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 *♦• 
 
 The Schorr-Waite Marking Algorithm Revisited 
 
 by 
 Nachum Dershowitz 
 
 November 19 79 
 
 IHE LIBRARY OF THE 
 
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 "RBANA-CHAMPAIGN 
 
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 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 
 
The Schorr-Waite Marking Algorithm Revisited* 
 
 by 
 
 Nachum Dershowitz 
 
 November, 1979 
 
 *This research was partially supported by the National Science Foundation 
 under grants MCS-77-22830 and MCS-79-04897. 
 
Digitized by the Internet Archive 
 
 in 2013 
 
 http://archive.org/details/schorrwaitemarki999ders 
 
Introduction 
 
 There has been a recent wave of interest (Duncan and Yelowitz 
 [1979], Gerhart [1979], Gries [1979], Topor [1979]) In proving the correct- 
 ness of Schorr and Waite's [1967] algorithm for marking all the nodes in a 
 directed graph that are accessible from a given node (see also Knuth [1968] , 
 Section 2.3.5). In this paper we show how, by developing that algorithm in a 
 systematic manner (along the lines of Knuth [1974]) from the standard 
 depth-first search paradigm (see, for example, Tarjan [1972]), a simple 
 proof of correctness follows naturally. 
 
 First Version 
 
 Given a finite directed (not necessarily strongly connected) graph 
 
 G, the goal is to set an "active" bit a(v) to true for those nodes v in G 
 
 that are accessible from some given node p. It is assumed that initially 
 
 a(v) is false for all veG. Each node v has a given outdegree d(v) with 
 
 pointers G (v) ,G (v) , . . . ,G, , N (v) to neighboring nodes, 
 i l d(v) 
 
 The following recursive program accomplishes that task by perform- 
 ing a depth-first search of the graph: 
 
 call Mark(p) 
 procedure Mark(v): 
 if ~a(v) 
 
 then a(v) :■ true 
 
 for c :■ 1 to d(v) 
 
 do call Mark(G (v) ) 
 c 
 
 od 
 fi . 
 
- 2 - 
 
 The program marks the nodes accessible from p by first marking p and then 
 recursively calling the marking program for each of the neighbors of p. 
 
 To prove that this recursive program is correct, four things must 
 be shown: 
 
 a) executing the program leaves the structure of 
 the graph G unchanged; 
 
 b) only nodes accessible from p are activated; 
 
 c) the program terminates; 
 
 d) upon termination, all nodes accessible from p are active. 
 That (a) G is unchanged is obvious from the fact that the program 
 
 contains no assignment to pointers in G. By induction on the computation it 
 
 can be seen that (b) Mark is only called for accessible nodes: The argument 
 
 p of the initial call is accessible. Furthermore, assuming that Mark(v) is 
 
 called for an accessible node v, then the calls Mark(G (v) ) are to the like- 
 
 c 
 
 wise accessible neighbors of v. 
 
 The termination property (c) is proved by structural induction 
 (Burstall [1969]) on the number of unactivated nodes in G: Since in execut- 
 ing Mark(v) one previously inactive node in G, viz. v, is activated before 
 calling Mark(G (v)), the inductive hypothesis asserts that each such call to 
 Mark for a neighbor of v will terminate. Since Mark(v) calls Mark(G (v) ) 
 only a finite number of times, Mark(v) must terminate as well. 
 
 Lastly, the following line of reasoning demonstrates that (d) all 
 accessible nodes are activated: Clearly, the program never deactivates a 
 node. Thus, after any execution of Mark(v), v must be active. Now, since 
 all nodes are initially inactive, if upon termination a(v) holds for any 
 node v, then it must be on account of the assignment a(v) !■ true , after 
 
- 3 - 
 
 which the for- loop was executed and consequently a(G (v) ) must also hold for 
 all the neighbors G (v) , 1 < c < d(v), of v. In particular, a(p) holds upon 
 termination of Mark(p); it follows (by induction on the length of an access 
 path) that a(v) holds for all nodes v accessible from p. 
 
 Intermediate Version 
 
 As is well-known, the underlying idea of the Schorr-Waite algo- 
 rithm is to stack the arguments of the recursive calls within the graph G 
 itself. The value of the loop counter c for a call Mark(v) is stored in a 
 field c(v) that can take on values from 1 to d(v) + 1. (These c(v)'s could 
 be placed on a stack, or stored within the nodes v.) Let u point to the head 
 of the stack of arguments; the continuation of the stack from a node v is 
 stored in the c(v)-th pointer from v. 
 
 We are led to the program: 
 
 (u,v) := (null .p) 
 call Mark 
 procedure Mark: 
 if ~a(v) 
 
 then a(v) := true 
 
 for c(v) ;» 1 to d(v) 
 
 do (v,u,G c ^ (v)) :- (G c ^j (v),v,u) 
 call Mark 
 
 (G c(u) (u),v,u) : ' (v,u,G c(u) (u)) 
 od 
 
 fi . 
 
 What were implicit assignments v :» p and v :■ G (v) in the calls of the 
 
- 4 - 
 
 first version are now explicit. The stack u is set initially to some unique 
 "null" pointer (actually the initial value of u does not affect the computa- 
 tion); before the recursive call, the node v is added to the top of the 
 
 stack u and the c-th pointer from v, G . . (v) , is made to point to what was 
 
 c (v) r 
 
 the top of the stack. 
 
 The proof of the correctness of this program is the same as that 
 of the previous version except that to prove that the structure is un- 
 changed, that hypothesis is combined with termination in the inductive argu- 
 ment: One may assume that the recursive call to Mark returns with the values 
 of u, v, and G as they were before the call. Also, the counter c(v) is 
 unaffected by the recursive call, since Mark can never be called again with 
 v inactive. Thus, one need only verify that the effect of the multiple as- 
 signment 
 
 (v,U,G c(v) (v)) := (G c (v) (v),V,u) 
 is undone by the subsequent assignment 
 
 < G C ( U ) (u),v,u) := (v,u,G c(u) (u)). 
 
 But that is easy to see, e.g. the original value of u was in fact stored in 
 
 what was G . . (v) and is now G , . (u) . 
 c(v) c(u) 
 
 Final Version 
 
 In this section we derive a somewhat more efficient marking algo- 
 rithm from the previous version by a series of program transformations. 
 Each transformation is "correctness-preserving" in that the transformed pro- 
 gram is guaranteed to be correct if the untransformed program was. (See, 
 for example, Gerhart [1975] for methods of demonstrating this property.) 
 
 As a first step, note that in the event that Mark is called for an 
 active node, that call is essentially vacuous and all the stacking before 
 
- 5 - 
 
 the call and unstacking after are superfluous. This suggests separating out 
 that case by replacing the body of the for- loop with 
 
 C : " G c(v) (v) 
 If ~a(t) 
 
 then (v,u,G c , . (v) ) := (t,v,u) 
 
 call Mark 
 
 (G c(u) (u),V,u) := (v » u » G c (u) (u)) 
 fi . 
 
 With this change, Mark can never be called for an active node. Therefore, 
 that test at the beginning of Mark may be eliminated. (Similarly, if the 
 node has no outgoing edges, then the call does no more than set the active 
 bit, and that case, too, could have been separated out.) 
 
 Finally, one may remove the recursion from the program and replace 
 it with two goto 's in the standard manner. To detect the return from the 
 outermost call, one need only test for a null stack. Thus, our final mark- 
 ing algorithm is: 
 
- 6 - 
 
 (u,v) := ( null .p) 
 Mark: 
 
 a(v) := true 
 
 for c(v) := 1 to d(v) 
 
 do t :-G c(v) (v) 
 
 If ~a(t) 
 
 then (v,u,G (v)) := (t,v,u) 
 goto Mark 
 cont'd: 
 
 (G c(u) (u) ' V ' u) := (v » U ' G c(u) (u)) 
 
 f_l 
 
 od 
 if u * null then goto cont'd _fi . 
 
 This "final" version can be further modified according to one's 
 particular needs and tastes. It can, for example, be adapted to the case 
 where all nodes have outdegree zero or two (as in LISP) by unwinding the 
 f or- loop. 
 
 Conclusion 
 
 We have shown how the sophisticated Schorr-Waite marking algorithm 
 may be developed from a straightforward depth-first marking algorithm by a 
 short sequence of relatively simple modifications. Other (more or less de- 
 tailed) derivations of this or related algorithms are pursued in Soule 
 [1977], Griffiths [1979], and Lee, Gerhart, and deRoever [1979]; some of 
 those references go on to develop more involved algorithms. 
 
- 7 - 
 
 We have also seen how the correctness of the last version follows 
 without difficulty from the (virtually obvious) correctness of the original 
 version. Developing the program in steps has made it easier, in our view, 
 to understand and validate the final result. 
 
 Acknowl edgment 
 
 I thank Ed Reingold for suggesting that a simpler proof of 
 correctness ought to be possible. 
 
- 8 - 
 
 References 
 
 1. R. M. Burstall [Feb. 1969], Proving properties of programs by structural 
 induction , Computing J., vol. 12, no. 1, pp. 41-48. 
 
 2. A. G. Duncan and L. Yelowitz [July 1979], Studies in abstract /concrete 
 mappings in proving algorithm correctness , Proc. Sixth Colloq. on Automata, 
 Languages and Programming, Graz, Austria, pp. 218-229. 
 
 3. S. L. Gerhart [Jan. 1975], Correctness-preserving program 
 transformations , Proc. Second Symposium on Principles of Programming 
 Languages, Palo Alto, CA, pp. 54-66. 
 
 4. S. L. Gerhart [Summer 1979], Marking algorithms , in Program Construction 
 International Summer School, F. L. Bauer and M. Broy, eds., Springer-Verlag, 
 pp. 472-492. 
 
 5. D. Gries [Mar. 1979], The Schorr-Waite graph marking algorithm , Acta In- 
 formatica, vol. 11, no. 3, pp. 223-232. 
 
 6. M. Griffiths [Summer 1979], Development of the Schorr-Waite algorithm , 
 in Program Construction International Summer School, F. L. Bauer and M. 
 Broy, eds., Springer-Verlag, pp. 464-471. 
 
 7. D. E. Knuth [1968], The Art of Computer Programming, vol. 1: Fundamen- 
 tal Algorithms, Addison-Wesley, Reading, MA. 
 
 8. D. E. Knuth [Dec. 1974], Structured programming with go to statements , 
 Computing Surveys, vol. 6, no. 4, pp. 261-301. 
 
 9. S. Lee, S. L. Gerhart, and W. P. deRoever [Jan. 1979], The evolution of 
 list-copying algorithms , Proc. Sixth ACM Symposium on Principles of Program- 
 ming Languages, San Antonio, TX, pp. 53-67. 
 
 10. H. Schorr and W. M. Waite [Aug. 1967], An efficient machine-independent 
 procedure for garbage collection in various list structures , Comm. ACM, vol. 
 
 10. no. 8, pp. 501-506. 
 
 11. S. Soule [Nov. 1977], A note on the nonrecursive traversal of binary 
 trees , Computer Journal, vol. 20, no. 4, pp. 350-352. 
 
 12. R. Tarjan [June 1972], Depth-first search and linear graph algorithms , 
 SIAM J. Computing, vol. 1, no. 2, pp. 146-160. 
 
 13. R. W. Topor [Mar. 1979], The correctness of the Schorr-Waite list 
 marking algorithm , Acta Informatica, vol. 11, no. 3, pp. 211-221. 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 UIUCDCS-R- 79-999 
 
 4. Tide and Subtitle 
 
 The Schorr-Waite Marking Algorithm Revisited 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 November 39 79 
 
 6. 
 
 7. Author(s) 
 
 Nachum Dershowitz 
 
 8. Performing Organization Rept. 
 No. 
 
 9. Performing Organization Name and Address 
 
 Dept. of Computer Science 
 University of Illinois 
 222 Digital Computer Lab 
 Urbana, IL 61801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract/Grant No. 
 
 NSF-MCS- 77-22 830 
 
 12. Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, D.C. 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 The Schorr-Waite graph marking algorithm is developed in a 
 stepwise fashion. A simple proof of correctness of the final version 
 follows naturally. 
 
 17. Key Words and Document Analysis. 17a. Descriptors 
 
 program correctness, marking algorithms, program transformation, Schorr-Waite 
 algorithm, stepwise refinement. 
 
 17b. Identifiers/Open-Ended Terms 
 
 17c. COSATI Field/Group 
 
 18. Availability Statement 
 
 FORM NTIS-35 ( 10-70) 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 13 
 
 22. Price 
 
 USCOMM-OC 40329-P71 
 
UNIVERSITY OF ILLIHOIS URIANA 
 
 3 0112 001342416