M. Potkonjak and J. Rabaey, "Maximally fast and arbitrarily fast implementation of linear computations," IEEE/ACM International Conference on Computer-Aided Design (ICCAD), pp. 304-308, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... transformation that is implemented in most software compilers [3, 4] In the domain of high level synthesis, CSE has been used for throughput improvement [5] for optimizing multiple constant multiplications [6, 7] and as an algebraic transformation for operation cost minimization [8, 9] 10] and [11] present the converse of CSE, namely, common subexpression replication, whereby a redundant operation is inserted to aid scheduling. A compiler transformation called partial redundancy elimination (PRE) 12] inserts copies of operations present in only one conditional branch into the other ....

M. Potkonjak and J. Rabaey. Maximally fast and arbitrarily fast implementation of linear computations. In International Conference on CAD, 1992.

.... for throughput improvement [29] for optimizing multiple constant multiplications [30, 31] and as an algebraic transformation for operation cost minimization [32, 33] A converse of CSE, namely, common sub expression replication has been proposed to aid scheduling by adding redundant operations [34, 35]. Partial redundancy elimination (PRE) 36] inserts copies of operations present in only one conditional branch into the other conditional branch, so as to eliminate common sub expressions in subsequent operations. The authors in [32, 37] propose doing CSE at the source level to reduce the effects ....

M. Potkonjak and J. Rabaey. Maximally fast and arbitrarily fast implementation of linear computations. In International Conference on CAD, 1992.

....from the set is often done in computationaily intensive way. Interestingly several such basic building blocks components are already readily available. For example, methods such as block processing [Par89] and scatter look ahead [Par89] maximally fast implementation of linear programs [Pot91b], and heuristic and optimum techniques for simultaneous of latency and throughput [Sri94] are examples of such transformational scripts. The knowledge based user driven transformation ordering is supported by the following high level synthesis software and knowledge infrastructure. Interestingly ....

....in reducing the search space. A transformation is enabling, if its application does not improve objective function, but makes feasible application of another transformation which significantly improve the quality of design. Enabling and disabling effects among transformations is summarized in [Whi90, Pot91b] 3 Asymptotic analysis information. Often it is of interest to analyze the effectiveness of transformations as the size of target problem instance increases. This analysis is not of just theoretical interest because there is a simple way to increase the size of any ASIC computation on infinite ....

M. Putkonjak, J. Rabaey: "Maximally Fast and Arbitrarily Fast Implementation of Linear Computations", Technical Report, NEC USA, Princeton, NJ, 1991.

....Also using transformations for direct area optimisation has not yet attracted much attention. Local resource utilisation optimisation has been done using transformations steered by stochastic techniques [7] Sometimes area optimisation is a secondary goal in throughput optimisation [8]. Because most transformations are well known [9, 10, 11] selecting a set of transformations is relatively simple. It is however extremely important to reduce the set to a minimal size to limit the search space. This requires an appropriate model. A transformation can be executed when all its ....

....Elementary transformations A small set of elementary transformations is defined for the area minimisation problem. An important aspect of this set is that most transformations have a reverse. An example is common subexpression replication, which is the reverse of common subexpression elimination [8]. The set of currently supported transformations is listed below. eliminate and replicate. Eliminate is common subexpression elimination. An example is c = a b ; d = a b = c = a b ; d = c . Replicate is the reverse of eliminate. collectR and distributeR. CollectR is a transformation based ....

M. Potkonjak, J. Rabaey, "Maximally Fast and Arbitrarily Fast Implementation of Linear Computations, " in Proceedings of ICCAD-92, pp. 304-308, Nov. 1992.

....we support power optimizing transformations, unlike compiler related optimization techniques for which speed is the primary objective. In high level synthesis, transformations for throughput and power optimization for data flow intensive (DFI) behavioral descriptions is a well researched problem [8], 9] 10] 11] 12] 13] Transformations for CFI behaviors are considered in [14] 15] 16] In [14] loop merging and loop unrolling are used, followed by a greedy application of tree height reduction and constant propagation. In [15] procedure inlining, redundant code elimination, and ....

M. Potkonjak and J. Rabaey, "Maximally fast and arbitrarily fast implementation of linear computations," in Proc. Int. Conf. Computer-Aided Design, pp. 304--308, Nov. 1992.

....we needed to look for behavioral and RTL optimizing transformations capable to reduce the critical path delays. Amid possible transformations, we studied module swapping (i.e. replacing the module with a faster (larger) implementation) redundant module assignment[20] and tree height reduction[21]. In the experiment, these transformations been performed manually in a way to lower the critical path. Figure 7(c) exemplifies the result of applying the transformation for the subgraph depicted by bold lines in the Figure 7(a) Figure 7(d) characterizes three feasible solutions derived after ....

M.Potkonjak and J.Rabaey, "Maximally Fast and Arbitrarily Fast Implementation of Linear Computations", Proc. IEEE ICCAD, 1992, pp.304-308.

....max as the additive operator, and arithmetic addition as the multiplicative operator. In recent years two fundamental properties of implementations of linear computations have been identified. The first is that an arbitrary linear computation can be implemented at an arbitrarily high throughput [Pot91]. The second is that there is a fundamental limit on the joint optimization of throughput and latency, the two metrics of speed. Srivastava and Potkonjak [Sri94] showed that it is impossible to reduce latency and increase throughput to an arbitrary level at the same time, and provided analytical ....

....systems can be reduced, both independently and in conjunction with throughput improvement, and to develop techniques for doing so. Chandrakasan et al. Cha92b] made an attempt to utilize the approach of achieving an arbitrarily high throughput implementation of linear computation described in [Pot91] to lower power by trading off increased throughput with reduced voltage at the cost of additional hardware. Unfortunately, the hardware overhead was so high that it overshadowed the savings due to voltage throughput trade off. In several linear filter examples the overall power consumption ....

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M. Potkonjak, J. Rabaey, "Maximally Fast and Arbitrarily Fast Implementation of Linear Computations", Technical Report, NEC USA, Princetion, NJ, 1991.

....approaches. Also in the array synthesis community (see most Professor at the Katholieke Universiteit Leuven y This research has been partly sponsored by the E.C. projects ESPRIT 2260 (SPRITE) and HCMERBCHRXCT930382 (Vision algorithms and optical computer architectures) of the papers in [17]) the emphasis has been on the distribution (projection scheduling) of nested loop operations on a regular array, and not on the memory organisation for irregularly nested loops on weakly parallel processors. This includes work on regular or piece wise regular partitioning [17] which is oriented ....

....of the papers in [17] the emphasis has been on the distribution (projection scheduling) of nested loop operations on a regular array, and not on the memory organisation for irregularly nested loops on weakly parallel processors. This includes work on regular or piece wise regular partitioning [17] which is oriented to the uniform distribution of arithmetic operations in parallel processors, usually for a given latency, and which does not incorporate memory costs. In order to solve this problem better, an effective model and methodology are proposed to derive an optimized architecture ....

M.Potkonjak, J.Rabaey, "Maximally Fast and Arbitrarily Fast Implementation of Linear Computa tions", Proc. IEEE Int. Conf. Comp. Aided Design, Santa Clara CA, pp.304--308, Nov. 1992.

....in the subpart. We assume that when there is an arc from a subpart X to a subpart Y, every output and state of Y depend on all inputs and states of X. The number of operations per input sample is initially 2081(We explain how the number of operations is calculated in a maximally fast approach [9] in Figure 3) Using the technique of [15] which unfolds the entire computation, the number can be reduced to 725 with an unfolding factor of 12. Our approach optimizes each subpart separately. This separate optimization is enabled from isolating the subparts using pipeline delays. The Figure 4 ....

....such as throughput, latency, area, power, permanent and temporal fault tolerance, and testability [10] Interestingly, the power of transformations is most often focused on secondary metrics such as parallelism, instead on the primary metrics such as the number of operations. Potkonjak and Rabaey [9] addressed the minimization of the number of multiplications and additions in linear computations in their maximally fast form so that the throughput is preserved. Potkon7 jak et al. 10] presented a set of techniques for minimization of the number of shifts and additions in linear computations. ....

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M. Potkonjak, J. Rabaey, "Maximally fast and arbitrarily fast implementation of linear computations," International Conference on Computer-Aided Design, pp. 304-308, 1992.

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M. Potkonjak and J. Rabaey, "Maximally fast and arbitrarily fast implementation of linear computations," IEEE/ACM International Conference on Computer-Aided Design (ICCAD), pp. 304-308, 1992.

....and throughput, clearly indicate that efficient application of transformations often has significantly higher impact on the quality of the final implementation than other high level synthesis tasks, like scheduling and assignment. Techniques using transformations to optimize area [33] throughput [32], throughput and latency [36] power [5] and transient [19] and permanent [16] fault tolerance report Manuscript received September 1, 1994; revised January 25, 1995. This paper was recommended by Guest Editors W. Maly and Y. Zorian. The authors are with C C Research Laboratories, NEC USA, ....

....have been successfully used in a number of computation related areas, including compilers, databases, VLSI algorithms, parallel algorithms, and computer architecture. Transformations have been successfully used in high level synthesis for optimization of variety of goals [51, 161, 19] 21] [32], 331, 36] Recently, a new transformation technique was developed which increases the complexity of the behavioral description while reducing the structural complexity of the resulting datapath [34] Application of the new transformation technique to reduce the partial scan overhead for ....

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M. Potkonjak and J. Rabaey, "Maximally fast and arbitrarily fast implementation of linear computations," in IEEE Int. Conf. ComputerAided Design, 1992, pp. 304-308.

.... S2 [n ; 1] 4 X[n ; 1] S2 [n] 3 S1 [n ; 1] 5 S2 [n ; 1] X[n ; 1] Y [n] 2 S1 [n ; 1] 3 S2 [n ; 1] # addition = 5, # multiplication = 6 # operations = # addition # multiplication = 11 Figure 2. A simple example for calculating the number of operations in a maximally fast procedure [7] AB CD E D D D D D D D 7 9 8 6 10 Figure 3. Amotivational example after the isolation step We stress here that the assumptions are not necessary for our approach. We assume that each subpart is linear and dense, which means that every output and state in a subpart are linear ....

....the subpart. We assume that when there is an arc from a subpart X to a subpart Y, every output and state of Y depend on all inputs and states of X. The number of operations per input sample is initially 2081 (We illustrate how the number of operations is calculated in a maximally fast procedure [7] using a simple linear computation with 2 states and 1 output which is described in Figure 2) Using the technique of [10] which unfolds the entire computation, the number can be reduced to 725 with an unfolding factor of 12. Our approach can optimize each subpart separately,which is enabled from ....

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M. Potkonjak, J. Rabaey, "Maximally Fast and Arbitrarily Fast Implementation of Linear Computations, " IEEE International Conference on Computer-Aided Design, pp. 304-308, 1992.

....reduction in the number of operations is the key to power minimization. 3 Related Work Power minimization efforts across all levels of design abstraction process are surveyed in [10] Parhi and Messerschmitt [6] presented optimal unfolding of linear DSP computations. Potkonjak and Rabaey [7] addressed the minimization of the number of multiplications and additions in linear computations in their maximally fast form so that the throughput is preserved. Sheliga and Sha [9] presented an approach to minimizing the number of operations in linear computations. Srivastava and Potkonjak [11] ....

....the previous states. SCCs are further classified as either linear or nonlinear. Minimization of the number of operations for linear computations is NP complete [9] We have adopted an approach of [11] for the optimization of linear sub parts, which uses unfolding and the maximally fast procedure [7]. We note that instead of maximally fast procedure, the ratio analysis by [9] can be used. 11] has provided the closed form formula for the optimal unfolding factor with the assumption of dense linear computations, which means that every output and state are linear combinations of all inputs and ....

M. Potkonjak and J. Rabaey. Maximally fast and arbitrarily fast implementation of linear computations. ICCAD, pages 304--308, 1992.

....using the original . D1 D2 D3 IN2 IN1 IN3 IN4 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 M1 M2 M3 M4 M5 C1 C2 C3 C4 D1 D2 D3 OUT IN5 Critical path reduction Distributivity for reduction of number of multiplications Simplification of linear computations [Pot92] Cut variables GC Figure 1: Trade offs related to selection of cut variables. The developed DfD technique poses a number of optimization tasks. Using the following motivational example, we show that de bugging of an optimized behavioral specification can be performed efficiently. For brevity ....

....for critical path reduction, number of multiplications can be reduced by applying the distributivity rule to multiplications and , and addition . area optimization) and all operations in the remaining shaded area can be optimized using the fast linear computation procedure[Pot92]. We define an important concept that enables effective DfD. A golden cut is defined as a subset of variables in the source code, which should be correct [Hen82] in the optimized program. A complete golden cut is a golden cut with the property that all user variables and primary outputs ....

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M. Potkonjak and J. Rabaey. Maximally fast and arbitrarily fast implementation of linear computations. ICCAD, pp.304-8, 1992.

.... domain on enhancing performance by adding dummy hardware units was proposed by Patel [Pat76] In logic synthesis, the replication of logic gates is used in several projects [Cri91, Hwa92] The replication of arithmetic operations for critical path reduction was proposed in high level synthesis [Lob91, Pot92]. Differences in domain, goal, and probabilistic nature of phenomena in computer network and distributed computing, and static (compile time) nature of research presented here, is very significant. The only similarity, besides the same conceptual goal and means, is the adapted name (hot potato) ....

M. Potkonjak, J. Rabaey: "Maximally Fast and Arbitrarily Fast Implementation of Linear Computations ", ICCAD-92, pp. 304-308, November 1992.

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M. Potkonjak, J. Rabaey: "Maximally Fast and Arbitrarily Fast Implementation of Linear Computations", ICCAD-92, pp. 304-308, 1992.

....e.g. Dey et al. 1992] However, there is a strong experimental evidence that they are most effective at the highest levels of abstractions, such as system and in particular behavioral synthesis. Transformations only received widespread attention in high level synthesis [Ku and Micheli 1992; Potkonjak and Rabaey 1992; Walker and Camposano 1991] Comprehensive reviews of use of transformations in parallelizing compilers, stateof the art general purpose computing environments, and VLSI DSP design are given in [Banerjee et al. 1993] Bacon et al. 1994] and [Parhi 1995] respectively. The approaches for ....

....the throughput and latency optimization problems, where the bottlenecks can be easily identified and well quantified. Finally, the idea of enabling and disabling transformations has been recently explored in a number of compilation [Whitfield and Soffa 1990] and high level synthesis papers [Potkonjak and Rabaey 1992; Srivastava and Potkonjak 1996] Using this idea several very powerful transformations scripts have been developed, such as one for maximally and arbitrarily fast implementation of linear computations [Potkonjak and Rabaey 1992] and joint optimization of latency and throughput for linear ....

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Potkonjak, M. and Rabaey, J. 1992. Maximally fast and arbitrarily fast implementation of linear computations. In International Conference on Computer-Aided Design (1992). 304--308.

.... The techniques include peephole optimization [McK65] pre defined script based optimization [Bra84] theory based ordering [Wol91] generic probabilistic techniques [Pot94, Cha95] bottleneck identification and elimination approaches, and approaches using enabling and disabling transformations, [Whi90, Pot92, Hua93]. The key novelties of this work are 1) a combination of quantitative and qualitative design advice is given, 2) design advice that is specific to the design at hand is given, 3) the designer remains an integral part of the design process, 4) the approach is extendible to include new ....

.... this case, R ij a 1 t ij ImmedImprovement ij 1 t ij EnablingPotential ij a 2 t ij ( a 3 t ij Infeasibility ij = suggested actions are pipelining, constant multiplication replacement with additions and shifts, time loop unfolding, and the maximally fast script [Pot92]. For each action it provides feedback to guide the designer s selection. Information regarding pipelining, for example, includes a bound on critical path improvement attainable by pipelining; in this case, it can reduce the critical path from 12 clock cycles to the iteration bound of 6 clock ....

Potkonjak, M. and Rabaey, J. Maximally fast and arbitrarily fast implementation of linear computations. ICCAD, 304308, 1992.

.... , ffl number of multiplications can be reduced by applying the distributivity rule to multiplications M1 and M2 and addition A5 (area optimization) and ffl all operations in the remaining shaded area can be optimized according to the transformations for achieving fast linear computation [Pot92]. We define an important concept that enables effective design for debugging. A golden cut is defined as a subset of variables in the source code, which should be correct [Hen82] in the optimized program. A complete golden cut Vcut is a golden cut with the property that all user variables and ....

.... of a computation [Kir99] D1 D2 D3 IN2 IN1 IN3 IN4 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 M1 M2 M3 M4 M5 C1 C2 C3 C4 D1 D2 D3 OUT IN5 Critical path reduction Distributivity for reduction of number of multiplications Simplification of linear computations [Pot92] Cut variables GC Figure 1: An example of trade offs involved in selection of cut variables such that optimization potential of the computation is not impacted. Variables of an example complete golden cut GC (output of M3, M5, and A6) are depicted in Figure 1. Using the variables in GC and the ....

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M. Potkonjak and J. Rabaey. Maximally fast and arbitrarily fast implementation of linear computations (circuit layout CAD). IEEE/ACM International Conference on ComputerAided Design, pp.304-8, 1992.

.... it was realized that the scheduling in the traditional behavioral synthesis usually does not have high impact on the quality of the final implementation [49] and that other synthesis optimization tasks, such as transformations, usually make greater differences in the final results [5] 16] 22] [32] [33] Moreover, it was reported that the available scheduling algorithms often produce optimal results [6] 36] 39] The focus of behavioral synthesis research, therefore, shifted from scheduling to other issues of synthesis that were found to be 3 more beneficial if addressed. From the system ....

M. Potkonjak and J. Rabaey. Maximally fast and arbitrarily fast implementation of linear computations. In Proc. ICCAD '92, pages 304--308, Santa Clara, CA, 1992. IEEE Intl. Conf. Computer-Aided Design.

....latency and throughput, even when no assumption are made about the initial state and values of the coefficients. Until now all approaches which were able to improve throughput to an arbitrary extent were based on a combination of unfolding of the computation with block processing or interleaving [33, 21, 34]. However, this comes at the expense of a proportional degradation in latency. This long standing Latency and Sample Period Bottleneck is broken by employing a novel combination of unfolding with On Arrival Processing where input samples are processed as soon as they arrive, and a provably ....

....LTI system, such as a large fraction of DSP systems, our method can achieve a latency of for all , and a latency of at . These numbers are far superior to the fastest reported results in literature even in the specific case of the popular 5 th order elliptical filter benchmark ( and reported in [34] for ) Similarly, for any 2 input LTI system, such as many servo controllers, our method guarantees that a latency of can be achieved for all , and that a latency of can be achieved at . 7. We would like to emphasize that the results in observations 4 and 5, are independent of , the number of ....

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M. Potkonjak and J. Rabaey. "Maximally Fast and Arbitrarily Fast Implementation of Linear Computations." In Proceedings of IEEE International Conference on Computer-Aided Design, pages 304--308, November 1992.

....are most often results of extensive experimentation. While the initial application of transformations in high level synthesis was mainly based on the use of compiler like strategies [McF92] recently emphasis has been shifted to target statistically validated objective functions which model area [Pot92], fault tolerance overhead [Gue93] and power [Cha92] Transformations used in all domains are mainly based on a very few common principles. All currently used syntax transformations can be classified into three groups: control flow transformations, transformations for alternation of the ....

.... which target changes in the organization of computations the most widely used in substitution of constant multiplications by shifts and additions [Rab91] Dataflow transformations encompass two large groups: algebraic transformations [Fis88] and redundancy manipulation transformations [Fis88, Pot92]. Algebraic transformations is probably the most widely studied class of transformations and are based on exploration of algebraic axioms and theorems in a variety of algebraic structures. Typical examples include commutativity, associativity, distributivity, the inverse and the zero element laws. ....

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M. Potkonjak, J. Rabaey: "Maximally Fast and Arbitrarily Fast Implementation of Linear Computations", IEEE International Conference on Computer-Aided Design, Santa Clara, CA, pp. 304-308, November 1992.

.... static scripts [2] exhaustive search based generate and test methods [8] algebraic ordering of linear loop control flow transformations [13] probabilistic search techniques [3] bottleneck removal methods [4] and microscopic and special domain enabling effect based techniques [11]. This paper addresses several design metrics, establishes new insights in enabling disabling effects with respect to the design metrics, develops a method for quantifying these effects, and provides a technique for automatic script development and validation. 3 Preliminaries 3.1 Computational ....

....path (R) 6. pipelining with k pipeline stages for critical path (P) 7. loop unfolding by a factor k (U) 8. constant multiplication substitution with additions subtractions and shift (CM) All transformations subroutines which use associativity also use integrated inverse element law transformation [11, 12]. The initial selection of transformations was mainly influenced by the availability of robust software implementations capable of handling numerous real life designs. The final selection was mainly dictated by their eventual effectiveness and run time efficiency as well as their power to fully ....

[Article contains additional citation context not shown here]

M. Potkonjak and J. Rabaey. Maximally fast and arbitrarily fast implementation of linear computations. In International Conference on Computer-Aided Design, pages 304-- 308, 1992.

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M. Potkonjak and J. Rabaey. Maximally fast and arbitrarily fast implementation of linear computations. In Proceedings of the International Conference on Computer Aided Design, pages 304--8, Santa Clara, Calif., Nov. 1992.

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M. Potkonjak and J. Rabaey, "Maximally Fast and Arbitrarily Fast Implementation of Liner Computations", Proc. of ICCAD, 1992.

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