the con- dition q < n - 1, we consider the case of sequence ending in 1 or 2. This yields:

F(p, q, n) = F(p - 1, q, n) + F(p, q - 1, n - 1).

Here, the condition p < n - 1 must be imposed to satisfy the inductive hy- pothesis. Finally if p = q = n - 1, this would be the only combination of p l's and q 2's satisfying (1) and hence the theorem is true trivially.

J. Gilmore refers to the paper Sorting, trees and measures of order, by W. H. Burge, Information and Control, vol. 1 (1958), pp. 181-197, which is concerned with finding the sorting method which takes the least time to keep a large quantity of data in an orderly array for easy reference. Burge finds that the optimal strategy for sorting a set of data depends upon the amount of order (in the in- formation-theoretic sense) already existing in the data; and he shows how to find an optimal sorting strategy given a measure of this order.

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