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1978, Communications of the ACM, vol. 21, pp. 847–857. [1]8.3 HeapsortWhile usually not quite as fast as Quicksort, Heapsort is one of our favoritesorting routines. It is a true “in-place” sort, requiring no auxiliary storage. It is anN log2 N process, not only on average, but also for the worst-case order of input data.In fact, its worst case is only 20 percent or so worse than its average running time.It is beyond our scope to give a complete exposition on the theory of Heapsort.We will mention the general principles, then let you refer to the references [1,2] , oranalyze the program yourself, if you want to understand the details.A set of N numbers ai , i = 1, .

. . , N , is said to form a “heap” if it satisfiesthe relationaj/2 ≥ ajfor 1 ≤ j/2 < j ≤ N(8.3.1)Here the division in j/2 means “integer divide,” i.e., is an exact integer or elseis rounded down to the closest integer. Definition (8.3.1) will make sense if youthink of the numbers ai as being arranged in a binary tree, with the top, “boss,”node being a1 , the two “underling” nodes being a2 and a3 , their four underlingnodes being a4 through a7 , etc. (See Figure 8.3.1.) In this form, a heap hasevery “supervisor” greater than or equal to its two “supervisees,” down throughthe levels of the hierarchy.If you have managed to rearrange your array into an order that forms a heap,then sorting it is very easy: You pull off the “top of the heap,” which will be thelargest element yet unsorted.

Then you “promote” to the top of the heap its largestunderling. Then you promote its largest underling, and so on. The process is likewhat happens (or is supposed to happen) in a large corporation when the chairmanof the board retires. You then repeat the whole process by retiring the new chairmanof the board. Evidently the whole thing is an N log2 N process, since each retiringchairman leads to log2 N promotions of underlings.Well, how do you arrange the array into a heap in the first place? The answeris again a “sift-up” process like corporate promotion.

Imagine that the corporationstarts out with N/2 employees on the production line, but with no supervisors. Nowa supervisor is hired to supervise two workers. If he is less capable than one ofhis workers, that one is promoted in his place, and he joins the production line.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.

To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).You could, in principle, rearrange any number of additional arrays along withbrr, but this becomes wasteful as the number of such arrays becomes large. Thepreferred technique is to make use of an index table, as described in §8.4..

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