336 Chapter 8.Sorting 22 You could,in principle,rearrange any number of additional arrays along with brr,but this becomes wasteful as the number of such arrays becomes large.The preferred technique is to make use of an index table,as described in 88.4. CITED REFERENCES AND FURTHER READING: Sedgewick,R.1978,Communications of the ACM,vol.21,pp.847-857.[1] NUMERICAL 8.3 Heapsort While usually not quite as fast as Quicksort,Heapsort is one of our favorite sorting routines.It is a true "in-place"sort,requiring no auxiliary storage.It is an N 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. 3②州 Press. 需 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 [1.21,or analyze 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 satisfies the relation a/2≥aj for1≤j/2<j≤N (8.3.1) 61 Here the division in j/2 means "integer divide,"ie.,is an exact integer or else is rounded down to the closest integer.Definition(8.3.1)will make sense if you think of the numbers a;as being arranged in a binary tree,with the top,"boss,"node being a1,the two "underling"nodes being a2 and a3,their four underling nodes being a4 、三 througha7,etc.(See Figure 8.3.1.)In this form,a heap has every"supervisor"greater Numerica 10521 than or equal to its two"supervisees,"down through the levels of the hierarchy. 431 If you have managed to rearrange your array into an order that forms a heap, E Recipes then sorting it is very easy:You pull off the "top of the heap,"which will be the largest element yet unsorted.Then you "promote"to the top of the heap its largest underling.Then you promote its largest underling,and so on.The process is like North what happens(or is supposed to happen)in a large corporation when the chairman of the board retires.You then repeat the whole process by retiring the new chairman of the board.Evidently the whole thing is an N log2 N process,since each retiring chairman leads to log2 N promotions of underlings. Well,how do you arrange the array into a heap in the first place?The answer is again a"sift-up"process like corporate promotion.Imagine that the corporation starts out with N/2 employees on the production line,but with no supervisors.Now a supervisor is hired to supervise two workers.If he is less capable than one of his workers,that one is promoted in his place,and he joins the production line. After supervisors are hired,then supervisors of supervisors are hired,and so on up336 Chapter 8. Sorting Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). } } } You could, in principle, rearrange any number of additional arrays along with brr, but this becomes wasteful as the number of such arrays becomes large. The preferred technique is to make use of an index table, as described in §8.4. CITED REFERENCES AND FURTHER READING: Sedgewick, R. 1978, Communications of the ACM, vol. 21, pp. 847–857. [1] 8.3 Heapsort While usually not quite as fast as Quicksort, Heapsort is one of our favorite sorting routines. It is a true “in-place” sort, requiring no auxiliary storage. It is an N 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], or analyze 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 satisfies the relation aj/2 ≥ aj for 1 ≤ j/2 < j ≤ N (8.3.1) Here the division in j/2 means “integer divide,” i.e., is an exact integer or else is rounded down to the closest integer. Definition (8.3.1) will make sense if you think 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 underling nodes being a4 through a7, etc. (See Figure 8.3.1.) In this form, a heap has every “supervisor” greater than or equal to its two “supervisees,” down through the 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 the largest element yet unsorted. Then you “promote” to the top of the heap its largest underling. Then you promote its largest underling, and so on. The process is like what happens (or is supposed to happen) in a large corporation when the chairman of the board retires. You then repeat the whole process by retiring the new chairman of the board. Evidently the whole thing is an N log 2 N process, since each retiring chairman leads to log2 N promotions of underlings. Well, how do you arrange the array into a heap in the first place? The answer is again a “sift-up” process like corporate promotion. Imagine that the corporation starts out with N/2 employees on the production line, but with no supervisors. Now a supervisor is hired to supervise two workers. If he is less capable than one of his workers, that one is promoted in his place, and he joins the production line. After supervisors are hired, then supervisors of supervisors are hired, and so on up