Abstract:
A practical definition for determining an upper bound on information content is provided and used to reduce the widths of operators and edges of data flow graphs. A top down procedure for systematically pruning data flow graphs is described. The result is shown to enhance the mergeability of subgraphs and provide reduced data path widths. This may result in lower area, power requirements and other benefits as readily understood in the field of circuit design.

Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
     This application claims benefit of U.S. Provisional Application No. 60/298,536 filed Jun. 15, 2001, which is incorporated by reference in its entirety. This application is related to co-pending U.S. application Ser. No. 10/173,066, filed Jun. 17, 2002. This application is also related to co-pending U.S. application Ser. No. 10/173,477, filed Jun. 17, 2002, and to co-pending U.S. application Ser. No. 10/172,941, filed Jun. 17, 2002. 
    
    
     BACKGROUND 
     The number and complexity of datapath operations implemented in various kinds of systems, particularly those on integrated circuit chips, has increased considerably over the years. This is especially true in chips used for graphics, communication, and multimedia processing applications, which have employ parallel implementation of signal processing algorithms such as fast Fourier transforms, finite impulse response filters and other DSP algorithms. 
     One perennial need in this field is in the optimization of datapath operations to minimize area, power requirements, and delay. Current techniques are limited in scope, permitting only the merging of individual datapath operators such as adders, multipliers, and shifters. For example, datapath-intensive register transfer level (RTL) designs require synthesis techniques that yield optimized implementations of groups of datapath operators instead of individual operators. 
     One useful technique is operator merging, which refers to clustering of multiple datapath operators so that they can be synthesized together as a unit. In particular, designers and researchers have explored synthesizing a cluster of datapath operators as a sum of addends using carry-save adders and Wallace trees. For example, synthesis of the sum of product expression a*b+c*d using traditional synthesis requires 2 multipliers and an adder. Such an implementation has 2 carry-propagate adders on any input-to-output path. Operator merging can implement such an expression using only one carry-propagate adder by reducing the partial products of the multipliers in a single carry-save reduction tree (CSA-tree). 
     An algorithm for operator merging to achieve datapath synthesis has also been proposed which first partitions a data flow graph into clusters of datapath operators and then synthesizes each cluster using a CSA-tree, that is, a combination of a reduction tree of carry-save adders and a final adder. 
     The effectiveness of operator-merging in improving performance of netlists for datapath intensive designs has been demonstrated. Research has also focussed on the optimal implementation of synthesizing clusters of datapath operators as sums of addends using carry-save adders and bit-oriented Wallace trees. Such work has further supported the usefulness of operator merging. 
     The problem of optimization of datapaths is a deep problem and will continue to demand attention from researchers. There is thus a continuing need for improvements in the various approaches. 
     SUMMARY OF THE INVENTION 
     Partitioning a data flow graph into clusters is a preliminary step in the optimization of datapaths. Operator merging maximizes the mergeability of operators to permit larger and fewer clusters to be defined by optimization procedures. Each cluster representing a sum of addends is associated with the burdensome delay and area of a final carry-propagate adder. Partitioning of datapaths into larger numbers of small clusters generally means more timing delay and area of the resulting netlist. In contrast, increased merging may provide reductions in the number of carry-propagate adders and consequently reduced critical path delay. 
     In the present specification, several techniques are proposed for partitioning data flow graphs into clusters. In particular, the techniques allow safe reduction in the bitwidths of datapath operators used in designs. This allows the first pass of synthesis to generate faster and smaller netlists. They also reduce the amount of work at the gate-level logic optimization step required to meet timing and area constraints. Further, the proposed method of partitioning a data flow graph into maximal mergeable clusters also defines criteria for safe partitioning of data flow graph and these may be used in problem scenarios other than operator merging. For example, they may be used for rebalancing computation graphs consisting of associative operators. 
     Safe clustering of data flow graphs (DFGs) is characterized in terms of required precision and information content of signals. This characterization is applicable to DFGs that have both signed and unsigned extensions of signals. Note that signed extension refers to adding higher significant bits by replicating the sign bit and unsigned refers to adding higher significant bits by adding zeros. The basic formulas and processes, based on notions of required precision and information content of a signal, are used to define safe, functionality-preserving, transformations on the DFGs, which allow the transformed graph to have potentially smaller widths (bitwidths) of datapath operators and potentially greater mergeability of datapath operators. Efficient algorithms for computation of required precision and upper bounds on information content and the related DFG transformations are proposed. These algorithms may be combined in an iterative procedure for partitioning a graph into maximal safe clusters. 
     The inventions will be described in connection with certain preferred embodiments, with reference to the following illustrative figures so that it may be more fully understood. With reference to the figures, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of the preferred embodiments of the present invention or inventions only, and are presented in the cause of providing what is believed to be the most useful and readily understood description of the principles and conceptual aspects of the invention or inventions. In this regard, no attempt is made to show structural details of the invention in more detail than is necessary for a fundamental understanding of the invention or inventions, the description taken with the drawings making apparent to those skilled in the art how the several forms of the invention or inventions may be embodied in practice. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1A illustrates a simple data flow graph (DFG) with a bottleneck that prevents merging according certain criteria. 
     FIG. 1B illustrates the DFG of FIG. 1B highlighting what may be deduced to be non-mereable features. 
     FIGS. 2A and 2B illustrate the transformation of a DFG similar to that of FIGS. 1A and 1B into a DFG that may be deduced to be mergeable by virtue of the elimination of the purported bottleneck. 
     FIG. 3 is a flow chart illustrating a method for enhancing the mergeability of a DFG by identifying required precision and transforming the DFG accordingly. 
     FIG. 4A illustrates a DFG for purposes of discussing information content. 
     FIG. 4B illustrates the DFG of FIG. 3A after transforming to reduce bitwidths to a minimum required to preserve information content. 
     FIG. 5 is a flow chart illustrating a method for transforming a DFG based an upper bound on information content. 
     FIG. 6A illustrates a DFG for purposes of discussing constraints on apparent mergeability that arise from topology features. 
     FIG. 6B illustrates the DFG of FIG. 4B after transforming it to change the topology to remove the apparent constraint. 
     FIG. 7 is a flow chart illustrating a procedure for Huffman rebalancing of the topology of a DFG to permit greater reduction in bitwidths and improved mergeability. 
     FIG. 8 is a flow chart illustrating an iterative procedure for determining maximal clusters. 
     FIG. 9 is a generic illustration of media that may be used for storing algorithms. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Operations of a data flow graph (DFG) may include width extension of a signal, which is the padding the most significant bit (MSB) side of the signal with multiple copies of a fixed bit to obtain a new signal of larger bitwidth. If the padding is done with a zero bit, the extension may be said to be unsigned. If it is done with the current MSB of the original signal, the extension may be said to be signed. For example, 00011 and 11111 are obtained from the two bit signal  11  by a five bit unsigned and five bit signed extension respectively. 
     As used in the instant specification, a DFG, which includes datapath operators, is a directed acyclic connected graph where nodes represent inputs, outputs and datapath operations. The term “edges” is used to identify the flow paths for data between operators. The interface of an edge with its source or destination node is referred to as a port. A port may be an input (or output) port representing an interface of an edge with its destination (resp. source) node. Each input (or output) node may have one output (resp. input) port. Each operator node may have one output port and one or two input ports depending on whether the datapath operator on the node is unary or binary. The following quantities may be defined for the nodes and edges in a DFG: 
     Each operator node N may have a width value w(N), which is a positive integer. For an input (or output) node, represents the bitwidth of the input (resp. output) signal represented by the node. For an operator node, it represents the number of bits used to represent the operands and/or result of the operation labeling the node. 
     Each edge e has a width value w(e), which is a positive integer. For an edge, the width represents the number of least significant bits of the result of the operation at the source node, which may be used as input by the operation at the destination node of the edge. 
     Each edge e may be labeled with a binary attribute t(e) called the signedness of the edge. The signedness is either signed or unsigned. The binary bits {0, 1} may be used to represent the signedness types “unsigned” and “signed,” respectively. 
     Let N 1  and N 2  be the source and destination nodes of an edge e. Let their widths be w(N 1 ), w(N 2 ) and w(e) respectively. If w(e)≦w(N1), then a signal defined by w(e)—many least significant bits of the result of N 1 , may be said to be carried by e. If w(e)&gt;w(N 1 ), then e would carry a signal obtained by extending the result of N 1  to w(e) width. The type of extension may be determined by the signedness of the e. Similarly if w(N 2 )≦w(e), the signal defined by w(N 2 )—many least significant bits of the signal carried by e may be used as an input operand by the operator at the destination node. If w(N 2 )&gt;w(e) and implementation of the operator at N 2  requires an extension of its operand, then a w(N 2 ) bit extension (whose signedness is determined by signedness of e) of the signal carried by e may be used as the input operand. 
     Referring to FIG. 1A, the idea of merging of datapath operators may be illustrated with a simple example. A DFG  100  has inputs A and B linked by edges  140  and  150 , respectively, to an operator N 1  illustrated at  125 . DFG  100  has inputs C and D linked by edges  145  and  160 , respectively, to an operator N 2  illustrated at  130 . Operators N 1  and N 2    125  and  130  are illustrated as addition operators but could any of a variety of types of operators. The bitwidths of edges  140 ,  145 ,  150 , and  160  are each equal to 8. The widths of operators N 1  and N 2    125  and  130  are equal to 9. While an output edge  155  has a bitwidth of 9, which corresponds to the output of operator N 2    130 , that of an output edge  165 , which corresponds to the output of operator N 1    125 , is equal to 7 so the output of node N 1    125  is obtained by truncating a 9 bit result to 7 bits by the operator N 1    125 . Furthermore on the edge  165 , the truncated value may be sign-extended to 9 bits to be used as an operand for the operator N 3    135 . The output edge  170  of the operator N 3    135  indicated at  170  has a bitwidth of 10, corresponding to a result R. 
     Observe that because the truncated value carried by the edge  165  may be sign-extended to 9 bits, to be used as an operand for operator N 3    135 , the output  190  of the DFG  100  is not directly expressible as sum of addends derived from input signals. Therefore, the whole of the DFG  100  could not be in the same cluster. That is, it is not mergeable. Referring now to FIG. 1B, the maximal merging possible in the DFG  100  is identified by broken lines  105  and  110  surrounding the mergeable extents. The situation where a signal is truncated and then subsequently extended in the downstream computation creates a mergeability bottleneck and forces a boundary that limits merging. 
     The following two essential conditions may be identified as being required for a set of datapath operators in a DFG to be identified as a cluster: 
     1. The subgraph formed by the operators is a connected induced subgraph with a unique output. 
     2. The value of the output signal, at the unique output, is definable as a mergeable function of inputs to the cluster. 
     For example, this function may be a sum of products of signals derived from inputs. Note that an addend may be said to be derived from an input signal if it is obtained by truncation, extension or 2&#39;s complement of the input signal of products of signals derived from inputs. Note also that since a product operation can be implemented as sum of multiple partial products, a sum of products of signals may be viewed as a sum of addends, where the partial product of inputs form the addends. 
     Referring to FIGS. 2A and 2B, a DFG  101  is similar to that of FIGS. 1A and 1B, except for a difference in the width of an output edge  171 , which is 5 bits in FIG. 2A rather than 10 as in FIGS. 1A and 1B. Since only 5 LSBs of the final sum 191 need to be generated, the required precision of every in the DFG  101  is only 5 bits. This is because the higher significance bits are superfluous. Hence no extension is required on the edge  165  and the bottleneck of FIGS. 1A and 1B may be seen to be avoidable by appropriate transformation of the DFG  101 . Thus, the entirety of the DFG  101  is mergeable. The DFG  101  may be transformed to the DFG  200 , which has smaller respective widths of edges  240 ,  250 , (which correspond to edges  140  and  150 ), edges  245  and  260  (which correspond to edges  145  and  160 ), operators N 4  and N 5    225  and  230  (which correspond to operators N 1  and N 2    125  and  130 ) and edges  265  and  255  (which correspond to edges  165  and  155 ) compared to the DFG  101 . The transformed DFG  200  may then be analyzed using prior art mergeability algorithms and clusters identified and merged. 
     Note that, although in this example the width of the output signal  191  is used to transform the width of the operators of the DFG  200 , the width of any node or edge inside the DFG can also be used to transform the widths of nodes and edges in the fan-in cone of the given node (or edge, respectively). Essentially, a procedure may be followed in which, working backward from output to input, where an operator and/or its inputs are wider than required given the width of the output, the operator and its inputs are pruned. For example, if an 8-bit-wide operator with inputs whose widths are also 8 bits, has an output that is only 6 bits wide, the operator and its inputs may be pruned to 6 bits, which is the minimum precision required for the output. Any additional width results in the operator ignoring MSBs of the inputs, so they are pruned in advance. Then the pruned inputs are followed to their respective outputs and the same process is followed again for each operator, pruning along the way. Note, the procedure may not hold for all operator types, for example shift and rotate operators. 
     The following procedure is preferably recursive and, as suggested above, applied in bottom up fashion, i.e. the ports on the output nodes form the base case. For an input or output port p, a required precision r(p) for the signal entering or leaving the port, respectively, is defined by the following rules: 
     For input port p of an output node N: r(p)=w(N). 
     For input port p of a non-output node N: r(p)=min {r(p o ); w(N)}. Here p o  is the output port of N. 
     For output port p o  of a node N: 
     
       
           r ( p   o )=max e outedges(N) (min{ w ( e ); r ( p   d )}) 
       
     
     Here p d  denotes the input port at the destination node of edge e. 
     Referring to FIG. 3, a procedure for implementing the above in a design for a circuit may be defined as follows. In step S 10 , a DFG is defined to represent a proposed circuit. In step S 15 , a new port in the DFG is identified. Preferably, the nodes of a DFG are processed in reverse topological order. As stated, the ports are traversed in bottom-up fashion with the outputs taken first. In step S 20 , a new directed path from the port to an output node is identified. Note that the directed path may be confined to the immediate fan-out region, or a selected number of levels of such, of the node. 
     In step S 25 , the minimum width of any node or edge on that path is determined. Then, in step S 30 , the required precision is taken as the maximum of this value over all of these directed paths. If the required precision of a signal is n, it means, not more than n least significant bits of the signal are needed to completely define the signals at every output node in the fan-out cone of the port. The remaining higher order bits of the signal get truncated by some intermediate operation or explicit truncation and the corresponding bits on subsequent paths may be regarded as superfluous. In step S 35 , if the last directed path from the current port has been followed, step S 40  is executed, if not, steps S 20 -S 30  are repeated for a new one, until all are followed out and a required precision determined for each. In step S 40 , if the last port has been traced, step S 45  is executed, if not, steps S 20 -S 35  are repeated for a new one, until all are followed out and a required precision determined for each directed path therefrom. 
     In step S 45 , the DFG is transformed according to the new required precision values by applying each to a corresponding operator and edge. A transformation that changes the widths of nodes and edges in a DFG such that 
     
       
           w ( n )=min{ w ( n ); r ( p   o )} and  w ( e )=min{ w ( e ); r ( p   d )} 
       
     
     where p o  is the output port of node n and p d  is the destination port of edge e preserves the functionality of the DFG. In step S 50 , mergeable clusters may be identified based on the transformed DFG and in step S 55 , the transformed DFG may be used as a basis for the design of a logic circuit, as an exemplary application of the method. 
     As demonstrated by the examples given, analysis of required precision of a DFG graph can potentially reduce the required width of operators and operands and thereby expose the mergeability of operators to algorithms for identifying clusters. 
     Referring now to FIGS. 4A and 4B, a simple example of a DFG  300  has inputs A 1 , B 1 , C 1 , and D 1  applied through edges  340 ,  350 ,  345  and  360  to operators N 7  and N 8    325  and  330 , respectively. Outputs of operators N 7  and N 8    325  and  330  are applied through edges  365  and  355  to an operator N 9    335 , whose output is applied through edge  310  to an operator N 10    395 , whose output at edge  370  is a result  390 . Note that the edge  310  appears at first inspection as a potential boundary of merging (i.e., a bottleneck), because it is sign-extending an 8 bit truncated sum. However since A 1 , B 1 , C 1 , and D 1  all have narrow bitwidths, the 8-bit result of nodes N 7    325  and N 8    330  are simply sign extensions of 4 bit sums. Tracing the consequences of this observation one level further, the result of N 9    335  is, functionally, a sign-extension of 5 bit sum. This means, the combination of the widths of node N 9    335 , edge  310  and node N 10    395  does not required a sign-extension of a truncated result as may first appear. In fact, the operand entering N 9    335  via edge  365  is a sign extension of 5 bit sum. As a result, DFG  300  may be replaced with a functionally equivalent graph  301 , which has smaller widths for operators N 7 ′ and N 9 ′  326  and  336  and edges  366 ,  311 , and  356 . Further, output R  390  may be expressed as sum of sign-extensible inputs A 1 , B 1 , C 1 , and D 1  and the entire graph is, thus, mergeable. 
     The example illustrates that essential content of information in the result of every operator node may be transformed, in some situations, to allow the merging of operators that otherwise seem unmergeable. Also, as noted in the context of preceding example, the same analysis also allows a reduction in the widths of datapath operators that are working on operands with low information content. An algorithm is described below for defining and exploiting an upper bound on the information content of signals at every port of a DFG. This information content results may then be used to prune the widths of nodes and edges in the DFG safely. 
     The information content of a signal in a DFG may be defined as the tuple &lt;i,t&gt; of the smallest possible non-negative integer i and an extension type t {0; 1} (i.e. unsigned, signed) such that for all possible values of the inputs to the DFG, the signal is a t-extension of its i many least significant bits. For a port p, &lt;i(p), t(p)&gt; may denote the information content of the signal entering (or leaving) the port if the port is an input (resp. output) port. Intrinsic information content of a node may be defined as the information content of its result signal in terms of the information content of its operands, assuming the operation at the node is done without any loss of information. For example, intrinsic information content i int  of addition of operands with information contents &lt;m 1 , 0&gt; &lt;m 2   , 0&gt; is &lt;max{m   1 , m 2 }+1, 0&gt;, again, the value 0 for t a signedness of unsigned. The problem of determining the first component of information content of signals in an arbitrary DFG with +, − and × operators is nondeterministic polynomial-hard (NP-hard), which means it is essentially intractable. But, while computing the exact value (say &lt;i, t&gt;) of information content is hard, a heuristic for efficiently computing an upper bound on information content i.e. a &lt;i′, t′&gt; where i′≧i such that the signal is a t′-extension of its i′ many least significant bits, is still possible. 
     The notation î(p) (similarly î(N) and î int (N)) may be used to denote upper bounds on the information content &lt;i(p), t(p)&gt; of a port. If the upper bounds on intrinsic information content of inputs of binary operators of addition (+), subtraction (−), multiplication (×), and unary minus(− u ), are denoted by &lt;i 1 , t 1 &gt;, &lt;i 2 , t 2 &gt; then: 
     
       
           î   int (+)=&lt;max{ i   1   ,i   2 }+1, t   1   |t   2 &gt;; 
       
     
     
       
           î   int (−)=&lt;max{ i   1   ,i   2 }+1,signed&gt;; 
       
     
     
       
           î   int (×)=&gt; i   1   +i   2   ,t   1   |t   2 &gt;; 
       
     
     
       
           î   int (− u )=&lt; i   1   +i   2 ,signed&gt;. 
       
     
     Note that the vertical bar refers to a Boolean OR operation so that if any input is signed, then the output information content is signed. 
     Information content of a signal at the output edge of an operator node may depend on the width of the operator node and information content of the input operands of the operator node. As a consequence, the information content of signals are preferably computed in a given DFG in a top-down order; i.e. starting at input nodes and finishing at output nodes. 
     Referring to FIG. 5, a procedure for optimizing a design for a logic device begins with the definition of a DFG S 110  and identifying an next operator node S 115  in an output-to-input sequence. In steps S 120  and S 125 , propagating information content across an operator node, information content for the output port of the nodes are computed based on the information content of the inputs ports of the operator node. The information content at the output port of a node is the smaller of the intrinsic information content of the node and its width. If at step S 130  the last operator node has been identified and its output port information content determined, step S 135  is executed. If not, steps S 115 -S 125  are repeated for each. 
     At step S 135 , an edge is identified in the DFG. In steps S 140  and S 145 , propagating information content across an edge, information content for the destination port of the edge is computed based on the information content for the source port of the edge. For propagating information content across an edge, if the signedness of the information content and the edge are the same, then the magnitude of the information content across the edge is the smaller of upper bound on i and w e . In the scenario where the signedness type t of the information content at the source port differs from signedness type t(e), when t=unsigned and t(e)=signed, if there is a strict extension of the information content across the edge (i.e. w(N 1 )&gt;upper bound on i and w(e)&gt;upper bound on i), then the first component of the information content is upper bound on i and the signedness is unsigned. Even though the edge is signed, in this case, the data going into the destination node can be regarded as unsigned because it will always have zeros in the most significant bits beyond the upper bound on i least significant bits. If, at step S 150 , the last edge has been identified and its information content determined, step S 155  is executed. If not, steps S 135 -S 145  are repeated for each edge. 
     Information content upper bound is used to reduce the widths of nodes and edges in the DFG at step S 155 , when widths exceed the information content. In step S 157 , to maintain compatible connections between a pruned subgraph and its inputs and outputs, a new type of operator may be defined and added to reconcile the interfaces, as required. This operator is referred to here as an extension node. The extension node may have the following two attributes: width and signedness (denoted by w(N) and t(N) for node N), may be defined such that the result of extension operation is: 
     (i) if w(N)&gt;w(e in ) (where e in  is the unique input edge of the node), then result is a w(N) bit extension of the signal at the destination port of e and the type of extension is same as t(N). 
     (ii) if w(N)≦w(e in ), then result is the w(N) many least significant bits of the signal on destination port of e. 
     If the intrinsic information content of an operator node N is &lt;i, t&gt; and w(N)&gt;i, then a transformation can be done without changing functionality of the DFG. This transformation begins by decreasing the width of N to i. Then, all the outedges of N may be removed. The output port of N is then connected to a new extension node and the removed outedges of N connected to the output port of the new extension node. The width and signedness type of the edge connecting N and the new extension node is &lt;w(N), X&gt; (where X means either of signed or unsigned); the width and signedness type of the new extension node are w(N) (old value) and t respectively. If the information content at the destination port of an edge in a DFG is &lt;i, t&gt;, the width and sign type of the edge can be changed to i and t without changing the functionality of the DFG. The width transformations above are preferably performed while evaluating the information content in topological order from inputs towards outputs. 
     In step S 160 , mergeable clusters are identified and merged (i.e., the DFG is repartitioned) and in step S 162 , if new extension nodes are added from a previous iteration, the information content is propagated across the extension nodes by returning to step S 115  and iterating. 
     There are situations, in which, a safe rebalancing of a subgraph of a DFG, can allow tighter (i.e. smaller) values of upper bounds on information content of signals. This may allow for potentially greater merging and smaller widths of operators. For example, consider the DFG shown in FIG. 6A, which, as in earlier examples, could be part of a bigger DFG. 
     In a DFG  400 , inputs A 2 , B 2 , C 2 , and D 2  are applied through edges  440 ,  450 ,  445  and  460  to operators N 11 , N 12  and N 13    425 ,  430 , and  435 , respectively. Output of operators N 11    425  is applied through edge  465  to operator N 12    430 , whose output is applied through edge  455  to operator N 13    435 , whose output at edge  470  is a result  490 . Note that the operators N 11 , N 12  and N 13    425 ,  430 , and  435  form a skewed tree. The algorithm for computing information content would compute &lt;7, 0&gt; as the upper bound on information content of the output signal R  590 . 
     However, the DFG  400  shaped as a skewed tree may be rebalanced as illustrated at  500  in FIG.  6 B. Here, DFG  500  has inputs A 2 , B 2 , C 2 , and D 2  applied through edges  540 ,  550 ,  545  and  560  to operators N 14  and N 15    525  and  530 , respectively. Outputs of operators N 14  and N 15    525  and  530  are applied through edges  565  and  555  to an operator N 16    535 , whose output at edge  570  is a result  590 . In the DFG  500 , the upper bound computed would be &lt;6, 0&gt;. Note that a rebalancing of a subgraph in a DFG did not alter its functionality. Therefore once a subgraph has been identified as safely rebalanceable, the upper bounds on the output of the subgraph can be computed using a more balanced ordering of operations in the graph. Note also that actual rebalancing of the nodes and alteration of the graph is not required. The only requirement is to define a more balanced ordering of operators to compute tighter upper bounds. 
     Preferably, subgraphs should be rebalanced only if doing so is safe. A cluster obtained from mergeability analysis is a safely rebalanceable subgraph (for example, the subgraphs enclosed by boundaries  105  and  110  in FIG.  1 B), because the output of a cluster is expressible directly as sum of products of input signals. If a DFG consisting of addition, subtraction, multiplication and unary minus operators and a cluster exists such that its unique output is expressible as a sum of constant multiples of addends. For example, e.g. z=5*b−4*d+3*f) is a safely rebalanceable subgraph because each constant integer product is equivalent to multiple addends coming from the same signal (e.g. 5*b is b+b+b+b+b and −4*d is (−d)+(−d)+(−d)+(−d)). Therefore, the output can be viewed as sum of addends derived from input signals. 
     After identifying clusters using an initial mergeability analysis, the information content of the output of the clusters can be computed by rebalancing them. Further, if this recomputation leads to reduction in the value of the width component of information content, further merging of operators should be attempted. 
     A computational problem exists which is how to compute tighter upper bounds on information content of a cluster representing a sum of constant multiples of inputs. An algorithm employs Huffman Rebalancing to take an expression representing a sum of constant multiples of input signals and compute an upper bound on the integer value of information content of the output signal using an optimal ordering of operations. The following is a definition of the proposed algorithm. 
     The input to the algorithm is an expression representing a sum of constant multiples of input signals. The upper bounds on information contents of the input signals are assumed to be known. The output is an upper bound on information content of the output signal of the expression. Referring to FIG. 7, in step S 210 , a DFG is defined. In step S 220 , first, a priority heap structure H of integers is created. For each term c*i in the expression (where c is an integer constant and i is an input signal, c copies of the numeric value of information content of i are placed in the heap. Next, the following procedure, represented in pseudocode, is performed on the value in the heap, H. 
     
       
         
               
             
               
               
             
               
             
               
               
             
               
             
           
               
                   
               
             
             
               
                 While ( H has more than one value) { 
               
             
          
           
               
                   
                 min1 = extractMin(H); 
               
               
                   
                 min2 = extractMin(H); 
               
               
                   
                 InsertValue( H , max{min1,min2}+1); 
               
             
          
           
               
                 } 
               
             
          
           
               
                   
                 return extractMin( H ); /* Return the single remaining value in H .*/ 
               
             
          
           
               
                 End Algorithm 
               
               
                   
               
             
          
         
       
     
     The above procedure computes the upper bound on information content, which is the best possible among all possible orderings of operations. Among all possible orderings of operations in an expression representing sum of constant multiples of inputs, the ordering defined by the Huffman Rebalancing algorithm gives the tightest possible upper bound on information content of expression result. If the Huffman rebalancing results in a change in topology, merging should be reattempted otherwise the procedure may be terminated—step S 230 . The other procedures for bitwidth reduction based on required precision and required information content may be applied as well in the procedure of FIG. 7 immediately between steps S 230  and S 215 . 
     Referring now to FIG. 8, the overarching problem of partitioning a DFG into clusters may employ each of the above measures in a single algorithm for computing maximal clusters based on the analyses of required precision and information content. The algorithm illustrated in FIG. 8 involves an iterative bottom-up traversal (outputs to inputs) of the DFG and identifies break nodes i.e. every operator node N such that N is not mergeable with at least one of the operators at the destination of its outedges. This defines a partitioning of the graph into clusters, which are connected components obtained by removing those outedges of every break node whose destination nodes are not operator nodes. Assuming that a DFG has been transformed based on analysis of required precision and information content, an operator node N of the DFG, is a break node if one or more of following conditions hold: 
     1. Safety Condition 1: For some outedge of the operator node N, the destination node of the outedge is an extension node. 
     2. Safety Condition 2: Let p i ; . . . ;p m  be the destination ports of outedges of the operator node. Let r(p i ) denote the required precision of signal for each p i . Then min {i int (N); max{(p i ); . . . ;r(p m )}}≦w(N). 
     3. Synthesizability Condition 1: For some outedge of N, the destination node has multiplication operator. 
     4. Synthesizability Condition 2: There is a node N′ such that every directed path starting at N goes through N′ and there are no break nodes between N and N′ on any of these paths. 
     Synthesizability condition 2 ensures that every cluster has a unique operator node providing outputs; synthesizability condition 1 ensures that this unique output is expressible as sum of products of inputs to the cluster. Then each cluster can be synthesized as a sum of addends. 
     If the algorithm for information content computation encounters an extension node, created by the previous iteration of information content computation, it needs to propagate information content across the extension node. If N is an extension node and &lt;i, t&gt; are upper bounds on information content at its input port and e is the inedge of N, then an upper bound &lt;i 0 , t 0 &gt; on the output port of N can be defined as follows. 
     (i) if ((t==t(N)) OR ((t==unsigned) AND (t(N)==signed))) then i o =min {i; w(N)}; to=t(N); 
     (ii) if ((t==signed) AND (t(N)==unsigned)) then i o =min {w(e); w(N)}; t o =t(N); 
     After initial computation of required precision and information content, the algorithm for maximal merging enters an iterative mode. Every iteration defines a partitioning based on current values of information content and uses current set of clusters to compute tighter upper bounds on the information content of the output signals of clusters. Whenever the value of information content of the output signals of any cluster change, another iteration of cluster definition is done with the anticipation that smaller information content could lead to more mergeability and result in bigger and fewer clusters. This way the algorithm converges to a partitioning with maximal safe clusters. 
     A simple procedure for implementing the above method is outlined in FIG.  8 . First, a DFG is defined for some target circuit design (S 315 ). Next, in steps S 315  and S 322 , the DFG is pruned responsively to required precision and information content upper bounds. Preferably, this may be done using the algorithms defined above or parts thereof. Next, in step S 326 , mergeable subgraphs may be identified in the DFG. Next, in step S 335 , the potentially mergeable subgraphs are rebalanced and upper bound on information content determined. Steps S 322 -S 335  are repeated the first time S 345  is encountered and if information content remains unchanged afterward, the process is terminated otherwise, steps S 322 -S 335  are repeated again until the information content upper bounds remains unchanged for all clusters. Note that only a subset of clusters need be handled as required by the loop defined above. Note also that the required precision step S 315  may be omitted and the benefit of information content and rebalancing obtained without it. Also, other techniques for rebalancing, determining information content, and/or required precision may be substituted in the process of FIG.  8 . 
     The DFG partitioning algorithm was implemented and tested as a DFG optimization and datapath operator-merging step in the BuildGates synthesis tool of Cadence Design Systems. Datapath intensive RTL test cases were used and experimental data collected on the performance of the algorithm. These were compared with results obtained using an older implementation of cdfg partitioning algorithm. The older algorithm did mergeability analysis using criteria similar to “leakage of bits” approach and without doing any transformations based on information content and required precision. 
     Using the TSMC 0.25-micron technology cell library, two types of performance data were collected: 
     (i) Longest path delay and area of the netlists obtained after synthesis but before any timing driven gate level logic optimization. 
     (ii) Runtime of timing driven gate level logic optimization done on netlists obtained from synthesis. 
     Tables 1 and 2 respectively present the above two types of data from five datapath-only test cases. To highlight the impact of operator merging in datapath synthesis, Table 1 also includes the data obtained using a synthesis flow which does not do any operator merging. When the non operator-merging based flow was used, the runtimes of logic optimization were much larger than those with operator-merging based flows; so runtime was not included in Table 2. To further compare of the quality of the final netlists generated using old and new merging algorithm, Table 2 includes the data on final longest path delay and final area after timing driven logic optimization. All delay numbers are in nanoseconds and the area numbers are scaled down by a factor of 100. Note that to collect data for both tables, we set the arrival times at all inputs in each test case to 0. 
     
       
         
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Test cases 
                 D 1   
                 D 2   
                 D 3   
                 D 4   
                 D 5   
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Del. (ns) 
                 No mg 
                 14.47 
                 18.01 
                 33.59 
                 29.23 
                 25.89 
               
               
                   
                 Old mg 
                 13.04 
                 11.97 
                 29.90 
                 28.13 
                 25.89 
               
               
                   
                 New mg 
                 12.73 
                 11.07 
                 29.27 
                 16.97 
                 15.57 
               
               
                   
                 % red. 
                 2.38 
                 7.52 
                 2.11 
                 39.67 
                 39.86 
               
               
                 Area (unit) 
                 No mg 
                 93.8 
                 79.3 
                 1866 
                 490 
                 279 
               
               
                   
                 Old mg 
                 91.7 
                 66.6 
                 501 
                 397 
                 225 
               
               
                   
                 New mg 
                 90.3 
                 66.6 
                 476 
                 43 
                 33.3 
               
               
                   
                 % red. 
                 1.53 
                 0 
                 5 
                 89.2 
                 85.2 
               
               
                   
               
             
          
         
       
     
     
       
         
               
               
               
               
               
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 Test cases 
                 D 1   
                 D 2   
                 D 3   
                 D 4   
                 D 5   
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Target delay (ns) 
                 5.0 
                 4.0 
                 21.0 
                 10.5 
                 14.0 
               
             
          
           
               
                 Opt time 
                 Old mg 
                 470 
                 1031 
                 26 
                 118 
                 21 
               
               
                 delay (sec) 
                 New mg 
                 6.8 
                 208 
                 17 
                 2.2 
                 1.3 
               
               
                   
                 % red. 
                 98.5 
                 79.8 
                 34.6 
                 98.1 
                 93.8 
               
               
                 End Del 
                 Old mg 
                 4.99 
                 4.35 
                 20.7 
                 10.5 
                 13.9 
               
               
                   
                 New mg 
                 4.99 
                 3.98 
                 20.9 
                 9.1 
                 12.2 
               
               
                 End Area 
                 Old mg 
                 161 
                 155 
                 377 
                 609 
                 259 
               
               
                   
                 New mg 
                 142 
                 118 
                 363 
                 44 
                 35 
               
               
                   
               
             
          
         
       
     
     Test case D 1  and D 2  were created using multiple addition operations, which are potentially mergeable. These addition operations did not have any redundant widths in RTL code, so the first pass of information-analysis leads to clusters that are not distinguishable from those created by the old merging algorithm. However, the post-clustering information analysis based on optimal reordering of operations, which is done by the second or subsequent iteration of the new merging algorithm, allows the inference of smaller information content for output signals of clusters. This allows the second or subsequent iterations to merge the set of clusters created in previous iteration into bigger and fewer clusters. This reduction in number of clusters, leads to the better longest path delay and area values after initial synthesis. Since there were no apparent redundant widths in RTL, the gains seen after the initial synthesis do not seem as large as D 4  and D 5 . Nevertheless during timing driven logic optimization, we see considerable advantages of creating larger clusters, and see significantly smaller runtimes. 
     Test cases D 4  and D 5  were created with a great deal of redundancy in the bit widths of intermediate paths in RTL, to test the effect of information-analysis based width reduction on timing and area of netlists. In these test cases, the merging algorithm was able to prune the redundant widths to the minimum required, and this in turn helped in reducing the number of clusters created. As a result, significant reduction in longest path delay and area after the initial synthesis was noted. This also translates to drastic reduction in the runtime of the timing driven logic optimization for these two test cases, as seen in Table 2. 
     Test case D 3  represented a sum of products of sum computation, where information-based-analysis allowed the new merging algorithm to prune with widths of outputs of products and merge them with the final addition. 
     The above results demonstrate the benefits of using analyses of required precision and information content of signals in DFGs for operator merging based datapath synthesis. 
     Referring to FIG. 9, any of the methods, algorithms, or techniques presented may be embodied in software and stored on media  600  according to known techniques. 
     Although the foregoing invention has been described by way of illustration and example, it will be obvious that certain changes and modifications may be practiced that will still fall within the scope of the appended claims. The devices and methods of each embodiment can be combined with or used in any of the other embodiments. For another example, the concepts of required precision, information content, the related transformations, and the partitioning algorithms described below are applicable to data flow graphs (DFGs) that have datapath operators other than addition, subtraction, unary minus and multiplication e.g. comparators and shifters. However, for the sake of clarity the discussion is limited to examples involving +, − and × operations. 
     The following references are hereby incorporated by reference as if fully set forth herein in their entirety. 
     D. A. Huffman, A method for the construction of minimum-redundancy codes, Proceedings of the IRE, 40(9), 1952, pp. 1098{1101. 
     T. Kim, W. Jao, S. Tjiang, \Arithmetic Optimization using Carry-Save-Adders”, Proceedings of the 35th Design Automation Conference, 1998, pp.433-438. 
     A. R. Omondi, \Computer Arithmetic Systems: Algorithms, Architectures and Implementations”, Prentice-Hall International Series in Computer Science, 1998. 
     J. Um, T. Kim, C. L. Liu, \Optimal Allocation of Carry-Save-Adders in Arithmetic Optimization” Proceedings of International Conference on Computer Aided Design, 1999, pp.410-413. 
     J. Um, T. Kim, C. L. Liu, “A Fine-Grained Arithmetic Optimization Technique for High-Performance/Low-Power Data Path Synthesis” Proceedings of the 37th Design Automation Conference, 2000, pp.98-103. 
     C. S. Wallace, \A suggestion for a fast multiplier” IEEE Trans. Electron. Comput., February 1964, vol EC-13, pp.14-17. 
     N. Weste, K. Eshraghian, “Principles of CMOS VLSI Design—A System Perspective” Addition Wesley Publishers, 1985. 
     M. Willems, V. Bursgens, H. Keding, T. Grotker, H. Meyr, “System Level Fixed-Point Design Based on an Interpolative Approach”, Proceeding of the 34th Design Automation Conference, 1997, pp. 293-298.