Comparison of Decision Diagrams for Multiple-Output Logic Functions

Transcription

1 Comparison of Decision Diagrams for Multiple-Output Logic Functions sutomu SASAO, Yukihiro IGUCHI, and Munehiro MASUURA Department of Computer Science and Electronics, Kyushu Institute of echnology Center for Microelectronic Systems, Kyushu Institute of echnology Department of Computer Science, Meiji University Abstract his paper shows four different methods to evaluate multiple-output logic functions using decision diagrams: SBDD, MBDD, BDD for characteristic functions (CF), and BDDs for ECFNs (Encoded Characteristic Function for Non-ero outputs) Methods to compute average evaluation time for each type of decision diagrams are presented By eperimental analysis using benchmark functions, the number of nodes and average evaluation time are compared Evaluation using BDDs for ECFN outperforms those using MBDDs, BDDs for CF, and SBDDs with respect to the sie of BDDs and computation time he sies of BDDs for ECFNs are smaller than ones of corresponding MB- DDs (Multi-erminal BDDs), BDDs for CFs (Characteristic Functions), and SBDDs (Shared BDDs) Introduction Various kinds of BDDs eist to represent multiple-output logic functions Among them, an MBDD (multi-terminal BDD), an SBDD (shared BDD), and a BDD representing the characteristic function (BDD for CF) are popular For a BDD for CF or an MBDD, the evaluation time is O n m, where n denotes the number of input variables, and m denotes the number of output variables For an SBDD, the evaluation time is O n m BDDs for CFs and MBDDs are suitable for high speed evaluation Unfortunately, the sies of these BDDs tend to be too large hus, we have to resort to SBDDs, which require longer evaluation time In [6], a new data structure, a BDD for ECFN (encoded characteristic function for non-ero outputs) is introduced In this paper, we show that by using BDDs for ECFNs, logic evaluation can be more than two times faster than by using SBDDs Also, the sie of the memory is smaller than by using SBDDs his method can be useful for logic simulation [, ] and embedded system [] Function Evaluation using BDDs Another method to represent a logic function is the branching program Fig shows a method to convert a BDD into a branching program For a given logic function, construct a binary decision diagram (BDD), as shown in Fig (a) hen, replace each non-terminal node by an if then else statement he result is a branching program, as shown in Fig (b) hen, by implementing this program by a computer, we can evaluate the logic function he time to evaluate a logic function for a given input is proportional to the number of non-terminal nodes that appear in the path from the root node to the terminal nodes -edge -edge v v v 5 f v v v (a) BDD v 5 : if( ) goto v ; else goto v ; v : if( ) goto v ; else goto v ; v : if( ) goto v ; else goto v ; v : if( ) goto v ; else goto v ; v : return(); v : return(); (b) Branching program generated from BDD Figure : BDD and branching program -edge -edge v v v 5 f v v v f (,,, )= v v f (,,,)= Figure : Function evaluation using BDD Eample Consider the BDD in Fig When is applied, f is evaluated, as shown in Fig Note that this involves a traverse across three edges his figure also shows that the minimum path length is two, while the maimum path length is four As shown in the above eample, the evaluation time depends on the input values o estimate the evaluation time of different DDs, we introduce a metric average path length, which measures the average evaluation time over all possible combinations We measure the average evaluation time by the average path length in the BDD We assume that each variable occurs as a with the same probability as a hat is, at any node, a -edge is as likely to be traversed as a -edge Definition he node traversing probability, denoted by P v i, is the probability of traversing the node v i when a BDD is traversed from the root node to a terminal node he edge traversing probability, denoted by P e i

2 %&'!! # $ v = 6 v v 5 f!"! # $ v v 6 v Figure : Average path length of BDD (P e i ) is the probability of traversing the edge (the edge) from the node v i Since, the probabilities that and occur are assumed to be equal and (, P e i ) P e i ) P v i *( Lemma he node traversing probability is equal to the sum of the edge traversing probabilities of the incoming edges heorem he average path length is equal to the sum of the node traversing probabilities of the non-terminal nodes Eample Let us calculate the average path length of the BDD in Fig P v 5, and we have P e 5 P e 5, ( P v, ( P e, P e - ( P v - P e 5 ) P e ( ( ( P e P e ( P v / ( P e P e - ( 6 hus, the average path length of the BDD is given by P v 5 ) P v ) P v ) P v ( ( ( ( 6 65 BDDs for Multiple-Output Functions In this part, we introduce MBDDs, SBDDs, and BDDs for CFs to represent multiple-output logic functions Evaluation using MBDDs Fig shows an eample of an MBDD representing a -variable -output logic function Since each terminal node stores all output values, the output values are obtained by just traversing the MBDD from the root node to a terminal node When the outputs are stored in -bit words, up to output values of a terminal node can be evaluated by a single memory access, and up to 6 output values can be evaluated within two memory accesses In general, the evaluation time is O n m( O n m Similar to the case of BDDs, the average evaluation time is equal to the average path length Since many outputs are evaluated by one traversal of the MBDD, the evaluation is fast Unfortunately, the number of terminal nodes can be up to m, and the number of nodes will be too large to be stored in a memory for many practical applications f = f = f f f f f = f = Figure : Average path length of the MBDD f = v v f = f = v f = v Figure : Eample of SBDD Evaluation using SBDDs An SBDD is obtained from the set of BDDs representing multiple-output functions, by sharing nodes among BDDs to reduce the number of nodes For an n-variable m-output function, m BDDs are shared o evaluate a multiple-output function for each output, traverse the edges from the root node to a constant node according to the values of the input variables hus, the evaluation time is O n m Fig is the SBDD corresponding to the MBDD in Fig o evaluate all the output values, we have to traverse all the BDDs for f, f, f, and f from the root node to the constant nodes, sequentially he average path length of an SBDD is the sum of the average path lengths of the individual BDDs in Fig, instead of the SBDD in Fig Eample he average path length of the SBDD shown in Fig is he BDD in Fig represents the multiple-output function by using auiliary variables and In this case, the average eval- f = v v f = f = v f = v :9 Figure : Average evaluation time of SBDD 9 69

3 v 6 v 5 (,,, )=(,,,) v f f False False f f f Figure : BDD for multiple-output function using auiliary variables able : Derivation of CF from the truth table of multipleoutput function (b) (a) f f f f CF f f CF f f uation time is 656=65 Evaluation using BDDs for CFs For fast evaluation of multiple-output functions, characteristic functions (CFs) are useful [,, 7] he CF for an n-variable m-output logic function is an n m -variable - output logic function, where CF = if and only if the combinations of the inputs and the outputs are valid Eample able (a) shows the truth table of the twooutput function f, f able (b) is the truth table for the corresponding CF For eample, since the combination f f eists in able (a), the value of CF in (b) the corresponding row is Since the combination f f ) does not eist in able (a), the value of CF in (b) the corresponding row is For an n-variable m-output function, the number of rows in the truth table for CF is n; m Among them, n rows have s and the remaining rows have s By using the BDD for CF, we can evaluate a multipleoutput logic function quickly Consider the multiple-output function from Figs and he BDD for CF for the multiple-output function is shown in Fig 5 Note that the non-terminal nodes having indices f, f, f, and f correspond to auiliary variables In a BDD for CF, the auiliary variable f i can appear only after all the input variables that depend on f i appear Also, either the -edge or the -edge f f rue f f f Figure 5: Evaluation of function using BDD for CF of each auiliary variables is connected to the constant node he net eample shows a method to evaluate a logic function by using a BDD for CF Eample (Function evaluation using BDD for CF) Using the BDD for CF in Fig 5, obtain the output values for the input ) he inde of the root node v 6 is Since the value of is, proceed to the -edge to node v 5, whose inde is Since the value of is, proceed to the -edge to reach the node v, which corresponds to auiliary variable f When we arrive at an auiliary variable, we have to proceed along either the -edge or the -edge If we proceed along the -edge and arrive at the terminal node, then the output value is non-ero, that is, the output is In this case, backtrack and proceed along the -edge On the other hand, if we proceed along the -edge and arrive at the terminal node, then the output value is non-, that is, the output is In this case, we have to backtrack and proceed along the -edge In this eample, we proceed along the -edge first and arrived at the terminal node So we have to backtrack to the -edge, because we obtained the value f If we proceed along the -edge first, we will not arrive at the terminal node, and we obtain the value f In this case, we can reduce the search time for the backtrack In this eample, when we arrive at an auiliary variable, we first proceed along the -edge Similarly, we search the remaining parts, and by traversing edges, we can determine that the output values are f f f f If the probabilities of taking values and are the same for all f i, then the evaluation time does not depend on the order of traversal As shown in Eample, the average evaluation time using a BDD for CF is obtained from the average path length of the BDD However, when we arrive at a node v i that corresponds to an auiliary variable, we have to modify the algorithm as follows: At the node v i that corresponds to an auiliary variable, either the -edge or the -edge is connected to the terminal node Let the rue

4 K 6 f f f f f f f 6 f 6 f f 6 6 Figure 6: Average path length in BDD for CF probability taking values and be the same for f i hen, the probability of traversing the edge that is connected to the constant is one half of the probability P v i Also, the other edge is always traversed hus, the sum of the passing probability for both edges is 5 P v i Eample Fig 6 shows the method to obtain the average evaluation time of the BDD for CF shown in Fig 5 It is calculated as ' 6 ' 6 ' 6 ' 6 6 < < ) In this way, BDDs for CFs evaluate a logic function in O n m time, which is faster than the corresponding SB- DDs However, when the value of n m is large, the number of nodes is ecessive, and we cannot construct the BDD for CF Function Evaluation using BDDs for ECFNs A new method to represent multiple-output functions, using a BDD for ECFN (encoded characteristic function for non-ero outputs) [6], is faster and requires smaller amount of memory than SBDDs his section shows the properties of the BDD for ECFN ECFN and Output Encoding Problem An ECFN represents the mapping: F : B n = B u > B, where log ma F CB CB ED iff f vf, where DHG v D is an integer represented by the binary vector D Eample he ECFN for the four-output function shown in Fig is F f f f f ECFNs can be used in FPGA design, logic emulation, embedded system, etc [5] A BDD for ECFN is considered as a generaliation of an SBDD Definition and Definition For an m-output function f i i EC m I mj, the ECFN is F b um uj il b um uj 6 E b f i where D able : Encoding Methods for Four-output Function Encoding Encoding Encoding f f f f f f f f f f f f Auiliary variables Input variables SBDD BDD for ECFN Auiliary variables and input variables Figure : SBDD ; and BDD for ECFN b uj b uj 6E b is a binary representation of the integer i, and u N? log ma,, EE, uj are auiliary variables In the above definition, the integer i is represented by a binary vector D using the natural encoding However, different encodings can simplify the representation Eample Consider the four-output function F f f f f, where f, f, f, and f he ECFN F generated by Encoding in able is F f f f f In this case, we have F he ECFN F generated by Encoding in able is F f f f f In this case, we have F his eample shows that encodings influence the sie of the representation Optimiation of BDDs for ECFNs he BDD shown in Fig can be considered as a special case of a BDD for ECFN From now on, such a BDD will be simply denoted by an SBDD ; As shown in Fig, in an SBDD ;, the auiliary variables appear above the input variables However, in a general BDD for ECFN, the auiliary variables and the input variables can be mied together By optimiing the ordering of the input and the auiliary variables, the BDD can be minimied Definition he number of nodes in the BDD (including both non-terminal and terminal nodes) is denoted by nodes(bdd) heorem nodes BDD for ECFN O nodes SBDD ;

5 Auiliary variables Figure : BDD for ECFN considering variable ordering Encoding Encoding Figure : Optimiation of the output encoding Eample Let the BDD in Fig be a BDD for ECFN When we optimie the ordering of the variables, the BDD for ECFN shown in Fig is obtained In this way, BDDs for ECFNs can be made smaller than corresponding SBDD ; s By optimiing both the output encoding and the variable ordering, we can obtain the BDD for ECFN having fewer nodes than the corresponding SBDD ; Eample Fig shows the BDDs for ECFNs that represent F and F in Eample Note that the numbers of nodes are 6 and 7, respectively hey use Encodings and in able his eample shows that output encodings influence the sie of the BDDs Fast Evaluation using BDDs for ECFNs By using the BDD for ECFN in Fig, we can save memory and evaluate functions faster than the corresponding SBDD ; For eample, when we apply the input P to the BDD in Fig, we can see that all the outputs f, f, f, and f are On the other hand, if we use the SBDD ; in Fig, we need to traverse the BDD four times his shows that a BDD for ECFN often evaluates several outputs at one traversal Eample 5 Consider the BDD for ECFN shown in Fig Note that not all the auiliary variables appear in a path from the root to a terminal node Consider applying the input to the BDD for ECFN shown in Fig By traversing five paths Q, we can evaluate eight outputs In fact, in the path, f is evaluated; in the path, f is evaluated; in the path, f and f are evaluated; in the path, f and f 6 are evaluated; in the path, f 5 and f 7 are evaluated From here, we will show a fast method to evaluate a BDD for ECFN During BDD traversal, when we encounter an auiliary variable, by searching both subtrees for -edge and -edge, we can evaluate all the outputs efficiently Path Path Path Path Path Figure : Method to traverse BDD In order to evaluate all the outputs for an input, we need an efficient algorithm to traverse the BDD he following algorithm produces the values of multiple-output logic functions in a compact form Note that the algorithm is more complicated than that of BDD for CF Algorithm (Evaluation of Multiple-Output Function using a BDD for ECFN) eval ER u S? log ma ; eval root u ; epand the parenthesis of the output list; evalp p lastinde ER if p, non-terminal node ER if p, node for an input variable UR if V p > indew, evalp p > edge lastinde ; else evalp p > edge lastinde ; else if p, node for an auiliary variable R currentinde X inde of the current node; printf lastindej currentindej ; printf ; evalp p > edge currentinde ; evalp p > edge currentinde ; printf ; else R /* terminal node */ printf currentinde ; printf ; printf (the value of current terminal node); printf ; return : he average evaluation time is equal to the average path length, which can be obtained similarly to the case of BDDs Change the values of to,,,6e, and, and obtain the sum of the lengths 5

6 = (a), (d), (b), 6 = = = (e) (c), = Figure 5: Average path length of BDD for ECFN of paths from the auiliary variables to the terminal nodes Eample 6 Figs 5(a) (e) illustrate a method to obtain the average evaluation time of the BDD for ECFN in Fig he average evaluation time is calculated as Eperimental Results For selected MCNC benchmark functions, we constructed MBDDs, BDDs for CFs, and SBDD ; s able 5 compares the average number of nodes and evaluation time of decision diagrams o construct BDDs for ECFNs, we used the encoding method in [5] In the table, Name denotes the function name, In denotes the number of inputs, Out denotes the number of outputs, Node denotes the number of nodes, and Eva denotes the average evaluation time In the columns BDD for ECFN, Min denotes the case where the auiliary variables and the input variable are mied to reduce the BDDs 6 Summary In this paper, we presented four different methods to represent multiple-output functions by using BDDs We compared the sies and average evaluation time of the BDDs We eperimentally showed that BDDs for ECFNs require fewer nodes and require relatively short time to evaluate logic functions A future research area includes efficient method to find good encodings for ECFNs Acknowledgments his research is partly supported by the Grant in Aid for Scientific Research of he Japan Society for the Promotion 9 able 5: Average number of nodes and evaluation time of decision diagrams Name In Out MBDD BDD for SBDDY BDD for CF ECFN Nod Eva Nod Eva Nod Eva Nod Eva ape ape duke e eep k mainpla mark mise opa pdc rckl seq shift spla t table ts vg dn dn dn parc Ratio Nod: Number of terminal and non-terminal nodes Min: When the auiliary variables can be any place Eva: Average evaluation time Ratio: Average of ratios when the value for SBDD is of Science (JSPS), and the akeda Foundation Prof Jon Butler improved English presentation References [] P Ashar and S Malik, Fast functional simulation using branching programs, ICCAD, pp -, Oct 995 [] F Balarin, M Chiodo, P Giusto, H Hsieh, A Jurecska, L Lavagno, A Sangiovanni-Vincentelli, E M Sentovich, and K Suuki, Synthesis of software programs for embed-ded control applications, IEEE rans CAD, Vol, No 6, pp -9, June 999 [] P C McGeer, K L McMillan, A Saldanha, A L Sangiovanni- Vincentelli, and P Scaglia, Fast discrete function evaluation using decision diagrams, ICCAD 95, pp -7, Nov 995 [] R Rudell, Dynamic variable ordering for ordered binary decision diagrams, Proceedings of the IEEE International Conference on Computer-Aided Design, pp -7, Santa Clara, CA, November 99 [5] Sasao, Compact SOP representations for multiple-output functions: An encoding method using multiple-valued logic, International Symposium on Multiple-Valued Logic, Warsaw, Poland,, pp 7- [6] Sasao, M Matsuura, Y Iguchi, and S Nagayama, Compact BDD Representations for Multiple-Output Functions, VLSI-SOC, Montpellier, France, Dec -5, [7] C Scholl, R Drechsler, and B Becker, Functional simulation using binary decision diagrams, ICCAD 97, pp -, Nov 997 [] S Yang, Logic Synthesis and Optimiation Benchmark User Guide Version, MCNC, Jan 99 6