Chapter 8. Sorting http://www.nr.com 8.0 Introduction or call This chapter almost doesn't belong in a book on numerical methods.However, some practical knowledge of techniques for sorting is an indispensable part of any 1-800-872- Cambridge good programmer's expertise.We would not want you to consider yourselfexpert in NUMERICAL RECIPES IN numerical techniques while remaining ignorant of so basic a subject. In conjunction with numerical work,sorting is frequently necessary when data server (either experimental or numerically generated)are being handled.One has tables compu or lists of numbers,representing one or more independent (or "control")variables. Press. C:THEA and one or more dependent(or "measured")variables.One may wish to arrange these data,in various circumstances,in order by one or another of these variables Alternatively,one may simply wish to identify the "median"value,or the "upper quartile"value of one of the lists of values.This task,closely related to sorting, is called selection. SCIENTIFIC Here,more specifically,are the tasks that this chapter will deal with: 6 .Sort,i.e.,rearrange,an array of numbers into numerical order. Rearrange an array into numerical order while performing the corre- sponding rearrangement of one or more additional arrays,so that the correspondence between elements in all arrays is maintained. .Given an array,prepare an index table for it,i.e.,a table of pointers telling r Numerical Further which number array element comes first in numerical order.which second. and so on. Recipes Given an array,prepare a rank table for it,ie.,a table telling what is the numerical rank of the first array element,the second array element, (outside ecipes and so on. Select the Mth largest element from an array. North Software. For the basic task of sorting N elements,the best algorithms require on the order of several times N log2 N operations.The algorithm inventor tries to reduce the constant in front of this estimate to as small a value as possible.Two of the America). visit website best algorithms are Ouicksort(88.2),invented by the inimitable C.A.R.Hoare,and machine Heapsort (88.3),invented by J.W.J.Williams. For large N(say 1000),Quicksort is faster,on most machines,by a factor of 1.5 or 2;it requires a bit of extra memory,however,and is a moderately complicated program.Heapsort is a true "sort in place,"and is somewhat more compact to program and therefore a bit easier to modify for special purposes.On balance,we recommend Quicksort because of its speed,but we implement both routines. 329
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 machinereadable 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). Chapter 8. Sorting 8.0 Introduction This chapter almost doesn’t belong in a book on numerical methods. However, some practical knowledge of techniques for sorting is an indispensable part of any good programmer’s expertise. We would not want you to consider yourself expert in numerical techniques while remaining ignorant of so basic a subject. In conjunction with numerical work, sorting is frequently necessary when data (either experimental or numerically generated) are being handled. One has tables or lists of numbers, representing one or more independent (or “control”) variables, and one or more dependent (or “measured”) variables. One may wish to arrange these data, in various circumstances, in order by one or another of these variables. Alternatively, one may simply wish to identify the “median” value, or the “upper quartile” value of one of the lists of values. This task, closely related to sorting, is called selection. Here, more specifically, are the tasks that this chapter will deal with: • Sort, i.e., rearrange, an array of numbers into numerical order. • Rearrange an array into numerical order while performing the corresponding rearrangement of one or more additional arrays, so that the correspondence between elements in all arrays is maintained. • Given an array, prepare an index table for it, i.e., a table of pointers telling which number array element comes first in numerical order, which second, and so on. • Given an array, prepare a rank table for it, i.e., a table telling what is the numerical rank of the first array element, the second array element, and so on. • Select the Mth largest element from an array. For the basic task of sorting N elements, the best algorithms require on the order of several times N log2 N operations. The algorithm inventor tries to reduce the constant in front of this estimate to as small a value as possible. Two of the best algorithms are Quicksort (§8.2), invented by the inimitable C.A.R. Hoare, and Heapsort (§8.3), invented by J.W.J. Williams. For large N (say > 1000), Quicksort is faster, on most machines, by a factor of 1.5 or 2; it requires a bit of extra memory, however, and is a moderately complicated program. Heapsort is a true “sort in place,” and is somewhat more compact to program and therefore a bit easier to modify for special purposes. On balance, we recommend Quicksort because of its speed, but we implement both routines. 329
330 Chapter 8.Sorting For small N one does better to use an algorithm whose operation count goes as a higher,i.e.,poorer,power of N,if the constant in front is small enough.For N 20,roughly,the method of straight insertion (88.1)is concise and fast enough. We include it with some trepidation:It is an N2 algorithm,whose potential for misuse(by using it for too large an N)is great.The resultant waste of computer time is so awesome,that we were tempted not to include any N2 routine at all.We will draw the line,however,at the inefficient N2 algorithm,beloved of elementary computer science texts,called bubble sort.If you know what bubble sort is,wipe it from your mind:if you don't know,make a point of never finding out! 三 For N 50,roughly,Shell's method ($8.1),only slightly more complicated to program than straight insertion,is competitive with the more complicated Quicksort on many machines.This method goes as N3/2 in the worst case,but is usually faster. See references [1.2]for further information on the subject of sorting,and for detailed references to the literature. 的 CITED REFERENCES AND FURTHER READING: Knuth,D.E.1973.Sorting and Searching.vol.3 of The Art of Computer Programming(Reading. MA:Addison-Wesley).[1] Sedgewick,R.1988,Algorithms,2nd ed.(Reading,MA:Addison-Wesley),Chapters 8-13.[2] 、当%a 令 Press. 9访 8.1 Straight Insertion and Shell's Method OF SCIENTIFIC Straight insertion is an N2 routine,and should be used only for small N. say 20. The technique is exactly the one used by experienced card players to sort their cards:Pick out the second card and put it in order with respect to the first:then pick out the third card and insert it into the sequence among the first two;and so on until 、兰学% COMPUTING (ISBN 188810920 the last card has been picked out and inserted. 10621 void piksrt(int n,float arr[]) Sorts an array arr[1..n]into ascending numerical order,by straight insertion.n is input;arr is replaced on output by its sorted rearrangement. uurggoglrion 43106 f Numerical Recipes int i,ji float a; (outside for (j=2;j0&&arr[i]a){ Look for the place to insert it. arr[i+i]sarr[i]; 1--； arr[i+1]=a; Insert it. What if you also want to rearrange an array brr at the same time as you sort arr?Simply move an element of brr whenever you move an element of arr:
330 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 machinereadable 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). For small N one does better to use an algorithm whose operation count goes as a higher, i.e., poorer, power of N, if the constant in front is small enough. For N 0 && arr[i] > a) { Look for the place to insert it. arr[i+1]=arr[i]; i--; } arr[i+1]=a; Insert it. } } What if you also want to rearrange an array brr at the same time as you sort arr? Simply move an element of brr whenever you move an element of arr: