Compatible Class Encoding in Roth-Karp Decomposition for Two-Output LUT Architecture

Transcription

1 Compatible Class Encoding in Roth-Karp Decomposition for Two-Output LUT Architecture Juinn-Dar Huang, Jing-Yang Jou and Wen-Zen Shen Department of Electronics Engineering, National Chiao Tung Uniersity, Hsinchu, Taiwan, R.O.C. Abstract. How to select the lambda set?. How to encode the compatible classes? Roth-Karp decomposition one of the most popular techniques for LUT-based FPGA technology mapping because it can decompose a node into a set of nodes with fewer numbers of fanins. In th paper, we show how to formulate the compatible class encoding problem in Roth-Karp decomposition as a symbolic-output encoding problem in order to exploit the feature of the two-output LUT architecture. Based on th formulation, we also deelop an encoding algorithm to minimize the number of LUT's required to implement the logic circuit. Experimental results show that our encoding algorithm can produce proming results in the logic synthes enironment for the two-output LUT architecture.. Introduction Field Programmable Gate Arrays (FPGA's) are modern logic deices which can be programmed by the users to implement their own logic circuits. Because of the short turnaround time compared with that of the standard ASIC process, they become increasingly popular in rapid system prototyping recently. Many FPGA architectures hae been proposed and the Look-Up Table(LUT)-based architecture the most popular one. It consts of many configurable LUT's and each LUT can implement any k-input function. Besides, there another similar LUT-based FPGA architecture which implements not only single-output functions but also two-output functions. A LUT of th architecture can implement either one k-input function or two (k-)-input functions with totally k inputs. For example, in Xilinx XC000 architecture[], k equal to. Many algorithms deeloped for LUT-based FPGA technology mapping hae been proposed in preious studies[- ]. Most of these algorithms first decompose the gien Boolean network to be k-feasible. A Boolean network said to be k-feasible if all nodes in the network are k-feasible, and a node said to be k-feasible if the number of its fanins no more than k. Hence, the corresponding circuit can be directly realized by an one-to-one mapping between nodes and LUT's. If there are some nodes that are not k-feasible, they then need to be decomposed to be a set of k-feasible nodes. Many proposed decomposition techniques, such as AND-OR decomposition, cofactoring, djoint decomposition, if-then-else DAG, communication complexity reduction[], and Roth-Karp decomposition[], are widely used here. In th paper, we The algorithm proposed in [] proides a heurtic to choose a good lambda set. Another algorithm proposed in [0] formulates Problem as a symbolic-input encoding problem. Howeer, both of these two algorithms only consider the singleoutput LUT architecture. In th paper, we propose a new formulation for Problem and deelop a new compatible class encoding algorithm which can fully exploit the feature of the two-output LUT architecture. Th paper organized as follows. Section describes the compatible class encoding problem in Roth-Karp decomposition. In Section, our new encoding algorithm which addresses the two-output LUT architecture gien in detail. Section shows experimental results and the concluding remarks are gien in Section.. Compatible Class Encoding Definition.: Let X and Y be two sets of binary ariables, X Y = ; B x and B y are Cartesian products spanned by X and Y, respectiely. Then, gien a completely specified function F : B x B y B, we say that x B x are compatible with respect to F, denoted as x, if y B y, (x, y) and (x, y) B x B y such that F(x, y) = F(x, y). + Each element B x called a l minterm. All mutually compatible λ minterms can be grouped together to form a compatible class, and all compatible classes are pairwely djoint. Theorem.: Gien two functions α : B x W and G : W B y B, such that (x, y) B x B y, F(x, y) = G( α (x), y) () holds if and only if x B x, α (x ) = α (x ) x () + α a function with binary inputs and a symbolic output. X called the bound (l) set. Y called the free (m) set. Property.: The number of the admsible alues in W, W, must be no less than the number of compatible classes in X. + From Property., minimum W equal to the number of compatible classes in X and can be obtained by redefining Eq. () as: only focus on Roth-Karp decomposition. Gien the LUT-based FPGA as the target architecture, two interesting problems in Roth-Karp decomposition should be noticed: x B x, α (x ) = α (x ) x (')

2 In order to implement α by binary logic, the symbolicoutput encoding for W has to be performed. At least t = The admsible alues of W are defined to be the id's of the compatible classes. Thus, α maps each λ minterm to the id of log w binary-output functions, α,..., and α t, are required to encode α. Hence, Eq. () can be rewritten as: F(x, y) = G'(α (x) (x),..., α t (x), y) (') It clear that the djoint Roth-Karp decomposition can be used to reduce the number of fanins of a function under the condition of t < X. From the aboe dcussion, we find that Roth-Karp decomposition only requires that λ minterms belonging to the same compatible class are encoded with the same code, i.e., if x, then α (x ) = α (x ). In other words, to hae a correct Roth-Karp decomposition, we only need to assign a unique code to each compatible class and do not hae to care what the code. Howeer, different encoding combinations for the compatible classes will result in different α,..., α t, and G'. There a compatible class encoding strategy being proposed in [0]. It models the compatible class encoding problem as the classical symbolic-input encoding problem. Many exting techniques are then used to encode the compatible classes for minimizing the literal counts of G'. Because it assumes that the better decomposition quality can be obtained if the resulting G' simple, i.e., has smaller number of literals. Howeer, th strategy only concentrates on singleoutput functions. Furthermore, it did not properly take adantage of the feature of either the single-output or the twooutput LUT architecture. To exploit the property of the two-output LUT architecture, we formulate the compatible class encoding problem as: To find a set of compatible class encoding patterns to encode as many α functions to be independent of at least one of their the compatible class it belongs. In th example, ( log ) binary-output functions α, and α, are needed to implement α. Suppose these symbolic alues are randomly encoded without any strategy; for instance, each symbolic alue encoded with its binary bit pattern. Then, the encoding of α and the resulting Boolean functions are shown below: class id α α α xx α = xxxx xx xx α = xxx + xxx + xxx α = xx + xxx + xxx + xxx + xxx + xxx Obiously, three LUT's are required to implement these α functions since all of them depend on all four input ariables. Example : As in Example, a better encoding of α used here. The encoding of α and the resulting Boolean functions are shown below: input ariables as possible. class id α α α Thus, two α functions which are independent of at least one of their input ariables can be merged into a two-output k-lut In fact, our formulation of th encoding problem a kind of 0 0 xx the symbolic-output encoding because we encode the symbolic output ariable W into binary-output encoding functions α, α,..., and α t.. Our Compatible Class Encoding Algorithm For the easy illustration, the LUT used in th section either a -LUT or a two-output -LUT. Example : Gien a compatible class encoding function α with a set of inputs X as the λ set and a symbolic output ariable W: α : B x W, where X= { x,x,x,x } and W ={0,,,,} xx W α = xx + xx xx xx α = xxx + xxx α = xxxx We find that α independent of x and x, and α independent of x, respectiely. Therefore, α and α can be merged into a two-output LUT. Two instead of three LUT's are thus required to implement these α functions. Now we gie some definitions and derie some properties from them. Based on these definitions and properties, our compatible class encoding algorithm then constructed.

3 Definition.: {(0:)}. After dcarding two useless ISx's ( and W) and An independent set with respect to an input ariable x, exploiting the equialence relation among dichotomies, the size denoted as ISx, defined as a set of class id's such that there of Set_Dx gien by Property.: exts a binary-input/output function α being independent of x Property.: where Set _ MIS Set_ ISx x ON the ON - set of α, α = { m m a λ minterm where α( m) ISx} Set_ Dx =.+ = + Example : Property.: In Example, {0, } an ISx. Because we can find a A dichotomy Dx = (l:r) can be used to make the encoding function α = xx + xx as illustrated in Example satfies function α i independent of x if the i-th bit of the code of the Definition.. compatible class c equal to: { Definition.: if the id of, 0 if the id of c c r l. An ISx a minimum independent set with respect to the + input ariable x, denoted as MISx, if and only if Example 6: T, T ISx and T T no lo nger an ISx. + In Example, there exts a dichotomy, D = (0:) Set_Dx. If α encoded by D: Example : In Example, {0, } and {,, } are two MISx 's by class id α α α Definition Property.: Gien two λ minterms m and m' where m' the same with - - xx m except that it complemented at the position of ariable x Let α (m) = k and α (m') = k', then k and k' must belong to the same MISx by Definition.. + α = xx + xx Property.: It clear that α independent of x under th encoding Two MISx's are either identical or djoint, and the union pattern. set of all MISx's W. + Because we do not care which ariable x that α i From Property. and., we can easily deelop an independent of, dichotomies from different Set_Dx's can be algorithm to find all MIS's for a gien ariable x. merged into a set Set_D. Then, we formulate the compatible Example : class encoding problem to satfy the following three encoding The set, Set_MISx, containing all MIS's with respect to constraints: each ariable x of Example shown below, respectiely.. Each compatible class must be encoded with a dtinct code. Set_ MIS = {{0,},{,,}} Set x _ MIS x = {{0,},{,,}}. Only minimum number(t) of encoding bits are allowed to Set_ MIS = {{0,},{,,}} Set x _ MIS be used. x = {{0,,,,}}. Use as many dichotomies for the encoding as possible. Property.: Constraint gien to satfy the definition of Roth-Karp Each combination of an arbitrary number of MISx's decomposition. Constraint gien to minimize the number represents an ISx. So the size of the set Set_ISx containing all of LUT's used to implement α functions. Constraint gien ISx's equal to Set _ MIS x. + to exploit the feature of the two-output LUT architecture. Thus, Set_ISx can be deried from Set_MISx by applying Property.. For example, Set_ISx in Example {, {0, }, {,, }, {0,,,, }}. Notice that and {0,,,, } (W) are dcarded since they are useless for encoding purpose described later. At th point, we introduce a terminology, dichotomy, which first used for symbolic-input encoding in []. The notion of dichotomy slightly modified here for conenience. Definition.: A dichotomy with respect to x, Dx, gien by an ordered pair, denoted as (l:r), where l an ISx and r W- ISx. + Definition.: Two dichotomies, D = (l :r ) and D = (l :r ), are equialent if (l = l and r = r ) or (l = r and r = l ). + From Definition. and., a set of dtinct, i.e., nonequialent, dichotomies with respect to x, Set_Dx, can be generated from ISx. For instance, Set_Dx in Example While merging Set_Dx's into Set_D, if two equialent dichotomies, D and D, are from two different Set_Dx and Set_Dx, respectiely, then α i encoded by either D or D independent of both x and x. In th case, the number of fanins of α i can be further reduced and the number of interconnection nets required for routing also reduced. Therefore, a dichotomy which independent of the most inputs should be chosen for encoding first. A procedure, Exhaustie_Search_Encoding, deeloped to encode the compatible classes under three constraints described aboe. It first tries to find the maximum number(t) of dichotomies for encoding to satfy Constraint and. If it fails, it reduces the number of encoding dichotomies by and tries again. Th procedure guarantees to get the optimum encoding solution, i.e., the maximum number of dichotomies can be found to encode α functions without iolating Constraint and.

4 Example 7:. Experimental Results By reexamining Example, we find that two dichotomies The algorithm described aboe has been implemented in (0:) and (0:) can be found by SIS enironment which deeloped by UC Berkeley[]. Exhaustie_Search_Encoding to encode α and α, respectiely. Besides, our algorithm integrated into a ersion of Roth-Karp α then encoded to satfy Constraint. Therefore, the decomposition[], which has the lambda set selection strategy, to enhance its performance. In order to inestigate the quality encoding result identical with that illustrated in Example. of our encoding algorithm, denoted as Algorithm, two The time efficiency of th procedure not that good. experiments are conducted oer a large set of MCNC and Suppose there are d dichotomies being used to encode t bits, ISCAS benchmark circuits to compare the results with those of then the worst case, in which no one can be used for encoding, another two ersions of Roth-Karp decomposition. Algorithm d d d takes C has neither the lambda set selection strategy nor the compatible t + Ct + L + C iterations when d > t. We find that the class encoding strategy and used in m-pga[]. Algorithm number of iterations dramatically increases as d increases. has only the lambda set selection strategy and implemented Thus, a more efficient algorithm should be deeloped for the in []. The target architecture the Xilinx XC000 FPGA practical usage. Consider an example that the numbers of the which can implement either one -input function or two -input compatible classes and dichotomies are 6 and, respectiely. t functions with totally inputs as described aboe. The initial then equal to in th case. The search space of finding the networks of one experiment are two-leel circuits and are optimum encoding solution for th example shown in Fig.. obtained by performing the SIS script: Any path from node S to the nodes of the third leel in the tree collapse represents a possible combination of three dichotomies. simplify -d -m nocomp Similarly, any path from node S to the nodes of the second The initial networks of another experiment are multi-leel leel in the tree represents a possible combination of two circuits and are obtained by performing the SIS standard multileel optimization script. After obtaining the initial networks, dichotomies. S the same mapping script: C iterations xl_k_decomp /*arious algorithms applied here */ xl_partition -tm C iterations xl_coer xl_merge -l C iterations used in both experiments. Th script first decomposes the Fig. : The search space of finding the maximum number of dichotomies for the encoding. network to be -feasible. Algorithm,, and are applied here. Thus, each node can be implemented by a -LUT. Then, it tries to merge pairs of nodes which can be implemented by two-output -LUT's as many as possible. Both experiments run on a SUN SPARC workstation and the results are shown in Table I and II, respectiely. In th example, iterations are required to find the optimum solution in the worst case. Suppose that the first dichotomy (0:), we find that it impossible to dtinguh class ~ no matter what the remaining dichotomies are because only two more bits are allowed to use. Thus, the subtree rooted at the first dichotomy can be pruned without affecting the search of the optimum solution. Th pruning algorithm illustrated in Fig.. Pruned =(0:) S Fig. : An example of the pruning strategy on the search space of the dichotomies. Hence, the pruning algorithm can be formally described as: At any node of the search space of the dichotomies, if all the compatible classes cannot be partitioned into sets whose sizes remaining bits are no more than by using the current dichotomy, then the subtree rooted at th node pruned. Th pruning algorithm also guarantees to get the optimum encoding solution and executes more efficiently than the procedure Exhaustie_Search_Encoding. Table I shows the results on 6 two-leel Benchmarks. On aerage, Algorithm requires % and 0% fewer LUT's than that of Algorithm and, respectiely. Moreoer, Algorithm also ery time-efficient. It only requires % and 90% CPU time than that of Algorithm and, respectiely. Table II shows the results on 6 multi-leel benchmarks. We find that Algorithm and produce almost the same results. On aerage, they both require % fewer LUT's than that of Algorithm, and also take ~0% less CPU time than that of Algorithm in th experiment. These two experiments show that our encoding strategy can proide greater improement for circuits starting with two-leel forms but no significant improement for circuits starting with multi-leel forms. It because that node functions of the multi-leel circuits optimized by SIS are simpler and hae fewer inputs than node functions of the two-leel circuits. The similar reason has also been suggested in [0]. Therefore, our encoding strategy does not make much difference for multileel circuits after choosing a good lambda set for decomposing functions.. Conclusions In th paper, we dcuss the compatible class encoding problem in Roth-Karp decomposition for two-output LUT

5 architecture. We show how to formulate th problem as a symbolic-output encoding problem. Based on th formulation, we also deelop an encoding algorithm and integrate it into a ersion of Roth-Karp decomposition with the lambda set selection strategy. Experiment results show that our new encoding algorithm can efficiently use fewer LUT's to implement circuits starting with two-leel forms for the twooutput LUT architecture. By inestigating the optimization strategy of current logic synthes systems, our new encoding technique ery useful in both two-leel and multi-leel logic synthes systems targeting for the two-output LUT-based FPGA. Table I : The experimental results of two-leel circuits. Algorithm Algorithm Algorithm CKT name #in #out #LUT time #LUT time #LUT time xp sym alu apex b clip count duke e fm mex References mex [] Xilinx Inc., 00, Logic Drie, San Jose, CA-9, The rd Programmable Logic Data Book. rd [] R. Murgai, Y. Nhizaki, N. Shenoy, R. K. Brayton, and A. sao Sangioanni-Vincentelli, "Logic Synthes for Programmable Gate zml Arrays," in Proc. 7th Design Automation Conf., June 990, pp Total [] R. Murgai, N. Shenoy, R. K. Brayton, and A. Sangioanni-Vincentelli, "Improed Logic Synthes Algorithms for Table Look Up Architectures," in Proc. Int. Conf. Computer-Aided Design, No. 99, pp [] R. J. Franc, J. Rose, and K. Chung, "Chortle : A Technology Mapping Program for Lookup Table-Based Field Programmable Gate Arrays," in Proc. 7th Design Automation Conf., June 990, pp [] R. J. Franc, J. Rose, and Z. Vranesic, "Chortle-crf : Fast Technology Mapping for Lookup Table-Based FPGA's," in Proc. 8th Design Automation Conf., June 99, pp.7-. [6] K. Karplus, "Xmap : A Technology Mapper for Table-Lookup Field Programmable Gate Arrays," in Proc. 8th Design Automation Conf., June 99, pp.0-. [7] N. Woo, "A Heurtic Method for FPGA Technology Mapping Based on the Edge Vibility," in Proc. 8th Design Automation Conf., June 99, pp.8-. [8] D. Filo, J. C. Yang, F. Mailhot, and G. D. Micheli, "Technology Mapping for a Two-Output RAM-based Field-Programmable Gate Arrays," in Proc. European Design Automation Conf., Feb. 99, pp.-8. [9] Y. T. Lai, M. Pedram, and Sarma B. K. Vrudhula, "BDD Based Decomposition of Logic Functions with Application to FPGA Synthes," in Proc. 0th Design Automation Conf., June 99, pp [0]R. Murgai, R. K. Brayton, and A. Sangioanni-Vincentelli, "Optimum Functional Decomposition Using Encoding," in Proc. st Design Automation Conf., June 99, pp.08-. []W.-Z. Shen, J.-D. Huang, and S.-M. Chao, "Lambda Set Selection in Roth-Karp Decomposition for LUT-Based FPGA Technology Mapping," in Proc. nd Design Automation Conf., June 99, pp []TingTing Hwang, Robert M. Owens, Mary J. Irwin, and Kuo Hua Wang, "Logic Synthes for Field-Programmable Gate Arrays," in IEEE Trans. on Computer-Aided Design, Oct. 99, pp []J. P. Roth, and R. M. Karp, "Minimization Oer Boolean Graphs," in IBM Journal of Research and Deelopment, April 96, pp.7-8. []S. Yang and M. Ciesielski, "Optimum and Suboptimal Algorithm for Input Encoding and Its Relationship to Logic Minimization," in IEEE Trans. on Computer-Aided Design, Jan. 99, pp.-. []R. K. Brayton, R. Rudell, A. Sangioanni-Vincentelli, and A. R. Wang, "MIS : A Multi-Leel Logic Optimization System," in IEEE Trans. on Computer-Aided Design, No. 987, pp Normalized Normalized Table II : The experimental results of multi-leel circuits. Algorithm Algorithm Algorithm CKT name #in #out #LUT time #LUT time #LUT time xp sym symml alu alu apex apex b bw C C clip count des duke e fm mex mex mex rd rd rot sao g zml Total Normalized Algorithm : R.-K. decomposition in SIS. Algorithm : R.-K. decomposition with λ set selection strategy[]. Algorithm : Algorithm with compatible class encoding strategy.