Patent Publication Number: US-6988253-B1

Title: Methods, apparatus and computer program products that perform layout versus schematic comparison of integrated circuits using advanced pin coloring operations

Description:
CROSS-REFERENCE TO PRIORITY APPLICATION 
   This application is a divisional of U.S. application Ser. No. 10/279,337, filed Oct. 24, 2002, now U.S. Pat. No. 6,799,307, which is a divisional of U.S. application Ser. No. 09/505,499, filed Feb. 17, 2000, now U.S. Pat. No. 6,499,130, the disclosures of which are hereby incorporated herein by reference. 

   FIELD OF THE INVENTION 
   This invention relates to the design, layout, testing and manufacture of microelectronic circuits and systems, and more particularly to apparatus and methods for verifying microelectronic circuits and systems prior to manufacture. 
   BACKGROUND OF THE INVENTION 
   Present day ultra-large-scale integration (ULSI) circuits may include hundreds of thousands or millions of interconnected active electronic devices on an integrated circuit chip. The large capital investment required to fabricate and test large scale integrated circuits prior to sale to a customer and the difficulty, expense and loss of goodwill associated with reworking and replacing integrated circuits which fail to operate as planned, have increased the need to accurately characterize the electrical behavior of integrated circuits prior to their manufacture. 
   Moreover, now that submicron and deep-submicron (0.5 μm and below) technologies have begun to dominate silicon chip manufacturing and the prospect of million-plus-gate chips operating at clock rates of 100 MHZ has become a reality, fundamental changes have had to be made to conventional integrated circuit design methodologies and the electronic design automation (EDA) tools based thereon. To meet the challenges posed by such large-scale circuits, techniques have been developed to represent integrated circuit designs at various levels of abstraction. According to these techniques, an integrated circuit design may be represented by an electrical schematic containing devices and nets interconnecting the devices and by geometric layout data that describes patterns of regions or elements to be formed in and/or on an integrated circuit substrate (e.g., wafer). 
   Techniques for managing highly integrated circuit designs include hierarchical design techniques. Using such techniques, a particular design is partitioned into functional cells and cells-within-cells, etc., so that at a given level of hierarchy the design may be analyzed as a set of cells and their respective interconnections, without being concerned with all the details of the contents of the cells (e.g., subcells within each cell). 
   These hierarchical techniques can be essential to the efficient performance of computer-assisted integrated circuit design verification. Such verification may include operations to perform layout versus schematic comparison (LVS) using computer-based design tools. As will be understood by those skilled in the art, tools to perform layout versus schematic comparison may include extraction software to extract a layout netlist from geometric layout data. An extracted layout netlist is then compared to an electrical schematic netlist to determine functional equivalence between the original integrated circuit schematic and the integrated circuit layout. One difficulty associated with the performance of these operations may be caused by a dissimilarity in the labeling of nets and devices in the extracted layout netlist relative to the electrical schematic netlist. 
   Conventional methods for determining correspondence between an electrical schematic netlist and a layout netlist are described in U.S. Pat. No. 5,249,133 to Batra entitled “Method for the Hierarchical Comparison of Schematics and Layouts of Electronic Components”; U.S. Pat. No. 5,463,561 to Razdan entitled “High Capacity Netlist Comparison”; and U.S. Pat. No. 5,243,538 to Okuzawa et al. entitled “Comparison and Verification System for Logic Circuits and Method Thereof.” Another conventional method for determining correspondence includes operations to represent the electrical schematic netlist and the layout netlist as a schematic graph and a layout graph, respectively. Each of these graphs may be represented as bipartite graphs having vertices (also referred to herein as “nodes”) that represent devices and nets within their respective netlists. LVS software is then used to determine an isomorphism between the bipartite graphs. 
   The unambiguous determination of isomorphism between two arbitrary graphs may be a computationally intractable problem. To address this problem, heuristic methods for identifying graph isomorphisms with acceptable reliability and efficiency for ULSI designs have been developed. One generally established heuristic method is an iterative graph-coloring method described in articles by C. Ebeling and O. Zajicek entitled “ Validating VLSI Circuit Layout By Wirelist Comparison ,” Proceedings of ICCAD, pp. 172–173 (1983); and by C. Ebeling entitled “ Gemini II: A Second Generation Layout Validation Program ,” IEEE ICCAD-88, Digest of Technical Papers, pp. 322–325, Nov. 7–10 (1988), the disclosures of which are hereby incorporated herein by reference. As described in these articles, an integer node value (color) is assigned to each node of a bipartite graph of the electrical schematic netlist and the extracted layout netlist, based on a graph invariant such as “number of nearest neighbors” (i.e., number of adjacent nodes/vertices). Each node is iteratively recolored as a function of the colors of its neighbors, until the maximum number of unique colors is achieved (i.e., an equilibrium state is achieved). Because these operations are independent of labeling, equivalent schematic and layout netlists generally will be represented by the same set of colors. A one-to-one correspondence may then be achieved by simply matching up each node in the schematic graph with a node in the layout graph that has the same color. 
   Unfortunately, some circuits may exhibit symmetry that may cause different nodes to receive the same color because the “neighborhoods” associated with these nodes are similar. When two or more nodes have the same color, ambiguities in selecting matching nodes may arise. Typically, this situation is handled by making a guess as to which ones of the nodes in the schematic graph correspond to the same colored nodes in the layout graph, then assigning new colors to the matched nodes and then recoloring. If the guess was incorrect, a number of nodes may fail to match when the matching is applied at the next level of hierarchy, even though an alternate guess might have resulted in a complete one-to-one mapping. 
   For example, the AND-OR-INVERT (AOI) cell of  FIG. 1  exhibits a number of symmetries with respect to input A because input A may be independently swapped with input B or input A may be swapped with input C if and only if input B is also swapped with input D. Similar symmetries also exist with respect to inputs B, C and D.  FIG. 2  illustrates an original electrical schematic (S 1 ) of the AOI cell and an extracted layout schematic (L 1 ) of the AOI cell.  FIG. 3  illustrates an original electrical schematic (S 2 ) which contains the AOI cell S 1  of  FIG. 2  as a child cell and an extracted layout schematic (L 2 ) which contains the AOI cell L 1  of  FIG. 2  as a child cell. As will be understood by those skilled in the art, verification of the schematics of  FIG. 3  will only be concerned with the mapping of ports (W,X,Y,Z) of the AOI cell L 1  of  FIG. 2  to the ports (Q,R,S,T) of the schematic S 2  of  FIG. 2 . However, because the symmetry of the design may cause the ports of each AOI child cell in  FIG. 2  to acquire the same color when the schematic and layout graphs of the AOI child cells have been colored to an equilibrium state, a conventional LVS tool may make an arbitrary mapping which may be incorrect (e.g., Q→W, R→Y, S→X, T→Z). A consequence of this arbitrary mapping may be manifested at the next level of hierarchy. 
   For example, as illustrated by  FIG. 3 , an incorrect choice in the mapping of S 1  to L 1  (i.e., the child cells) may cause S 2  and L 2  (i.e., the parent cells) to be reported as nonequivalent after a coloring algorithm has been performed on the schematic and layout graphs at the parent level. Here, devices D 1 –D 4  are distinct devices that are connected between the ports of the AOI “child” cell and the ports of the “parent” cell. Thus, LVS software may report a mismatch between an original electrical schematic netlist and an extracted layout netlist, even though it is possible to make assignments among symmetric nodes that will result in a match. Typically, a consequence of this limitation in LVS software is that the software user must manually intervene by providing the LVS software with specific assignments to resolve ambiguities due to symmetry. Since it is not always clear where an erroneous guess was made, such manual intervention may be time consuming. For large and highly symmetric designs such as memories and gate arrays, these limitations may significantly reduce the utility of conventional LVS tools including hierarchical LVS tools using general-purpose graph isomorphism algorithms. In particular, the high degree of symmetry associated with large memories may force numerous arbitrary matchings or guesses to be made between layout and schematic. These guesses may preclude matching in the parent cells of the memories if the parent cells use permutations of the memory ports that are valid by virtue of the symmetries, but are inconsistent with the guesses. 
   To address some of these limitations associated with conventional verification tools, an LVS software tool  100  has been developed to determine equivalency between an integrated circuit schematic and an integrated circuit layout, using the operations  102 – 114  illustrated by the flow diagram of  FIG. 4 . This LVS software tool may be embodied in a commercially available product from the assignee of the present application, Avant! Corporation of Fremont, Calif. This software product, which is marketed under the tradename Hercules™, is more fully described in an instruction manual by the same name, Release 2.1, January (1997), the disclosure of which is hereby incorporated herein by reference. In particular, the LVS software tool of  FIG. 4  can perform the operations of generating a hierarchical electrical schematic netlist having at least one parent cell and a plurality of child cells in the parent cell, Block  102 , and extracting a corresponding integrated circuit layout as a hierarchical layout netlist, Block  104 . An operation is also performed to generate at least one color symmetrizing matrix corresponding to a child cell in the schematic netlist, Block  106 . Here, the child cell may have a number of symmetries which, when taken alone or in combination, may result in a number of electrically equivalent permutations of the child cell. As illustrated by Block  108 , operations are then performed to generate schematic and layout graphs of the parent cells in the schematic and layout netlists, respectively. These graphs are similar to the above-described bipartite graphs. The nodes in the schematic graph are then colored and a first color symmetry vector is generated for a child cell in the schematic graph. Similarly, the nodes in the layout graph are colored and a second color symmetry vector is generated for a child cell in the layout graph, Block  110 . 
   An operation is then performed to determine an equivalency between the colors of the nodes in the schematic and layout graphs based on a selected permutation of the child cell in the layout graph, Block  112 , and then an operation is performed to determine a vector equivalency between a product of the color symmetrizing matrix and the first color vector and a product of the color symmetrizing matrix and the second color vector, Block  114 . Finally, a membership test is automatically performed at Block  116  to determine whether the selected permutation of the child cell can be derived from the valid symmetries associated with that child cell. As described in a textbook authored by G. Butler, entitled  Fundamental Algorithms for Permutation Groups , Springer-Verlag, p. 144 (1991), a Furst-Hopcroft-Luks version of a Schreier-Sims method may be performed. Unfortunately, although the software tool of  FIG. 4  typically requires no human intervention and works well with most designs exhibiting symmetry, the automatic performance of membership test to validate the accuracy of the matched layout and schematic may incur an unduly large computational expense and limit the applicability of the above software to large integrated circuit designs having large degrees of symmetry. 
   Thus, notwithstanding the above described attempts, there continues to be a need to provide verification tools which have the capability of automatically resolving ambiguities in symmetric circuits. Such tools should be conservative in the identification of graph isomorphism, in the sense that if any ambiguities remain after the verification operations are performed, a nonisomorphism result should be generated and the circuits should be designated as non-equivalent even if they may be equivalent. This is because the penalty for erroneously identifying equivalent circuits as nonequivalent (i.e., manual intervention by the user) is far less onerous than the penalty for misidentifying non-equivalent circuits as equivalent (i.e., the expense of prototyping and manufacturing an incorrect design). 
   SUMMARY OF THE INVENTION 
   It is therefore an object of the present invention to provide improved methods, apparatus and computer program products that can perform post-layout verification of microelectronic integrated circuits. 
   It is also an object of the present invention to provide improved methods, apparatus and computer program products that can perform layout versus schematic comparison of integrated circuits. 
   It is another object of the present invention to provide methods, apparatus and computer program products that can perform layout versus schematic comparison of hierarchical integrated circuits having symmetrical cells therein. 
   These and other objects, features and advantages can be provided, according to embodiments of the present invention, by preferred methods, apparatus and computer program products for determining equivalency between integrated circuit schematics and integrated circuit layouts. These methods, apparatus and computer program products are particularly useful as layout versus schematic (LVS) comparison tools when handling hierarchical designs that exhibit high degrees of overall symmetry and/or have large numbers of cells therein having symmetrical properties. These LVS tools preferably perform the operations of coloring a graph of an integrated circuit to an equilibrium state and then identifying a “provisional” swap group of nodes having the same color within the colored graph. The colored graph may be a colored layout graph or a colored schematic graph of the integrated circuit. The colored graph may be a colored bipartite graph having both net and device nodes therein. This “provisional” swap group of nodes corresponds to pins of a device that may be swappable by virtue of the fact that the nodes associated with the pins have the same color after a graph coloring operation has been performed. A new (e.g., unique) color is then assigned to a node within the identified swap group. An operation is then performed to recolor the layout graph (with the new colored node) until another equilibrium state has been achieved. During this recoloring operation, the node receiving the assigned color may be held at the assigned color. This recoloring operation may include operations to repeatedly partition nets and devices in alternating sequence until an equilibrium colored state is achieved. At this point, a determination of whether the swap group of nodes are independently swappable can be made by evaluating whether all the nodes within the swap group, with the exception of the node that received the assigned color, have the same color. If they do, then all the nodes in the swap group can be treated as independently swappable nodes. 
   Preferred embodiments of LVS comparison tools may also perform an operation of identifying at least one connected group within the swap group. This operation preferably comprises determining a first connected group of nodes as nodes within the swap group that are linked together (and to the node receiving the assigned color) by other nodes that have non-unique colors within the graph. Then, after the graph has been recolored to an equilibrium state, an operation can be performed to determine whether the nodes within the first connected group that did not received the assigned color all have the same color. If so, these nodes within the first connected group can be treated as being independently swappable (i.e., as their own swap group), even if all the nodes within the “provisional” swap group are not independently swappable. 
   Once nodes that belong to respective connected groups have been identified, subsequent or concurrent operations may also be performed to generate a first vector of first colors corresponding to a first swap group of nodes associated with a first child cell in an integrated circuit schematic graph having symmetric pins that include independently swappable pins and dependently swappable pins. Operations may also be performed to generate a second vector of second colors corresponding to a second swap group of nodes associated with a second child cell in an integrated circuit layout graph. These graphs may be graphs of first and second parent cells containing the first and second child cells, respectively, and nets connected to input nodes of the respective child cells. To resolve symmetries identified by the presence of duplicate colors within the first and second vectors, an assignment of a new color is made to a node within the first swap group of nodes in the schematic graph and a corresponding assignment is made to a node within the second swap group of nodes in the layout graph. This new color is preferably a unique color that is not within the first vector. This assignment of a new color to a node of a child cell is, when viewed at the parent level, actually an assignment of a color to a net that may be connected to one or more pins of a child cell(s). At the parent level, this net may be treated as a single node of the graph having a single color and each pin of a child cell that is connected to the net may be treated herein as having an effective “color” equal to the color of the net. Operations are then performed to generate a third vector of third colors corresponding to the first swap group by determining a new color for one of the nodes within the first swap group using a coloring operation that is a function of a device value of the first child cell and a color of another node in the first swap group. A fourth vector of fourth colors corresponding to the second swap group are also generated by determining a new color for one of the nodes within the second swap group using a coloring operation that is a function of a device value of the second child cell and a color of another node in the second swap group. 
   In particular, an embodiment of the operation to generate a third vector includes generating a new color for a first node in the first swap group using a coloring operation that is a function of a device value of the first child cell and a respective color of each of the other nodes in the first swap group. To reduce computational expense, this node/pin coloring operation is independent of the colors of nodes that are dependently swappable with the first node. Similarly, the operation to generate a fourth vector includes generating a new color for a second node within the second swap group using a coloring operation that is a function of a device value of the second child cell and a respective color of each of the other nodes in the second swap group. Again, to reduce computational expense, this node coloring operation is independent of the colors of nodes that are dependently swappable with the second node. More preferably, the operation to generate a third vector comprises generating a new color for a node within the first swap group using the following relationship: New Color=Existing Node Color−(Device Value)×(Σ Colors of Other Nodes in Respective Swap Group). An identical relationship is also preferably used to generate a fourth vector. 
   Preferred coloring operations performed by LVS tools of the present invention may also include initially determining a first vector of first colors corresponding to a first swap group of nodes associated with a child cell in an integrated circuit layout graph and then assigning a unique node color not within the first vector of colors to a first node within the first swap group. Operations are then performed to generate a second vector of colors corresponding to the first swap group by generating a new color for the first node using the following relationship: New Color=Unique Node Color−(Device Value of Child Cell)×(Σ Colors of Other Nodes in the First Swap Group). Preferably, this relationship does not require the computational expense associated with conventional operations to generate a product of a color symmetrizing matrix and a vector of node colors for a child cell. 
   Operations to perform layout versus schematic comparison may also include operations to color a schematic graph of a parent cell to an equilibrium state and then recolor nets connected to first and second child cells having the same device value. The operation to recolor the nets preferably uses a net coloring operation that recolors a first plurality of symmetric pins of the first child cell and recolors a second plurality of symmetric pins of the second child cell. Operations are then performed to generate distinct device values for the first and second child cells by determining a first product of the colors of the recolored first plurality of symmetric pins and a second product of the colors of the recolored second plurality of symmetric pins. These operations to determine distinct device values are preferably performed in order to prevent an incorrect mapping of these child cells in the schematic and layout during subsequent device coloring operations. These operations preferably include determining a new device color for one cell (e.g., AOI cell) in the schematic by evaluating a product of the symmetric pin colors. In particular, the new device value for the child cell can be determined using the following relationship: New Device Value=Old Device Value+π(Colors of Symmetric Pins of Cell). Alternatively, the new device value may be determined as the product of the colors of the symmetric pins. As determined by the inventors herein, merely determining an algebraic sum of the colors of the symmetric pins may not distinguish the AOI cells during subsequent coloring operations. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is an electrical schematic of an AND-OR-INVERT (AOI) circuit. 
       FIG. 2  is an electrical schematic of an AOI cell (S 1 ) and a schematic of an AOI cell (L 1 ). 
       FIG. 3  is an electrical schematic of a parent cell (S 2 ) containing the cell S 1  of  FIG. 2  and an extracted schematic of a parent cell (L 2 ) containing the cell L 1  of  FIG. 2 . 
       FIG. 4  is a flow diagram of operations performed by a layout versus schematic (LVS) comparison tool according to the prior art. 
       FIG. 5  illustrates a functional block diagram of microelectronic integrated circuit synthesis, layout, verification and manufacturing tools according to the present invention. 
       FIG. 6  illustrates a general hardware description of post-layout verification tools according to the present invention. 
       FIG. 7  illustrates general operations performed by post-layout verification tools, methods and computer program products according to the present invention. 
       FIG. 8  is a flow diagram of operations performed by a layout versus schematic (LVS) comparison tool according to the prior art. 
       FIG. 9  is a partial flow diagram illustrating operations to detect absence of spurious symmetries in color symmetrizing matrices, according to the flow diagram of  FIG. 8 . 
       FIG. 10  illustrates a graphical representation of the symmetrical AOI circuit of  FIG. 1 . 
       FIG. 11  illustrates operations to determine a first product of the color symmetrizing matrix (for the AOI circuit of  FIG. 1 ) and a first color symmetry vector corresponding to an AOI child cell in a schematic graph, a second product of the color symmetrizing matrix and a second color symmetry vector corresponding to an AOI child cell in a layout graph configured as a good permutation, and a third product of the color symmetrizing matrix and a third color symmetry vector corresponding to an AOI child cell in a layout graph configured as a bad permutation. 
       FIGS. 12A and 12B  illustrate electrical schematics of parent cells having the same symmetries as respective child cells therein. 
     FIGS.  13 A–F and  14 A–F illustrate a sequence of bipartite schematic and layout graphs corresponding to the AOI child cell of  FIG. 1 , respectively. 
     FIGS.  15 A–C and  16 A–C illustrate a sequence of bipartite schematic and layout graphs corresponding to the parent cell of  FIG. 3 , respectively. 
       FIGS. 17A–D ,  18 A–D and  19 A–D illustrate a sequence of schematic graphs, correct layout graphs and incorrect layout graphs of a parent cell of  FIG. 3 , respectively. 
       FIG. 20  is a flow diagram of operations performed by a layout versus schematic (LVS) comparison tool according to a first embodiment of the present invention. 
       FIG. 21  illustrates operations to determine the presence of swappable nodes of a bipartite graph, in accordance with the embodiment of  FIG. 20 . 
       FIG. 22  is a flow diagram of operations performed by a layout versus schematic (LVS) comparison tool according to a second embodiment of the present invention. 
       FIG. 23  illustrates application of preferred coloring operations to a first exemplary circuit. 
       FIG. 24  illustrates application of the coloring operations of  FIG. 22  to a second exemplary circuit. 
       FIG. 25  illustrates application of preferred coloring operations to a third exemplary circuit. 
       FIG. 26  illustrates application of preferred coloring operations to a fourth exemplary circuit. 
       FIG. 27  illustrates application of preferred coloring operations to a fifth exemplary circuit. 
   

   DESCRIPTION OF A PREFERRED EMBODIMENT 
   The present invention now will be described more fully hereinafter with reference to the accompanying drawings, in which preferred embodiments of the invention are shown. This invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art. The operations of the present invention, as described more fully hereinbelow, may be performed by an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product on a computer-readable storage medium having computer-readable program code embodied in the medium. Any suitable computer-readable medium may be utilized including hard disks, CD-ROMs or other optical or magnetic storage devices. Like numbers refer to like elements throughout. 
   Referring now to  FIG. 5 , operations performed by a preferred computer aided microelectronic integrated circuit synthesis, layout, verification and manufacturing system  10  will be generally described. The system may include four general subsystems: an integrated circuit functional specification and logic synthesis system  200 , an integrated circuit layout system  300 , a post-layout verification system  400  and an integrated circuit manufacturing system  500 . The microelectronic integrated circuit functional specification and logic synthesis system  200  may include circuit libraries or macros for such elements as logic gates, flip-flops, registers, memory elements, drivers and buffers, for example. The functional specification and logic synthesis system  200  typically provides a vehicle for generating and displaying an electrical schematic of an integrated circuit and producing a computer readable representation of the schematic as a design netlist. The microelectronic integrated circuit layout system  300  typically provides a vehicle for generating a physical layout by placing and routing an electrical circuit schematic generated by the functional specification and logic synthesis system  200 . A wiring program may also be provided with the layout system  300  for automatically determining the placement of the wiring interconnect nets between active device elements within the microelectronic integrated circuit. 
   A verification system  400  is also preferably provided for performing an independent verification of the physical layout to ensure compliance with the requirements of the functional specification and logic synthesis system  200  as well as the manufacturing system  500 . Accordingly, the verification system  400  is typically referred to as a “post-layout” verification system and is typically employed near the end of the design process. In addition to acting as an independent verification of the operability and correctness of the layout of the circuit design, the verification system  400  may provide means by which changes and optimizations can be performed. As will be understood by those skilled in the art, various other types of analyses such as timing analysis and circuit/logic simulation may be performed to check whether the specifications and requirements of the first two subsystems  200  and  300  are satisfied. After verification, the physical layout is forwarded to the manufacturing system  500  to produce the integrated circuit. The microelectronic circuit manufacturing system  500  may generate the required masks, and may control the manufacturing tools necessary to fabricate the integrated circuit on a semiconductor wafer, for example. 
   It will be understood by those having skill in the art that the integrated circuit functional specification and logic synthesis system  200 , microelectronic circuit layout system  300  and various parts of the microelectronic integrated circuit manufacturing system  500  may be realized in whole or in part as software modules running on a computer system. Alternatively, a dedicated stand-alone system with application specific integrated circuits for performing the above described functions may be provided. The general design and operation of the functional specification and logic synthesis system  200 , layout system  300  and manufacturing system  500  are well known to those having a skill in the art and need not be described further herein. 
   Referring now to  FIG. 6 , a general hardware description of a computer workstation containing, among other things, the integrated circuit post-layout verification system  400  of  FIG. 5  will be described. As shown by  FIG. 6 , the workstation  20  preferably includes a computer  15  containing a post-layout verification software sub-system  400  running thereon. The post-layout verification system  400  may accept a computer representation of the electrical schematic and layout of an integrated circuit via a file  19 , disk input  23  or data bus  27 . A display  13  and a printer  17  are also preferably provided to assist in verifying the layout and design of the integrated circuit. The hardware design of the above described components  13 ,  17 ,  19 ,  27  and  23  is well known to those having skill in the art and need not be described further herein. As one example, the post-layout verification system  400  may be installed on a computer system such as an UltraSPARC 5 workstation manufactured by Sun Microsystems Computer Corporation (see, http://www.sun.com). 
   Referring now to  FIG. 7 , there is illustrated a general overview of operations that are preferably performed by embodiments of post-layout verification systems  400  of the present invention. In particular, the post-layout verification system  400  is illustrated as performing a plurality of operations  410 ,  430 A– 430 B and  440 – 450 . For the sake of clarity, these operations are illustrated and described as discrete operations, however, in practice many of these operations may be combined and performed in parallel. Additional operations may also be performed by the post-layout verification system  400 . The first operation  410  is performed by a design rule checking (DRC) tool which runs comprehensive hierarchical design rule checks on a geometric database containing the physical layout of an integrated circuit to be verified. The operations performed by the design rule checking tool include, among other things, checking for area, grid, length, size, spacing, corner, enclosure, intersection and overlap errors, for example. The second operations  430 A– 430 B are performed by a preferred hierarchical layout versus schematic (LVS) tool which extracts a hierarchical netlist from the geometric layout database and compares it to the electrical schematic netlist. Layout and schematic logic agreement is then verified at hierarchical matching points, for example. 
   The third operation  440  is performed by a layout parameter extraction (LPE) tool. This tool may, among other things, perform an initial operation of modeling each of a plurality of interconnect nets within the integrated circuit layout in order to obtain estimates of the parasitic resistance and capacitance of the nets. An exemplary LPE tool is described in detail in U.S. Pat. No. 5,896,300 to Raghavan et al., entitled “Methods, Apparatus and Computer Program Products for Performing Post-Layout Verification of Microelectronic Circuits by Filtering Timing Error Bounds for Layout Critical Nets”, assigned to the present assignee, the disclosure of which is hereby incorporated herein by reference. Once this layout parameter extraction operation  440  is complete, modeling and simulation operations  450  are performed. The results obtained from these modeling and simulation operations may also be passed back to the layout parameter extraction tool  440 . As illustrated by the reverse upward sequence of arrows in  FIGS. 5 and 7 , the performance of post-layout verification may necessitate redesign of the integrated circuit by the functional specification and logic synthesis system  200  and/or modifying the physical layout using the layout system  300 . Finally, in the event the integrated circuit is verified for design rule compliance and the layout versus schematic (LVS) comparison and modeling and simulation operations yield acceptable results, the physical layout is forwarded to the manufacturing system  500  to produce the integrated circuit. 
   To further exemplify the characteristics of state-of-the-art LVS tools that may be embodied in post-layout verification systems and provide further background to the subject matter of the present invention, the operations illustrated by the flowcharts of  FIGS. 8 and 9  will be described. Such LVS tools are more fully described in detail in U.S. Pat. No. 6,009,252 to Lipton, entitled “Methods, Apparatus and Computer Program Products for Determining Equivalencies Between Integrated Circuit Schematics and Layouts Using Color Symmetrizing Matrices”, assigned to the present assignee, the disclosure of which is hereby incorporated herein by reference. It will be understood that each block of the flowchart illustrations, and combinations of blocks in the flowchart illustrations, can be implemented by computer program instructions. These computer program instructions may be provided to a processor or other programmable data processing apparatus to produce a machine, such that the instructions which execute on the processor or other programmable data processing apparatus create means that implement the functions specified in the flowchart block or blocks. These computer program instructions may also be stored in a computer-readable memory that can direct a processor or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means that implement the functions specified in the flowchart block or blocks. Accordingly, blocks of the flowchart illustrations support combinations of means that perform the specified functions, combinations of steps that perform the specified functions and program instruction means that perform the specified functions. 
   In particular, the operations  420  illustrated by  FIGS. 8–9  include operations to determine equivalency between hierarchical integrated circuit schematics and hierarchical integrated circuit layout designs using color symmetrizing matrices and operations to detect absence of spurious symmetries in the color symmetrizing matrices. These operations may be performed to reduce the computational expenses associated with the automatic performance of complex membership tests to determine whether permutations of cells in a hierarchical integrated circuit layout design can be derived from valid symmetries of the respective cells therein. 
   The operations include representing the integrated circuit schematic as a hierarchical schematic netlist having at least one child cell therein and a parent cell containing the at least one child cell, Block  421 . An operation is also performed to extract the integrated circuit layout as a hierarchical layout netlist, Block  422 . This operation may be performed using conventional techniques. As described more fully hereinbelow with respect to an exemplary portion of an integrated circuit containing an AND-OR-INVERT (AOI) cell therein as a child cell, a color symmetrizing matrix corresponding to the child cell is also generated, Block  423 . The child cell may also comprise a device having AOI-type swappable symmetry with respect to its input pins (which have classes, but do not receive colors) and input nodes (which receive colors). Input nodes connected to swappable input pins are referred to herein as “swappable nodes”. A bipartite schematic graph of the parent cell in the hierarchical schematic netlist is also generated along with a bipartite layout graph of the parent cell in the hierarchical layout netlist, Block  424 . The nodes in the schematic graph are then colored and a first color symmetry vector for the child cell therein is generated from the colors of the “input” nodes of the child cell in the graph, Block  425 . These “input” nodes of the child cell in the graph correspond to the “net” nodes of the graph that are connected to the input pins of the child cell. Similarly, the nodes in the layout graph are colored and a second color symmetry vector for the child therein is generated, Block  426 . An equivalency between the colors of the nodes in the schematic graph and the colors of the nodes in the layout graph is then determined, Block  427 . 
   Based on this equivalency, an operation may then be performed to determine a vector equivalency between a product of the color symmetrizing matrix and the first color symmetry vector and a product of the color symmetrizing matrix and the second color symmetry vector, Block  428 . If these vectors are equivalent, an equivalency between the integrated circuit schematic and the integrated circuit layout may be present. However, a possibility may still exist that with respect to the corresponding symmetric child cells in the schematic and layout, isomorphism (i.e., one-to-one correspondence) between the schematic and layout is not present. 
   To address this possibility, an operation may be performed to detect the absence of a spurious symmetry in the color symmetrizing matrix. If an absence is detected, the presence of the vector equivalency will unequivocally establish the one-to-one correspondence with respect to the child cell being analyzed. Therefore, the need to perform a computationally expensive membership test, for example, to determine whether a selected permutation can be derived from valid symmetries can be successfully eliminated. In particular, rather than automatically performing a complex membership test which may require considerable amounts of computational expense to confirm equivalency (particularly for large integrated circuits such as memories and gate arrays which have high degrees of symmetry), an operation is performed to detect absence of a spurious symmetry in the color symmetrizing matrix, Block  429 , by initially checking whether the number of actual symmetries of the color symmetrizing matrix is equivalent to the number of different symmetrical port configurations or permutations of the child cell undergoing analysis (i.e., the number of desired symmetries), Blocks  429 A– 429 B. As illustrated by Blocks  429 C and  429 D, if the number of actual symmetries of the color symmetrizing matrix is greater than the number of desired symmetries, a membership test will need to be performed to verify equivalency. If, on the other hand, the number of actual and desired symmetries are determined to be the same which, in practice, has been found to be true in the vast majority of cases, the computationally expensive membership test can be eliminated altogether. These operations are repeatedly performed for all cells, at each level of hierarchy. 
   Further description of the operations of  FIGS. 8 and 9  will now be provided by illustrating performance of these operations on an integrated circuit containing an AND-OR-INVERT (AOI) cell therein as a child cell. In particular, schematic and layout netlists are initially generated for corresponding schematic and layout designs of a hierarchical integrated circuit. Here, the layout netlist may be generated by extracting the layout design using conventional techniques well known to those skilled in the art. The schematic and layout netlists are then abstracted into respective bipartite graphical representations where the nodes of the graph represent devices or nets. For example,  FIG. 10  illustrates a graphical representation of the symmetrical AOI circuit of  FIG. 1 . The graphical schematic and layout representations are provided with an initial partitioning of nodes which distinguishes between nets and devices, and also between external nets (e.g., input nodes A, B, C, D and output port Z) and internal nets (N 1  and N 2 ). As will be understood by those skilled in the art, graphical automorphism detection operations are then performed on the graphical representations. Here, the purpose of the initial partitioning is to constrain the automorphism operations to find symmetries that do not involve swapping between partitions. Moreover, because the ability to swap devices and nets is an artifact of the graphical abstraction and not a real possibility, it may be excluded. Swapping of input nodes and internal nets may also be excluded because hierarchical analysis is only concerned with symmetries among the input nodes. 
   Using conventional techniques such as those described in the aforementioned Ebeling articles, a conventional iterative coloring process can then be applied to the nodes of the schematic and layout graphs. As will be understood by those skilled in the art, the establishment of unique colors at the nodes of the schematic and layout graphs typically permits matching of the nodes. Nodes that may be swapped by a valid symmetry will also have identical colors (although not all nodes with identical colors are necessarily swappable). However, in order to distinguish between good and bad permutations, a set (e.g., vector) of color values will need to be established using a conventional coloring function (F) that depends not just on the neighbors of a respective node, but on all the nodes that share a symmetry with it. Here, a coloring function is desired that results in the same set of colors when applied to valid permutations and a different set of colors when applied to invalid permutations. In particular, when the layout is a valid permutation of the schematic, the coloring function will result in the same set of colors for the layout and schematic, and these colors will be related by the permutation. On the other hand, when the layout is not a valid permutation of the schematic, the coloring function will result in different sets of colors. 
   A preferred coloring function (F) may be characterized algebraically, where:
           n≡vector of color values   P g , P b ≡good &amp; bad permutations     n′≡new color values     n′≡F( n)
 
Require:
   F ( P   g (   n ))= P   g (   n ′)= P   g ( F (   n ))   F ( P   b (   n ))≠ P   b (   ′n )  P   b ( F (   n )) 
Assume F and P can be represented by matrix multiplication:
   F (   n )= F     n      P (   n )= P     n    
Require ∀ n:
   FP   g     n=P   g   F     n    ( FP   b     n ) i ≠( P   b   F     n ) i          

   Note that the condition that (FP b   n) i ≠(P b F n) i , is more stringent than the condition that FP g   n=P g F n because not only must the two sides of the equation be unequal, but each component of each side must be unequal. With this understood, a simplification may be performed:
 
 FP   g   =P   g   F  
 
 FP   b   ≠P   b   F  
 
Finally:
 
 P   g   −1   FP   g   =F  
 
 P   b   −1   FP   b   ≠F  
 
   These equations describe a matrix F which is symmetric with respect to the permutation represented by P g , but not symmetric with respect to P b . Thus, the desired function is a matrix which exhibits symmetry with respect to all legal permutations and violates symmetry with respect to all illegal permutations. 
   As described above, this preferred matrix is a color symmetrizing matrix (CSM). An exemplary CSM which encodes the symmetries of the AOI cell of  FIG. 1  is provided as: 
       CSM   =     [           0   ⁢             11       13       13           11         0   ⁢             13       13           13       13         0   ⁢             11           13       13       11         0   ⁢               ]         
 
   As illustrated by  FIG. 11 , respective products of the CSM and (i) a vector of colors for the input nodes (pins) of the schematic AOI, (ii) a correctly permuted vector of colors associated with the input nodes (pins) of the layout AOI and (iii) an incorrectly permuted vector of colors associated with the input nodes of the layout AOI, results in two vectors (486, 514, 350, 245) and (245, 350, 486, 514) which have the same set of color values and are thereby equivalent, and one vector (454, 426, 438, 277) which is different. The vectors which agree correspond to the schematic and layout based on a “good” permutation of the AOI cell. Based on these operations, a determination of vector equivalency for all cells at each level of hierarchy, Block  428  of  FIGS. 8 and 9 , can be used to determine LVS equivalency. 
   A preferred approach to form the above CSM includes building the matrix in the most restrictive way possible so that the matrix is guaranteed to have all the desired symmetries. The matrix may be constructed by starting with an empty (zero) matrix (with dimensions equal to the number of interdependent input nodes: e.g., a 4×4 matrix for an AOI cell) and inserting an arbitrary number somewhere into the matrix. Prime numbers are preferably used to minimize the probability of collisions. This prime number is then copied into every entry whose indices can be related to the indices of the original entry by some combination of legal permutations. The result is a matrix that is symmetric with respect to all legal permutations. If there are any empty entries in the matrix remaining, a new and different number is inserted into an empty entry, and the process is repeated. This procedure is continued until there are no empty entries. At the end of any given iteration, the CSM matrix is provided with the desired symmetries, however, by iterating until the matrix is full, it is possible to minimize the probability of spurious symmetries. 
   A preferred method of counting the number of symmetries of the CSM is based on a version of the Schreier-Sims algorithm. This version is more fully described in Chapter 10 of the aforementioned textbook by G. Butler entitled “ Fundamental Algorithms for Permutation Groups ” and in Chapter 14 of a textbook by P. Cameron entitled “ Combinatorics ” Cambridge University Press (1994), the disclosures of which are hereby incorporated herein by reference. In particular, this preferred method includes determining a number N equal to the index of the last row of the CSM matrix (N=4 for the CSM matrix corresponding to the AOI cell). The orbit of N is then determined. Here, an orbit is defined as the set of indices (include N itself) which N may be mapped into by any legal permutation or combination of permutations. Next, any permutations which involve moving N are ignored so that attention is restricted to the subgroup that is the stabilizer of N. Then, counting backwards from N, the next non-unique row is found and N is then set to this index. These steps are repeated until the subgroup is of the order one. 
   Unfortunately, it is not possible to guarantee that such a matrix will not have “spurious” symmetries which might enable an incorrectly permuted vector of colors to be reported as correct. If a spurious symmetry is present in the CSM, an LVS comparison tool may still make a correct mapping, but it is not forbidden from making an incorrect mapping. However, as described in the &#39;252 patent to Lipton, an incorrect mapping can be prevented by determining the number of actual symmetries of the CSM (i.e., the order of the permutation group which leaves the matrix invariant), Block  429 A in  FIG. 9 , and then comparing this number with the number of desired symmetries of the particular child cell being analyzed. If these two numbers are equal, the CSM matrix will be guaranteed to have all desired symmetries and no spurious symmetries. Thus, the determination of vector equivalency at Block  428  of  FIGS. 8 and 9  will unequivocally establish equivalency between the schematic and layout. If the number of actual symmetries of the CSM is greater than the number of desired symmetries (the number of actual symmetries of the CSM should never be less), a further check will need to be performed to verify LVS equivalency. This further check includes performing a membership test, preferably using a Schreier-Simms algorithm, to verify that the final mapping is a valid combination of the original symmetries. 
   As will be understood by those skilled in the art, the Schreier-Simms algorithm is a standard method which can be used to determine whether an arbitrary permutation may be expressed by some combination of generated permutations. If the mapping fails this test, the schematic and layout are reported as nonequivalent, but if the mapping passes this test, the schematic and layout will be reported as equivalent notwithstanding the fact that the number of actual symmetries of the CSM was greater than the number of desired symmetries associated with the symmetric cell under analysis. Thus, the LVS comparison tools of  FIGS. 8–9  can be used to perform post-layout verification of larger and more highly integrated circuits because the need to always perform computationally expensive membership tests to determine whether a selected permutation of a symmetric cell can be derived from valid symmetries, can be frequently eliminated. 
   A conventional LVS tool may have limited capability to propagate symmetries by inferring symmetries of the parent cell from the symmetries of a child cell therein. An example of a parent cell which exhibits the same symmetries as a child cell therein is illustrated by  FIG. 12A . Accordingly, an LVS tool may not be able to take full advantage of the node symmetries of the cell of  FIG. 12A  when analyzing a schematic and layout containing such a symmetric cell. Symmetries may, however, be readily inferred when the symmetric nodes of the child cell (e.g., AOI cell) are connected directly to the input nodes of the parent cell, with no intervening devices, as illustrated by the parent cell of  FIG. 12B . An LVS tool preferably takes these latter symmetries into account so that the symmetries of a child cell may be propagated to a parent cell when the LVS tool is evaluating a grandparent cell containing the parent cell. 
   A general analysis of a simple two-level integrated circuit design is now provided to demonstrate how a conventional method of arbitrarily breaking symmetry in a child cell may result in an erroneous mismatch of the parent cells and how a method of encoding the symmetries of a child cell using the color symmetrizing matrix (CSM) will automatically resolve symmetries in the parent cell. First, a partition function P is generated as:
 
 P   NET =[node value+3Σ(neighbor node values)+137Σ(pin class)]mod 256
 
 P   DEVICE =[node value+3Σ(neighbor node values)+137(device class)]mod 256
 
   These partition functions are chosen to depend on the local structure of the bipartite schematic and layout graphs illustrated by  FIGS. 13–14 , respectively, and be independent of labeling. The coefficients of “3” and “137” are arbitrary, and are chosen to minimize the likelihood of collisions (i.e., the possibility that different sets of input values might accidentally result in the same function value). The modulus of “256” is used here for convenience, but is preferably selected as large as possible to minimize collisions (typically the largest unsigned integer permitted by the hardware performing the LVS operations).  FIGS. 13 and 14  illustrate how the method of symmetry breaking results in a valid mapping from schematic nodes to layout nodes A→D′, B→C′, C→B′ and D→A′. Node values are shown next to each node. Each node value identifies the color of a net (connected to a pin of a device) or the color of a device and the terms “node value” and “color” are used interchangeably herein. Numbers in parentheses are pin classes when adjacent to an edge, or device classes when adjacent to a device node. 
   Referring now specifically to  FIGS. 13A and 14A , the nodes of the graphs are initialized by assigning node values/colors as equal to the number of adjacent nodes. After the graphs are initialized, a graph recoloring operation may be performed using a sequence of alternating net and device coloring operations. For example, as illustrated by  FIGS. 13B and 14B , a device coloring operation may be performed by partitioning devices. The node value of 152 for the AND1 and AND2 devices and the node value of 36 for the NOR device are obtained using the device partition function P DEVICE :
 
152=[3+3(1+1+2)+137(1)]mod256
 
36=[3+3(2+2+1)+137(2)]mod256
 
   Following the device partition of  FIGS. 13B and 14B , a net coloring operation is performed by partitioning the nets, as illustrated by  FIGS. 13C and 14C . The node value of 82 for the nets A, B, C and D, the node value of 209 for the nets N 1  and N 2  and the node value of 127 for the net OUT are obtained using the net partition function P NET :
 
82=[1+3(152)+137(1)]mod256
 
209=[2+3(152+36)+137(2+1)]mod256
 
127=[1+3(36)+137(2)]mod256
 
   At this point, the symmetries of the bipartite graphs are manifested by the similar node values/colors, and it is apparent that further partitioning will not resolve symmetries. When this occurs, the graphs can be treated as being in respective equilibrium states, because further net and device coloring operations will not result in a change in the number of distinct colors. Because of the symmetry, a guess can be made. That is, a matching is arbitrarily imposed between two nodes of the same value, and they are given a new unique (arbitrary) value. In particular, in  FIGS. 13D and 14D , a selection is made between AND1 in the schematic and AND2 in the layout and their node values are set to 103. It would also have been acceptable to make the selection between AND2 in the schematic and AND1 in the layout, or any matching among the other nonsingular partitions (i.e., between (A,B,C,D) and (A′,B′,C′,D′) or between (N 1 , N 2 ) and (N 1 ′, N 2 ′)). Referring now to  FIGS. 13E and 14E , another net partition is performed and once again it is apparent that no further partitioning will reduce the symmetry. Finally, in  FIGS. 13F and 14F , the remaining matches (i.e., two net assignments) necessary to a complete mapping from schematic to layout are made. 
   Referring now to FIGS.  3  and  13 – 16 , an illustration is provided on how to use of a conventional method of comparing a parent cell schematic with a parent cell layout (that is valid by virtue of the symmetries of the AOI cell) may result in apparent non-equivalence. At the parent level illustrated by  FIGS. 15–16 , the AOI child cell is represented as a single node with four input nodes (the output node has no bearing on the results in this case, and is omitted for simplicity). The input nodes of the AOI child cell are differentiated (consistent with the schematic and layout graphs of  FIGS. 13E and 14E , respectively) by the assignment of a distinct pin class to each pin, as illustrated. Matching pins in the schematic and layout are assigned the same class. The input nodes A, B, C and D of the AOI are connected to four distinct devices (D 1 –D 4 ), each of which has a different device class, as illustrated by  FIGS. 15A and 16A . 
   Referring now to  FIGS. 15B–15C  and  FIGS. 16B–16C , a single device partition followed by a single net partition (using the above partition functions) is sufficient to attain unique values for all nodes. The apparent non-equivalence, manifested in  FIGS. 15C and 16C  by differing node values between the schematic and layout, is due to the fact that the pin classes (as assigned) do not exhibit the node symmetries of the AOI child cells. As will be understood by those skilled in the art, this is a consequence of an unfortunate choice when an arbitrary match was assigned in  FIGS. 13D and 14D . 
   Referring now to  FIGS. 17A–17D ,  18 A– 18 D and  19 A– 19 D, a graphical illustration of the use of a color symmetrizing matrix (CSM) to resolve symmetries, is provided. In particular, in  FIGS. 17A ,  18 A and  19 A, the bipartite graphs corresponding to a schematic of the device of  FIG. 3 , a correct layout of the device of  FIG. 3  and an incorrect layout of the device of  FIG. 3 , are initialized. However, unlike the graphs of  FIGS. 15–16  (where different classes were assigned to each pin), all pins that are related by a symmetry and may be interchanged by some valid permutation (i.e., share the same orbit), are assigned the same class. In the method illustrated by  FIGS. 15–16 , such assignment may result in forbidden permutations being evaluated as correct. However, the LVS tools described in accordance with the &#39;252 patent to Lipton use color symmetrizing matrices (CSM) to prevent forbidden permutations from being evaluated as correct. 
   Referring now to  FIGS. 17A–17D , a partitioning of the schematic graph of the device of  FIG. 3  is provided, with the CSM being applied to the input nodes of the AOI (which are arranged as a vector) as a final step. In particular, in  FIG. 17A  the schematic graph is initialized, in  FIG. 17B  a device partition is performed and in  FIG. 17C  a net partition is performed. The CSM shown below is then applied to a vector of colors corresponding to the pins of the AOI in  FIG. 17C  which are arranged from A–D as: [180, 79, 234, 133]. Using a modulus of 256, as described above, a new vector of colors is generated as [8, 95, 222, 53] in  FIG. 17D . 
           [           0   ⁢             11       13       13           11         0   ⁢             13       13           13       13         0   ⁢             11           13       13       11         0   ⁢               ]     ⁡     [         180           79           234           133         ]       ⇒     [         8           95           222           53         ]         
 
   Referring now to  FIGS. 18A–18D , a partitioning of the correct layout graph is provided, with the CSM being applied to the pins of the AOI (which are arranged as a vector) as a final step. This correct layout graph is equivalent to the schematic by a valid permutation of A→D′, B→C′, C→B′ and D→A′. In particular, in  FIG. 18A  the correct layout graph is initialized, in  FIG. 18B  a device partition is performed and in  FIG. 18C  a net partition is performed. The CSM is then applied to a vector of colors corresponding to the pins of the AOI in  FIG. 18C  which are arranged from A–D as: [133,234, 79, 180]. Using a modulus of 256, a new vector of colors is generated as [53, 222, 95, 8] in  FIG. 18D . This vector is equivalent to the corresponding vector associated with the schematic graph of  FIG. 17D . 
           [           0   ⁢             11       13       13           11         0   ⁢             13       13           13       13         0   ⁢             11           13       13       11         0   ⁢               ]     ⁡     [         133           234           79           180         ]       ⇒     [         53           222           95           8         ]         
 
   Referring now to  FIGS. 19A–19D , a partitioning of the incorrect layout graph is provided, with the CSM being applied to the input nodes of the AOI (which are arranged as a vector) as a final step. This incorrect layout graph is not equivalent to the schematic because the following invalid permutation of A→A′, B→C′, C→B′ and D→D′ was applied. Here, in  FIG. 19A  the incorrect layout graph is initialized, in  FIG. 19B  a device partition is performed and in  FIG. 19C  a net partition is performed. The CSM is then applied to a vector of colors corresponding to the input nodes of the AOI in  FIG. 19C  which are arranged from A–D as: [180, 234, 79,133]. Here, all graphs of  FIGS. 17C ,  18 C and  19 C exhibit the same set of node values, so there is no indication that the incorrect layout is incorrect. Yet, by the above CSM and a modulus of 256, a new vector of colors is generated as [210,128,189, 107] in  FIG. 19D . The resulting vector of  FIG. 19D  is clearly not equivalent to the corresponding vector associated with the schematic graph of  FIG. 17D  or the layout graph of  FIG. 18D . 
           [           0   ⁢             11       13       13           11         0   ⁢             13       13           13       13         0   ⁢             11           13       13       11         0   ⁢               ]     ⁡     [         180           234           79           133         ]       ⇒     [         210           128           189           107         ]         
 
   Accordingly, after application of the CSM, the correct layout and schematic still share a common set of node values, but the incorrect layout contains node values that do not appear in the schematic. Thus, by setting the classes of the related ports equal, the application of the CSM may distinguish between good and bad permutations. 
   Referring now to  FIGS. 20–21 , preferred operations performed by embodiments of a layout versus schematic (LVS) comparison tool  430 A according to a first embodiment of the present invention will be described. These operations include coloring a graph of an integrated circuit to an equilibrium state and then identifying a “provisional” swap group of nodes having the same color within the colored graph, Blocks  430 A 1 ,  430 A 2 . The colored graph may be a colored layout graph or a colored schematic graph of the integrated circuit. For example, as illustrated by  FIG. 21 , the colored graph may be a colored bipartite graph having both net and device nodes therein. This “provisional” swap group of nodes corresponds to pins of a device that may be swappable by virtue of the fact that the nodes associated with the pins have the same color after a graph coloring operation has been performed. A new (e.g., unique) color is then assigned to a node within the identified swap group, Block  430 A 3 . An operation is then performed to recolor the layout graph (with the new colored node) until another equilibrium state has been achieved, Block  430 A 4 . During this recoloring operation, the node receiving the assigned color may be held at the assigned color. This recoloring operation may include operations to repeatedly recolor nets and recolor devices in alternating sequence until an equilibrium colored state is achieved. These recoloring operations may be performed using conventional partitioning functions, as described above. At this point, a determination of whether the swap group of nodes are independently swappable can be made by evaluating whether all the nodes within the swap group, with the exception of the node that received the assigned color, have the same color, Block  430 A 5 . If they do, then all the nodes within the swap group can be treated as independently swappable nodes. 
   The preferred LVS comparison tool  430 A may also perform an operation of identifying at least one connected group within the swap group. This operation preferably comprises determining a first connected group of nodes as nodes within the swap group that are linked together (and to the node receiving the assigned color) by other nodes that have non-unique colors within the graph. Then, after the graph has been recolored to an equilibrium state, an operation can be performed to determine whether the nodes within the first connected group that did not receive the assigned color all have the same color. If so, these nodes and the assigned node within the first connected group can be treated as being independently swappable (i.e., as their own swap group), even if all the nodes within the “provisional” swap group are not independently swappable. 
   These preferred operations will now be described more fully hereinbelow with respect to  FIG. 21  which illustrates a bipartite graph of a dual-AOI cell. As will be understood by those skilled in the art, this graphical illustration of a cell may be represented within a layout and/or schematic netlist of an integrated circuit using conventional techniques. The illustrated dual-AOI cell has eight (8) input ports A–H and comprises four (4) two-input AND gates, two (2) NOR gates and an OR gate, connected as illustrated. The node values C1–C7 represent the colors of the device and net nodes after the cell has been colored to an equilibrium state. Upon reaching an equilibrium colored state, the input ports A–H may be designated as having equal colors C1. The output OR gate is represented as having a color C6 which is unique to the cell. The numbers represented in parentheses represent potential pin and device classes and nodes N 1 –N 7  represent internal or output nets of the dual-AOI cell having respective colors. 
   Based on the equivalent colors C1, input ports A–H are identified as exhibiting symmetry and as being potentially independently swappable as a result of this symmetry. In addition, nodes A–D can be identified as being within a first connected group since these nodes can be linked together by other nodes that have non-unique colors within the graph of the cell (i.e., nodes other than the device node OR which has a unique color). Similarly, nodes E-H can be identified as being in a second connected group since these nodes can be linked together by other nodes that have non-unique colors within the graph of the dual-AOI cell. 
   As illustrated by Block  430 A 3  of  FIG. 20 , once a provisional swap group of input nodes having the same color has been identified, a new color is assigned to a node within the swap group. This new color is preferably unique to all the nodes within the graph. Conventional recoloring operations may then be performed until a new equilibrium state has been achieved for the graphs. This graph recoloring operation may include performing a sequence of device and net recoloring operations, while simultaneously retaining the color of the assigned node as the assigned color. For example, in the event the input node H of the provisional swap group of nodes A–H is assigned the unique color C8 and a conventional graph coloring operation is performed, the input nodes A–D will retain their color as C1, the input nodes E–F will be recolored as C10 and input node G will be recolored as C9. A determination of whether the swap group of nodes are independently swappable can then be made by evaluating whether all the nodes A–G within the provisional swap group, with the exception of node H which received the assigned color, have the same color. Because nodes A–G do not all have the same color after the graph recoloring operation, nodes A–H can be confirmed as not being independently swappable. Moreover, because nodes E-G within the second connected group also do not have the same colors, the second connected group of nodes can be confirmed as not being independently swappable. Additional operations to assign a new color to a node within the first connected group of nodes A–D (followed by recoloring) can then be performed as described above to similarly confirm that nodes A–D are not independently swappable. Operations may then be performed to treat the dual-AOI cell of  FIG. 21  as having unresolvable symmetries and flatten the cell within the next higher level of hierarchy. 
   Applying the operations described above with respect to  FIGS. 20–21  to the simpler AOI cell of  FIG. 10  will result in a determination that device node NOR 1 is a unique node and that nodes A–B within a first connected group are independently swappable and nodes C–D within a second connected group are independently swappable. These conclusions in combination with an original recognition that at least some degree of symmetry is present between nodes A–D (by virtue of the fact that these nodes will exhibit the same colors after an equilibrium colored state has been achieved), enables the first and second connected groups to be treated as having dependent swappability whereby if node A is swapped with node C, node B must be swapped with node D to maintain equivalency. 
   Referring now to  FIGS. 22–26 , preferred operations performed by a layout versus schematic (LVS) comparison tool  430 B according to a second embodiment of the present invention will be described. These operations include coloring a schematic graph of a parent cell to obtain a first set of net colors for a first plurality of nets in the schematic graph, Block  430 B 1 . Next, a new color is assigned to a first net in the first plurality of nets, Block  430 B 2 . This first net is connected to a first pin of a child cell within the parent cell. The new assigned color is preferably unique to the other colors in the first set. A net recoloring operation is then performed, Block  430 B 3 . This net recoloring operation preferably includes recoloring the first set of net colors by determining a new color for the first net using a coloring operation that is a function of the assigned color, a color of the child cell (i.e., device value of the child cell) and a color of each of the first plurality of nets that are electrically connected to pins of the child cell that are independently swappable with the first pin. In particular, the recoloring operation preferably comprises determining effective “colors” associated with each of the pins of the child cell, using the preferred coloring operation, and determining a new color for the first net based on the effective “colors” of each of the pins in the child cell that are connected to the first net. For example, the new color for the first net may be determined by computing a sum of the effective “colors” of each of the pins in the child cell that are connected to the first net, after the recoloring operation has been performed. As described herein, pins (i.e., ports) of a child cell will be treated as having effective “colors”, where the “color” or each pin corresponds to the color of the net that is connected to the pin. Accordingly, even if three swappable pins of a child cell (e.g., AOI cell) are connected together by a single net and therefore would be represented in a layout or schematic graph by a single node corresponding to the net, these pins will be treated herein for purposes of coloring as each having effective “colors” which may during the course of recoloring be unique. 
   The operations performed by the LVS tool  430 B of the present invention will now be described more fully with respect to the parent cells illustrated by  FIGS. 23–27 . For example,  FIG. 23  illustrates a plurality of exemplary parent cells. The labels “S”, GL” and “BL” refer to schematic, good layout and bad layout, respectively. As illustrated, each of these parent cells has a six-input AOI child cell and six discrete devices D 1 –D 6  therein. Using the operations described above with respect to FIGS.  10  and  20 – 21  at the child level, pins A–C of the AOI child cell (which represent three input pins of an AND logic gate) can be treated as being independently swappable with each other and pins D–F can also be treated as being independently swappable with each other. Pins A–C are also dependently swappable with pins D–F. Each of the six devices D 1 –D 6  is connected to a respective pin of the AOI child cell by a respective net. These hierarchical integrated circuit schematic and layout designs (containing the parent cell, the devices, the AOI child cell, nets, etc.) are preferably described by schematic and layout netlist files and may be represented within these netlist files as schematic and layout graphs (e.g., bipartite graphs). The schematic and layout graphs corresponding to the respective designs may be colored until an equilibrium colored state is achieved. For purposes of illustration, the nets connecting devices D 1 –D 6  to input pins A–F may receive respective colors that are different from each other. In particular, the input nodes or pins A–F of the schematic AOI cell (having a device value (i.e., color) equal to 2) may receive the distinct colors {1,2,3,4,5,6}. Similarly, the pins A–F of the good layout parent cell and the bad layout parent cell may receive the distinct colors {3,2,1,4,6,5} and {1,2,6,4,5,3}. In contrast to the examples described below, the determination of distinct net colors typically precludes the need to assign a new net color prior to performing a net recoloring operation.  FIG. 23  illustrates these colors as respective equivalent pin vectors or sets S, GL and BL. Because the vectors are equivalent, the bad layout parent cell BL cannot be readily distinguished at this stage from the good layout parent cell GL (relative to the parent cell S). However, by performing a preferred net recoloring operation on each of the color vectors, the good layout GL can be distinguished from the bad layout BL. With respect to the schematic color vector S, the new color for pin A is determined using the following expression:
 
 NPC=OPC−DV (Σ Colors of Other Independently Swappable Pins)
 
where NPC represents the new pin color, OPC represents the old pin color and DV represents the device value (e.g., color) of the corresponding child cell. This preferred expression can require significantly less computational expense than a prior art technique of determining updated node colors of a cell using a color symmetrizing matrix (CSM) and the following expression:
 
( NPC )=( OPC )+( CSM )( OPC ) DV  
 
   Moreover, when recoloring a particular pin of a child cell, for example, the preferred expression does not require a determination of a product of the device value and a color of a pin that is merely dependently swappable with the pin being recolored. Thus, when evaluating child cells having both independently swappable pins and dependently swappable pins (e.g., a cell having AOI-type pin symmetry), a pin may be recolored by evaluating the colors of independently swappable pins but not the colors of the dependently swappable pins. When evaluating child cells having symmetric pins that are all independently swappable (e.g., a multi-input NAND gate), the preferred expression does not require the generation of a CSM and the computational expense of determining a product of the CSM and a vector of node colors. 
   Based on the above preferred expression, the pins of the schematic graph (and the separate nets connected to each pin) can be recolored as:
 
new color of pin  A= 1−2(2+3)=−9
 
new color of pin  B= 2−2(1+3)=−6
 
new color of pin  C= 3−2(1+2)=−3
 
new color of pin  D= 4−2(5+6)=−18
 
new color of pin  E= 5−2(4+6)=−15
 
new color of pin  F= 6−2(4+5)=−12
 
   Performing this net recoloring operation on the color vectors for the good layout GL and bad layout BL readily illustrates an inequality between the new color vector for the bad layout BL and the new color vector for the schematic S, as illustrated by  FIG. 23 . Based on this inequality, the bad layout can be detected. 
   Referring now to  FIG. 24 , another exemplary parent cell design is provided having a pair of identical devices D 1  which drive inputs of a six-input AOI child cell. Identical equilibrium state color vectors for nets N 1  and N 2  in the schematic, good layout and bad layout can be achieved using conventional graph coloring techniques. Identical device values (DV) are also present by virtue of the symmetry of the pin connections to the devices D 1 . A new color (1) is then assigned to net N 2  in the schematic, good layout and bad layout vectors. Alternatively, the new color may have been assigned to net N 1 . This assignment step is necessary because at least two nets identified by each vector have the same colors. Preferably, only a net having the same color as another net within a vector should be assigned a new color. Vectors illustrating effective pin “colors” may then be determined readily by expanding the net vectors, as illustrated. The effective colors of the pins are then modified using a recoloring operation that is a function of the device value of the AOI cell and the colors of the other pins that are independently swappable with the pin being recolored (but is independent of the colors of pins that are dependently swappable). A new net color is then determined for each net by adding the colors of all pins connected to a respective net together to obtain a sum. As illustrated, the new net colors for nets N 1  and N 2  readily identify the bad layout as being incorrect notwithstanding the fact that the initial coloring operations failed to identify a difference between the equilibrium state net color vectors. 
   Referring now to  FIG. 25 , another exemplary parent cell design is provided having two pairs of identical devices D 1  and D 2  which drive inputs of a six-input AOI child cell. The equilibrium state color vectors for nets N 1 –N 4  in the schematic, good layout and bad layout parent cells are identical with nets N 1  and N 3  having identical colors and nets N 2  and N 4  having identical colors. A new color (3) may then be assigned to net N 1  in the parent cells. Effective pin colors are then determined by expanding the net vectors, as illustrated. The preferred recoloring operation is then performed on each of the pins. The new net vectors are then determined by adding back the pin colors. For example, the new color for nets N 1  are determined by looking at pin A (−1) in the schematic, pin B (−1) in the good layout and pin A (−1) in the bad layout. The new color for nets N 2  are determined by adding the colors of pins B and C in the schematic, adding the colors of pins A and C in the good layout and adding the colors of pins B and E in the bad layout. The resulting net vectors illustrate the presence of the bad layout since the color vector for the bad layout does not match the color vector for the schematic. Moreover, no further net assignments are required between the schematic and good layout since the colors of the nets illustrated by the color vector for the schematic do not overlap (i.e., the colors of each of the nets N 1 –N 4  in the schematic are unique). 
   Referring now to  FIG. 26 , an exemplary parent cell design is provided having two 4-input AOI child cells and two devices D 1  and D 2 . Because the two devices are different, the AOI child cells are illustrated as having different device values (i.e., different colors on a graph). The original equilibrium state color vectors for nets N 1 –N 4  in the schematic, good layout and bad layout parent cells are illustrated as being identical with all nets having the same original colors. An assignment of a new color (3) to net N 1  is then made, as illustrated. This net assignment may be arbitrary. To limit unnecessary computational expense, a net receiving an assigned color may retain its assigned color throughout a net coloring operation. The effective colors of the pins are then determined. In particular, the effective pin colors are used along with the respective AOI device colors during a preferred pin recoloring operation. This preferred pin coloring operation may be performed as a discrete numerical operation within a net coloring operation. The pin coloring operation is also a function of the device value of the corresponding child cell and the colors of the other pins that are independently swappable with the pin being recolored. This pin coloring operation is preferably independent of the colors of other pins that are dependently swappable with the pin being recolored. The new colors of the nets may then be determined by adding back the new pin colors determined by the pin recoloring operation. As illustrated, the net color vector associated with the bad layout can be distinguished from the net color vector associated with the schematic. Nonetheless, because the net color vectors for the schematic and good layout still have duplicate colors (−8), a new net color (10) is assigned to net N 4  (or, alternatively, net N 3 ). Next, the operations of determining pin colors and then recoloring the pins are repeated. The recolored pins are then combined, if necessary, to determine the final net colors. As illustrated, the net colors are unique and the color vectors are identical. This result confirms the equivalency between the schematic of the parent cell and the good layout of the parent cell. 
   Referring now to  FIG. 27 , another exemplary parent cell design is provided having two 4-input AOI child cells. As illustrated, the schematic and layout devices values (SDV 1 , SDV 2 , LDV 1 , LDV 2 ) for the AOI child cells are initially identical. The original equilibrium state color vectors for nets S 1 -S 4  in the schematic parent cell and nets L 1 -L 4  in the good layout and bad layout parent cells are illustrated as being identical with all nets having the same original colors (5). An assignment of a new unique color (7) can be made arbitrarily to net S 1  in the schematic and nets L 3  in the good layout and bad layout, as illustrated. To limit unnecessary computational expense, a net receiving an assigned color retains its assigned color throughout a net coloring operation. 
   Effective pin colors are then determined for the pins of the AOI child cells in the schematic and layouts. Using the identical device values of (3), the pins of the AOI child cells are recolored. New net colors for nets S 2 –S 4 , L 1 –L 2  and L 4  are then determined by combining the pin colors. For example, since net S 2  of the schematic is connected to pin B of the top AOI cell and pin C of the bottom AOI cell, the new color for net S 2 =(−16)+(−10). The determination of the new net colors reveals a lack of equivalency between the schematic and the bad layout. The determination of the new net colors also reveals two nets in the schematic and two nets in the good layout that have identical colors. Accordingly, another arbitrary net assignment is made to net S 2  and net L 2 . For purposes of illustration only these nets may receive a new net color of 3 (although a color that is unique to the entire schematic and layout graphs is preferred). Based on these new net assignments, updated pin colors are determined for the pins of the top and bottom AOI child cells. The updated pin colors are then recolored using the illustrated operations as applied to the good layout. New net colors for nets S 3  and S 4  in the schematic and nets L 2  and L 4  in the layout are then determined. These new colors readily reveal an equivalence between the schematic and good layout. No further net assignments are necessary since each of the colors of the schematic net vector are unique. Moreover, because the last set of pin colors reveal an equivalency between the top AOI cell in the schematic and the bottom AOI cell in the good layout, operations are then performed to prevent an incorrect mapping of these child cells in the schematic and good layout during subsequent device coloring operations. These operations preferably include determining a new device color for the top AOI cell in the schematic by evaluating a product of the symmetric pin colors of pins A–D. As illustrated, the new device values for the AOI cells in the schematic and layout can be determined using the following relationship: New Device Value=Old Device Value+π(Colors of Symmetric Pins of Cell). Thus,
 
 SDV   1   new =3+(−2)(−18)(58)(34)=70,995
 
 SDV   2   new =3+(−47)(85)(63)(−29)=7,298,868
 
 LDV   1   new =3+(63)(−29)(85)(−47)=7,298,868
 
 LDV   2   new =3+(−2)(−18)(58)(34)=70,995
 
   As determined by the inventors herein, merely determining an algebraic sum of the colors of the symmetric pins will not distinguish the AOI cells during subsequent coloring operations since:
 
(−2)+(−18)+(58)+(34)=72=(−47)+(85)+(63)+(−29)
 
   In the drawings and specification, there have been disclosed typical preferred embodiments of the invention and, although specific terms are employed, they are used in a generic and descriptive sense only and not for purposes of limitation, the scope of the invention being set forth in the following claims.