Method and apparatus for interconnect-driven optimization of integrated circuit design

[0002] 1. Field of the.Invention

[0003] The present invention relates to a tool for integrated circuit design. In particular, the present invention relates to a tool for optimizing the physical design of a standard cell-based integrated circuit for performance.

[0004] 2. Discussion of the Related Art

[0005] A standard cell-based integrated circuit is designed using a library of building blocks, known as “standard cells.” Standard cells include such elements as buffers, logic gates, registers, multiplexers, and other logic circuits (“Macros”).

[0006] FIG. 1ashows a typical design process or “flow” 100 that an integrated circuit designer would use to design a standard cell-based integrated circuit. As shown in FIG. 1a, at step 101, the designer provides a functional or behavioral description of the integrated circuit using a hardware description language. In addition, the designer specifies timing and other performance constraints (109) with which the integrated circuit must comply. Then, at step 102, the designer selects a standard cell library to implement the design. Typically, the standard cells in the library are designed to the requirements of a target manufacturing technology. Often, each cell is also characterized to provide performance parametric values such as delay, input capacitance and output drive strength.

[0007] At step 103, the designer uses a “synthesis tool” to create from the functional or behavioral description a functionally equivalent logic gate-level circuit description known as a “netlist.” The elements of the netlist are instances of standard cells selected by the synthesis tool from the standard cell library in accordance with functional requirements and the performance constraints. At this stage, the synthesis tool uses the characteristic parametric values of each standard cell and a model of input and output loads (“wire load model” or “WLM”) to attempt to meet performance requirements.

[0008] At step 104, a “place and route” tool creates a “physical design” by placing the standard cell instances of the netlist onto the “silicon real estate” and routes conductor traces (“wires”) among these standard cell instances to provide for interconnection. Typically, the placement and routing of these standard cell instances are guided by cost functions, which minimize wiring lengths and the area requirements of the resulting integrated circuit.

[0009] At step 105, with the wires of the integrated circuit having been routed at step 104, a more accurate set of parasitic impedance values in the wires can be extracted. Using the extracted parasitic impedance values, a more accurate timing analysis can be run at step 106 using a static timing analyzer (STA). If the physical design meets timing constraints, the design process is complete (step 108). Otherwise, steps 103-106 are repeated after appropriate modifications at step 107 are made to the netlist and the performance constraints.

[0010] Design process 100 suffers from a number of disadvantages. First, WLM is a crude model based on statistics. Because of the inaccurate model, a designer typically uses an “80^thpercentile WLM” (i.e., 80% of the nets will have a capacitance less than predicted by the WLM). As a result, the drivers for many nets are unnecessarily large, while other driver are too weak. Additionally, designers tend to provide 30% or more additional safety margins to accommodate other inaccuracies in the design flow. Such over-design represents inefficiencies in both silicon area and performance. Second, under this typical method, whenever a non-trivial modification is made to the design to meet a performance requirement, the design is re-synthesised, re-placed and re-routed, which are very time-consuming and costly steps, even when timing is met in a majority of nets. Typically, at each iteration, the physical design undergoes major changes that may introduce new sub-optimal nets requiring another iteration of synthesis, placement and routing to correct.

[0011] The inefficiency in the prior art method results in both high cost and long development time in engineering, time-to-market and manufacturing.

[0012] The present invention provides methods and systems for optimizing a post-layout design without requiring re-synthesis. In these methods and systems, accurately extracted timing information from the physical design drives transformation of the physical design, thereby avoiding the inaccuracy of wire load models of the prior art. Further, methods and systems of the present invention apply local transformations to the physical design, thereby maintaining substantial integrity (i.e., validity and accuracy) in the interconnect models during the transformation process. Accurate models of parasitic impedance can be obtained using an asymptotic waveform evaluation technique.

[0013] According to one embodiment of the present invention, one method for post-layout optimization of an integrated circuit includes: (a) providing a logic description of the integrated circuit; (b) synthesizing from the logic description a netlist of the integrated circuit using instances of cells from a standard cell design library; (c) placing and routing the instances to provide a physical design of the integrated circuit; (d) extracting from the physical design models of parasitic impedance of interconnect in the physical design; and (e) optimizing the physical design by modifying the physical design according to the models of parasitic impedance. Under that method, in one embodiment, the optimization iteratively (a) identifies, using a static timing analyzer, locations in the physical design where timing violations occur and (b) applies one or more local transformations to the physical design to correct the timing violation.

[0014] In one implementation, the method performs a forward sweep and a backward sweep of the physical design to compute a required signal arrival time and a latest signal arrival time, respectively.

[0015] In accordance with another aspect of the present invention, a library analysis step provides characterization of the standard cell library to allows accurate timing and load driving ability analyses. In particular, one method enables a cell to be selected from a library to perform a given logic function and to drive a given load capacitance. That method includes: (a) dividing the cells in the library into groups, such that cells within each of the groups perform substantially the same logic function; (b) within each group, assigning to selected cells each an operating range of loads; and (c) selecting a cell by matching the logic function and the given load capacitance to the operating range of the cell. In one implementation, the operating range of loads to a cell in the library are assigned according to a metric relating an area of the cell to a delay of the cell. In one implementation, each group contains not only cells performing the given function, but also combinations of such cells and buffers of appropriate drive strengths, and combinations of cells providing a complementary logic function and inverters.

[0016] According to another aspect of the present invention, a method of the present invention includes: (a) extracting from the physical design parasitic models of interconnect in the physical design; and (b) applying optimization steps, each optimization step transforming the physical design to achieve a desired performance based on area or delay. In one embodiment, the optimization steps are applied in order of potential intrusiveness to the physical design. Thus, the present invention allows the less complex modifications to be accomplished first. Typically, a large portion of the potential optimization can be achieved by these minimally intrusive modifications to the physical design, leaving the physical design to be substantially optimized even before the more intrusive optimization steps are applied.

[0017] In one implementation, an initial optimization step identifies in the physical design a cell instance mismatched to an output load driven by the cell instance; and replaces the cell instance by a second cell instance matched to the output load. Then, a second optimization step computes a potential improvement in slack for each cell instance in the physical design, selects from the physical design cell instances having the largest potential improvements in slack, and applies transformations to the selected cell instances.

[0018] In that second optimization step, a bidirectional combinational total negative slack (BCTNS) ranking method of the present invention is used. The BTCNS ranking method identifies “hot spots” in the physical design, which are locations where performance improvements with the highest potential impact. The BTCNS method includes: (a) performing a forward sweep and a backward sweep of the physical design to provide for each cell instance a forward priority value and a backward priority value; (b) calculating an equivalent priority value based on the forward priority value and the backward priority value; and (c) ranking cell instances in the physical design according to the equivalent priority value.

[0019] Following the second step of optimization, a third optimization according to a metric based on a path-based algorithm (e.g., a critical path algorithm). The path-based optimization can be used to correct hold and set-up time violations. In that method, the last optimization step identifies in the physical design a cell instance meeting timing requirements but mismatched to an output load driven by the cell instance, and replaces the cell instance by a second cell instance matching the output load and having a smaller silicon area.

[0020] In one implementation, the method of the present invention takes advantage of a static timing analyzer capable of performing incremental timing analysis, and an extraction tool capable of performing incremental extraction of parasitic impedance in the interconnect.

[0021] The local transformations in the present invention include cell instance upsizing, cell instance downsizing, node off-loading, input swapping and logic duplication.

[0022] In one embodiment of the present invention, a system for post-layout design optimization, includes: (a) a library interface for access to a standard cell library; (b) a timing analyzer interface for accessing a static timing analyzer; (c) a design tool interface for accessing a place and route design tool; (d) a design database for storing a physical design of an integrated circuit composed of instances of standard cells from the standard cell library. The system provides routines for traversing the instances in accordance with predetermined orders, a control program for obtaining timing information of the instances from the static timing analyzer, a control program for applying local transformations of the instances guided by the timing information.

[0023] The present invention is better understood upon consideration of the detailed description below and the accompanying drawings.

[0024] FIG. 1ashows a typical design flow 100 that an integrated circuit designer would use to design a standard cell-based integrated circuit.

[0025] FIG. 1bshows design flow 150, in accordance with one embodiment of the present invention.

[0026] FIG. 2 shows design flow 200, representing the operations of step 109bof FIG. 1b, in one embodiment of the present invention.

[0027] FIG. 3 is an overview of optimization tool 300 in one embodiment of the present invention.

[0028] FIG. 4 is flow diagram 400 representing library analysis step 201 of FIG. 2.

[0029] FIG. 5ashows the drive strengths of cells 501-504.

[0030] FIG. 5bshows, in the operating range of interest (i.e., 0 to 2 pf), process flow 400 found cells 501-503 which cover the entire operating range with their individual operating ranges 0 to C₁, C₁to C₂, and C₂to 2pf.

[0031] FIG. 6 shows a network model 600 used in STA 308.

[0032] FIG. 7 is a flow diagram 700 that illustrates the operations of delay calculator 307.

[0033] FIG. 8 is a flow diagram 800 showing the operations of Phase 1 optimization, according to one embodiment of the present invention.

[0034] FIG. 9ais a flow diagram 900 providing an overview of the optimization steps in Phase 2A.

[0035] FIG. 9bis a flow diagram 900 providing an overview of the optimization steps in Phase 2B.

[0036] FIG. 10 is a flow diagram 1000 showing the operations of BCTNS sort step 904 of FIG. 9a.

[0037] FIG. 11ashows cell instance 1101 with its output “effective load” modeled by capacitor. 1102 (C_L) and input and output signal transition times 1104, 1105 and 1106, as computed by delay calculator 307.

[0038] FIG. 11bshows assumed operating conditions necessary to achieve a largest possible delay improvement of cell instance 1101.

[0039] FIG. 12ais a flow diagram 1250 showing the operations of backward propagation of PV values at step 1008 of FIG. 10.

[0040] FIG. 12bshows a backward column PV table initialization step 1200, used in output pin initialization step 1253 of FIG. 12a.

[0041] FIG. 12cshows a flow diagram 1280 that sets forth the steps for backward propagation of values of a PV table to a divergence point.

[0042] FIG. 12dshows a flow diagram 1260 that illustrates the steps for backward propagation of values of a PV table to a merged point.

[0043] FIG. 13ashows backward propagation of PV values over a parasitic model that is driven by multiple input terminals.

[0044] FIG. 13bshows backward propagation of PV values over a cell instance having multiple input terminals.

[0045] FIG. 13cshows backward propagation of PV values from multiple output terminals of a parasitic model to a single input terminal.

[0046] FIG. 14ais a flow diagram 1450 showing forward propagation of PV values at step 1009 of FIG. 10.

[0047] FIG. 14bshows a forward column PV table initialization step 1400, used in input pin initialization step 1453 of FIG. 14a.

[0048] FIG. 14cshows a flow diagram 1480 that sets forth the operations for forward propagation of values of a PV table to a divergence point.

[0049] FIG. 14dshows a flow diagram 1460 that illustrates the steps for forward propagation of values of a PV table to a merged point.

[0050] FIG. 15ashows forward propagation of PV values over a parasitic model that is driven by multiple input terminals.

[0051] FIG. 15bshows forward propagation of PV values over a cell instance having multiple input terminals.

[0052] FIG. 15cshows forward propagation of PV values from a single input terminal of a parasitic model to multiple output terminals.

[0053] FIG. 16 shows flow diagram 1600, which illustrates the steps for computing EPV for each cell in the cluster.

[0054] FIG. 17 shows flow diagram 1700, which illustrates the operations for optimization step 907 (i.e., cell downsizing).

[0055] FIG. 18 shows flow diagram 1800, which illustrates the operations for optimization step 908 (i.e., cell upsizing).

[0056] FIG. 19 shows flow diagram 1900, which illustrates the operations for optimization step 909 (i.e., node off-loading).

[0057] FIG. 20 is a flow diagram 2000, which provides an overview of the optimization steps in Phase 3.

[0058] FIG. 21 shows flow diagram 2100, which illustrates “input swapping” optimization step 2011 of Phase 3.

[0059] FIG. 22 shows flow diagram 2200, which illustrates “logic duplication” optimization step 2012 of Phase 3.

[0060] FIG. 23 provides an example of circuit optimization by logic duplication.

[0061] FIG. 24 shows flow diagram 2400, which illustrates a buffer insertion technique for addressing hold time violations.

[0062] FIG. 25 shows flow diagram 2500 which illustrates a process for reducing overall silicon area

[0063] The present invention provides a design tool and a method for optimizing a standard-cell based integrated circuit after placement and routing are performed, without requiring complete re-synthesis of the integrated circuit design. The present invention optimizes the integrated circuit design based on accurate extraction and modeling of the interconnect network.

[0064] FIG. 1bshows an overview of design flow 150 in one embodiment of the present invention. Unlike the prior art, in the present invention, the integrated circuit design steps of synthesis, initial placement and initial routing are not re-iterated. Instead, modifications to the physical design are performed incrementally. After completing HDL description, synthesis, place and route, extraction and timing analysis steps 101b-106b, which can be substantially the same as corresponding steps 101-106 of FIG. 1a, the timing problems uncovered by timing analysis step 106bare addressed by an interconnect optimization step 109b. Step 109bfixes some or all of the timing problems using the local transformation techniques described below. These local transformations are realized at step 110bby providing incremental place and route directives to the corresponding place and route tools. At step 111b, an incremental extraction of parasitic impedance is performed on the revised physical design. Process flow 150 then returns to timing analysis step 106bto determine if the revised physical design meets all timing requirements. If not, step 109b, 110band 111bare repeated.

[0065] FIG. 2 shows in further detail step 109bof FIG. 1b. As shown in FIG. 2, at step 201; a standard cell library (e.g., a “.lib” file of a format supported by Synopsys Corp.) is analyzed and characterized. Under step 201, cells are classified according to their logic functions (e.g., NAND gates of different drive strengths are grouped), and each cell's operating characteristics (e.g., drive strength at each output terminal and capacitance at each input terminal) are estimated, as explained in further detail below. The results are included in an augmented library file (in a suitable format, such as Copernicus Library Format or “CLF”).

[0066] At step 202, the design database is prepared for receiving an input netlist. The design database provides data structures, described in additional detail below, for facilitating the optimization steps in FIG. 2. The synthesized, placed and routed physical design is then read into the database. The design is typically provided, for example, in the LEF and DEF file formats supported by Candence Design Systems Inc. In addition, timing and other constraints (expressed, for example, in an industry standard format, such as those formats used in the “Primetime” tool from Synopsys, Inc. or the “DesignCompiler” tool from synopsys Inc.) are also read into the database.

[0067] At step 203, parasitic impedance models (“parasitic” models) of interconnect wires are incorporated into the database. Parasitic models are provided by parasitic extractor 204, which can be implemented by, for example, the extraction tool “Columbus”, which is available from Frequency Technology, Inc., Santa Clara, Calif. The parasitic models are incorporated into the initial netlist. Such parasitic models can include such circuit elements as resistors, capacitors and inductors.

[0068] At step 205, a clock tree analysis is performed by clock tree analyzer 206 to identify clock signals and clock signal paths. Clock tree analyzer 206 can be provided internally, or by an external clock tree analyzer (e.g. “Cartier” from Frequency Technology, Inc.) interfaced to the design tool of the present invention. The extracted clock information is incorporated into the design database.

[0069] At step 207, based on the clock analysis, the extracted parasitic models, the operational characteristics of the cell instances in the physical design, and the performance constraints of the physical design, an initial timing analysis is performed. In this embodiment, the initial timing analysis is performed by a static timing analyzer (STA), which is described in further detail below. In this static timing analysis step, the “slack” of each electrical terminal, or “pin,” is calculated. On a pin, the term “slack” refers to the time difference between the latest signal arrival time and a required signal arrival time. A cell instance can also be assigned a slack, which is typically the least slack selected from the cell instance's input and output terminals.

[0070] Based on the slack values, the design tool of the present invention provides one or more optimization steps. To simplify presentation, only optimization steps 208 and 209 are explicitly shown in FIG. 2. In one embodiment of the present invention, four optimization steps (identified below as Phases 1-4 and described in further detail below) are provided in the design tool. In each optimization step, the physical design is modified by a number of local transformations—i.e., each transformation affects only a small number of closely related cell instances and nets. In one embodiment, the local transformations are reported and implemented by providing incremental placement and routing directives to a placement and routing tool (e.g., steps 210-212). At the end of each optimization step, a static timing analysis is performed, using the same STA mentioned above. If the timing constraints are met, further optimization is not necessary.

[0071] As mentioned above, in one embodiment, four optimization steps (“phases”) are provided. In one embodiment, described below, the first three phases are arranged in such a manner that each phase has a potential for resulting in greater modification to the post-layout circuit than the previous phase (i.e., increasing “intrusiveness”). In the first phase (“Phase 1”), which is a “clean-up” optimization step, the physical design is inspected for load-driver mismatches. A load-driver mismatch occurs when a driver drives a load outside of the driver's optimal range. In Phase 1, to correct a load-driver mismatch, a cell instance can be upsized or down-sized to meet the required timing constraints (i.e., the mismatched cell instance can be replaced by a logically equivalent cell instance with more or less drive strength, or longer or shorter propagation delay).

[0072] In the second phase (“Phase 2”), “hot spots” are identified in the physical design. A “hot spot” is a cell with a potential timing improvement that can result in a substantial improvement in timing performance both locally and along signal paths that include this cell. In one embodiment, Phase 2 consists of two phases, referred to below as Phase 2A and Phase 2B. Phase 2A is based on a “total negative slack” calculation at each terminal. Negative slack at a terminal is the amount of time by which the expected signal arrival time at the terminal fails to meet the required arrival time, taking into consideration all timing paths leading to the terminal. “Total negative slack (or “TNS”)” at a terminal is the cumulative negative slacks over all timing end points of interest. An endpoint having a positive slack is ignored. More detailed information regarding TNS can be obtained, for example, from Synopsys Inc. Depending on the nature of the hot spot, one or more local transformations can be applied to realize the timing improvements.

[0073] Because only local transformations are applied at Phases 1 and 2, the resulting modified physical-design does not require re-synthesis. In many physical designs, a very high percentage of all timing violations can be corrected by the local transformations of Phases 1 and 2. Thus, optimization of these physical designs can be achieved without reiteration of the time-consuming re-synthesis, placement and routing loop, thereby reducing the cost of an integrated circuit design.

[0074] In the embodiment mentioned above, in addition to Phases 1 and 2 described above, a third phase (“Phase 3”) also applies local transformations to minimize worst negative slake (WNS), in signal paths. In a first part of a fourth phase (“Phase 4A ”), “hold” timing violations in signal propagation paths are corrected. A “hold” time violation occurs when a signal transition at a clocked element (i.e., a sequential element, such as a flip-flop) occurs prior to the previous logic value of the signal is latched by the clocked element. A “setup” timing violation occurs when the clocked element latches a signal prior to the signal's arrival.

[0075] Finally, in the second part of the fourth phase. (“Phase 4B”), the physical design is examined to minimize overall silicon area, by downsizing appropriate cell instances.

[0076] In the present invention, because highly accurate parasitic models are used in the optimization steps, a more aggressive design style can be used. For example, a 50% WLM target can be set in the synthesis step, so as to leave a larger portion of the timing violations to be corrected by the optimization steps. Under such an arrangement, over-design in the final physical design is reduced, resulting in a lower silicon area and a more timing-efficient integrated circuit. Because the present invention applies local transformations, rather than relying on a global re-synthesis, changes to the placed and routed physical design are incremental and minimally intrusive. Physical design optimization can therefore be achieved much more quickly than in the prior art.

[0077] FIG. 3 is an overview of optimization tool 300 in one embodiment of the present invention. Design tool 300 includes an overall control program 301, which controls and sequences the process flow 200 in FIG. 2, for example. In optimization tool 300, design database (“design graph”) 305 contains the data structures representing the physical design at all times. Some examples of objects in design graph 305 includes:

[0078] a. “Macro”—a representation of a standard cell in the standard cell library;

[0079] b. “MacroPin”—a terminal of a Macro;

[0080] c. “Timing Arc”—a data structure representing the propagation delay between two MacroPins;

[0081] d. “Node”—an instance of a Macro;

[0082] e. “NodePin”—a terminal in a Node;

[0083] f. “Net”—a net connecting two or more NodePins;

[0084] g. “TransformFactory”—a data structure representing a collection of Net and Node changes to transform the design graph; and

[0085] h. “Transform”—an instance of a transformation in a TransformFactory.

[0086] To import the placed and routed physical design and the timing and performance constraints, interfaces 310-314 are provided. Interfaces 310-314 each translate design data or constraints expressed in an industry standard data format to internal data structure of design graph 305.

[0087] The physical design can be exported to an external tool to perform further design activities, such as to perform incremental placement and routing, or to perform more accurate extraction of parasitic impedance. Interface 304 translates selected data structures of design graph 305 into industry standard formats accepted by the external tool.

[0088] Algorithms 315 include routines for traversing design graph 305, thus allowing application programs in optimization tool 300 to extract information in design graph 305 in specified orders. Some examples of such routines include routines for returning a cluster, a cell, a net or a path in depth-first, breadth-first or another ranked order. (A cluster is a group of combinational logic elements between two clocked elements in common or related clock domains.) Specifically, algorithms 315 provide routines for a “forward sweep” and a “backward sweep” of a cluster. These operations are explained in further detail below. Algorithms 315 provide an internal interface between functional modules (e.g., transformation routines 309, described below) and design graph 305.

[0089] FIG. 3 shows four functional modules: optimization module 302, transform module 309, STA 308 and delay calculator 307. Delay calculator 307, which is described in further detail below, computes a delay in a given net using an “asymptotic waveform evaluation” (AWE) method. STA 308 performs both the initial static timing analysis (e.g., step 207 of FIG. 2) and the static timing analysis after each optimization step (e.g., steps 208-209), as mentioned above. To compute delay at a net, STA 308 invokes delay calculator 307. Transformation module 309 includes all programs for transforming a Node. During an optimization step, transformation module 309 invokes STA 308 to evaluate each applicable transformation. Optimization module 302 includes all programs for executing the optimization steps (e.g. Phases 1-5). Optimization module 302 invokes transformation module 309 to implement local transformations.

[0090] Library analysis step 201 in FIG. 2 computes the appropriate operational output load ranges for the standard cells of each logic function. FIG. 4 is a flow diagram 400 representing library analysis step 201 of FIG. 2 in one embodiment. As shown in flow diagram 400 of FIG. 4, a user provides a desired relative “delay to area” tradeoff ratio (“α_desired”) 402, the basic driver 403 of the given technology (typically, a small buffer cell in the library), a “load increment” ΔC_Lvalue 404 (i.e., the finest load capacitance resolution for the library analysis), and the standard cell library file (.LIB) 405, including all performance characterization data. A relative “delay to area” tradeoff ratio (denoted α_i,j,k) is used to control cell selection. α_i,j,kis a measure of the delay advantage gained by replacing cell i by cell j under the condition of an output load k. Generally, a lower α_i,j,kresults in a design optimized towards higher speed performance. Conversely, a higher α_i,j,kresults in a design optimized towards reducing silicon area.

[0091] At step 401, the standard cells are grouped according to logic functions (e.g., NAND, OR, NOR, AND, XOR, etc.). Standard cells included in the same logic function group are interchangeable with respect to logic function. Two cells belong to the same function group if they have the same number of input and output terminals or “pins”, perform the same logic function and provide, at each output pin, the same output “sense”—i.e., negative or positive logic. In addition, among the logic function groups., groups that perform “complementary” logic functions (e.g., AND and NAND) are identified. Standard cells in complementary logic function groups are interchangeable by the insertion of an inverter. Step 401 further identifies:

[0092] (a) buffers, inverters, and primary input and output cells (i.e., registers, flip-flops and other state elements) in the cell library;

[0093] (b) for each state element, clock signal terminals and the timing requirement between the clock terminal and each input or output terminal of the state element;

[0094] (c) for each cell, the area of the cell, the drive strength—i.e., delay as a function of load—of each output terminal and the loading of each input terminal; and

[0095] (d) for each combinational logic cell, a propagation delay.

[0096] After the function groups are identified, library analysis step 201 examines all function groups individually (i.e., step 406 of FIG. 4). For each function group (selected at step 408), a zero-load cell delay is calculated for each standard cell within the function group (step 409). In the following, the delay for a standard cell i driving an output load C_Lis denoted by D(i, C_L). Under this convention, the zero-load cell delay for cell i is denoted D(i, 0). The zero-load cell delay D(i, 0) of a given standard cell i can be obtained, for example, using delay calculator 307 of FIG. 3 by evaluating the standard cell's delay response when driven by basic driver 403 with an ideal rising or falling transition. In one embodiment,-the standard cell's delay responses are estimated for both rising and falling transitions. Delay calculator 307 is discussed in further detail below.

[0097] At step 410, the mean value D_m(0) of all zero-load delays in a function group and the mean area A_mof all cells in that function group are computed. At step 411, the cells in the function group are sorted according to their drive strengths (e.g., in order of increasing area). The next steps (i.e., steps 412-421) find the operating ranges of the cells in the function group. The operating range of each cell is defined between a “low load” operating point (C_LL) and a “high load” operating point (C_HL)

[0098] The smallest driver (i.e., the driver with the smallest area) is examined first (step 412). This smallest driver is assigned a C_LLof 0 pf (step 413) Beginning with a trial C_HLvalue of ΔC_L, the C_HLof the cell is found iteratively by calculating, at step 415, the α_i,j,k's between the current cell i and all other stronger drive cells (j being the running index for these stronger drive cells) under the condition of an output load value k=C_HL. After each iteration of steps 415 and 416, the value of C_HLis increased by ΔC_L(step 417).

[0099] An α_i,j,kis calculated according to the following equation: ${\alpha }_{i,j,k}=\frac{\frac{D\left(i,k\right)-D\left(j,k\right)}{D_m\left(0\right)}}{\frac{A\left(j\right)-A\left(i\right)}{A_m}}$

[0100] in which, D(i,k) and D(j,k) are respectively the delays of cells i and j under a load k, D_m(0) is the mean value of all zero-load delays for cells in logic function group, A(i) and A(j) are the areas of cells i and j, and A_mis the mean area of all cells in the function group, as mentioned above.

[0101] If a cell j is found such that α_i,j,kexceeds α_desired, the current C_HLis the “high load” operating point for cell i (step 418). Cell j, which has the largest α_i,j,kthat exceeds α_desired, is selected (step 419) as the cell to operate in the next operating range, with a C_LLvalue assigned the current C_HLvalue (step 420), and an initial C_HLequaling the current C_HLplus ΔC_L(step 414). The next function group is selected (step 406) after all the cells in the present function group providing coverage for the operating ranges of interest are identified (as determined by step 421). Library analysis step 201 completes after all function groups are processed (step 407).

[0102] FIGS. 5aand 5billustrate the results of applying process flow 400 to compute the operating ranges for standard cells in a NAND group. FIG. 5ashows the drive strengths of standard cells 501-504.

[0103] As shown in FIG. 5b, using process flow 400, the operating range of interest, zero to 2 pf, are found covered by standard cells 501-503, with operating ranges (0,C₁), (C₁,C₂), and (C₂, 2) pf.

[0104] As mentioned above, in one embodiment, timing analysis is provided by STA 308 of FIG. 3. STA 308 can be called upon to compute path delays in circuits that can include state elements and combinational logic elements. In one embodiment of the present invention, cell instances in the design database that are inserted or modified since the last timing analysis are marked. Incremental timing analysis is achieved by computing timing for these marked instances and instances whose timing is affected by such marked instances. Suitable techniques for providing this incremental timing analysis capability can be found, for example, in “An Algorithm for Incremental Timing Analysis,” by Lee et al., published in The Proceedings of the32^ndACM/IEEE Design Automation Conference(1995). FIG. 6 shows a network model 600 used in STA module 308. The signal arrival time at input terminal 602 is provided by an “entry delay” relative to a clock signal 606, based on an assumption that the input signal is driven by an output driver of an upstream state element 604. Similarly, the required signal arrival time at output terminal 603 is provided by an “exit delay”, relative to clock signal 612, based on an assumption that the output signal is fed into an input terminal of second state element 605. Entry and exit delays are computed from clock terminals identified by a clock analysis step, such as clock analysis step 206 of FIG. 2. To accommodate interacting clocks, clock skews and offset between clocks are modeled in STA 308.

[0105] STA module 308 can use a primary input terminal, a clock terminal in a state element, or a terminal with user-specified constraints as a start timing point. Similarly, STA 308 can use a primary output terminal, a terminal with a defined setup time or a terminal with user-specified constraints as a timing end point.

[0106] Circuit 601 includes clusters 607 and 610, which are each a combinational circuit that couples an output terminal of a first state element and an input terminal of a second state element. Cluster 607 is a combinational circuit between flip-flops 604 and 608, and cluster 610 is a combinational circuit between flip-flops 608 and 609. Timing within a cluster is calculated “stage” by “stage” using, for example, delay calculator 307, which is mentioned above. A stage begins at the input terminals of a driver cell instance providing output signals, and ends at the input terminals of receiver cell instances receiving the driver cell instance's output signals. Instead of delay calculator 307, commercial timing calculators, such as “PrimeTime”, from Synopsys Corporation, or the “Central Delay Calculator”, from Cadence Design Systems can also be used.

[0107] To allow signal timing through a stage to be calculated, STA 308 requires (a) pin-to-pin cell delays from the cell library, which can be estimated, for example, in library analysis step 201 of FIG. 2, as mentioned above, and (b) interconnect parasitic models, which can be extracted, for example, by parasitic extraction step 204 of FIG. 2, as mentioned above. STA 308 also accepts from a user a list of false paths, which guides the timing analysis and allows more accurate results. STA 308 computes (a) for each input and output terminal, a “worst” slack value, (b) for each cell instance, a cell delay, and (c) at each output~terminal of a cell instance, an “effective load” model.

[0108] To perform a timing analysis, STA 308 performs a “forward sweep” and a “backward sweep.” In a forward sweep, STA 308 starts from the start timing points and traverses cell instances and parasitic models level by level (i.e., using the well-known critical path method, or “CPM”) to compute a “latest arrival time” (LAT) at each terminal. LAT is the longest cumulative delay to the current pin relative to a timing start point. (The LAT at a timing start point is the “entry delay.”) During a forward sweep, the timing of a cell instance or parasitic model is computed only after the timing for all cell instances driving the input terminals of the cell instance or parasitic model are computed. The timing data associated with a forward sweep are: (a) the LAT at each input terminal; (b) the input transition time used to compute the delay at each input terminal; and (c) pin-to-pin delay between any input terminal of the cell instance or parasitic model to any output terminal of the cell instance or parasitic model.

[0109] In a backward sweep, STA 308 starts at the timing end points and traverses cell instances and parasitic models level by level to compute a “required arrival time” (RAT) at each terminal. RAT is the longest cumulative delay from the current pin relative to a timing end point. (RAT at a timing end point is the “exit delay.”) During a backward sweep, the RAT is computed only after computing RATs for all cell instances connected to the output terminals of the cell instance. When both RAT and LAT are available at a terminal, a “slack” value for the terminal—defined as the difference between the required arrival time and the latest arrival time (RAT-LAT)—is computed. If the slack is negative, i.e., the expected latest arrival time is later than the required arrival time, a timing violation is detected. Where multiple slacks can be computed at a terminal, the smallest slack among the multiple slacks (which may be negative) is selected as the “cell slack”.

[0110] To compute a delay, delay calculator 307 uses a graph of the stage, parasitic models representing the interconnect wires between the output terminals of the driver cell instance and the input terminals of the receiver cell instances, and input time transitions at all input terminals of the driver cell instance. Delay calculator 307 outputs delay and transition times for both positive- and negative-going transitions at each. output terminal of the driver cell instance and at each input terminal of the receiver cell instances. In addition, as mentioned above, an effective load model is provided to each output terminal of a cell instance.

[0111] FIG. 7 is a flow diagram 700 that illustrates the operations of delay calculator 307. As shown in FIG. 7, at step 701, for each input terminal of a receiver cell, a capacitance is obtained from the cell library to represent the capacitance load at the input terminal of the receiver cell instance. Next, at step 702, using AWE techniques, the parasitic model of the interconnect wires between the output terminal of the driver cell instance and an input terminal of a receiver cell instances is combined with the input capacitances at the receiver cell instances to create a reduced-order model. In one embodiment, as shown in FIG. 7, a π-model is provided using the first three moments of the driving point admittance. A suitable method for creating a π-model from the first three moments is described, for example, in “An explicit re-circuit delay approximation based on the first three moments of the impulse response,” by Tutuianu et al., IEEE Design Automation Conference, 1996. Higher accuracy can be achieved using higher order models.

[0112] At step 703, the size of the “effective load” capacitor C_effis iteratively derived by equating the average current from the reduced-order model with the single capacitor model. Also, during this step, using the input transition time (“slew rate”) at each input terminal of the driver cell, a gate delay gate and an output transition time or slew rate at an output terminal of the driver cell instance are computed.

[0113] Finally, at step 704, using the reduced-order model of step 702, and the output transition times computed at step 703, the input transition time at each input terminal of the receiver cell instances is calculated. In one embodiment, the input transition times are obtained using a Newton-Raphson iteration scheme on the π-model mentioned above.

[0114] As discussed above, after the initial timing analysis of step 207 (FIG. 2) is completed, Phase 1 optimization of step 208 is performed. FIG. 8 is a flow diagram 800 showing the operations of Phase 1 optimization, according to one embodiment of the present invention. As shown in FIG. 8, Phase 1 begins at step 801 by receiving a netlist annotated with performance constraints and slack values from initial timing analysis step 207. (In the following, a netlist including slack values and performance constraints is referred to as an “annotated netlist”.) After the appropriate routines in algorithms 315 are invoked to set up a “backward sweep” traversal of the netlist (step 802), each cell instance encountered during the backward sweep is examined (step 803). At step 805, if the cell slack is determined to be non-negative, i.e., no timing violation has occurred at that cell, the cell is skipped over. However, if the cell slack is determined to be less than zero, the effective load C_effof the cell instance is then examined to determine if C_effis within the operating range of the cell instance. If C_effis within the operating range of the cell instance, nothing further is done for that cell instance. Otherwise, i.e., if C_effis not within the operating range of the cell instance, one of the following local transformations is invoked at step 807: (i) replacing the current cell instance by a larger cell instance in the same function group with an operating range covering C_eff; (ii) inserting a buffer that has an operative range covering C_eff, or (iii) replacing the current cell instance by a combination of an instance of a cell in the complementary function group and an inverter with a drive covering C_eff.

[0115] After algorithms 315 complete the “backward sweep” traversal of the netlist discussed above, timing is recomputed at step 820. Then, a second “backward sweep” is set up at step 810. Again, each cell instance encountered during the backward sweep is examined (step 811). At step 813, if the cell slack is determined to be negative, i.e., a timing requirement violation has occurred, the cell instance is skipped over. Skipping over this cell instance avoids creating a worse a timing violation as a result of a downsizing step or a buffer elimination step. Downsizing and buffer elimination are local transformations that can be applied at this second backward sweep. However, if the cell slack is determined to be non-negative, i.e., no timing violation has occurred, the effective load C_effof the cell is examined to determine if C_effis within the operating range of the cell instance. If C_effis within the operating range of the cell instance, nothing further is done for that cell instance. Otherwise, i.e., if C_effis not within the operating range of the cell instance, one of the following transformations is invoked at step 815: (i) replacing the current cell instance by a smaller cell instance in the same function group with an operating range covering C_eff; or (ii) removing a buffer, so as to allow the drive strength of the previous cell instance to directly drive C_eff, or (iii) replacing the current cell instance by a combination of a cell instance in the complementary function group and an inverter with a drive operating range covering C_eff.

[0116] At Phase 2A, optimization is performed using a “bidirectional combinational total negative slack” (BCTNS) algorithm. FIG. 9ais a flow diagram 900 showing an overview of the optimization steps in Phase 2A. As shown in flow diagram 900, the physical design is first partitioned into clusters at step 901. (In the following, a netlist that has its logic circuits partitioned into clusters is referred to as a “cluster-partitioned netlist”). Optimization under the BCTNS algorithm proceeds on a cluster by cluster basis (i.e., repeating steps 902-910), until all clusters are optimized (step 912). For each cluster, the cells within the cluster are first ranked by BCTNS sorting step 904 in descending order of worst BCTNS values. A user-specified number of cells are then selected one by one in the sorted order (steps 905 and 906) for optimization. BCTNS values are recomputed after each optimization pass.

[0117] BCTNS sorting step 904 is illustrated by flow diagram 1000 of FIG. 10. As mentioned above, prior to Phase 2A, STA 308 annotates slack values on the physical design. Then, at steps 1002-1004, the BCTNS algorithm computes a “potential improvement” (PI) value for each cell in a given cluster. PI is computed according to the circuit models shown in FIGS. 11aand 11b. FIG. 11ashows cell instance 1101 with its output “effective load” modelled by capacitor 1102 (C_L) and input and output signal transition times 1104, 1105 and 1106, as computed by delay calculator 307 in the manner described above. As computed by delay calculator 307, the delay between an input terminal of cell instance 1101 and output terminal 1107 is denoted D_current.

[0118] To compute PI, the largest possible delay improvement is assumed to be achievable by replacing cell instance 1101 by the optimal driver as in the function group. FIG. 11bshows the assumed operating conditions necessary to achieve PI. In FIG. 11b, cell instance 1101 is replaced by cell instance 1151, which is the largest driver in cell instance 1101's function group with each input terminal driven by basic drivers with an ideal step waveform. Delay calculator 307 then computes, the delay D_bestunder the conditions of FIG. 11b. PI is defined as the difference between D_currentand D_best.

[0119] After the PIs for all the cell instances in the cluster are computed, the largest PI (PI_max) and the least PI (PI_min) obtained for the cluster are identified (step 1005). For each terminal in the cluster, a data structure is created to represent a three-column table (“Priority Value” or PV table) having a user-specified number T_Sof rows (step 1013). The value r*Δ_pI, where Δ_PIis (PI_max−PI_min)/T_s, fills column 1 of each row r of the PV table, denoted by “PV(r,1)” (step 1007). At step 1008, according to the method illustrated in FIGS. 12a-12dand described below, column 2 (“PV(r, 2)”) of each PV table in the cluster is filled by backward propagation of PV values from the output terminals of the cluster. At step 1009, according to the method illustrated in FIGS. 14a-14dand described below, column 3(“PV(r, 3)”) of each PV table in the cluster is filled by forward propagation of PV values from input terminals of the cluster. Then, at step 1010, using columns 2 and 3 of the PV table of each cell and the associated PI value, and the size of the cell, an “equivalent priority value” (“EPV”) is computed for each cell according to flow diagram 1600 of FIG. 16. At step 1011, BCTNS sort step 904 for the cluster is complete after the cells in the cluster are ranked in decreasing EPV order.

[0120] Backward propagation of PV values step 1008 of FIG. 10 is illustrated by flow diagram. 1250 of FIG. 12a. As shown in flow diagram 1250, backward propagation of PV values begins from a timing-annotated cluster (step 1251). Algorithms 315 routines for traversing the cell instances of the cluster are initialized at step 1252. Then, at step 1253, a backward column initialization step 1253 fills column 2 of the PV table for each output terminal of the cluster, as discussed below in conjunction with FIG. 12b. Subsequently, at steps 1254-1257, a backward sweep traces from the output terminals of the cluster stage by stage back to the input terminals of the cluster. In this embodiment, at each stage, the backward sweep first propagates PV values at the output terminals of the parasitic interconnect model to the input terminal or terminals of the parasitic model (step 1256), and then continues to propagate the PV values at these input terminals of the parasitic interconnect model over the cell instance to the input terminals of the cell instance (step 1257). For each stage, the nets of the input terminals of the stage are taken as the output terminals of the stage become “ready”. A net is said to be “ready” in this context after the values in the second column (i.e., PV(r, 2)) of its PV table are filled. Backward propagation of PV values is complete when all ready nets are traversed.

[0121] Flow diagram 1200 of FIG. 12billustrates backward column initialization step 1253. As shown in backward column initialization flow diagram 1200, initialization step 1253 begins at step 1201 with a timing-annotated cluster, as discussed above. In flow diagram 1200, steps 1202-1211 fill column 2 of the PV table of every output terminal (i.e., the timing end points) of the cluster. For each row r of the PV table for a cell instance, the slack S for the output terminal is added to potential improvement value PV(r, 1) in the first column of the same row r (step 1206) to provide an improved slack value S′ for that output terminal. If improved slack value S′ is greater than 0 (i.e., timing is met by this improvement), the improved slack value is set to 0 (steps 1207 and 1208). Otherwise, an incremental improvement value ΔS which equals the difference between the improved slack and the current slack (i.e. ΔS=S′−S) is obtained at step 1209. The backward PV value of row r (i.e., PV(r, 2)) for that output terminal is provided as the incremental improvement value for the corresponding PI value of column 1 of the PV table (i.e., ΔS*PV(r, 1)).

[0122] As discussed above, PV values are propagated at steps 1256 and 1257 by a backward sweep over parasitic interconnect models and over cell instances, respectively. When a parasitic model is driven by multiple input terminals, as illustrated by FIG. 13a, or is driven by an output terminal of a cell instance having multiple input terminals, as illustrate by FIG. 13b, the values in the PV table of the output terminal are propagated to divergence points. For example, in FIG. 13a, the values of the PV table of output terminal 1301 are propagated backwards to the PV tables of input terminals 1302 and 1303. Similarly, in FIG. 13b, the values in the PV table of output terminal 1304 of cell 1307 are backward propagated to the PV tables of input terminals 1305 and 1306 of cell instance 1307. If multiple output terminals of a parasitic model are driven by a single input terminal of the parasitic interconnect model, as illustrated in FIG. 13c, the values of the PV table of output terminals of the parasitic interconnect model (e.g., output terminals 1321, 1322 and 1323) are propagated to a merge point at terminal 1324. In this embodiment, different procedures are provided for backward propagation of the values of a PV table for propagating to divergence and merge points, as illustrated by the flow diagrams in FIGS. 12cand 12d, respectively.

[0123] FIG. 12cshows a flow diagram 1280 that illustrates the steps for backward propagation of values of a PV table to a divergence point. As shown in FIG. 12c, at step 1281, a running index i is initialized to zero. Index i indicates the current input terminal of the parasitic model or the cell instance (“parent”) whose column 2 of the PV table is to be filled. For example, if the parasitic model has three input terminals, then index i runs from 0 to 2. Steps 1282, 1283 and 1284 step through each input terminal of a parasitic model or a cell instance to fill in the rows of the PV table of the input terminal one by one. For each row to be propagated from the PV table of the output terminal of parasitic model or cell, the slacks of other input terminals of the parasitic model or cell are also considered. Index k, which is initialized at step 1285, is another running index for traversing the same input terminals of the cell instance or parasitic model. Thus, at step 1285, PV(row, 2) (i.e., the current row in the PV table of the current input terminal) is initialized to zero. For each input terminal k (kept track of by step 1286), the slack s(k) of input terminal k and PV(row, 1) of output terminal k times PI are summed to provide an improved slack s′ (k) (step 1287). If the improved slack s′ (k) exceeds 0, the improved slack s′ (k) is set to 0 (steps 1288 and 1289). The total slack improvement r (i.e., r=s′−s) is obtained by accumulating (step 1290) the slack improvements of all input terminals. At step 1292, PV(row, 2) of input terminal i provided the ratio of its slack improvement to the total slack improvement r (i.e., (s′ (i)−s(i))/r). Thus, PV(row, 2) represents a measure of the relative contribution of slack improvement among the input terminals, given the propagated PV(row, 2) of the output terminal.

[0124] FIG. 12dshows a flow diagram 1260 that illustrates the steps for backward propagation of values of a PV table to a merged point. As shown in flow diagram 1260, for row: i of the PV table of the input terminal of the cell instance or parasitic model, entry PV(i, 2) is assigned the sum of all corresponding PV(i, 2) in the PV tables of the output terminals (“children”) of the cell instance or parasitic model, as steps 1263, 1265 and 1266 iterate over all rows of the PV table of the input terminal of the cell instance or parasitic model.

[0125] Forward propagation of PV values step 1009 of FIG. 10 is illustrated by flow diagram 1450 of FIG. 14a. In this embodiment, forward and backward propagation steps are substantially identical. Thus, flow diagram 1450 of FIG. 14ais substantially identical to flow diagram 1250 of FIG. 12a. To avoid repetition, a detailed description of flow diagram 1450 is omitted. For the same reason, the descriptions of the following flow diagrams are also omitted: (a) flow diagram 1400 of FIG. 14b, which illustrates forward column initialization step 1453 of FIG. 14a; (b) flow diagram 1460 of FIG. 14c, which illustrates forward propagation of PV values to a divergence point; and (c) flow diagram 1460 of FIG. 14d, which illustrates forward propagation of PV values to a merge point.

[0126] Similarly, the examples of divergence points and merged points in FIGS. 15a, 15band 15c, which are substantially similar to FIGS. 13a, 13band 13cabove (except for the direction of propagation) are also not described to avoid excessive repetition. FIGS. 15a, 15band 15cillustrate, respectively, forward propagation (a) when a parasitic model is driven by multiple input terminals, (b) when an output terminal of a cell instance has multiple input terminals, and (c) from multiple input terminals of a parasitic model to a single output terminal (“merge point”).

[0127] As discussed above with respect to FIG. 10, subsequent to forward propagation of values in the PV tables in step 1010, EPV values are computed. FIG. 16 shows flow diagram 1600 that illustrates the steps for computing EPV for each cell in the cluster. As shown in FIG. 16, at step 1601, the PI of a cell instance C is identified. From column 1 of the PV table of each output terminal of cell instance C, at step. 1602, the value PI is used to identify two rows containing the closest values to PI. A backward PV (“BPV”) value is then obtained by interpolation between corresponding PV values in column 2 of the PV table of the output terminal. Similarly, at step 1603, a forward PV (“FPV”) value is obtained by interpolation between PV values in column 3 of the PV table. At step 1605, the EPV for the output terminal of cell instance C is provided by the product of PI and the sum of BPV and FPV (i.e., EPV=BPV+FPV). Subsequent to computing EPV for all cell instances, the cell instances of the cluster are ranked in decreasing EPV order, as discussed above, at step 1011 of FIG. 10.

[0128] Returning to FIG. 9a, using the operating ranges computed for the cells of the standard cell library above, optimization steps 907, 908 and 909 in this embodiment apply, when appropriate, to downsizing, upsizing and node-offloading operations, respectively. FIG. 17 shows flow diagram 1700, which illustrates the steps for optimization step. 907 (i.e., cell downsizing). As shown in flow diagram 1700, at steps 1701 and 1702, the cell instances (“children”) driven by a driver cell instance A are examined one by one. Slacks SA and S denote the slacks at output terminals of driver cell instance A and at child instance i, respectively (steps 1703, and 1714). The running index i keeps track of which of children cell instance is being examined. If the slack S of child cell instance i (i.e., the slack on the output terminal of child cell instance i) is less than a predetermined threshold value S_m(step 1704), no further action on child cell i is performed. The running index i is incremented at step 1712 to select the next child cell instance. Otherwise, slack S of child cell instance i is greater than threshold value S_m, the operating range of child cell instance i is checked (step 1705). If the load driven by cell instance i is not within cell instance i's operating range, cell instance i is replaced by an instance of a cell C_optof the function group of cell instance i (step 1715). Otherwise, i.e., if the load driven by cell instance i is within cell instance i's operating range, no further action is taken with respect to cell instance i. The running index i is incremented at step 1712 to select the next child cell instance.

[0129] After replacement at step 1715 by an instance of cell C_opt, STA 308 is called at step 1707 to recompute timing in the local cluster. The recomputed slacks SA′ and S′ of driver cell instance A and replaced cell instance i are calculated at steps 1708 and 1716, respectively. A timing improvement in driver cell instance A, denoted by ΔS=SA′−SA, is computed at step 1709. If S′ exceeds S_mand ΔS exceeds a predetermined minimum slack amount S_d, substitution of cell instance i by the instance of cell C_optis made permanent (step 1711). The process returns to step 1702 for the next child cell instance, until all children cell instances are considered.

[0130] FIG. 18 shows flow diagram 1800, which illustrates the operation of optimization step 908 (i.e., cell upsizing). As shown in flow diagram 1800, at steps 1809, 1820, 1825 and 1821, for a given cell instance C with an output load of L, the cell library is searched to determine whether there exists (i) an optimal cell C_oin the same function group as cell instance. C with an operating range encompassing load L and (ii) a combination C_cwithin the same function group as cell instance C and buffer having an operating range encompassing load L (steps 1808-1809). If neither optimal cell C_onor combination C_cexists, no transformation is available (step 1826). Otherwise, if only one local transformation is available (i.e., if either optimal C_oor combination C_cexists, but not both exist), the local transformation is applied (steps 1822 and 1823).

[0131] However, if both transformations are available (i.e., if both optimal C_oand combination C_cexists), steps 1801-1807 first examine every cell instance (“parent cell instance”) C_pthat drives an input terminal of cell instance C. Running index i, which is initialized at step 1801, indicates which parent cell instance is currently under consideration. At step 1803, the load L_idriven by each parent cell instance C_pis examined to determine if load L_iis within the optimal operating range of parent cell instance C_p(steps 1803-1806). If the load is not within the optimal operating range of parent cell instance C_p, an optimal cell C_o′ is identified and substitutes for cell instance C_p. The process returns to step 1802 until all parent cell instances are examined.

[0132] Then, at step 1812, the following quantities are computed: (i) the sum S_oof worst negative slack WNS1 at the input terminals of cell instance C and “delta” slack S1 (i.e., the slack between the input terminal of cell instance C having the worst negative slack and the output terminal of cell instance C), on the basis of an instance of cell C_osubstituting for cell instance C; (ii) the sum S_cof worst negative slack WNS2 and delta slack S2, on the basis of an instance of C_csubstituting for cell instance C; and (iii) the difference ΔS between S_oand S_c. S_orepresents the slack at the output terminal of cell instance C_o, if cell C_osubstitutes in cell instance C. Similarly, S_crepresents the slack at the output terminal of the instance of combination C_c, if that instance of combination C_csubstitutes for cell instance C. The difference ΔS=S_o−S_cis calculated at step 1812 to determine which of these substitutions minimizes negative slack at the output terminal (steps 1814 and 1815). If ΔS>0, then cell C_ois selected to replace cell instance C (step 1815). Otherwise, combination C_cis selected to replace cell instance C (step 1814). The selected transformation is applied at step 1816. The transformations made to parent nodes (i.e., step 1806) are then reversed at step 1819.

[0133] FIG. 19 shows flow diagram 1900, which illustrate the operations for optimization step 910 (i.e., node off-loading). Flow diagram 1900 provides for node off-loading of a cell instance C driving multiple input terminals as an output load. At steps 1902-1904, the slack S_cof the output terminal of cell instance C, the topology of the net N driven by the output terminal, and the branches B on net N (i.e., the parasitic models between the output terminal of cell instance C and the input terminals of children cell instances) are obtained. At step 1905, the impedance L_iof each branch of net N is computed and the resulting impedances are ranked in decreasing order of load. At steps 1907, 1908, 1909, the slack S of each branch B_i(i being the running index indicating the current branch) is examined. If S is greater than or equal zero, that branch is not be off-loaded, and the process returns to step 1907. Otherwise, the cell library is searched for a buffer BUF (step 1913) whose operating range matches load value L of branch B_i. Timing is then recomputed at steps 1914-1916 for cell instance C to obtain an updated slack S_c, assuming that buffer BUF is inserted to drive branch B_i. A slack improvement ΔS=S_c−S_cis computed at step 1917. If slack improvement ΔS does not exceed a predetermined threshold S_min, no further processing is needed (step 1921). Otherwise, at step 1919, BUF is inserted into branch Bi and local timing at cell instance C is recomputed after the buffer insertion. The process then returns to step 1903, after an incremental timing analysis on cell instance is performed by STA 308 to determine if further optimization of net N is possible.

[0134] At step 910, STA 308 performs a static timing analysis to determine the effectiveness of the optimization steps. If timing is improved by a predetermined threshold amount, at step 911, the BCTNS algorithm of step 904 is re-run to rerank the cells in the cluster. Otherwise, i.e., timing improvement does not exceed the predetermined threshold amount, no further optimization is attempted on the present cluster. The next cluster is then selected upon return to step 903.

[0135] After all optimization steps (e.g., steps 907-909) shown in flow diagram 900 are carried out, Phase 2B optimzation steps can be run. Phase 2B is illustrated by flow diagram 950 of FIG. 9b. As shown in flow diagram 950, in Phase 2B, optimization is performed on a timing-annotated netlist partitioned into clusters (step 951). Processing is carried out cluster by cluster (steps 952, 953, 961). For each cluster, a backward sweep traverses the cluster Node by Node (steps 954-956). (As discussed above, a Node is a macro in the physical design). For each Node with a negative output slack, optimization steps 957, 958 and 959 are carried out. Steps 957, 958 and 959 are respectively substantially identical to the downsizing step 907, upsizing step 908 and node off-loading step 909 discussed above. After all Nodes in the cluster are traversed, timing in the cluster is recomputed (step 960). The process then returns to step 952 for operation on the next cluster, until all clusters are processed (step 961).

[0136] Phase 3 is illustrated by flow diagram 2000 of FIG. 20. In Phase 3, as shown in flow diagram 2000, optimization is performed on a timing-annotated netlist partitioned into clusters (2001). Processing is carried out cluster by cluster (steps 2002-2003 ). A backward sweep traverses the identified path Node by Node and a potential improvement (PI) is calculated. The Nodes are then sorted by PI (steps 2004, 2005 and 2016). In each cluster, at step 2004, STA 308 is called to identify a path with the worst negative slack (steps 2005-2007). For each Node, optimization steps 2008-2012 are carried out. Optimization steps 2008-2010 are respectively substantially identical to the downsizing step 907, upsizing step 908 and node off-loading step 909 discussed above. Optimization steps 2011 and 2012, corresponding to optimization involving “input swapping” and “logic duplication” are discussed in further detail below. After all Nodes in the cluster are traversed, timing in the cluster is recomputed (step 2014), if a Node is optimized during the traversal of the path. The process returns to step 2004 to identify a path in the cluster having the worst negative slack. However, if no node is optimized during the last iteration within the cluster, the process returns to step 2002 for operation on the next cluster, until all clusters are processed (step 2015).

[0137] FIG. 21 shows flow diagram 2100 which illustrates “input swapping” optimization step 2011. Input swapping optimization step 2011 examines cell instances or Nodes whose slack performance can be improved by swapping input terminals. To consider a Node C for input swapping (step 2101), the slack S_cof Node C′s output terminal is obtained (step 2102). An input terminal I_targetof the Node C is identified on a path that is being considered for optimization. From the cell library, intrinsic input-to-output delay D_ibetween each equivalent input terminal of Node C and the output terminal of Node C is obtained (step 2105). The least D_minof these input-to-output delays, which is equivalent to the input-output pin pair and corresponding to input terminal I_min, is selected. At step 2109, D_minis compare to the intrinsic delay D_target, between input terminal I_targetand the output terminal of Node C. If D_minsubstantially equals or exceeds D_target, no optimization can proceed, since swapping I_min, with I_targetdoes not result in a significant improvement (step 2119). The local timing on Node C, children of Node C, and other cell instances also receiving signals from input terminals I_targetand I_minare recomputed, assuming input terminals I_targetand I_minare swapped (steps 2110 and 2111). Under such an assumption, the slack S_c′ at the output terminal of Node C is recomputed. A slack improvement value ΔS=S_c′−S_cis calculated at step 2113. If ΔS is greater than predetermined threshold value S_min, input terminals I_minand I_targetare swapped (i.e., the driver that previously drives input terminal I_minis now coupled to drive input terminal I_targetand the driver that previously drives input terminal I_targetis now coupled to drive input terminal I_min.

[0138] FIG. 22 shows flow diagram 2200 that illustrates “logic duplication” optimization step 2012 of Phase 3.

[0139] FIG. 23 provides an example of an optimizing step using logic duplication. As shown in FIG. 23, in logic circuit 2300, cell instance 2301 drives input terminals of cell instances 2302-2307. Logic duplication is applied to logic circuit 2300 to provide logic circuit 2350. In logic circuit 2350, an additional cell instance 2308, which is identical to cell instance 2301, is provided. Cell instance 2308 is driven by the same input signals as cell instance 2301. Cell instance 2308, however, drives cell instance 2307, which is severed from the output terminal of cell instance 2301. If cell instance 2307 is a cell in a critical path, by duplicating cell instance 2301 in cell instance 2308 and appropriately sizing cell instance 2308, the signal delay in the critical path can be reduced.

[0140] As shown in flow diagram 2200, for each cell instance C considered for logic duplication (step 2201), the slack S_cat the output terminal of cell instance C and the parasitic network representing the net N at the output terminal of cell instance C are obtained (steps 2202-2204). Using STA 308, a critical path Node N_ccan be identified. At step 2205, a circuit topology can be created in which Node N_cis severed from net N. A new cell instance C′, which is an instance in cell instance C′s function group is then provided in this new circuit topology to drive N_c(steps 2207-2208). At step 2210, local timing is then computed for this new circuit topology, which includes cell instances C and C′, their “children” cell instances, and “sibling cell instances” (i.e., cell instances sharing common input terminals with cell instances C and C′). After local timing is computed, at step 2211, the slack S_c′ at the output terminal of cell C′ is calculated (step 2211). A slack improvement value ΔS=S_c′−S_cis calculated at step 2212. If ΔS is less than predetermined threshold value S_min(step 2213), no modification of net N is performed (steps 2216 and 2217). Otherwise, the new circuit topology replaces net N. Local timing is then recomputed (step 2215).

[0141] While Phases 1, 2 and 3 described above optimize the physical design from the point of view of meeting setup time, Phase 4A addresses hold time violations. FIG. 24 shows flow diagram 2400, which illustrates a buffer insertion technique for addressing hold time violations. Hold time violations usually result from clock skews or phase differences between common or related clocks. As a result of a hold time violation, a new signal transition may arrive at a state element before the previous signal can be latched. To avoid a hold time violation, the process of flow diagram 2400 inserts one or more buffers to lengthen the signal path. The process of flow diagram 2400 begins at step 2401 with a timing-annotated netlist that is cluster partitioned. The clusters in the netlist are examined one by one. Within each cluster, STA 308 is called to identify one by one signal paths with a hold time violation (steps 2402-2404). In a path identified with a hold time violation, the amount H indicating the extent of the hold time violation is calculated (step 2406). From this value H, an equivalent number N of basic driver delays is calculated (step 2407). A basic driver delay is the delay of the smallest driver in the cell library. The end point P_tof the signal path having the hold time violation is then identified (step 2408). (P_tis an input terminal to a state element.) N serially connected basic buffers are then inserted between P_tand the input terminal of the state element at the end of the signal path. Timing is then recomputed for the cluster (step 2411). The process then returns to step 2404 to process the next signal path with a hold time violation. Phase 4A completes when all paths with hold time violations in all clusters are processed.

[0142] At Phase 4B, each cell instance is with positive slack is examined to ensure that the required silicon area is reduced. A process for implementing Phase 4B is illustrated in FIG. 25 by flow diagram 2500. The process of flow diagram 2500 begins at step 2501 with a timing annotated netlist that is partitioned into clusters. The process of flow diagram 2500 traverses the netlist cluster by cluster and, within each cluster computes PI for each cell instance, traversing from the output terminals of the cluster (steps 2503-2505. Each Node with a positive slack is examine to determine if the output load is within the operating range of the Node, by identifying the cell N_owhose optimal operating range (e.g., using an α that favors area over speed) encompasses the output load (step 2506-2508). If the current Node is not optimal, the current Node is replaced by an instance of No (step 2509). The process then returns to step 2504 for the next Node, until all Nodes in the cluster are traversed. Timing for the cluster is recomputed after traversal of all Nodes in a cluster (step 2510). Phase 4B completes after all clusters in the netlist are traversed (step 2512)

[0143] The above detailed description is provided to illustrate specific embodiments of the present invention and is not intended to be limiting of the present invention. Numerous modification and variations within the scope of the present invention are possible. The present invention is set forth in the following claims.