7. Algorithms and Heuristics in VLSI Design
7.3 Heuristics for Optimizing OBDD-Size — Variable Reordering 147
Most of the reordering heuristics published so far were general purpose algo- rithms independent on the application where they were used. This assured them universality, but on the other side, a lot of useful information was ig- nored. This section is devoted to an approach that uses the meaning of the functions represented by OBDDs in a particular application for speeding up the computational process. The main idea is to focus minimization on that part of the OBDD that represents the functions used in the next steps of the computation. We call these functions key functions and the corresponding subOBDDs key OBDDs. Obviously, the set of key functions is dynamically changing during the computation. If the size of the rest of the OBDD remains manageable, we achieve a gain for two reasons: the minimization of a part of a OBDD can be performed faster than for the entire OBDD, and secondly, the particular OBDD operations are faster. The latter is caused by the fact that the operations are performed over the key functions that have due to the approach smaller OBDD representation than they would have had if any usual reordering strategy aimed to the minimization of the whole OBDD would have been used.
Although the definition of the key OBDDs is dependent on an application, it requires only a small extension of the interface of the application software to the OBDD package used. The main part of the reordering can be implemented in the OBDD package itself. In this sense, the proposed method is universal and can be used in diverse applications. In the following we will describe an application of sampling to symbolic model checking.
7.3.1 Sample Reordering Method
Random sampling is a technique successfully used in several hard discrete problems. The idea to use sampling for the variable order problem arises naturally from the character of the problem.
The difficulty of the dynamic reordering problem does not arise from the size of the search space, but from the fact that the quality of the found solution (i.e., the OBDD size) can only be determined by constructing the resulting OBDD.
The first application of sampling to variable reordering was presented by Meinel and Slobodov`a in [7.28]. The basic idea can be summarized in a few sentences: A part of the OBDD is chosen as a representative of the OBDD and the minimization problem is solved for this part. The new order found as a feasible (or even optimal) solution for the sample is extrapolated and applied to the entire OBDD. If the attempt is evaluated as successful, i.e., the reduction of the OBDD size achieved a given threshold value, the algorithm terminates. Otherwise further attempts with new samples are undertaken, until success, or the number of allowed attempts is exhausted.
Obviously, the choice of a sample has an influence on the variable order found. It can be done in random manner (like in many sampling strategies) or by use of some structural and semantic properties of the OBDD under consideration. In our approach, we use randomness, but we are targeting to key OBDDs, i.e., we chose random fractions of the key OBDDs. Randomness substitutes a missing information, assures a balance between the key OBDDs and the rest of the OBDD, and avoid repetition of the same sample choice.
There are several important implementation details that may play an im- portant role on the success of the heuristic, e.g., how to minimize the sample, how to extrapolate the compute order, or how to rebuild the whole OBDD with respect to the new order. In the following we describe an implementation of the Sample Reordering in the CUDD package [7.29].
LetInitialSizebe the OBDD size at the start of the reordering. A sample of the size
SamplePortion×InitialSize,
is chosen from the OBDDs whose roots are passed by the application/user. If no roots are given random sampling is chosen. The chosen sample is copied to a new OBDD and then reordered by means of Sifting. The new variable order is derived from the new variable order of the sample. The variables that do not occur in the sample are kept on their old positions. All other variables are moved according to their positions in the new variable order of the sample. Rebuilding of the entire OBDD with respect to the new order is done by subsequent movement of each variable to its new position. We monitor the size of the OBDD during this reconstruction. If there is a better order with respect to the corresponding OBDD size than the target order, we shuffle variables back to this order. This is also the case of an unsuccessful attempt when the new order is worse than the original one. The process of the rebuilding is interrupted if the size of the OBDD grows over a given threshold:
ChangeOrderBound×InitialSize,
This may happen even if the targeted order is better than the original but the peak size is too big.
The CUDD package has a user option for grouping of variables. Variables in a group should be always kept together. This is useful for some applications where the meaning of variables is known and can be used as a navigation in the search for a good order, e.g., the couple of present and next state variable in the coding of finite state machines. If such groups are defined, they are respected by the new order, too. The rebuilding procedure moves the variables of the same group together. Also the candidates for a better order that could be found during the rebuilding process are required to fulfill the group restrictions.
If the new size of the OBDD is less than
ExpectedReduction×InitialSize,
the reordering is considered to be successful. Otherwise, a second attempt with a new sample is allowed.
7.3.2 Speeding up Symbolic Model Checking with Sample Sifting Sample Sifting is a good candidate for speeding up model checking. But, a successful application of the sampling method to model checking is challeng- ing: Any branch-and-bound algorithm has a trade-off between computation time and quality of the result. In model checking the problem arises from the fact that a poor order results in larger OBDDs that require more computa- tion time. Also larger OBDDs lead to earlier and more time consuming calls to variable reordering. Thus, the trade-off multiplies and there are usually not enough calls to variable reordering to compensate these effects. Never- theless, a successful sampling strategy for symbolic model checking can be implemented, if the following points are taken into account:
Sample Size. The size of the sample is the most important parameter of sample sifting. Choosing a smaller sample will reduce the computational over- head for copying the sample. But even more important: The accelerating effect of sample sifting results from the fact that only a small OBDD is reordered, also resulting in smaller intermediate OBDD sizes during the re- ordering. The smaller the sample is, the faster the reordering performs. But, the sample cannot be chosen arbitrarily small, because in this case it does not represent the original OBDD’s properties sufficiently. The result of the reordering usually will be a poor ordering for the original OBDD. Thus, the size of the sample directly influences the quality of the computed order. To fulfill the quality requirements of model checking the sample has to be chosen larger than for combinatorial applications.
Method for Reordering the Sample. As stated above the time saved be sample sifting results from sifting a smaller OBDD. One may try to accelerate even this reordering, but this will usually result in variable orders of less quality. Instead, we suggest to reorder the sample even more by enlarging the search space, e.g., by allowing a larger growth of the OBDD during reordering.
This may compensate the quality losses resulting from reordering only a fraction of the OBDD.
Number of Attempts per Reordering. More than one sampling attempt per reordering might be a good idea for combinatorial application but not for model checking for the following reasons:
– Due to the small number of reorderings, several trials will compensate all the time savings, especially if larger samples are used.
– In some situations OBDD sizes grow despite of good variable orders. Here any reordering will fail.
While the above points are about reordering time, the following points deal with the choice of the sample that is crucial for the quality of the com- puted order.
Sample Without Semantical Information. If no external semantical in- formation is available one may at least use some structural information about the represented functions. One may use a random strategyRandom Sampling proposed by [7.28]: Starting from the top level of the OBDD nodes that are not representing projection functions (i.e., f = xi) are chosen randomly as roots of subOBDDs for the sample. This process is repeated level by level until the size requirements for the sample are fulfilled.
Another strategy is to chose the sample from the roots with the largest subOBDDs. Unfortunately, this strategy does not work well. Obviously, op- timizing the order of only a few OBDDs does not meet the requirements of all represented functions.
Sample with Semantical Information. One should make use of the se- mantical information about represented functions provided by the model checker. In [7.28] it is proposed to userecently-used-roots, i.e., roots involved in operations in the last steps of the computation.
In more detail: The roots resulted from the Boolean operations are pushed into a stack. Any garbage collection of the unreferenced nodes is completed by cleaning the stack. The size of the stack is bounded. Its capacity can be set according to the considered application and examples. The push operation into a full stack discards the bottom item. When the sample reordering is invoked, the sample is preferably built from the roots in the stack. If the OBDDs whose roots are in the stack do not suffice to cover the requirements on the size of the sample, we choose additional roots randomly.
Again, this strategy is not suitable for model checking, since the huge number of operations will result in a random choice of roots. In [7.19] it was shown how to utilize thekey functionsof FSM traversal like the transi- tion relation or the reachable state set to get a good sample for reordering.
Here, we userecently-used-important-roots, i.e., roots involved in elementary model checking operations like Exist-Abstract, Universal-Abstract and And- Abstract (see [7.17]), since state sets play a minor role in model checking.
If we cannot fulfill the size requirements for the sample by using important roots we fall back to Random Sampling. Using this strategy we obtain the best results for sampling.
Methods for Copying. In [7.28, 7.19] copying a fraction of an OBDD is done in the following way (postorder): The OBDD is traversed in DFS order and the nodes are copied to the sample whenever a node is backtracked. This is done until the required size of the sample is reached. This method copies at first the lower part of the OBDD. The resulting sample is a subfunction of the original OBDD. If only a small sample is chosen it will leave some variables of the upper part of the OBDD (see Figure 7.4a).
To avoid this, we use the following method (preorder): The OBDD is also traversed in DFS order. But, the nodes are copied to the sample when the node is visited the first time. This results in samples that include usually all variables and the outline of the sample is related to the outline of the original
a b
Fig. 7.4. a) Sample using postorder method b) Sample using preorder method
OBDD, i.e., from a level with many nodes a larger number of nodes is chosen for the sample. Applying this method results in unvisited edges that are set to the 1-sink. Thus, the resulting sample is modified but closely related to the original function (see Figure 7.4b). Our experience has shown that the preorder method works more stable and produces better results than the postorder method.
7.3.3 Experiments
We implemented our sifting strategies in the CUDD Package [7.29] (version 2.3.0). All experiments were performed on Intel PentiumPro 200MHz Linux Workstations with 250MByte datasize and CPU-time limited to 4 hours.
For all computations we used the common technique of grouping present- and next-state variables, i.e., a present-/next-state pair is always kept in adjacent levels. This on the one hand accelerates reordering and on the other usually results in better orders. We compare our results to the standard sifting method.
For our experiments we used the publicly available SMV-traces of Yang [7.34, 7.35]. SMV is the description language for the SMV-model checker [7.17]. Traces are recorded calls of OBDD operations done by the SMV model checker during the computation of a certain model. With the use of Traces, one is not restricted to use the underlying OBDD-package of SMV instead one can use any OBDD package and/or own algorithms. The underlying SMV-models come from different sources and represent a range from communication protocols to industrial controllers. Traces have become the reference benchmark set for reordering during model checking. We used those traces, that require less than 250MB of memory and less than 4 hours CPU-time.
The choice of Traces as benchmarks enables us to show that our strategies are applicable to any OBDD based model checking tool and are not restricted to a special model checker.
Figure 7.1 gives an overview of computation time, reordering effort and peaknodes of the models computed with standard sifting as reordering strat-
egy. During reordering grouping of present- and next-state variables was en- abled. The maximum allowed growth of the OBDD-size while sifting one variable was set to 20%.
The figure shows some evident differences of model checking in comparison to the OBDD application of combinatorial verification that mostly consists of symbolic simulation.. The number of variables (244 avg.) is comparable or even smaller than in combinatorial verification. The computation time is quite high (2044s avg.). The fraction of computation time, that is spent on reordering is extremely large (61% avg. of each reordering fraction), but only a few reorderings occur (4.7 avg.), while in combinatorial verification usu- ally many reorderings occur. The average size reduction over all reorderings (avg. Size Reduction) is not very high. This results from the fact, that some reordering attempts do not result in smaller OBDDs at all (size reduction<
5%). E. g. four reorderings during the computation offurnace17do no lead to smaller OBDDs, but one reordering drastically reduces the OBDD size (85%). Finally, the models are quite large (2.8 Mio. peaknodes avg.). Thus, most of them will not finish computation without reordering.
Results. Due to the random choice when copying a sample, for all experi- ments 10 single runs were performed.
For experiments we used the method Important Roots (IR) with sample size of 30% and 40%. For experimental results see Figure 7.2 and Figure 7.3.
All samples are chosen by using the preorder method. We were able to de- crease the average computation time up to 35% and the overall computation time up to 34%. The maximum improvement is 70%.
Since we obtained our results with a very loose coupling of the model checker to the OBDD-package, a tighter coupling to the model checker e.g., having exact knowledge about the represented functions would lead to even better results.