Cofactoring-Based Upper Bound Computation for Covering Problems

Transcription

1 TR-CSE-98-06, UNIVERSITY OF MASSACHUSETTS AMHERST Cofactoring-Based Upper Bound Computation for Covering Problems Congguang Yang Maciej Ciesielski May 998 TR-CSE Department of Electrical and Computer Engineering University of Massachusetts Amherst, MA 0003 Abstract This paper introduces an efficient technique to compute a tight upper bound for the unate and binate covering problems. It is known that a covering problem can be solved by finding a shortest path on a BDD representing a satisfiability formula. Our technique is based on finding an approximation to the shortest path by identifying a path on a BDD composed of as many negative edges as possible. This operation is performed by iterative cofactoring without actually building a BDD. The results show that our technique generates very tight upper bound, with more than half of the test cases resulting in the optimum solution in just a few iterations. I. INTRODUCTION Many combinatorial problems can be naturally transformed into equivalent covering problems. For example, vertex covering problem, well-known to be intractable [2], can be formulated as a unate covering problem. Two level logic minimization also relies on solving unate covering problem (UCP), while binate covering problem (BCP) is the key to state minimization of incompletely specified finite state machines. As a result, efficient solution to the covering problem continues to be an important research topic [3][4][5][6][7]. Although many techniques have been developed to solve the covering problems, they mainly fall into one of the two categories [9]: the branch-and-bound techniques, and the binary decision diagrams (BDD s) based graph traversal techniques []. Since the branch and bound method is presented very well in literature [0], here we give a brief overview of BDD-based graph traversal techniques. Interested reader is referred to an excellent survey on this topic presented in [9]. A covering problem can be formulated in one of the two equivalent forms, the constraint matrix form and the constraint equation form. While branch-and-bound methods typically work on matrix form, BDDbased traversal techniques work on the equation form. Given a Boolean formula Ü ½ Ü ¾ Ü Ò µ to be satisfied, and a cost associated with each variable, an input assignment is a satisfying assignment if µ ½. Typically such a formula is given as a product of clauses or, equivalently, as a Boolean function in a product of sums (POS) form. Finding a minimum cost assignment satisfying is the task of solving the covering problem. As formulated by Lin and Somenzi [], a covering problem can be solved by finding a shortest path on a BDD representing satisfiability function. A shortest path on a BDD is a path from root to leaf node ½ which has the least number of positive edges (each representing a variable in the positive phase). This technique is also applicable to a weighted covering problem by simply assigning weights to the graph edges. The shortest weighted path represents the minimum cost assignment which satisfies the covering constraints. Example: consider a Boolean formula in the POS form, derived from a covering matrix of [6].

2 TR-CSE-98-06, UNIVERSITY OF MASSACHUSETTS AMHERST 2 ¼ ½ µ ½ ¾ µ µ ½ µ ¼ ¾ µ ¾ µ () Its BDD, shown in Figure, has been generated by the CUDD package [2] using -autodyn option for variable ordering. Solid lines represent positive edges, dashed lines are negative edges, and the dotted lines are the complement edges. The BDD is traversed by breadth-first search from node ½ to the root. At each node, the shortest distance of that node to ½ is labeled, and either one or both edges of that node are marked if that edge lies on the shortest path to ½ from that node. Subsequently, a shortest path on a BDD can be readily obtained by traversing the labeled tree from the root to node ½. In Figure, there are two shortest paths, each of length 3, ¼ ¼ ¼ ½ ¾ and ¼ ¼ ½ ¾. Hence the minimum cost assignments satisfying are ¾ and ½ ¾. In this figure the marks are represented by the black bars. ¾ f 3 a0 3 3 a a a3 a3 a3 2 a2 a2 a2 a4 Fig.. BDD of Boolean Equation. The main advantage of this method is that it can be used to solve both the weighted and unweighted, unate and binate covering problems. Furthermore, a shortest path on a BDD can be found in linear time. The drawback of this method is the construction of the BDD itself, which prevents it from handling large problems. We found that in order to solve a covering problem, explicitly building a BDD is not needed if an upper bound of the solution is available. One can avoid building a portion of BDD by pruning it in a fashion similar to that used in branch and bound. The key to this method is to be able to find an upper bound efficiently. While some efficient heuristics were proposed to calculate the lower bound for branch and bound computation, to the best of our knowledge there are no efficient heuristics to compute an upper bound. Usually the upper bound is chosen as the best solution found so far in the process of searching for the optimum. Consider a binary decision tree for a covering problem with Ò ½ variables. Let us calculate the number Ä Ù Òµ which is the number of nodes left on that tree after all nodes with distance-to-root Ù have been pruned. For Ù ½, For Ù ¾, Ä ½ Òµ ½ ¾ Ò ½µ Ò ¾µ Ç Ò¾ µ (2) Ä ¾ Òµ ½ ½¾ ¾Ò Ò ¾ Ò ¾½µ Ç Ò µ (3) When Ù Ò, which is quite common in a well formulated covering problem, Ä Ù Òµ increases exponentially with Ù.

3 TR-CSE-98-06, UNIVERSITY OF MASSACHUSETTS AMHERST 3 Ä Ù Òµ Ç Ò Ù ½ µ (4) Therefore, by decreasing the upper bound Ù by, the number of the pruned nodes increases by a factor of Ç Òµ. This is the reason why the Limit Lower Bound Theorem of Coudert [6] is so efficient in solving the covering problems. Computing a tight upper bound for either BDD-based or branch-and-bound technique is the key to solving the covering problems efficiently. A. Contributions In this paper, we propose a new efficient technique to compute an upper bound for the unate and binate covering problems. Our technique is based on the following observation: consider the path from the root to node of a BDD which follows as many negative edges as possible. Typically the length of such a path is very close to the length of the shortest path on a BDD. Therefore, for a given variable order, we can obtain an upper bound by taking negative co-factor with respect to consecutive variables in the tree as long as the value of co-factor is not equal to zero. For example, in Figure, such a path is obtained by taking negative co-factors at ¼ and ½, and taking positive co-factors at ¾ and. Since the co-factors are computed in a top-down fashion, building a BDD is actually not needed. For all the tests that we conducted the heuristic based on this observation seems to be quite efficient. Later we provide a few additional heuristics which can further improve the results. In fact, for more than half of our test cases, the heuristic was able to find the optimum solution within just a few iterations. The proposed method applies equally to the unate and binate covering problems as long as the the weights of the negative edges can be assumed to be zero. Since a unate function can always be transformed into a positive unate function, without loss of generality we concentrate on a positive unate covering problem. Extension of this method to binate covering problem is straightforward, it requires only a minor modification of the algorithm. The paper is organized as follows. Section II gives some basic definitions. Section III shows the upper bound computation technique using iterative co-factoring. Section IV gives the detail of the algorithm. We conclude by listing a number of open problems and future research. Test Bench Size Opt. Random Ascend. (R C) Sol. Order Order coudert.t hach9.t ½¾ bbara.t ¾ dk52x.t ½ ex4inp.t ½ ¾ ¼ ex5inp.t ex6inp.t ¾ maincont.t ½¼ 7 9 opus.t ricks.t 5 6 TABLE I UPPER BOUND COMPUTATION. NUMBER GIVES THE OPTIMUM. II. PRELIMINARIES In this section, we give some basic definition needed to develop an efficient technique for the upper bound computation. Definition. For a given BDD, the weight of a path from the root to node ½ is defined as the sum of weights of all positive edges in that path. The shortest path on a BDD is the one with the least weight.

4 TR-CSE-98-06, UNIVERSITY OF MASSACHUSETTS AMHERST 4 Definition 2. The leftmost path on a BDD is defined as a path from the root to node which has the following property: at each node, a negative edge (cofactor) is taken as long as that edge is not connected to node 0. In Figure, ¼ ¼ ¼ ½ ¾ is an exmaple of the leftmost path. Definition 3. Let Ö Úµ denote the total number of occurances of variable Ú in the Boolean formula representing the covering (satisfiability) problem. For example, in Equation, Ö ½ µ. Test Bench Size Opt. Random Order Descending Order Ascending Order (R C) Sol. Min. Hit Iteration Min. Hit Iteration Min. Hit Iteration coudert.t 3 3 2/ 3 3/2 3 2/ hach9.t ½¾ 3 4 0/ 4 3/ 3 2/ bbara.t ¾ 7 8 6/2 7 3/2 7 4/ dk52x.t ½ 6 0 2/ 7 6/5 7 4/3 ex4inp.t ½ ¾ ¼ 5 6 3/2 9 5/2 5 4/3 ex5inp.t 4 5 5/3 4 5/4 4 4/3 ex6inp.t ¾ 4 6 2/ 5 5/3 4 2/ maincont.t ½¼ 7 2/ 9 4/2 8 4/2 opus.t 5 5 7/6 6 3/2 6 0/ ricks.t 5 7 4/3 7 5/4 6 4/ TABLE II UPPER BOUND COMPUTATION WITH DIFFERENT VARIABLE ORDERING. NUMBER GIVES THE OPTIMUM. III. COMPUTATION OF UPPER BOUND BASED ON ITERATIVE CO-FACTORING As mentioned in Section I, the length of the leftmost path on a BDD representing a Boolean satisfiability formula is very close to the optimum solution of the associated covering problem. Specifically, the set of positive variables on that path forms an assignment which satisfies the corresponding Boolean satisfiability function. Furthermore, the total weight of the positive edges on that path gives an upper bound for the covering problem. Intuitively this could be justified by observing that taking negative co-factor does not contribute to the total weight of the path, while taking a positive co-factor increases the path weight. In other words, the leftmost path is a good estimation of the shortest path, with its weight providing a good upper bound for the covering problem in question. Lemma. The upper bound obtained by the proposed technique includes all the essential variables in the satisfiability formula for a given covering problem. Proof: If a variable Ú is essential, it must be set to ½, otherwise the formula will not be satisfied. Obviously, negative edges of those nodes that are associated with essential variable Ú must be connected to node ¼. Therefore, positive co-factor must be taken at those nodes, and, as a result, Ú will appear in the positive form on the leftmost path to ½. ¾ We tested this upper bound computation technique on several examples using random variable ordering. The results are listed in Table I under the column Random Order. Test case coudert.t comes from Figure of [6], hach9.t comes from problem 4-9 of [0], the remaining ones come from [7]. The upper bound given by a random variable ordering is very close the optimum solution known from the literature. In the remained of this section, we provide additional heuristics to further improve the upper bound. A. Variable Ordering Heuristic The simplest variable ordering heuristic used to build a BDD is based on the descending order of variable cardinality (c.f. Definition 3). Under this ordering the variable with the largest cardinality is assigned to the

5 TR-CSE-98-06, UNIVERSITY OF MASSACHUSETTS AMHERST 5 root. For other variables, the larger the cardinality of a variable, the higher that variable is positioned in the tree. Usually, this variable ordering can generate a relatively small BDD. However, when dealing with the covering problem the variable order should be reversed. Under the reverse order the nodes in the lower part of a BDD (now with larger cardinality) will tend to have more connections to ¼ than the nodes in the upper part, which are more likely to connect to the internal nodes. Since we only count the variables at which a positive co-factor is taken, most of the positive variables of the shortest path come from the lower part of the BDD. Since variables with larger cardinality are more likely to be selected for the optimum covering, putting those variables in the lower part of the BDD increases their chance to be selected for a satisfying assignment. Therefore, the variable ordering based on ascending cardinality is more suitable for the upper bound computation. Lemma 2. The upper bound computation based on ascending cardinality ordering will never select dominated variables (columns). Proof : Consider a covering problem in matrix form where column (variable) Ú is dominated by column (variable) Ù. Hence, Ö Úµ Ö Ùµ. Furthermore, in the corresponding POS satisfiability formula variable Ù must appear in all the sum terms which include Ú. Under this ordering, at the time when Ú is evaluated, the co-factor with respect to Ù has not yet been computed. At this point we are free to take negative cofactor w.r.to Ú. Consequently, the dominated variable Ú will not be selected for the assignment. ¾ According to Lemma 2, the dominated columns in a covering matrix will be filtered out automatically by using ascending cardinality variable order. As expected, the ascending variable order produces better results in terms of the upper bound. This is shown in Table I in column Ascend Order. In four cases, the optimum results were obtained. B. Group Re-ordering Heuristic Although the variables with larger cardinality are more likely to be selected for the optimum assignment, not all of them will always be selected. To take this into account, we devised a new variable re-ordering heuristic. The idea behind this heuristic is to give the variables with lower cardinality a chance to be placed in the lower part of the BDD where they are more likely to be selected for an assignment. The proposed heuristic is based on iterative re-ordering of groups of variables. Initially the variables are ordered by ascending cardinality. After the first iteration, all the positive variables found in the cover (in the leftmost path) are moved to the top of the variable order, and the process for finding the leftmost path is repeated. The process is iterated until no improvement on the path weight (upper bound) is found. Usually, the total number of iterations is relatively small (less than 5). Convergence may not be guaranteed if a cyclic core occurs, in which case the best upper bound is chosen within the predefined iteration limit. The improvement of the upper bound computation by applying the group re-ordering heuristic is demonstrated in Table II. Three types of initial variable ordering were examined: random, descending, and ascending variable cardinality. For each ordering, the column Iteration lists two numbers: the first one gives the total number of iterations; the second is the iteration number for which the lowest upper bound was found first (For example, 5/2 means that the total number of iterations was 5, and there was no change in the upper bound value after the second iteration). The total number of iterations for most cases is less then, except for opus.t, for which the upper bound oscillates. This was the only case where the iteration limit was reached. The optimum solution was found in more than half of the test cases within a few iterations. IV. ALGORITHM AND IMPLEMENTATION The input to our algorithm is a function in product of sum (POS) form which can be derived directly from the covering matrix. Performing co-factoring on the POS function is nothing but a set operation. Consider a sum term (clause) Ë È Ñ ½ Ú, with variables Ú ordered according to index, global to all the clauses.

6 TR-CSE-98-06, UNIVERSITY OF MASSACHUSETTS AMHERST 6 When Ú ½, the first variable in the order, is evaluated, no other variables have been evaluated yet. Therefore, it is safe to set Ú ½ ¼ (remove Ú from the set), as this will not make Ë ¼. The same argument applies to the remaining variables in Ë except for the last variable, Ú Ñ, in that clause. Variable Ú Ñ must be set to, otherwise Ë would evaluate to 0. For a POS function, when a variable is set to ½, all sum terms which include that variable are automatically set to ½. Then those sum terms can be removed from the set. A. Data Structures A POS function is mapped onto a covering graph. This graph has two types of nodes, sum node and variable node. A sum node is associated with one sum term in the POS function, and a variable node is associated with a variable. This data structure is designed to handle fairly large covering problems. For very large problems (number of columns 20,000), implicit set enumeration technique [3] must be used. B. Algorithm upperbound(*pos) oldpovar = variable = ascendingsort(variable) 2 Ò ÛÈ ÓÎ Ö 3 ÁÒÙÑ Ö ½¼ 4 while( ÁÒÙÑ Ö) 5 foreach Ú (variable) 6 if(eval one(ú)) /* POS = */ 7 Ò ÛÈ ÓÎ Ö Ò ÛÈ ÓÎ Ö Ì Ú 8 break /* jump out this foreach loop */ 9 else 0 foreach s (Ú ÙÑ ) if( Ø Ø ÆÍÄÄ) 2 delete Ú 3 ÆÙÑÎ Ö /* if only one variable left,set it to ½ */ 5 if( ÆÙÑÎ Ö ¼) 6 Ò ÛÈ ÓÎ Ö Ò ÛÈ ÓÎ Ö Ì Ú 7 foreach ss (Ú ÙÑ ) 8 Ø Ø ½ 9 /* foreach Ú */ 20 if( Ò ÛÈ ÓÎ Ö ÓÐ È ÓÎ Ö ) 2 return oldpovar 22 else 23 oldpovar = newpovar 24 reordervar(newpovar) 25 Ò ÛÈ ÓÎ Ö 26 i /* while */ 28 Fig. 2. Algorithm for upper bound estimation. The algorithm is given in Figure 2. ÓÐ È ÓÎ Ö and Ò ÛÈ ÓÎ Ö are two sets which contain variables that satisfy the POS function. In most cases their cardinality is much smaller than Ò (the total number of variables). Function Ò Ò ËÓÖØ µ on line sorts variables by using cardinality as key. Function Ú Ð ÓÒ Úµ

7 TR-CSE-98-06, UNIVERSITY OF MASSACHUSETTS AMHERST 7 on line 6 returns TRUE if È ÇË Ú ½µ ½. Function Ö ÇÖ ÖÎ Ö µ on line 24 reorders variables by putting the variables in Ò ÛÈ ÓÎ Ö on the top of the variable list. The core of this algorithm is the foreach loop, lines 5 to 9, which constitutes a single iteration of the upper bound computation. In lines 6 to 8, if È ÇË Ú ½µ ½, the iteration is terminated. In lines 0 to 8, Ú is removed from all sum terms. If Ú is the last variable left in a sum term, set Ú ½, and set all sum terms which contain Ú to ½, and put Ú in set Ò ÛÈ ÓÎ Ö. C. Complexity Analysis Let Ö = sparsity of a covering matrix, Ñ = the number of sum terms, and Ò = the number of variables. Therefore, the total number of edges is ÖÑÒ, average number of variables in a sum term is ÖÒ, and average number of sum terms which contain a specific variable is ÖÑ. The cardinality of Ò ÛÈ ÓÎ Ö is Ô. At line 5, foreach is executed Ò times. At line 6, Ú Ð ÓÒ Úµ takes Ç Ñµ time to find out if È ÇË Ú ½µ ½. At line 0, foreach is executed ÖÑ times, and line 7 to 8 are only executed ÔÖÒ times. Therefore, the total run time of the code from line 5 to 9 is Ò Ñ ÖÑ ÔÖÒµ. Usually, Ö ½ and Ô Ò, the run time is Ç ÑÒµ. Since the while block on line 4 is executed only a few times, the overall complexity of this algorithm is still Ç ÑÒµ. The limit on the number of iterations is set to 0. V. CONCLUSION This paper introduces an iterative cofactoring-based technique to compute an upper bound for covering problems. The results show that our technique generates very tight upper bound. In more than half of the test cases, the optimum was found. For those which are not optimum, the computed upper bounds are only unit away from the optimum. Our technique will help both branch and bound and BDD-based technique to improve efficiency exponentially. Future research includes adopting implicit set operations, studying the characteristics of BDD under iterative group variable re-ordering, and developing an efficient branch and bound package based on the proposed upper bound computation. ACKNOWLEDGMENTS The authors would like to thank Luca P. Carloni in UC Berkeley for providing the test cases. REFERENCES [] Bill Lin and Fabio Somenzi. Minimization of Symbolic Relations. In Proc. of ICCAD-90, pages [2] M. Garey and D. Johnson, Computers and Intractability, Freeman, New York, 979. [3] R. K. Brayton, G. D. Hachtel, C. T. McMullen, A. L. Sangiovanni-Vincentelli, Logic Minimization Algorithms for VLSI Synthesis, Kluwer Academic Publishers, 984. [4] R. L. Rudell, A. L. Sabgiovanni-Vincentelli, Multiple Valued Minimization for PLA Optimization, in IEEE Trans. on CAD, CAD-5, pages , 987. [5] J. Rho, G. D. Hachtel, F. Somenzi, R. Jacoby, Exact and Heuristics Minimization of Incompletely Specified Finite State Machine, in Proc. of EDAC 90, 990. [6] O. Coudert, J. C. Madre, New Ideas for Solving Covering Problems, in Proc. of DAC 95, pages [7] E. I. Goldberg, L. P. Carloni, T. Villa, R. K. Brayton, A. L. Sangiovanni-Vincentelli, Negative Thinking by Incremental Problem Solving: Application to Unate Covering, in Proc. of ICCAD 97, pages [8] S. Devadas, A. R. Newton, Exact Algorithms for Output Encoding, State Minimization and Four-level Boolean Minimization. In Hawaii International Conference on System Science, pages , 990. [9] T. Villa, T. Kam, R. K. Brayton, A. L. Sangiovanni-Vincentelli, Explicit and Implicit Algorithms for Binate Covering Problems, in IEEE Trans. on CAD, Vol-6, pages , 997. [0] G. D. Hachtel, F. Somenzi, Logic Synthesis and Verification Algorithms, Kluwer Academic Publishers, 996. [] fabio/ [2] S. Minato, Zero-Suppressed BDDs for Set Manipulation in Combinatorial Problems, in Proc. DAC 93, page