# Research Article Engineering Change Orders Design Using Multiple Variables Linear Programming for VLSI Design

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2 2 VLSI Design Old register transfer level vailable spare cell list Logic synthesis New register transfer level DFT ECO ECO Figure 1: typical ECO design flow. Find out patch logic Patch logic Mapping Mapped patch logic Figure 2: Two-phase ECO design flow. Placement and routing Old net-list ECO editing New net-list Equivalent and timing check programming for VLSI design is presented. In Section 5, we discuss the advantage and disadvantage of the related works. Finally, we conclude this paper in Section Typical ECO Design Flow Before describing the proposed method, we address a typical manual ECO design flow in Figure 1. IC designers perform the change in register transfer level and verify fixed code matching the new specification at first. Then, old net-list is scanned to search the possible fix points. fter the possible fix points are searched, IC designers modify the net-list and check the functionally equivalent between new net-list and new register transfer level [10 14]. Next, we describe two-phase ECO design flow in Figure 2. To patch the logics of the modified circuit, we prepare available spare cell list. ccording to logic function, the modified circuit is mapped to specified logics. fter patching logic, equivalent check and timing check are performed to make sure that the new function met the new specification. However, there are some important problems that appear during patching logic. re there enough spare cells and types to satisfy the consumption of patch logic? How to estimate the quantity and logic types of ECO procedure? In order to solve this problem, we proposed an engineering change orders design using multiple variables linear programming for VLSI design in this paper. 3. Logic Transformation Before discussing the engineering change orders design using multiple variables linear programming, we addressed ECO logic transformation. Figure 3 describes an ECO problem with an equation out = ( + (BC) ). Figure 3(b) lists theavailablesparecells.ccordingtothelist,wediscover the available spare cells are not enough. In order to solve theproblem,weadoptanothermappingsolutionwithan equation out = ( BC) instead of the original equation in Figure 3(c). It requires one ND and one INV gate.the mapping solution in Figure 3(c) requires gates fewer than theavailablesparecellsandisconstructedwiththeavailable spare cells. However, most of spare cells only provide basic logical functions that include ND, OR, NOT, NND, and NOR. Half dder (H), Full dder (F), nd-or-inverter (OI), and Or-nd-Inverter (OI) can provide complex logical functions. We can adopt these logical cells to perform ECO function. For example, OI22 can be implemented by two NND and one ND cells in Figure 4. ccording to the existing resources of spare cells, we can resynthesize the changed function lists. 4. Multiple Variables Linear Programming for VLSI Design lthough logic transformation skill makes the ECO technology come true, a chip often does not own enough spare cells tomodifythefunctiontomeetanewspecification.how to allocate limited resource? We should estimate quantity of spare cells and logic transformation rule to perform optimal engineering charge orders. In Figure 5, it describes the engineering change orders design using multiple variables linear programming for VLSI design and relation of logic transformation. Logic isone kind of spare cells that can be transformed into Logic a or Logic b. Similarly, Logic B can be transformed into Logic a, Logic b, or Logic c. Logic C performseco function instead of Logic c or Logicd. Besides, Logic D is transformed into Logic c or Logicd. Equivalently, Logic E is transformed to Logic d or Logice toachieveeco function. We assume X 1, X 2, X 3, X 4,andX 5 are the number of spare cells, Logic,LogicB,LogicC,LogicD,andLogicE. Let Y 1,Y 2, Y 3, Y 4,andY 5 be the desired number of target cells, Logic a,logic b, Logic c, Logic d, and Logic e. Besides, a is the number of spare cells (Logic ) to be transformed into Logic a and b is the number of spare cells (Logic ) to be transformed into Logic b. Similarly, Ba, Bb, and Bc are the number of spare cells (Logic B) to be transformed into Logic a,logicb,andlogicc.inasimilar way, Cc and Cd are the number of spare cells (Logic C) to be transformed into Logic c and Logic d, Dc and Dd are the number of spare cells (Logic D) to be transformed into Logic c and Logic d, anded and Ee are the number of spare cells (Logic E)tobetransformedintoLogicd and Logic e.

3 VLSI Design 3 B C BC Type Spare cells ND 2 OR 2 INV 1 NND 1 B C (a) (b) (c) Figure 3: Example of an ECO problem. (a) EC equation: output = ( + (BC) ).(b)sparecells.(c)mapping:output=( BC). (a) (b) Figure 4: OI22 can be implemented by two NND and one ND cells. Logic a b Logic a Therefore, the restriction rule of the number of spare cells and transformed target cells in Figure 5 is written as follows: X 1 =a+b; X 2 =Ba+Bb+Bc; X 1 Y 1 Logic B X 2 Y 2 Logic C X 3 Y 3 Logic D X 4 Ed Y 4 Logic E Ba Bb Bc Cc Cd Dc Dd Ee Logic b Logic c Logic d Logic e X 5 Y 5 Figure 5: ECO design using multiple variables linear programming for VLSI design and relation of logic transformation. X 3 =Cc+Cd; X 4 =Dc+Dd; X 5 =Ed+Ee. Besides, the restriction rule of the engineering change orders design using multiple variables linear programming in Figure5 is written as follows: (1) a + Ba Y 1 ; (2) b + Bb Y 2 ; (3) Bc + Cc + Dc Y 3 ; (4) Cd + Dd + Ed Y 4 ; (5) Ed + Ee Y 5. (6) However, spare cells are not often enough; designer should balance the spare cell allocation to meet all requirements of desirable cells. We assume one case when Bb Y 2. In order to provide enoughsparecells,weshouldincreasethenumberofb to achieve b + Bb Y 2. Similarly, when Cc + Dc Y 3,weshouldincreaseBc number to meet Bc + Cc + Dc Y 3. Therefore, we define another restriction rule of the engineering change orders design which is written as follows: a = X 1 b; Ba = X 2 Bb Bc. (7)

4 4 VLSI Design Table 1: ECO methods comparison. Method Traditional ECO Proposed method Cell resource prediction Constraint based Multiple variables linear programming Predictive precision of patching logic number Normal precision High precision Balance between spare cells and target cells Low balance High balance Restriction rule Not define Define Resource optimization Not define Define Solution boundary Not define Define ccording to formulas (2) and(7), we can balance the number of b, Bb, andbc to achieve the target number Y 1. Consider the following: X 1 b+x 2 Bb Bc Y 1. (8) In a similar way, we define the restriction rule of the engineering change orders design which is written as follows: Bc = X 2 Ba Bb; Cc = X 3 Cd; Dc = X 4 Dd. ccording to formulas (4) and(9), we can balance the number of Bc, Cc, anddc to achieve the target number Y 3. Consider (9) X 2 Ba Bb+X 3 Cd+X 4 Dd Y 3. (10) We model the engineering change orders problems using multiple variables linear programming. ccording to the functions, we can understand the engineering change orders relation between supply and requirement. Then, designer can estimate and perform ECO using spare cell efficiently. 5. Discussion In this Section, we discuss the advantage and disadvantage of the related works. Table 1 shows ECO method comparison. The proposed approach designs a multiple variable linear programming ECO for VLSI design. Our method can predict cell resource accurately using multiple variable linear programming techniques. Traditional ECO is not to predict it well. Besides, our scheme provides a high accurate prediction of patching logic number to balance between spare cells and target cells. It is hard for traditional ECO method to do these. Moreover, we define restriction rule, resource optimization, and solution boundary of ECO problem to increase the efficiency of the proposed ECO method and provide a well solution. 6. Conclusion In this paper, we proposed an engineering change orders design using multiple variables linear programming for VLSI design. The paper discusses typical ECO design flow, logic transformation, and multiple variables linear programming for VLSI design. The presented scheme estimates the resource of spare cells and provides a well solution of ECO problems. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. cknowledgments This work was supported by the National Science Council of Taiwan under Grant nos. NSC E MY2 and E CC3. The authors gratefully acknowledge the Chip Implementation Center (CIC), for supplying thetechnologymodelsusedinicdesign. References [1] J.. Roy and I. L. Markov, ECO-system: embracing the change in placement, IEEE Transactions on Computer-ided Design of Integrated Circuits and Systems, vol.26,no.12,pp , [2] C. Tan and I. H. Jiang, Recent research development in metal-only ECO, in Proceedings of the 54th IEEE International Midwest Symposium on Circuits and Systems (MWSCS 11),pp. 1 4, ugust [3] Y.M.Kuo,Y.T.Chang,S.C.Chang,andM.Marek-Sadowska, Spare cells with constant insertion for engineering change, IEEE Transactions on Computer-ided Design of Integrated Circuits and Systems, vol. 28, no. 3, pp , [4] D. Brand,. Drumm, S. Kundu, and P. Narain, Incremental synthesis, in Proceedings of the IEEE/CM International Conference on Computer-ided Design, pp , [5] S. Huang, K. Chen, and K. Cheng, utofix: hybrid tool for automatic logic rectification, IEEE Transactions on Computer- ideddesignofintegratedcircuitsandsystems,vol.18,no.9,pp , [6] C. Lin, K. Chen, and M. Marek-Sadowska, Logic synthesis for engineering change, IEEE Transactions on Computer-ided Design of Integrated Circuits and Systems, vol.18,no.2-3,pp , [7] T. Shinsha, T. Kubo, Y. Sakataya, J. Koshishita, and K. Ishihara, Incremental logic synthesis through gate logic structure identification, in Proceedings of the IEEE/CM Conference on Design utomation, pp , Jun [8] G. Swamy, S. Rajamani, C. Lennard, and R. K. Brayton, Minimal logic re-synthesis for engineering change, in Proceedings of

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