Han, Ching-Chih

Lee, Chia-Hoang

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Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1988 Comments on Mohr and Henderson's Path Consistency Ching-Chih Han Chia-Hoang Lee Report Number: 88-733 Han, Ching-Chih and Lee, Chia-Hoang, "Comments on Mohr and Henderson's Path Consistency" (1988). Department of Computer Science Technical Reports. Paper 632. https://docs.lib.purdue.edu/cstech/632 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. COMMENTS ON MOHR AND HENDERSON'SPATH CONSISTENCY ALGORITHMChing-Chili HanChia-Hoang LeeCSD·1R·733January 1988Comments on Mohr and Henderson's Path Consistency AlgorithmChing-Chih HanandChia-Hoang LeeDepartment of Computer SciencePurdue UniversityWest Lafayette, m 47907ABSTRACTMohr and Henderson have presented new algorithms for arc andpath consistency in [1]. Though the underlying ideas of their algorithmsare correct, the path consistency algorithm PC-3 is in error. In this paperwe point out the errors in this algorithm and give a correct one. Thetime complexity and space complexity of the revised algorithm are alsoanalyzed.1. IntroductionIn a recently published paper by Mohr and Henderson (M-H), new algorithms forarc and path consistency were presented. We show that the algorithm for path con-sistency is in error and give a refined solution.The idea of the arc consistency algorithm introduced by Mackworth [2] is basedon the notion of support. Mohr and Henderson further made this support evident byusing some data slIUctures to record the relevant supporting information. They use acounter for each arc-label pair [(ij),b] to indicate the number of labels at node j thatsupport (are consistent with) the label b at node i. In addition, for each label c at nodej, the member (i,b) of set Sjc is the label b at node i that is supponed by label c at nodej. They also use a table, M, to keep track of which labels have been deleted from whichnodes, and a list, List, to control the propagation of constraints. This idea has also beenapplied to the path consistency algorithm PC-3 stralghtforwardly. However, the pathconsistency algorithm is not completely correct. In section 2, we will point out theerrors and give a counterexample. In section 3, we will give a correct path consistencyalgorithm PC-4. Note that a fonnal treatment of the concept of path consistency wasfirst provided by Montanari [4].This work is supported by Lhe Nauonal Sciencc FoundaUOll under the grant IRI-87020S3.- 2-2. A Counterexample to Algorithm PC-3Arc consistency algorilhms check the binary relation between each pair of nodesand delete any inconsistent labels from the admissible labeling set of each node. How-ever, in path consistency algorithms, we cannot delete labels from the admissible label-ing set of a node since we are considering the relations between two nodes instead ofthe labelings at a single node. To show that the algorithm PC-3 is not completelycorrect. let us examine the following example depicted in Figure 1. (Counter[(iJ).b.c]should be Counter[(iJ).k.b.c] on lines 26 and 27 of algorithm PC-3 in [I]) We wlll fol-low [2] and use matrix fonns to represent binary relations.N ={l.2.3}A ={b.c}E ={(1.2).(1.3).(2.3)}A, ={b.c}A 2 ={b.c}A 3 ={b.c}b cb [: ~]R 12 = cb cb [~ ~]R 13 = cb cb [~ ~]R2j = CTRij = RjiNote that matrix Rij is the relation between i and j whose rows correspond to the possi-ble labels for node i and columns to the possible labels for node j. (In algorithm PC-3Rij(b,c) is represented by R (i,h,i,c).) For the entries of Rij. we use 1 to represent trueand 0 to represent false. It is easy to see that (e, b, b) is the only solution of the aboveconstraint satisfaction problem. However, after the first iteration of the for loop fromline 2 to line 19 (see [I]). the data structures will be as follows:-3-M[l,b] = 1,M[2,b] = 1,All other M's are zero,A, = (cJ,A2 = (cJ,A, ~ (b, cJ,All S's are empty,List ~ ((1, b), (2, b)}.The two b's are removed from the sets A 1 and A 2 because there are no labels at node 3that can support the relation R(1, b, 2, b). But we know that (c, b, b) is the solution ofthe constraint satisfaction problem. If set A2 does not contain label b, then (e, b, b)cannot be a final solution of the constraint satisfaction problem. The error comes fromthe deletion of label b from set A2 at line 17. Even though there is no label at node 3that supports the binary relation R(1, b, 2, b), label b should not be deleted from set A2,since it may satisfy the constraints with other labels at node 1. We should, instead, justdelete the binary relation R(1, b, 2, bJ, i.e., set R(1, b, 2, b) equal to false.After algorithm PC-3 is finished, the final result will be:Al = ()A 2 ~ (cJA, = (cJThis is not a correct result since from it we can not find the solution (e, h, b) of theconstraint satisfaction problem. We will give a revised path consistency algorithm,PC~4. in the next section and analyze its time and space complexities.3. Revised Algorithm PC-4Figure 2 is the revised algorithm PC-4 for path consistency. In algorithm PC-4, Mis a table with index [i, b, j, c] (in fact M is a boolean matrix). Set Sibjc containsmembers of the form (k, d), where the binary relations Rik(b,d) and Rki(d,b) are sup-ported by the binary relation Rij(b,c). We also use counters with indices of the formrei, b, i,e), k]. Counter[(i, b, i, cJ, k] is the number of adolissible pairs (i, b)·(k, d) thatsupport the binary relation Rij(b,c), where d is any admissible label at node k. Notethat Counter[(i, b, i, c), k] is essentially always equal to Counter[(i, c, i, bJ, k]. If bothof them equal to 0, then there is no admissible label at node k that can support thebinary relation Rij(b,c). Hence, Rij(b,c) must be set to 0 (false).Step 11 M:=O; Sibj,:=Empty_set; List:=Empty; Counter:=O;2 for (i,J) e E do-4-3 for k = I, n do4 for b E Ai do5 for C E Aj such that Rjj(b, c)=l do6 begin7 Total:= 0;8 fordeAkdo9 if R",(b, d) = I and Rk/d, e)=1 Iben10 begin11 Total := Total + I;12 Append(Sibkd, (j, c»;13 Append(Sj'kd, (i, b»;14 end;IS if Total = 0 Iben16 begin17 M[i, b, j, e]:=I; MU, e, i, b]:=I;18 Ri/b, e):=O; Rji(e, b):=O;19 end else20 begin21 Counter[(i, b, j, c), k] :~ Total;22 Counter[(j, e, i, b), k] := Total;23 end;24 end;25 initialize List witb [(i,bJ,e)lM[i,bJ,e]=MU,e,i,b]~1 and i<;;J;Step 226 while List not Empty do27 begin28 choose (k, d, I, e) from List and remove it from List;29 for (j, c) e Skdl, do30 begin31 Counter[(k,dJ,e),n:~Counter[(k,dJ,e),n-l;32 Counter[(j,e,k,d),n:=Counter[(j,e,k,d),n-l;33 remove (j,e) from Skdl,;34 remove (k,d) from Sj,,,;35 if Counter[(k, d, j, c), n=o Ihen36 if M[k, d, j, e]=O Ihen37 begin38 M[k, d, j, e]:~I; MU, e, k, d):=I;39 Append(List, (k, d, j, c»);- 5-40 Rki<d, c) :~ 0; Rjk(e, d) :~ 0;41 end;42 end;43 for V, c) E SHd do44 begin45 Counter[(I,eJ,e),kj :=Counter[(I,eJ,e),k]-1;46 Counter[v,e,l,e),k] :=Counter[V,e,I,e),k]-1;47 remove v,e) from Skkd;48 remove (I,e) from Sj'kd;49 if Counter[(I, e, j, c), k]=O tben50 if M[I, e, j, e]=O tben51 begin52 M[I, e, j, e]:~1;Mfj, e, I, e]:=1;53 Append(List, (I, e, j, c));54 R'j(e, c) :~ 0; Rj/(e, e) := 0;55 end;56 end;57 endFigure 2. Algorithm PC-4The complexity analysis of the algorithm PC-4 is similar to that of the algorithmPC-3. The maximum number of times line 11 to line 13 will be executed is on theorder of n3 a3 since lEI is of order n2 and lAil = IAjl = IAkl 5: a (and k is from 1 to n inthe for loop). For step 2, there are two ways to analyze its time complexity. First,since there are at most O(n 3a 2 ) counters and each has a maximum value of a, line 31to 34 and line 45 to 48 can be executed at most order O(n 3a 3 ) times. (Remember thatthe index of Counter is of the form [(i, b. j, c), k]. There are O(n 3 ) different i's, j's,and k's and Oea2 ) different b's and e's.) Second, the while loop is executed at mostn2a 2 times since each edge can be put into List at most once for each different pair oflabelings of its two ends. The for loop is bounded by the size of SkdIe which is of orderna. So the total time for step 2 is O(n 3a 3). Therefore, the time complexity of thewhole algorithm is O(n3a 3).The space complexity of PC-4 is:Number of counters::;; O(n 3a 2 ),Sum of the size of the different sets Sibjc ::;; na x L !Ajl x !Ajl ::;; n3a 3(i,j)eNxN(Since each set Sibjc is of size na.)- 6-So the space complexity of the whole algorithm is O(n3a 3 ). Note that PC-4 has thesame time and space complexity as PC-3.4. ConclusionMohr and Henderson made an important observation which leads to the refinementof the arc and path consistency algorithms. However, they misuse labels at each nodefor the binary relations between a pair of nodes in the context of path consistency. Infact, one cannot delete the labels from the admissible labeling set of a node if there isany path inconsistency. Instead, one should remove the inconsistent relations by settingthe relations to zero. This is the main error in the algorithm PC-3, and we have fixed itin the algorithm PC-4. Furthermore. the time complexity and space complexity of PC-4are both O(n 3a3 ).5. AcknowledgementsThe authors would like to thank the anonymous referees for their constructiveremarks.REFERENCES1. R. Mohr and T.C. Henderson, Arc and Path Consistency Revisited. Artificial Intel-ligence 28 (1986) 225-233.2. Alan K. Mackworth, Consistency in Networks of Relations, Artificial Intelligence8 (1977) 99-118.3. Alan K. Mackworth and Eugene C. Freuder, The Complexity of Some PolynomialNetwork Consistency Algorithms for Constraint Satisfaction Problems, ArtificialIntelligence 25 (1985) 65-74.4. V. Montanari, Networks of Constraints: Fundamental Properties and Applicationsto Picture Processing, Inform. Sci. 7 (1974) 95-132.- 7 -Figure 1. A counterexample to algorithm PC-3.

Comments on Mohr and Henderson\u27s Path Consistency

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Comments on Mohr and Henderson\u27s Path Consistency

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