8.6 Determination of Equivalence Classes 345 k=jm)break; if (k !m&&heap[k]heap[k+1])k++; if (heap[j]<heap[k])break; swap=heap[k]; heap[k]=heap[j」； heap[j]=swap; j=k; 22.2 83g CITED REFERENCES AND FURTHER READING: granted for (including 19881992 Sedgewick,R.1988.Algorithms.2nd ed.(Reading.MA:Addison-Wesley).pp.126ff.[1] Knuth,D.E.1973.Sorting and Searching.vol.3 of The Art of Computer Programming(Reading. NUMERICAL MA:Addison-Wesley). RECIPES I 8.6 Determination of Equivalence Classes A number of techniques for sorting and searching relate to data structures whose details America Press. are beyond the scope of this book,for example,trees,linked lists,etc.These structures and their manipulations are the bread and butter of computer science,as distinct from numerical analysis,and there is no shortage of books on the subject. 9 In working with experimental data,we have found that one particular such manipulation, namely the determination of equivalence classes,arises sufficiently often to justify inclusion SCIENTIFIC here. The problem is this:There are N“elements”(or“data points'”or whatever),numbered 1,...,N.You are given pairwise information about whether elements are in the same equivalence class of"sameness,"by whatever criterion happens to be of interest.For example, you may have a list of facts like:"Element 3 and element 7 are in the same class:element 19 and element 4 are in the same class;element 7 and element 12 are in the same class,...." COMPUTING (ISBN 1992 Alternatively,you may have a procedure,given the numbers of two elements j and k,for deciding whether they are in the same class or different classes.(Recall that an equivalence relation can be anything satisfying the RST properties:reflexive,symmetric,transitive.This is compatible with any intuitive definition of"sameness.") Numerica 10621 The desired output is an assignment to each of the N elements of an equivalence class number,such that two elements are in the same class if and only if they are assigned the same class number. Recipes 43106 Efficient algorithms work like this:Let F()be the class or"family"number of element (outside j.Start off with each element in its own family,so that F(j)=j.The array F(j)can be interpreted as a tree structure,where F(j)denotes the parent of j.If we arrange for each family North Software. to be its own tree,disjoint from all the other "family trees,"then we can label each family (equivalence class)by its most senior great-great-...grandparent.The detailed topology of the tree doesn't matter at all,as long as we graft each related element onto it somewhere. Therefore,we process each elemental datum"j is equivalent to k"by (i)tracking j up to its highest ancestor,(ii)tracking k up to its highest ancestor,(iii)giving j to k as a new parent,or vice versa (it makes no difference).After processing all the relations,we go through all the elements j and reset their F(j)'s to their highest possible ancestors,which then label the equivalence classes. The following routine,based on Knuth[1],assumes that there are m elemental pieces of information,stored in two arrays of length m,lista,listb,the interpretation being that lista[j]and listb[j],j=1...m,are the numbers of two elements which (we are thus told)are related
8.6 Determination of Equivalence Classes 345 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). k=j m) break; if (k != m && heap[k] > heap[k+1]) k++; if (heap[j] <= heap[k]) break; swap=heap[k]; heap[k]=heap[j]; heap[j]=swap; j=k; } } } } CITED REFERENCES AND FURTHER READING: Sedgewick, R. 1988, Algorithms, 2nd ed. (Reading, MA: Addison-Wesley), pp. 126ff. [1] Knuth, D.E. 1973, Sorting and Searching, vol. 3 of The Art of Computer Programming (Reading, MA: Addison-Wesley). 8.6 Determination of Equivalence Classes A number of techniques for sorting and searching relate to data structures whose details are beyond the scope of this book, for example, trees, linked lists, etc. These structures and their manipulations are the bread and butter of computer science, as distinct from numerical analysis, and there is no shortage of books on the subject. In working with experimental data, we have found that one particular such manipulation, namely the determination of equivalence classes, arises sufficiently often to justify inclusion here. The problem is this: There are N “elements” (or “data points” or whatever), numbered 1,...,N. You are given pairwise information about whether elements are in the same equivalence class of “sameness,” by whatever criterion happens to be of interest. For example, you may have a list of facts like: “Element 3 and element 7 are in the same class; element 19 and element 4 are in the same class; element 7 and element 12 are in the same class, ... .” Alternatively, you may have a procedure, given the numbers of two elements j and k, for deciding whether they are in the same class or different classes. (Recall that an equivalence relation can be anything satisfying the RST properties: reflexive, symmetric, transitive. This is compatible with any intuitive definition of “sameness.”) The desired output is an assignment to each of the N elements of an equivalence class number, such that two elements are in the same class if and only if they are assigned the same class number. Efficient algorithms work like this: Let F(j) be the class or “family” number of element j. Start off with each element in its own family, so that F(j) = j. The array F(j) can be interpreted as a tree structure, where F(j) denotes the parent of j. If we arrange for each family to be its own tree, disjoint from all the other “family trees,” then we can label each family (equivalence class) by its most senior great-great-...grandparent. The detailed topology of the tree doesn’t matter at all, as long as we graft each related element onto it somewhere. Therefore, we process each elemental datum “j is equivalent to k” by (i) tracking j up to its highest ancestor, (ii) tracking k up to its highest ancestor, (iii) giving j to k as a new parent, or vice versa (it makes no difference). After processing all the relations, we go through all the elements j and reset their F(j)’s to their highest possible ancestors, which then label the equivalence classes. The following routine, based on Knuth [1], assumes that there are m elemental pieces of information, stored in two arrays of length m, lista,listb, the interpretation being that lista[j] and listb[j], j=1...m, are the numbers of two elements which (we are thus told) are related
346 Chapter 8.Sorting void eclass(int nf[],int n,int lista[],int listb[],int m) Given m equivalences between pairs of n individual elements in the form of the input arrays lista[1..m]and listb[1..m],this routine returns in nf [1..n]the number of the equiv- alence class of each of the n elements,integers between 1 and n (not all such integers used). int 1,k,ji for (k=1;k<=n;k++)nf [k]=k; Initialize each element its own class for(1=1:1<=m;1++)[ For each piece of input information... j=lista[l]; while (nf[j]!j)j-nf[j]; Track first element up to its ancestor k=listbl]: while (nf [k]!k)k=nf [k]; Track second element up to its ancestor. if (j !=k)nf[j]=k; If they are not already related,make them for (j=1;j<=nij++) Final sweep up to highest ancestors. while (nf[j]!nf [nf [j]])nf[j]=nf [nf [j]]; Alternatively,we may be able to construct a function equiv(j,k)that returns a nonzero (true)value if elements j and k are related,or a zero(false)value if they are not.Then we want to loop over all pairs of elements to get the complete picture.D.Eardley has devised a clever way of doing this while simultaneously sweeping the tree up to high ancestors in a manner that keeps it current and obviates most of the final sweep phase: void eclazz(int nf[],int n,int (*equiv)(int,int)) 9▣ Given a user-supplied boolean function equiv which tells whether a pair of elements,each in the range 1...n,are related,return in nf [1..n]equivalence class numbers for each element. 9 int kk,jj; nf[1]=1; IENTIFIC for (jj=2;jj<-n;jj++){ Loop over first element of all pairs. nf[ii]=ii; for (kk=1;kk<=(jj-1);kk++){ Loop over second element of all pairs. to dir nf [kk]=nf [nf [kk]]; Sweep it up this much. if ((*equiv)(jj,kk))nf [nf [nf [kk]]]=jj; Good exercise for the reader to figure out why this much ancestry is necessary! 1920 COMPUTING(ISBN for (jj=1;jj<=n;jj++)nf [jj]=nf[nf[jj]]; Only this much sweeping is needed finally. Numerical Recipes 021 43108 CITED REFERENCES AND FURTHER READING: (outside Knuth,D.E.1968,Fundamental Algorithms,vol.1 of The Art of Computer Programming(Reading. MA:Addison-Wesley).82.3.3.[1] North Software. Sedgewick,R.1988,Algorithms,2nd ed.(Reading,MA:Addison-Wesley),Chapter 30. Ame
346 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). void eclass(int nf[], int n, int lista[], int listb[], int m) Given m equivalences between pairs of n individual elements in the form of the input arrays lista[1..m] and listb[1..m], this routine returns in nf[1..n] the number of the equivalence class of each of the n elements, integers between 1 and n (not all such integers used). { int l,k,j; for (k=1;k<=n;k++) nf[k]=k; Initialize each element its own class. for (l=1;l<=m;l++) { For each piece of input information... j=lista[l]; while (nf[j] != j) j=nf[j]; Track first element up to its ancestor. k=listb[l]; while (nf[k] != k) k=nf[k]; Track second element up to its ancestor. if (j != k) nf[j]=k; If they are not already related, make them } so. for (j=1;j<=n;j++) Final sweep up to highest ancestors. while (nf[j] != nf[nf[j]]) nf[j]=nf[nf[j]]; } Alternatively, we may be able to construct a function equiv(j,k) that returns a nonzero (true) value if elements j and k are related, or a zero (false) value if they are not. Then we want to loop over all pairs of elements to get the complete picture. D. Eardley has devised a clever way of doing this while simultaneously sweeping the tree up to high ancestors in a manner that keeps it current and obviates most of the final sweep phase: void eclazz(int nf[], int n, int (*equiv)(int, int)) Given a user-supplied boolean function equiv which tells whether a pair of elements, each in the range 1...n, are related, return in nf[1..n] equivalence class numbers for each element. { int kk,jj; nf[1]=1; for (jj=2;jj<=n;jj++) { Loop over first element of all pairs. nf[jj]=jj; for (kk=1;kk<=(jj-1);kk++) { Loop over second element of all pairs. nf[kk]=nf[nf[kk]]; Sweep it up this much. if ((*equiv)(jj,kk)) nf[nf[nf[kk]]]=jj; Good exercise for the reader to figure out why this much ancestry is necessary! } } for (jj=1;jj<=n;jj++) nf[jj]=nf[nf[jj]]; Only this much sweeping is needed } finally. CITED REFERENCES AND FURTHER READING: Knuth, D.E. 1968, Fundamental Algorithms, vol. 1 of The Art of Computer Programming (Reading, MA: Addison-Wesley), §2.3.3. [1] Sedgewick, R. 1988, Algorithms, 2nd ed. (Reading, MA: Addison-Wesley), Chapter 30