MaxRPC algorithms based on bitwise operations

1. Jinsong Guo Jilin University, China MaxRPC algorithms based on bitwise operations

2. Background • Filtering techniques are used to remove some local inconsistencies in the search algorithms solving the instances of the Constraint Satisfaction Problem (CSP). • They can be used in a preprocessing step or during the search.

3. Background • A value (xj, b) is an AC-support of value (xi, a) if ((xi, a), (xj, b)) is allowed by cij. • An AC-support (xj, b) of (xi, a) is a PC-support of (xi, a) iff at least one PC-witness exists in the domain of each third variable xk which is constrained with both xi and xj. • A PC-witness of the value pair ((xi, a), (xj, b)) is a value which is consistent with both (xi, a) and (xj, b).

4. Background • maxRPC is a promising local consistency. • Stronger than AC, maxRPC removes not only the values that have no AC-supports, but also those that have no PC-supports. • A binary CN is max-restricted path consistent iff for all xi∈X, D(xi) is a non empty arc consistent domain and, for all a∈D(xi), for all xj∈X linked toxi, there exists b∈D(xj) s.t. ((xi, a), (xj, b)) ∈rel(cij) and for all xk ∈X linked to both xiand xj, there exists c∈D(xk) s.t. ((xi, a), (xk, c)) ∈rel(cik)∧((xk, c), (xj, b))∈rel(cjk).

5. Background • maxRPC1 fine-grained, relatively high space cost time complexity O(end3) space complexity O(end) • maxRPC2 coarse-grained, reducing the space cost (O(ed)) The time complexity is O(end3) too • They are not suited for use during search

6. Background • maxRPCrm make use of the residues, can be better suited for use during search • maxRPC2 and maxRPCrm both suffer from the overhead caused by the redundant constraint checks. • maxRPC3 and maxRPC3rm largely eliminate the redundant constraint checks

7. Background • When used during search, lmaxRPC which is the light version of maxRPC is used.

8. The inspiration of this paper • Local consistencies stronger than AC (such as maxRPC ) have two aspects: • advantage They can remove more inconsistent values than AC. (pruning efficiency) • disadvantage Enforcing or maintaining them cost more.(computational cost)

9. The inspiration of this paper • disadvantage > advantage Good • advantage < disadvantage Not preferable Thus, I thought: Why not find a way to overcome the disadvantage so that the advantage can be more effective?

10. Where does the disadvantage come from? • maxRPC removes not only the values that have no AC-supports, but also those that have no PC-supports. • The “Culprit” Verifying whether a value has the PC-supports.

11. overcome the disadvantage • The process of verifying whether a value (xi, a) has a PC-support in the domain of xj: • Step 1: Verifying whether (xi, a) has an AC-support in the domain of xj. • Step 2(If such an AC-support (xj, b) is found) : Verifying whether the value pair ((xi, a), (xj, b)) has one PC-witness in all the domains of each third variable xk linked to both xi and xj.

12. overcome the disadvantage • Speed up the process of Step 2. • I think that if the step 2 costs little, the time cost of enforcing or maintaining maxRPC may be close to AC.

13. searching for a PC-witness • When searching for a PC-witness for a pair of values ((xi, a), (xj, b)) in a third variable xk, all existing algorithms need to perform constraint checks to confirm whether a value (xk, c) satisfies ((xi, a), (xk, c))∈rel(cik) and ((xj, b), (xk, c))∈rel(cjk).

14. searching for a PC-witness • The differences at this process in the existing algorithms are from which value in D(xk)the search starts. • maxRPC2 and maxRPCrm start the search from scratch thus there are redundant constraint checks, maxRPC1 and maxRPC3 avoid these redundant constraint checks, and maxRPC3 requires only two constraint checks when LastACxi, a, xk or LastACxj, b, xk happens to be the PC-witness.

15. Our new method • In our method, bitwise operations are performed instead of constraint checks when searching for PC-witnesses. • Our method is based on the binary representation. • In the following introduction, we consider that the computer is equipped with a 32-bit processor.

16. the binary representation • The arrays of words are used to represent domains and constraints. • Each word is a sequence of 32 bits • An array of words can be regarded as a bit sequence.

17. the binary representation • Two 2-dimensional arrays bitdom and bitSup were introduced to respectively present domains and constraints. • length is used to denote the size of an array. • If the domain of xj has a size of d, the size of bitdom[xj] and bitSup[cij, xi, a] is d/32, i.e. both bitdom[xj] and bitSup[cij, xi, a] consist of d/32 words.

18. the binary representation • Each bit in bitdom[X] can be associated with the index of any value in the domain of X. When a bit is set to 1 (resp. 0), it means that the corresponding value is present in the domain (resp. absent from it).

19. the binary representation • bitSup[cij, xi, a] represents the binary representation of the supports of (xi, a) in cij. For each value in the domain of xj, there is a corresponding bit in bitSup[cij, xi, a] to mark whether this value is a support of (xi, a).

20. the binary representation • The new method tests 32 values simultaneously each time and aims at finding out whether there is at least one PC-witness instead of finding out which value is the PC-witness.

21. A new method for searching for PC-witness of ((xi,a),(xj,b)) in xk ZERO denotes a sequence of 32 bits all set to 0. AND is the bitwise operator that simultaneously performs a logical AND operation on 32 pairs of corresponding bits.

22. If the result (here it is called result.1) of “bitsup[Cik, xi, a][w] AND bitsup[Cjk, xj, b][w]” is not ZERO, there must be common supports for (xi, a) and (xj, b) in the initial domain of xk. • If the result of “ result.1 AND bitdom[xk][w]”is not equal to ZERO, there must be at least one PC-witness in the current domain of xk

23. A new method for searching for PC-witnesses • This method is very efficient, consider that xk has a domain with the size of 100, only 8 bitwise operations need to be performed in the worst case, and these 8 bitwise operations can test all the values in xk, in contrast, 8 constraint checks can test only 4 to 8 values in existing algorithms

24. Two new algorithms • Utilizing this new method, we proposed two coarse-grained algorithms called maxRPCbit and maxRPCbit+rm

25. maxRPCbit can be regarded as the variant of maxRPC2. • It transforms the step of searching for PC-witnesses through constraint checks into the new method exploiting bitwise operations.

26. maxRPCbit+rm can be regarded as the variant of maxRPC3rm. • It transforms the step of searching for PC-witnesses through constraint checks into the new method exploiting bitwise operations. • When a PC-support b of (xi, a)is found, it does not set LastACxi, a, xj to b as maxRPC3rmdoes. Instead, LastACxi, a, xj is set to the position of b in bitSup (b div 32) .

27. Experimental evaluation • We compared the new algorithms with the most efficient ones among existing algorithms. • Performances have been measured in terms of cpu time in seconds (cpu) and the number of constraint checks (#ccks). • Note that for the new algorithms, #ccks corresponds to the number of bitwise operations.

28. Average stand-alone performance

29. Average stand-alone performance

30. Average search performance

31. Some instances on which the search algorithms applying lmaxRPC outperforms MAC

32. Conclusions • Main job: overcoming the disadvantage of maxRPC(computational cost). • We have introduced a method based on bitwise operations to speed up the processes of searching for PC-witnesses. • Based on the new method, we presented two algorithms maxRPCbit and maxRPCbit+rm. • Significantly, our experiments show that maxRPC is a much more promising filtering technique than what we thought because lmaxRPCbit+rm outperforms MAC on many instances. • In the future, we will do further research to overcome the disadvantages of other local consistencies.

33. gjs.king@gmail.com By the way, I am a first-year graduate student in Jilin University of China. And this paper is the first one of mine. If you need a PHD student, who has a talent for CP and enjoys doing researches, could you please take me into consideration and contact me by the email? I won’t let you down~!

34. Thank you very much!I am sorry for my absence.Wish you have a good time in Perugia!