
Portions of this patent application contain materials that are subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document, or the patent disclosure, as it appears in the Patent and Trademark Office file or records, but otherwise reserves all copyright rights whatsoever. [0001]
BACKGROUND OF THE INVENTION

1. Field of the Invention [0002]

The present invention relates to automated routing tools for integrated circuit designs. [0003]

2. Description of the Related Art [0004]

With the advent of circuit feature sizes in the submicron range, integrated circuits involving millions of transistors on a single chip have become commonplace. Due to the sheer number of devices on a single chip, an entire industry has evolved specifically to supply the semiconductor industry with software and hardware tools to automate much of the process of integrated circuit design. [0005]

Design automation tools are computerbased tools that assist through automation of procedures that would otherwise be performed manually. Simulation of proposed design functionality and synthesis of integrated circuit logic and layout are two examples. Design verification tools are computerbased tools used to verify that circuit design or layout meets certain prescribed objectives. [0006]

Both design automation and design verification tools require computerreadable descriptions of the underlying circuit function and structure to operate. These computerbased descriptions vary from simple geometrical specification languages, such as Caltech Intermediate Form (CIF) to highlevel functional description languages such as VHDL (a hardware design language). Geometrical specification languages for integrated circuits allow computerreadable definition of the geometries of the mask layers required to fabricate an integrated circuit. These specification languages contain primitive structures such as wires and boxes to specify geometrical shapes and layout levels. Organizational constructs are also provided to allow placement and repetition of the geometrical structures. [0007]

Most design tools are hierarchical in nature and employ more than one type of routing algorithm for routing interconnections between circuit components. Most routing tools used for cellbased designs begin with the placement of circuit elements, cells and/or cell blocks. Placement can be manual or automated, and typically decisions are made about where connectors to the circuit elements, cells and/or cell blocks should be located. Placement also includes determining the placement and orientation of blocks relative to one another. Such decisions can be driven by considerations of circuit compaction, which affects circuit congestion (similar to traffic congestion), the number of interconnect lines running between the blocks, and so on. With gate array designs, there is no placement step because placement has been predetermined by the manufacturer. [0008]

The next step in completing the circuit design is typically a global routing step, which is an attempt to logically determine a path for each interconnection between cells in the entire design. Routing decisions are made based on the available avenues formed by the current placement of circuit elements and/or blocks, and are assigned in consideration of various costs, also referred to herein as constraints (e.g., to incur the shortest total length of interconnect lines between the connectors). Once the global router has assigned the general flow of interconnect lines, a detailed router attempts to make the interconnect lines fit the assignments made by the global router. [0009]

A set of two or more interconnected cells in a circuit design is referred to herein as a “net.” A “net list” is a set of statements in a geometrical specification language that specifies the elements of a circuit, such as transistors and gates, and their interconnections. Individual transistors are described, along with cells to which they connect. The net list allows creation of a circuit diagram based on the actual geometrical specification statements. The creation of the circuit diagram is referred to as “circuit extraction,” and the extracted circuit can be compared to the original circuit specified by the designer to determine differences. A difference usually indicates an error that must be corrected. [0010]

In addition to providing the details of circuit interconnection, circuit extraction is useful for calculating layout areas and perimeters for each integrated circuit layer at each node of the circuit. These layout areas and perimeters can be used to accurately calculate the parasitic capacitances and resistance that load the active devices. With accurate capacitances and resistances from circuit extraction, a design can be accurately simulated to ensure correct operation. Thus, circuit extraction is an essential design verification tool for accurate characterization of modem integrated circuits. [0011]

A typical analysis in designing a circuit involves developing a routing solution for routing interconnections between circuit components. The routing solution is then evaluated using a constraint engine to identify nets that do not meet specified criteria, such as minimum spacing between nets. Offending nets are manually rerouted, and the routing solution and constraint engine rerun. This process is referred to as “parasitic extraction.”[0012]

As more complicated designs are developed to achieve higher performance and higher reliability, the demands placed on routing tools increase. Most current routing tools, provided by Electronic Design Automation (EDA) vendors, are insufficient to achieve the quality of route desired without several iterations and design cycles. Furthermore, most routing tools are primarily concerned with minimal distance as a constraint on global routing and do not permit timing to be directly considered. [0013]

What is needed is a new global routing technique to achieve a high quality, highly reliable route in as few iterations as possible. The global router should provide the capability to handle timing, noise avoidance, shielding and cell (repeater or latch) insertion constraints. The global router should produce output that can be used by a commercially available detailed router to complete the routing. [0014]
SUMMARY OF THE INVENTION

The present invention includes an algorithm that performs the constraintsbased global routing step in the physical design of integrated circuits. The algorithm is based on finding routes for the entire circuit based on constraints being satisfied for the entire design. Initially, for each net, a set of possible routing solutions is determined based on applicable constraints. The possible solutions for the nets are combined to create a highlyconnected “intersection graph,” with each intersection graph node representing a net. Edges are added between intersection graph nodes for nets which “intersect,” indicating that the nets share a region on the circuit's floorplan. Weights, based on constraints, are added to the edges of the intersection graph. Using these weights, the intersection graph is “pruned” to eliminate edges, producing a sparse graph. The intersection graph is partitioned based on constraints and performance criteria. An optimal solution is determined for each partition. The optimal solutions for the partitions are then combined to produce a global routing solution. The global routing solution is provided to a detailed router, which completes the routing for the design. [0015]

In one feature, a method includes finding a solution set including at least one route solution for each net of multiple nets of an integrated circuit design. The method further includes creating an intersection graph using the solution sets for the nets and partitioning the intersection graph into multiple partitions. The method further includes identifying an optimal solution satisfying a constraint for each partition and using the optimal solution for each partition to complete a global routing for the design such that the global routing satisfies the constraint. [0016]

In another feature, a method includes finding a route solution for each net of multiple nets of an integrated circuit design. The method further includes creating an intersection graph for the nets using each route solution for each of the nets. The method further includes partitioning the intersection graph into multiple partitions and identifying an optimal solution satisfying a constraint for each partition. The method also includes using the optimal solution for each partition to complete a global routing for the design such that the global routing satisfies the constraint. [0017]

In another feature, a system includes finding means for finding a solution set comprising at least one route solution for each net of multiple nets of an integrated circuit design. The system further includes creating means for creating an intersection graph for the nets using the solution sets for the nets. The system further includes partitioning means for partitioning the intersection graph into multiple partitions. The system further includes identifying means for identifying an optimal solution satisfying a constraint for each partition and using means for using the optimal solution for each partition to complete a global routing for the design such that the global routing satisfies the constraint. [0018]

In yet another feature, a computer program product includes finding instructions to find a solution set comprising at least one route solution for each net of multiple nets of an integrated circuit design. The computer program product further includes creating instructions to create an intersection graph for the nets using the solution sets for the nets. The computer program product further includes partitioning instructions to partition the intersection graph into multiple partitions. The computer program product further includes identifying instructions to identify an optimal solution satisfying a constraint for each partition of the partitions and using instructions to use the optimal solution for each partition to complete a global routing for the design such that the global routing satisfies the constraint. The computer program product further includes a computerreadable medium to store the finding instructions, the creating instructions, the partitioning instructions, the identifying instructions, and the using instructions. [0019]

In another feature, a computer system includes a processor for executing instructions and a memory to store the instructions. The instructions include finding instructions to find a solution set comprising at least one route solution for each net of multiple nets of an integrated circuit design. The instructions further include creating instructions to create an intersection graph for the nets using the solution sets for the nets and partitioning instructions to partition the intersection graph into multiple partitions. The instructions further include identifying instructions to identify an optimal solution satisfying a constraint for each partition and using instructions to use the optimal solution for each partition to complete a global routing for the design such that the global routing satisfies the constraint. [0020]

In still another feature, a system comprises a finding module to find a solution set comprising at least one route solution for each net of multiple nets of an integrated circuit design and a creating module to create an intersection graph for the nets using the solution sets for the nets. The system further includes a partitioning module to partition the intersection graph into multiple partitions and an identifying module to identify an optimal solution satisfying a constraint for each partition. The system further includes a using module to use the optimal solution for each partition to complete a global routing for the design such that the global routing satisfies the constraint. [0021]

The foregoing is a summary and thus contains, by necessity, simplifications, generalizations and omissions of detail; consequently, those skilled in the art will appreciate that the summary is illustrative only and is not intended to be in any way limiting. Other aspects, inventive features, and advantages of the present invention, as defined solely by the claims, will become apparent in the nonlimiting detailed description set forth below. [0022]
BRIEF DESCRIPTION OF THE DRAWINGS

The present invention may be better understood, and its numerous objects, features and advantages made apparent to those skilled in the art by referencing the accompanying drawings. [0023]

FIG. 1 is a flow diagram of the automated integrated circuit design process. [0024]

FIG. 2 shows an algorithm to perform the constraintsbased global routing step in the physical design of integrated circuits. [0025]

FIG. 3[0026] a shows a minimum spacing between nets to ensure that no noise is transferred generally or to one or more specific nets.

FIG. 3[0027] b shows a circuit resulting from cell insertion, which must be taken into account when routing.

FIG. 4[0028] a is a diagram of a single metal layer of an integrated circuit design divided into gtiles.

FIG. 4[0029] b shows three layers of an integrated circuit design and relationships between gtiles in the three layers.

FIG. 5 shows an example of two paths, with one path spanning multiple layers of a circuit design. [0030]

FIG. 6 shows an example of an obstruction of an arbitrary rectilinear shape that can routed around in a global routing produced according to the present invention. [0031]

FIG. 7[0032] a shows a solution for a route between driver D_{1 }and sink S_{1 }of the gtile map shown in FIG. 5.

FIG. 7[0033] b shows driver D_{2 }of FIG. 5 with two sinks S_{2a }and S_{2b}.

FIG. 8 shows alternative routing solutions between the drivers and sinks of FIGS. 7[0034] a and 7 b.

FIG. 9 shows an example of an intersection graph in which nets N[0035] 1 and N2 of FIG. 8 are connected to each other and selectively to other nets N3, N4, N5 and N6.

While the invention is susceptible to various modifications and alternative forms, specific embodiments are shown by way of example in the Drawings and are described herein in detail. The Drawings and Detailed Description are not intended to limit the invention to the particular form disclosed. On the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the scope of the present invention as defined by the appended Claims.[0036]
DETAILED DESCRIPTION

For a thorough understanding of the subject invention, refer to the following Detailed Description, including the appended Claims, in connection with the abovedescribed Drawings. [0037]

Although the present invention is described in connection with several embodiments, the invention is not intended to be limited to the specific forms set forth herein. On the contrary, it is intended to cover such alternatives, modifications, and equivalents as can be reasonably included within the scope of the invention as defined by the appended Claims. [0038]

In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the invention. It will be apparent, however, to one skilled in the art that the invention can be practiced without these specific details. [0039]

References in the specification to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the invention. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments mutually exclusive of other embodiments. Moreover, various features are described which may be exhibited by some embodiments and not by others. Similarly, various requirements are described which may be requirements for some embodiments but not other embodiments. [0040]
Introduction

The present invention includes an algorithm that performs the constraintsbased global routing step in the physical design of integrated circuits. The algorithm is based on finding routes for the entire circuit based on constraints being satisfied for the entire design. Initially, for each net, a set of possible routing solutions is determined based on applicable constraints. The possible solutions for the nets are combined to create a highlyconnected “intersection graph,” showing nets which “intersect,” indicating that the nets share a node. Weights, based on constraints, are added to the edges of the intersection graph. Using these weights, the intersection graph is “pruned” to eliminate edges, producing a sparse graph. The intersection graph is partitioned based on constraints and performance criteria. An optimal solution is determined for each partition. The optimal solutions for the partitions are then combined to produce a global routing solution. The global routing solution is provided to a detailed router, which completes the routing for the design. [0041]

The present invention produces solutions which meet timing constraints and have overall less congestion than other possible solutions. During routing, a static twodimensional parasitic extraction and timing analysis is performed to minimize resistance and capacitance while meeting timing constraints. In one embodiment, a 3D parasitic extraction module can be used in conjunction with the present invention to enable threedimensional parasitic extraction between layers of the circuit design. [0042]

FIG. 1 is a flow diagram of the automated integrated circuit design process. Initially Place Circuit Components step [0043] 110 indicates that most routing tools used for cellbased designs begin with the placement of circuit elements, cells and/or cell blocks. Placement can be manual or automated, and typically attempts to make intelligent decisions about where connectors to the circuit elements, cells and/or cell blocks should be located, as well as how cells and/or blocks should be oriented and positioned relative to one another. Such decisions can be driven by considerations of circuit compaction, number of interconnect lines running between the blocks, and so on. With gate array designs, there is no placement step because placement has been predetermined by the manufacturer.

Global Routing step [0044] 120 shows that the next step in completing the circuit design is typically a global routing step, which logically determines the paths for each interconnection. These decisions are made based on the available avenues formed by the current placement of circuit elements and/or blocks, and are assigned in consideration of various costs (e.g., the shortest total amount of interconnect lines between the connectors).

Once the global router has assigned the general flow of interconnect lines, in Detailed Routing step [0045] 130, a detailed router takes over and attempts to make the interconnect lines fit the assignments made by the global router. The detailed router uses the number of tracks provided for each metal layer to assign a track for eachnode through which the net, provided as input by the Global Routing step 120, must be routed. As shown in Output Representative of Routed Circuit Design step 140, the output of the detailed router represents the complete routed circuit design.

FIG. 2 shows a flowchart of an algorithm to perform the constraintsbased global routing step in the physical design of VLSI circuits. This flowchart shows an embodiment of Global Routing step [0046] 120 according to the present invention. In Provide Floorplan Design and Constraints step 205, constraints and the output from Initially Place Circuit Components step 110 are provided to the global routing tool. The output from Initially Place Circuit Components step 110 includes a floorplan that initially places circuit components, obstructions and connectors or pins within the design. Examples of obstructions include power grids, blocks, and charge pumps. The present invention allows for routing around obstructions of any rectilinear shape, including designs having 45° and/or diagonal lines.

In 3D Graph Creation step [0047] 210, a threedimensional graph representing the circuit design is created. In Develop Feasible Solution Set for Each Net step 220, a feasible solution set for each net is developed. In Create Intersection Graph with Weights step 230, a graph is created including the intersections of the solution sets for each net. Weights are assigned to the edges in the intersection graph to represent the constraints. In one embodiment, the intersection graph can be “pruned” to convert the highlyconnected intersection graph into a more sparselyconnected graph.

In Partition Intersection Graph step [0048] 240, the intersection graph is divided into partitions. In Develop Solution for Each Partition step 250, an optimal solution for each partition is determined using, for example, linear programming techniques. When the partitions are solved one at a time, optimal solutions for each partition can be merged with previouslyobtained optimal solutions for partitions. The output of Develop Solution for Each Partition step 250, with a solution for each partition, is used as input for the detailed router, as shown in Provide Output as Input for Detailed Router step 260.

One of skill in the art will recognize that each of the steps of the flowchart of FIG. 2 can be performed by a module of a global routing tool, also referred to as a global router, or by a standalone module working in conjunction with the global routing tool. For example, Provide Floorplan Design and Constraints step [0049] 205 can be performed by a Floor Planning and Constraint Determination module. 3D Graph Creation step 210 can be performed by a 3D Graph Creation module, Develop Feasible Solution Set for Each Net step 220 can be performed by a Net Solution Set Generation module, and Create Intersection Graph with Weights step 230 can be performed by an Intersection Graph Creation module. Partition Intersection Graph step 240 can be performed by a Graph Partition module, and Develop Solution for Each Partition step 250 can be performed by an Optimization module, also referred to as an LP solver for embodiments using linear programming techniques. Provide Output as Input for Detailed Router step 260 can be performed by an Output Provider module. The global routing tool may receive input from a user via a User Interface module, or the global routing tool may be completely automated. Each of these modules may reside on the same computer system, or the modules may be distributed across nodes of a distributed software system.

It is within the scope of the invention that at least some of these modules may be run in parallel. For example, the development of feasible solutions for each net performed by Develop Feasible Solution Set for Each Net step [0050] 220 can be performed in parallel. In an example environment having 10 computer systems available for calculating solutions sets, a design having 1,000 nets may be divided into sets of 100 nets for each computer system. The ten computer systems can find the solutions in parallel, and the resulting feasible solution sets can be passed to Create Intersection Graph with Weights step 230. If Develop Feasible Solution Set for Each Net step 220 is performed in parallel, the design can be divided such that nets that have noise between them are placed in the same set to be solved by the same computer system. In this way, noise avoidance is taken into account in developing the feasible solution sets.

In one embodiment, the solutions for each partition are placed in order of priority and solved one at a time in Develop Solution for Each Partition step [0051] 250. The previous solutions for the partitions are then used to develop solutions for subsequent partitions.

Alternatively, Develop Solution for Each Partition step [0052] 250 may be performed in parallel. A parallel approach can identify partitions that do not interact with each other, also referred to as “zero interaction cliques,” for separate simultaneous solution.
Provide Floorplan Design and Constraints

The constraints for the global routing tool of the present invention are determined by design requirements. Examples of the types of constraints that can be used are listed below. [0053]

Timing Constraints. In at least one embodiment, the maximum path delay allowed for the routed net can be specified, for example, by a user of the global routing tool. In one embodiment, the maximum path delay is specified using the standard delay format. Standard Delay Format (SDF) is set forth in Standard Delay Format Specification, Version 3.0 (May 1995, Open Verilog International), which is incorporated by reference in its entirety. An SDF file is an ASCII text file containing a header section followed by one or more cell entries describing cells of the design. The header section contains information relevant to the entire file, such as the design name, tool used to generate the SDF file, parameters used to identify the design, and operating conditions. Each cell entry identifies part of the design (a “region” or “scope”) and contains data for delays, timing checks, constraints, and the timing environment. For example, the launch time and edge rate at the driver pins of the nets and the required arrival time and edge rate at each sink pin of a net can be specified as timing constraints. [0054]

Noise Constraints. In at least one embodiment, the user can specify nets that should be avoided and the amount of spacing to be allowed between nets. The user can also specify whether a specified percentage of a particular net is to be routed along a power or ground signal. [0055]

Shielding Constraints. In at least one embodiment, the user can specify the shielding requirements of a net while routing. The shielding width, spacing and percentage of the net to be shielded can be specified. For example, 50% of a particular net may need to be shielded, and two tracks can be allocated along 50% of the length of that net to provide the necessary shielding. [0056]

Cell Insertion Constraints. In at least one embodiment, cell insertion constraints can be specified by the user along with the original timing constraints. Limits can be specified for a maximum transition time or maximum delay allowed such that a repeater or latch can be inserted to overcome the limit. For example, in a path from a driver D[0057] 1 to a sink S1, a repeater R1 can be inserted between D1 and S1 such that there is a topological breakpoint in the path from D1 to S1. The path from D1 to S1 includes two subpaths: a subpath from D1 to R1 and a subpath from R1 to S1. In such a case, noise constraints must be considered independently for the subpath from D1 to R1 and the subpath from R1 to S1.

FIGS. 3[0058] a and 3 b show constraints that are taken into account by the global routing tool of the present invention. For example, to avoid noise between nets, a minimum spacing between nets can be established to ensure that no noise is transferred generally or to one or more specific nets, as shown in FIG. 3a. Furthermore, a given net, such as Net A 320, may need to be routed outside a given width of an adjacent power or ground rail, referred to as “shielding,” for inductance prevention.

FIG. 3[0059] b shows a circuit resulting from the insertion of repeater cell R1 into a path from driver D_{1 }to sink S_{1}. The original path from driver D_{1 }to sink S_{1 }is divided into two portions, A and A′. For example, a cell may be inserted to speed up timing to achieve a certain rise time to drive capacitance. The added cell R1 must be routed around during the routing process.
3D Graph Creation

To create a threedimensional graph of the design, the entire design area is divided into a grid, and each grid section is assigned a “gtile.” Gtiles are usually square and one node, also referred to herein as a gNode, of a gtile exists in a given area per layer. [0060]

FIG. 4[0061] a is a diagram of a single metal layer of an integrated circuit design divided into gNodes. Five gNodes labeled g12, g2, g3, g4 and g5 are shown. A given gNode, such as gNode g12, is considered to be connected to each of its neighbors. GNode g12 is connected to each of neighbors g2 through g5.

FIG. 4[0062] b shows three layers of an integrated circuit design and relationships between gNodes in the three layers. The five gNodes of FIG. 4a are shown as part of metal layer M2 in FIG. 4b. In addition, FIG. 4b shows gNode g11 in metal layer M1 and gNode g13 in metal layer M3. A collection of gNodes in a vertical relationship between metal layers of the circuit is referred to herein as a “gtile,” and the shaded portions of FIG. 4b correspond to a gtile g1. Each gtile has a gNode for each of its metal layers; for example, gtile g1 includes gNode 11 in metal layer M1, gNode 12 in metal layer M2, and gNode 13 in metal layer M3. All the gNodes of a gtile are connected to each other by a neighbor relationship.

In the threedimensional graph shown in FIG. 4[0063] b, vertical neighbors are gNodes of the same gtile, and horizontal neighbors are gNodes of the same metal layer but adjoining gtiles. The graph is referred to herein as a global routing graph or gGraph. A gNode has a maximum of six neighbors, four horizontal neighbors and two vertical neighbors.

3D Graph Creation step [0064] 210 also adds obstructions, which may be of various rectilinear shapes, into the design that must be routed around in forming interconnections between circuit components. These obstructions may represent power grids, blocks, and charge pumps. The threedimensional graph enables modeling of congestion of the interconnect lines between connectors, taking into account these obstructions.
Develop Feasible Solution Set for Each Net

For each gtile, the number of routing tracks available through the gtile is calculated. The calculation takes into account previously routed nets and designwide obstructions. If a neighbor is obstructed completely, no connection to that neighbor is made in the final circuit design. Pins from the netlist, if located in a certain gNode, are attached to that particular gNode. If a net will require more than one track to be routed on a particular metal layer, the number of tracks is taken into account when routing the net. [0065]

The global route for a net is defined by mapping the net onto a set of gtiles. The present invention is used to produce a unique gtile map, together with width and spacing, for the nets such that all the global and individual net constraints are met. [0066]

As earlier described with regard to FIG. 1, a floor planning module places circuit blocks of the design into sections. The floor planning module also identifies where the drivers and sinks are located by gNode. A global router identifies through which of the sections the gNodes and sinks will be connected; in other words, the global router develops a map at a coarse level. A detailed router uses information about the number of tracks for each metal layer to assign a track for each gNode through which the net must be routed (as determined by the global router). In some embodiments, the floor planning module is a thirdparty module provided by a vendor independent of the global routing tool vendor, and in other embodiments, the floor planning module can be a component of the global routing tool provided by the same vendor. [0067]

FIG. 5 shows an example of a floorplan that is output from the floor planning component This example floorplan includes two paths having multiple nodes of two metal layers. A first path includes a driver D[0068] _{1 }and a sink S_{1 }as pins in metal layer 3. A second path includes driver D_{2 }and a sink S_{2b }as pins in metal layer 2, and sink S_{2a }in metal layer 3. Obstruction O_{1 }appears in metal layer 2 and obstruction O_{2 }appears in metal layer 3.

An obstruction, such as obstruction O[0069] _{1 }or O_{2}, indicates that tracks in the gtile map are “removed,” or considered to be unavailable, such that a net should not be routed through the corresponding gNode of the gtile. As mentioned earlier, examples of obstructions include power grids, blocks, and charge pumps. The present invention allows for obstructions in the form of any rectilinear shape.

FIG. 6 shows an example of a rectilinear obstruction [0070] 610 that can be routed around when using the global routing tool of the present invention. Unlike currently available global routing tools, the present invention enables obstructions of any rectilinear shape to be routed around, including obstructions having diagonal lines such as obstruction 610.

FIG. 7[0071] a shows an example of a gNode map for a given metal layer. The gtile map is produced by the global routing tool of the present invention. Ultimately, the global routing tool produces a unique gtile map for each net that meets all constraints and for which routing is both feasible and minimal. The width of the net is taken into account for global routing purposes.

FIG. 7[0072] a shows a solution for a route between driver D_{1 }and sink S_{1 }of the gtile map shown in FIG. 5. The route begins at driver D_{1 }in row 3 column 1 of the gtile map and proceeds to row 3 column 2. The route then proceeds from row 3 column 2 to row 2 column 2. The route then proceeds through row 2 column 3 to row 2 column 4, which includes sink S_{1}.

As mentioned above, an obstruction, such as obstruction [0073] 702, indicates that tracks in the gtile map are “removed,” or considered to be unavailable, such that a net should not be routed through the corresponding gNode of the gtile. Any route excluding row 4, column 4, which contains obstruction 702, can be taken between driver D_{1 }and sink S_{1}.

In creating a feasible solution set for a net, each route placed into the solution set should be feasible and meet applicable constraints. If no solutions exist for routing the net, a design flaw exists and the floorplan must be changed. If one solution exists, that solution corresponds to the solution set. If several solutions exist, a subset of the solutions can be selected to form the solution set; for example, a maximum of three solutions may be selected to form the solution set based on, for example, metrics for timing quality and length. [0074]

In FIG. 7
[0075] b, driver D
_{2 }of FIG. 5 is shown, with two sinks S
_{2a }and S
_{2b}. A solution for a route between D
_{2 }and S
_{2a }is determined independently of the solution for a route between D
_{2 }and S
_{2b}. Three solutions are illustrated for the path from D
_{2 }to S
_{2a}, respectively labeled S
_{2a}(
1), S
_{2a}(
2) and S
_{2a}(
3). Similarly, three solutions are illustrated for the path from D
_{2 }to S
_{2b}, respectively labeled S
_{2b}(
1), S
_{2b}(
2) and S
_{2b}(
3). These solutions are then combined to provide the following 9 possible solutions for nets including D
_{2}, S
_{2a }and S
_{2b}:


S_{2a(1)}, S_{2b(1)}  S_{2a(1)}, S_{2b(2)}  S_{2a(1)}, S_{2b(3)} 
S_{2a(2)}, S_{2b(1)}  S_{2a(2)}, S_{2b(2)}  S_{2a(2)}, S_{2b(3)} 
S_{2a(3)}, S_{2b(1)}  S_{2a(3)}, S_{2b(2)}  S_{2a(3)}, S_{2b(3)} 


A set of global route solutions for each net is generated. In one embodiment, Mikami Line Search Algorithm is used for finding these solutions. For more information regarding Mikami Line Search algorithm, see K. Mikami & K. Tabuchi, [0076] A Computer Program for Optimal Routing of Printed Circuit Connectors, IFIPS Proceedings, H47:14751478, 1968, which is incorporated herein by reference.

The Mikami Line Search algorithm searches across obstacles and multiple metal layers to find routing paths. For each net, each of its sink pins is paired with the driver pin to form (driver, sink) pairs for each path. Each path is routed using the Mikami Algorithm and multiple solutions are generated per path. After all solutions have been found for each (driver, sink) pair, the solutions are merged to give solutions for the entire net. For example, if a net had two paths, then two (driver, sink) pairs are formed. If the search algorithm generated three possible solutions for the first pair and two possible solutions for the second pair, then six global route solutions are possible for the net. The delay and actual length for each path solution is computed and stored for further analysis in the routing process. [0077]

Mikami Tabuchi algorithm, a line probe algorithm, is designed to perform better than maze routing algorithms, both in terms of route search time and quality. Maze routing algorithms search grid nodes in a breadthfirst fashion, whereas a line probe algorithm searches line segments available within the threedimensional gmap. The time and space complexity of Mikami Tabuchi algorithm is O(L), where L is the number of line segments produced to complete the route search. Mikami Tabuchi algorithm finds a path between driver and sink, if such a path exists, and assures that the path found is the shortest path available on the given floorplan. Initially, the line probes are drawn from both driver and sink in mutually perpendicular directions. These lines extend until reaching the floorplan boundary or an obstacle on that metal layer. If these lines do not intersect, then at each grid point on these lines, perpendicular lines are drawn until they reach the boundary of the floorplan or hit an obstacle. This process is continued recursively until an intersection point is found between the list of lines belonging to the driver and sink. [0078]

To enable multilayer routing, the lines drawn from points on a previous line can be on multiple layers, thereby reducing the overall search time. In practice, new lines are not drawn from every grid point on the previous line. A spacing factor between grid points is used to reduce the exponential growth in number of lines generated during a complicated search. [0079]

More sophisticated algorithms, such as Steiner Treebased approaches, can be used to determine routes more suitable for multiterminal nets. For more information about Steiner trees, see C. Chieng, M. Sarrafzadeh & C. K. Wong, [0080] A Powerful Global Router Based on Steiner MinMax Trees, Proceedings of IEEE International Conference on ComputerAided Design, pp. 25, Nov. 710, 1989, which is herein incorporated by reference.

Steiner Treebased approaches are based on first finding the minimum spanning tree for the given set of pins on a net. A minimum spanning tree is a minimumweight tree in a weighted graph which contains all of the graph's vertices (here, all the pins). From the minimal spanning tree, a rectilinear Steiner Tree is produced. A Steiner Tree is a tree resulting when, given a set of vertices S in an undirected weighted graph, a minimal weight spanning tree of S∪Q is produced for some added Steiner vertices Q in the graph. A Steiner tree differs from the minimum spanning tree in that the set of Steiner vertices must be identified. That is, additional vertices may be used. A Steiner Tree is based on rectilinear projections of the edges of this minimum spanning tree. [0081]

Once a set of solutions for each net is produced, the solutions for each net can be pruned according to delay and length constraints. In one embodiment, Elmore Delay is computed for each path of the net. For more information about Elmore Delay, see W. C. Elmore, [0082] The Transient Analysis ofDamped Linear Networks with Particular Regard to Wideband Amplifiers, Journal of Applied Physics, vol. 19(1), 1948; and J. Rubenstein, P. Penfield, Jr., & M. A. Horowitz, Signal Delay in RC Tree Networks, IEEE Transactions On Computer Aided Design, CAD2:202211, 1983, each of which is incorporated herein by reference.

The Elmore Delay model is the most commonly used delay model in works on interconnect design when inductance and distributed resistance and capacitance (RC) effects don't dominate. Under the Elmore delay model, the signal delay from the driver s[0083] 0 to a given node i in an RC tree is calculated as follows:

t(s 0, i)=Sum_over_all_nodes_{—} k(R(k _{i})*C(k))

where [0084]

C(k) is the downstream capacitance from node i [0085]

R(k[0086] _{1}) is the resistance at node i

All nodes and sinks in the RC tree have a different delay. In general, the Elmore delay of a sink in an RC tree is a (loose) upper bound on the actual 50% delay of the sink under step input. [0087]

Elmore delay provides a simple closedform expression with greatly improved accuracy for measuring delay, when compared to other RC models. Furthermore, the Elmore delay calculation can be done in linear time. The Elmore delay model is not the most accurate delay model available, but the Elmore delay model has a high degree of fidelity: an optimal or nearoptimal solution according to the estimator is also nearly optimal actual delay for routing constructions and wiresizing optimization. Other delay models which are more accurate can be used, but runtime for these models is usually greater than that of the Elmore delay model. [0088]

Using the arrival time constraints, the delay violation for each path solution is computed using the following formula:[0089]

Delay Violation=Actual Delay−Arrival Time Constraint

A value of 0 for the delay violation indicates that the arrival time constraint is met. A positive value indicates that the arrival time constraint is not met, and a negative value means that slack is available for optimization of other nets. The net solutions where all the paths of the net meet the constraint (delay) are retained, and the rest of the solutions are omitted from further consideration for inclusion in the final global routing solution. This “pruning” of the solutions guarantees that the ultimate global routing solution meets the timing constraint. [0090]

In one embodiment, feasible solutions sets are determined in order of priority. Consider a design having twenty nets. When the solution for the first net is developed, multiple solutions are available. After the first ten nets' solutions are developed, available tracks for routing the eleventh net are limited. Assume that the eleventh net interacts with eight other nets, four of which have already been routed as part of the first ten nets. Noise should be avoided between the interacting nets, and the topology of the previously routed nets can be considered to be an obstruction for the route of the eleventh net. [0091]

In the example, assume that the eleventh net needs to be a distance of four tracks from the previously routed tenth net. A temporary obstruction can be constructed determined by the topology of the tenth net. The eleventh net can be routed to automatically avoid the temporary obstruction. When the routing for the eleventh net is complete, the temporary obstruction can be removed for routing of subsequent nets. This approach of routing nets in order of priority and using temporary obstructions enables spacing constraints and noise to be taken into account. [0092]

FIG. 8 shows an example of a design including three drivers and three sinks in a single metal layer. The method of the present invention will be applied to the example design of FIG. 8 in the following sections. [0093]

FIG. 8 includes three nets: N[0094] 1, beginning at driver D_{11 }and ending at sink S_{11}; N2, beginning at driver D_{21}; and ending at sink S_{21}; and N3 beginning at driver D_{31 }and ending at sink S_{31}. Three obstructions 810, 820 and 830 are also present. Note that tracks are free in rows 6 and 8 and columns 7 and 8 such that paths can be routed through the gNodes of rows 6 and 8 and columns 7 and 8 despite obstructions 810 and 830. Obstruction 820 completely obstructs column 6 rows 27 and row 7 columns 26, and no path can be routed in column 6 rows 27 or row 7 columns 26. Taking these obstructions into account, feasible paths can be identified using algorithms such as those described above.

For net N[0095] 1, three possible paths include path t_{11 }(also referred to as topology_{11}), t_{12 }and t_{13}. For net N2, two possible paths include t_{21 }and t_{22}. For net N3, two possible paths include paths t_{31 }and t_{32}. Note that paths t_{11}, t_{12 }and t_{13 }meet at sink S_{11}. For the design to include all three paths, a separate track must be available for each path.

The result of steps Create 3D Graph step [0096] 210 and Develop Feasible Solution for Each Net step 220 of FIG. 2 is summarized below.

3 nets: N[0097] 1, N2 and N3

Number of solutions for each net: N[0098] 1 has 3, N2 has 2, N3 has 2.
Create Intersection Graph with Weights

In Create Intersection Graph with Weights step [0099] 230 of FIG. 2, finding a best solution from one point to another can be performed using known techniques. In most global routing tools, a single best solution is considered for each net. Using the best solution for each net, the netlist is updated and a new net is identified from the set of solutions including only one best solution for each net.

In contrast, the present invention considers a set of solutions, rather than the single best solution, for each net. Each solution for a given net is determined independently of other nets that do not have noise interaction with the given net. As an initial condition, the set of feasible solutions is constrained by the total number of tracks available in view of obstructions, but not by other feasible solutions for other nets that may need to use those tracks. The best set of feasible solutions is determined to include only solutions that are feasible for the entire set of nets. [0100]

Based on these sets of solutions for each net, an intersection graph for the entire netlist is created. Each net is represented as an intersection graph node. If the gNodes forming the solutions intersect with solutions of another net, an undirected edge is formed between the two intersection graph nodes. These edges are weighted based on the interactions between the intersection graph nodes. In order to distinguish the nodes of gmap from intersection graph nodes, the term “gNode” or “node” is used for gmap nodes, and the term “intersection graph node” is used for the nodes of the intersection graph. [0101]

FIG. 9 shows an example of an intersection graph in which nets N[0102] 1, N2 and N3 of FIG. 8 are included. An edge is placed between intersection graph nodes of the intersection graph N1 and N2 because nets N1 and N2 share the gNodes including driver D_{21 }and sink S_{11}, as well as the gNodes in row 6 columns 3 and 4 and colunm 5 rows 1 through 6. An edge is placed between nodes of the intersection graph N1 and N3 because nets N1 and N3 share the gNodes including driver D_{11 }and D_{31}, as well as the gNodes including column 1 rows 1 through 6 and row 1 columns 1 through 4. sink S_{11}. An edge is placed between nodes of the intersection graph N2 and N3 because nets N2 and N3 share the gNodes including row 8 columns 1 through 7.

In one embodiment, the edges are weighted by the percentage of solutions of two nets that intersect and the average probable congestion over intersecting gNodes. For example, the weight on the edge between two nodes (representing 2 different nets in the design) can be a function of the number of intersecting solutions and congestion, as shown in the formula below:
[0103] $=y*\left({A}_{\mathrm{NaNb}}\right)+\left(1y\right)*{B}_{\mathrm{NaNb}}$

where[0104]

A _{NaNb}(representing Intersecting Solutions)=(#solutions intersecting between N _{a}&N _{b})*(Avg. over all intersecting pairs of the % of gNodes intersecting)

B _{NaNb}(representing Congestion)=(#nodesSharedAndCongested/#nodesShared between all solutions for N _{a}& N _{b})*Average Congestion on nodesSharedAndCongested

[0105] $\mathrm{where}\ue89e\text{\hspace{1em}}\ue89e\mathrm{congestion}=\left(\frac{\#\ue89e\text{\hspace{1em}}\ue89e\mathrm{tracks}\ue89e\text{\hspace{1em}}\ue89e\mathrm{used}}{\#\ue89e\text{\hspace{1em}}\ue89e\mathrm{tracks}\ue89e\text{\hspace{1em}}\ue89e\mathrm{Available}}\right)$

Assume that one or more obstructions in a particular gNode of the grnap of FIG. 8 consume all but one track running through each gNode including the obstructions. Therefore, a gNode of FIG. 8 including one or more obstructions has only one track available for routing. [0106]

Typically, a path running through a gNode needs one track. However, a path may be wider, requiring two or more tracks, for example, to avoid noise. A gNode is considered to be congested when the value of congestion, as calculated above, is greater than one. Therefore, when a given gNode of FIG. 8 has one or more obstructions and more than one path is routed through the given gNode, the given gNode is considered to be congested. [0107]

Using the percentage of solutions of two nets that intersect and the average probable congestion over intersecting gNodes, a single weight for each edge is determined. Calculations of weights for the intersection graph of FIG. 9, representing the design shown in FIG. 8, are shown below. [0108]

Referring to FIG. 8 and the paths for nets N
[0109] 1 and N
2, all three paths for net N
1 intersect with path t
_{21 }from net N
2. For example, path t
_{11}, having a length of ten gNodes, intersects with path t
_{21}, having a length of 19 gNodes. The intersection shares the six gNodes in column
5 rows
1 through
6. The percentage of gNodes intersecting between path t
_{11 }and path t
_{21 }is calculated below:
$\left({t}_{11}\ue89e\text{\hspace{1em}}\ue89e\mathrm{with}\ue89e\text{\hspace{1em}}\ue89e{t}_{21}\right)=\frac{1}{2}*\left(\frac{6}{10}+\frac{6}{19}\right)=0.46$

Similarly, path t
[0110] _{12}, having a length of ten gNodes, intersects with path t
_{21}, having a length of 19 gNodes, with the intersection including the three gNodes in column
5 rows
4 through
6. The percentage of gNodes intersecting between path t
_{12 }and path t
_{21 }is calculated below:
$\left({t}_{12}\ue89e\text{\hspace{1em}}\ue89e\mathrm{with}\ue89e\text{\hspace{1em}}\ue89e{t}_{21}\right)=\frac{1}{2}*\left(\frac{3}{10}+\frac{3}{19}\right)=0.23$

Path t
[0111] _{13}, having a length of ten gNodes, intersects with path t
_{21}, having a length of 19 gNodes, with the intersection including the five gNodes in row
6 columns
1 through
5. The percentage of gNodes intersecting between path t
_{13 }and path t
_{21 }is calculated below:
$\left({t}_{13}\ue89e\text{\hspace{1em}}\ue89e\mathrm{with}\ue89e\text{\hspace{1em}}\ue89e{t}_{21}\right)=\frac{1}{2}*\left(\frac{5}{10}+\frac{5}{19}\right)=0.38$

Using these calculations, the average over the three intersecting pairs of the percentage of gNodes intersecting is calculated below:
[0112] $\frac{1}{3}\ue8a0\left[0.46+0.23+0.38\right]=0.36$

This average is used as part of the formula for A[0113] _{NaNb}, representing intersecting solutions, to assign weights to the edges between the nodes of the graph of FIG. 9. The weight assigned to the edge between nets N1 and N2 is calculated as shown below:

w _{1}(N 1⇄N 2)=y*A _{N1N2}+(1−y)*B _{N1N2}

A _{N1N2}=number of solution pairs intersecting*Avg. over all intersecting pairs of the percentage of gNodes intersecting 3*(0.36)=1.08

In the formula for A[0114] _{N1N2}, representing congestion, the number of congested gNodes is used as part of the calculation. Recall that when a gNode has one or more obstructions, the obstructions are considered to occupy all but one track running through the corresponding gNode. As a result, gNodes with two paths and one or more obstructions in FIG. 8 are congested, including the gNodes in column 5, rows 2 through 6; row 6, columns 2 through 5; and row 8, columns 2 through 6.

B _{NaNb}(representing Congestion)=(#nodesSharedAndCongested/#gnodesShared between all solutions for N_{a}& N _{b})*Average Congestion on nodesSharedAndCongested

[0115] $\mathrm{where}\ue89e\text{\hspace{1em}}\ue89e\mathrm{congestion}=\left(\frac{\#\ue89e\text{\hspace{1em}}\ue89e\mathrm{tracks}\ue89e\text{\hspace{1em}}\ue89e\mathrm{used}}{\#\ue89e\text{\hspace{1em}}\ue89e\mathrm{tracks}\ue89e\text{\hspace{1em}}\ue89e\mathrm{Available}}\right)$

Ten gNodes are shared between nets N[0116] 1 and N2, including the six gNodes in column 5, rows 1 through 6, and the four gNnodes in row 6, columns 1 through 4. (Note that the gNode of row 6, column 5, was already counted in the five gNodes for column 5.) Of the ten gNodes shared by nets N1 and N2, the five gNodes of column 5, rows 2 through 6, are congested. Each of the five congested gNodes has a value for congestion=2 tracks used/1 track available=2. The average congestion for gNodes shared and congestion is therefore (5*2)/5=2.

Therefore, A
[0117] _{N1N2 }is calculated as shown below:
$\frac{5}{10}*\left(\left(5*2\right)/5\right)=1.0$

Using this result, the weight for the intersection graph edge between N
[0118] 1 and N
2 is calculated as shown below:
$\begin{array}{c}{w}_{1}\ue8a0\left({\mathrm{N1}}_{1}\leftrightarrow \mathrm{N2}\right)=y*{A}_{\mathrm{N1N2}}+\left(1y\right)\ue89e{B}_{\mathrm{N1N2}}\\ \begin{array}{c}{w}_{1}\ue8a0\left({\mathrm{N1}}_{1}\leftrightarrow \mathrm{N2}\right)=\ue89e\left(y=\mathrm{.4}\right)*\left({A}_{\mathrm{N1N2}}=1.08\right)+\\ \ue89e\left(\left(1y\right)=\mathrm{.6}\right)*\left({B}_{\mathrm{N1N2}}=1.0\right)\\ =\ue89e\mathrm{.4}*1.08+\mathrm{.6}\end{array}\\ {w}_{1}\ue8a0\left(\mathrm{N1}\leftrightarrow \mathrm{N2}\right)=1.032\end{array}$

As shown in FIG. 10, the weight of the edge between nets N[0119] 1 and N2 has a value of 1.032.

To calculate the weight for the edge between nets N[0120] 1 and N3, the calculations below are used:

w _{2}(N 1 _{1} ⇄N 3)=y*A _{N1N3}+(1−y)*B _{N1N3}

Between nets N
[0121] 1 and N
3, path t
_{31 }intersects with path t
_{11 }at the five gNodes in row
1 columns
1 through
5. Path t
_{31 }has fifteen gNodes, and path t
_{11 }has ten gNodes.
$\left({t}_{31}\ue89e\text{\hspace{1em}}\ue89e\mathrm{with}\ue89e\text{\hspace{1em}}\ue89e{t}_{11}\right)=\frac{1}{2}\ue89e\left(\frac{5}{15}+\frac{5}{10}\right)=0.42$

Between nets N
[0122] 1 and N
3, path t
_{32 }intersects with path t
_{12 }at the four nodes in column
1 rows
1
4. Path t
_{32 }has fifteen gNodes, and path t
_{12 }has ten gNodes.
$\left({t}_{32}\ue89e\text{\hspace{1em}}\ue89e\mathrm{with}\ue89e\text{\hspace{1em}}\ue89e{t}_{12}\right)=\frac{1}{2}\ue89e\left(\frac{4}{15}+\frac{4}{10}\right)=0.33$

Between nets N
[0123] 1 and N
3, path t
_{32 }intersects with path t
_{13 }at the six gNodes in column
1 rows
1
6. Path t
_{32 }has fifteen gNodes, and path t
_{13 }has ten gNodes.
$\left({t}_{32}\ue89e\text{\hspace{1em}}\ue89e\mathrm{with}\ue89e\text{\hspace{1em}}\ue89e{t}_{13}\right)=\frac{1}{2}\ue89e\left(\frac{6}{15}+\frac{6}{10}\right)=0.5$

Using these calculations, the average over the three intersecting pairs of the percentage of gNodes intersecting is calculated below:
[0124] $\frac{1}{3}\ue8a0\left[0.42+0.33+0.5\right]=0.42$

These figures are used to assign weights to the edges between the nets N[0125] 1 and N3 of the graph of FIG. 9, as shown below:

w _{2}(N 1⇄N 3)=y*A _{N1N3}+(1−y)B _{N1N3}

A _{N1N3}=number of solution pairs intersecting*Avg. over all intersecting pairs of the percentage of gNodes intersecting 3*(0.42)=1.26

Of the gNodes shared between nets N[0126] 1 and N3, no nodes are congested.

B _{N1N3}=(#nodesSharedAndCongested/#nodesShared)*Average Congestion on nodesSharedAndCongested

[0127] $\frac{0}{6}[\left(0\right]=0$ $\begin{array}{c}{w}_{2}\ue8a0\left(\mathrm{N1}\leftrightarrow \mathrm{N3}\right)=\ue89ey*{A}_{\mathrm{N1N3}}+\left(1y\right)\ue89e{B}_{\mathrm{N1N3}}\\ =\ue89e\mathrm{.4}*1.26+\mathrm{.6}*0\\ =\ue89e0.504\end{array}$

As shown in FIG. 10, the weight for the edge between nets N[0128] 1 and N3 has a value of 0.504.

To calculate the weight for the edge between nets N[0129] 2 and N3, the calculations below are used:

w _{3}(n _{2} ⇄n _{3})=y*A _{N2N3}+(1−y)*B _{N2N3}

Between nets N
[0130] 2 and N
3, only two paths intersect. Path t
_{31 }intersects with path t
_{21 }at the three nodes in row
1, columns
5 through
7. Path t
_{31 }has fifteen gNodes, and path t
_{21 }has nineteen gNodes.
$\left({t}_{31}\ue89e\text{\hspace{1em}}\ue89e\mathrm{with}\ue89e\text{\hspace{1em}}\ue89e{t}_{21}\right)=\frac{1}{2}\ue89e\left(\frac{3}{15}+\frac{3}{19}\right)=0.18$

Between nets N[0131] 2 and N3, path t_{32 }intersects with path t_{22 }at the nine nodes in column 1 rows 6 through 7 and row 8 columns 1 through 7. Path t_{32 }has fifteen gNodes, and path t_{12 }has nine gNodes.

Using these calculations, the average over the two intersecting pairs of the percentage of gNodes intersecting is calculated below:
[0132] $\begin{array}{c}\left({t}_{32}\ue89e\text{\hspace{1em}}\ue89e\mathrm{with}\ue89e\text{\hspace{1em}}\ue89e{t}_{22}\right)=\frac{1}{2}\ue89e\left(\frac{9}{15}+\frac{9}{9}\right)=0.8\\ \frac{1}{2}\ue8a0\left[0.18+0.8\right]=0.49\end{array}$

These figures are used to assign weights to the edges between the nets N[0133] 2 and N3 of the graph of FIG. 9.

w _{3}(N 2⇄N 3)=y*A _{N1N3}+(1−y)*B _{N2N3}

A _{N2N3}=number of solution pairs intersecting*Avg. over all intersecting pairs of the percentage of gNodes intersecting2*(0.49)=0.98

Twelve gNodes are shared between nets N[0134] 2 and N3, including the three gNodes in columns 1, rows 6 through 8; the six gNodes in row 8, columns 2 through 7; and the three gNodes in row 1, columns 5 through 7. Of these twelve shared gNodes, seven gNodes are congested, including the five gNodes in row 8, columns 2 through 6 and the two gNodes in column 7, rows 1 and 2. Recall that, even when a gNode has more than one obstruction, one track is considered to be available in this example. Therefore, the value of congestion for each of the seven gNodes is 2, because each of these seven gNodes has two paths used but only one track available.

B _{N2N3}=(#nodesSharedAndCongested/#nodesShared)* Average Congestion on nodesSharedAndCongested

[0135] $\frac{7}{12}*\frac{\left(7*2\right)}{7}=1.167$

The weight for the edge is then calculated as follows:
[0136] $\begin{array}{c}{w}_{3}\ue8a0\left(\mathrm{N2}\leftrightarrow \mathrm{N3}\right)=\mathrm{.4}*0.98+\mathrm{.6}*1.167\\ =0.392+0.7002\\ =1.0922\end{array}$

As shown in FIG. 10, the weight of the edge between nets N[0137] 2 and N3 has a value of 1.0922.

Depending on whether congestion or timing is the higher priority, this weight can be adjusted by adjusting the value of the variable y. In addition, if noise is a constraint, then a weighting factor for noise avoidance can be added to the intersection graph. [0138]

After all solution sets are placed, the intersection graph becomes a highlyconnected graph for an average congested design. As noted above, the highlyconnected intersection graph can be pruned to produce a sparse graph before proceeding. [0139]
Partition Intersection Graph

In Partition Intersection Graph step [0140] 240 of FIG. 2, the intersection graph produced by Create Intersection Graph with Weights step 230 optionally can be partitioned using known graph partitioning techniques. The purpose of the partitioning is to minimize the time necessary to find an optimal solution for each partition. Standard linear programming techniques, for example, can be used to perform the optimization, although the number of variables that can be used is limited. To solve the optimization problem within a reasonable computer processing time, the highlyconnected intersection graph is divided into partitions. The graph is partitioned such that an optimization module, such as a standard linear programming solver, can be used to analyze each partition in a minimal computer processing time.

Partitioning takes as input the intersection graph of nets and gives as output a set of partitions. Partitions can be made such that nets within a partition have a high degree of interaction between them, and nets in different partitions have a lower degree of interaction. [0141]

In one embodiment, an initial partition is made and then refined using Fiduccia Mattheyses (FM) refinement. For more information regarding FM refinement, see C. M. Fiduccia & R. M. Mattheyes, [0142] A Linear Time Heuristic for Improving Network Partitions, Proceedings of the 19th Design Automation Conference, pp. 175181, 1982. Initial partitioning uses a greedy heuristic algorithm to distribute nets between the partitions keeping the constraint of lower and upper bounds on the number of nets per partition. FM refinement performs kway partitioning using the FM heuristic.

A circuit netlist is usually modeled by a hypergraph G, where V is the set of cells (also called nodes) in the circuit, and E is the set of nets (also called hyperedges). The number of nodes is designated by the variable n and the number of nets by the variable e. Each net connects two or more cells in the circuit. A net n[0143] _{1 }is represented as a set of the cells are connected via the net. A twoway partition of G is two disjoint subsets V_{1 }and V_{2 }such that each cell v that is an element of V belongs to either V_{1 }or V_{2}. A net is said to be cut if it has at least one cell in each subset and uncut otherwise, and this concept is referred to as the cut state of the net. All the nets that are cut form a set called the cut set. The objective of a twoway partitioning is to find a partition that minimizes the size of the cut set (called the cutsize). Usually, a predetermined balance criterion exists for the size of the subsets V_{1 }and V_{2}; for example,

0.45<=V _{1} /V<=0.55, where i=1,2.

The FM algorithm starts with a random initial partition. Each cell u is assigned a gain g(u) which is the immediate reduction in cutsize if the cell is moved to the other subset of the partition:
[0144] $g\ue8a0\left(u\right)=\sum _{{n}_{i}\in E\ue8a0\left(u\right)}\ue89ec\ue8a0\left({n}_{i}\right)\sum _{{n}_{j}\in I\ue8a0\left(u\right)}\ue89ec\ue8a0\left({n}_{j}\right)$

where E(u) is the set of nets that will be immediately moved out of the cut set on moving cell u, and I(u) is the set of nets that will be newly introduced into the cut set. Put in another way, a net in E(u) has only u in u's subset, and a net in I(u) has all its cells in u's subset. c(n[0145] _{1}) is the weight (cost) of the net n_{1}, which is assumed to be unity unless otherwise specified.

The goal of FM is to move a cell at a time from one subset to the other subset in an attempt to minimize the cutsize of the final partition. The cell being selected for the current move is called the base cell. At the start of the process, the cell with maximum gain value in both subsets is checked first to see if its move will violate the balance criterion. If not, it is chosen as the base cell. Otherwise, the cell with maximum gain in the other subset is chosen as the base cell. The base cell, say u[0146] _{1}, is then moved to the other subset and “locked”—the locking of a moved cell is necessary to prevent thrashing (a cell being moved back and forth) and being trapped in a bad local minimum. The reduction in cutsize (in this case, the gain g(u_{1}) ) is inserted in an ordered set S. The gains of all the affected neighbors are updated—a cell v is said to be a neighbor of another cell u, if v and u are connected by a common net. The next base cell is chosen in the same way from the remaining “free” (unlocked) cells and the move proceeds until all the cells are moved and locked. Then all the partial sum

Sj=Σ[0147] ^{j} _{t=1}g(u_{t}), 1<=j<=n, are computed, and p is chosen so that the partial sum S_{p }is the maximum. This corresponds to the point of minimum cutsize in the entire moving sequence. All the cells moved after u_{p }are reversed to their previous subset so that the actually moved cells are {u_{1}, . . . , u_{p}} This entire process is called a pass. A number of passes are made until the maximum partial sum S_{p }is no longer positive. This is a local minimum with respect to the initial partition [V_{1}, V_{2}].

The objective function for the FM algorithm is the cut cost. The cut cost is the weight of the edges going from one partition to another partition. The time complexity for FM in a 2way partition is O(ct), where t is the number of pins and c is the constant number of FM passes (c=5, 7 typically). One kway iteration is achieved by doing several twoway iterations, i.e., by forming pairs of partitions and performing a twoway FM algorithm in each partition. After one kway iteration, new pairs are formed as needed. In one embodiment, about 5 to 7 kway iterations are performed. [0148]

In another embodiment, partitions are made using a multilevel partitioning approach. A multilevel partitioning algorithm has linear space and time complexity and delivers stable solutions of high quality. Unlike kway FM, the quality of the solution does not depend on a pseudorandom initial partitioned state of the input graph. Quality, runtime and scalability of the partitioning stage can be improved by using multilevel partitioning algorithms, as compared to the kway FM algorithm. [0149]

Multilevel partitioning is designed for kway partitioning of graphs and includes a coarsening phase, an initial partitioning phase, and an uncoarsening/refinement phase. In the coarsening phase, the nodes of the input graph are clustered/coarsened in multiple steps to generate a hierarchy of graphs that are monotonically decreasing in size. Once a coarsest graph is reached, then that graph is partitioned into k partitions. The uncoarsening and refinement occurs in lockstep at all graph levels generated. At each level, the partition assigned to the parent node is also assigned to the children nodes and a local refinement is done at that level to minimize the cut cost. [0150]

The constraints used during the partitioning are the upper and lower bounds on the number of nets per partition. The upper and lower bounds help ensure a balanced number of nets per partition and reduce run time for the optimization module. [0151]

In the example shown in FIGS. 9 and 10, the graph is partitioned into partition PI, including net N[0152] 1, and partition P2, including nets N2 and N3. The highest weight, w'=1.0922, represents the highest level of interaction between intersection graph nodes. The two intersection graph nodes with the edge having the highest weight are placed into the same partition. The resulting partitions are shown below:

P[0153] 1: N1

P[0154] 2: N2 and N3.
Develop Solution for Each Partition

An optimization module is used to find a unique solution for each net in each partition, such that the global congestion is minimized. In one embodiment, the optimization problem is formulated as a 01 integer programming problem such that an increase in the size of the area to be routed does not cause a large increase in the number of variables that must be considered. Rather, the number of variables is proportional to the number of nets being routed and to the number of solutions per net. By using an objective function that is a linear function of the number of solutions for each net, the optimization problem is less complex and takes less time to compute than would an objective function that is not a linear function. [0155]

Each variable represents a net solution and can take a value of 0 or 1. A value of 0 means that the solution is rejected, while a value of 1 means that the solution is accepted after global congestion resolution. For a given net, only one solution variable can have a value of 1, while all the remaining solution variables should have a value of 0. These constraints for the optimization problem are illustrated below:
[0156] $\mathrm{Constraints}\ue89e\text{\hspace{1em}}\ue89e\text{\hspace{1em}}\left(\begin{array}{c}x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,1\right)+x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,2\right)+\dots \ue89e\text{\hspace{1em}}\ue89ex\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,{k}_{1}\right)=1\\ x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,1\right)+x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,2\right)+\dots \ue89e\text{\hspace{1em}}\ue89ex\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,{k}_{2}\right)=1\\ \vdots \\ x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89ep,1\right)+x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89ep,2\right)+\dots \ue89e\text{\hspace{1em}}\ue89ex\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89ep,{k}_{p}\right)=1\end{array}\right)$

where x(Ni, j)ε[0, 1] and integer. Each solution x(Ni, j) has a value of either zero or one, and the sum of all solutions for a net equals one, indicating that only one solution is selected for each net. [0157]

The objective function for the optimization module is to minimize the weighted average of the net solutions for probable congestion and delay. Ideally, no gNode should be congested. Generally, the objective function to be minimized is shown below:
[0158] $\left(\begin{array}{c}\left[y*\mathrm{delay}\ue89e\text{\hspace{1em}}\ue89e\left(N\ue89e\text{\hspace{1em}}\ue89e1,1\right)\right]+\left[\left(1y\right)*\mathrm{congestion}\ue89e\text{\hspace{1em}}\ue89e\left(N\ue89e\text{\hspace{1em}}\ue89e1,1\right)\right]+\dots \ue89e\text{\hspace{1em}}\\ \left[y*\mathrm{delay}\ue89e\text{\hspace{1em}}\ue89e\left(N\ue89e\text{\hspace{1em}}\ue89e1,2\right)\right]+\left[\left(1y\right)*\mathrm{congestion}\ue89e\text{\hspace{1em}}\ue89e\left(N\ue89e\text{\hspace{1em}}\ue89e1,2\right)\right]+\dots \ue89e\text{\hspace{1em}}\\ \left[y*\mathrm{delay}\ue89e\text{\hspace{1em}}\ue89e\left(N\ue89e\text{\hspace{1em}}\ue89e2,1\right)\right]+\left[\left(1y\right)*\mathrm{congestion}\ue89e\text{\hspace{1em}}\ue89e\left(N\ue89e\text{\hspace{1em}}\ue89e2,1\right)\right]+\dots \ue89e\text{\hspace{1em}}\\ \vdots \end{array}\right)\hspace{1em}$

Each partition is provided to the optimization module for solution selection. The nets in a partition can be simultaneously solved, thus reducing overall routing time. Since there is no dependence between nets for these solutions, synchronization is not an issue. The original gGraph can be used by each net for finding its respective set of solutions. [0159]

Each partition may be assigned a priority to be solved with respect to the other partitions, thereby assigning a sequence in which the partitions are solved. Using a priority scheme, after one partition is solved, a gtile database can be updated with the current metal utilization in each gtile. Hence, the remaining metal resources available to the next partition are reduced and the next partition has lower metal resources per gtile. Metrics that can be used to prioritize the partitions include the following: [0160]

average delay violation; [0161]

total number of nets where all solutions are violating delay constraints; and [0162]

number of noisesensitive nets. [0163]

Partitions may be recombined when interaction between them is high. A decision whether to recombine nets requires a tradeoff between increasing the runtime of the optimization module versus improving the quality and feasibility of the solution. [0164]

An alternative constraint can be an upper bound on the number of congested gNodes in a set of gNodes included in the global route. Such a constraint allows the optimization module to perform within a reasonable run time and assumes the possibility that the detailed router can solve the congestion problem. [0165]

In the example of FIGS. 8, 9, and [0166] 10, net N1 has three possible solutions, labeled N1, 1; N1, 2; and N1, 3 below. Net N2 has two possible solutions, labeled N2, 1 and N2, 2. Net N3 has two possible solutions, labeled N3 , 1 and N3, 2. The constraint equations for this example appear as shown below:

[0167] $\mathrm{Constraints}\ue89e\text{\hspace{1em}}\ue89e\{\begin{array}{c}x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,1\right)+x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,2\right)+x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,3\right)=1\\ x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,1\right)+x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,2\right)=1\\ x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e3,1\right)+x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e3,2\right)=1\end{array}$

where x(N[0168] 1, j)=0 or 1.

The objective function is to minimize the delay violation for each solution and overall congestion on the floorplan. The weighted average of the % Delay Violation and % Additional Tracks Needed is taken, multiplied by the value of x(Ni, j) (which is 0 or 1), and multiplied by the probability that the particular solution represented by Ni, j will be selected.
[0169]  
 
   % Congestion 
 Relative  % Delay  (% Additional 
 Weights  Violation  Tracks Needed) 
 
 y_{dv }= 0.4  DV (N1, 1) = 0  Av.cong.(N1,1) = 100 
 y_{cong }= 0.6 =  DV (N1, 2) = 0  Av.cong.(N1,2) = 100 
 (1 − y_{dv})  DV (N1, 3) = 0  Av.cong.(N1,3) = 0 
  DV (N2, 1) = 10  Av.cong.(N2,1) = 100 
  DV (N2, 2) = 0  Av.cong.(N2,2) = 100 
  DV (N3, 1) = 20  Av.cong.(N3,1) = 0 
  DV (N3, 2) = 20  Av.cong.(N3,2) = 100 
 

For example, the percentage congestion value of 100 for (N
[0170] 1,
1) indicates that one track is available, and one more track is needed, so that a 100% increase in the number of tracks is needed. For the data presented above, the linear programming problem is set up as indicated below:
$\frac{x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,1\right)*\left[0*0.4+100*0.6\right]}{3}+\frac{x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,2\right)*\left[0*0.4+100*0.6\right]}{3}+\frac{x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,3\right)*\left[0*0.4+0*0.6\right]}{3}=20\ue89ex\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,1\right)+20\ue89ex\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,2\right)$ $\frac{x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,1\right)*\left[10*0.4+100*0.6\right]}{2}+\frac{x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,2\right)*\left[0*0.4+100*0.6\right]}{2}=\frac{x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,1\right)*10}{2}+\frac{x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,2\right)*6}{2}=32\ue89ex\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,1\right)+30\ue89ex\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,2\right)$ $\frac{x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e3,1\right)*\left[20*0.4+0*0.6\right]}{2}+\frac{x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e3,2\right)*\left[20*0.4+100*0.6\right]}{2}=\frac{x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e3,1\right)*8}{2}+\frac{x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e3,2\right)*68}{2}=4\ue89ex\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e3,1\right)+34\ue89ex\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e3,2\right)$

Therefore, the objective function to minimize is[0171]

20x(N 1,1)+20x(N 1,2)+32x(N 2, 1)+30x(N 2, 2)+4x(N 3, 1)+34x(N 3, 2).

Solving this equation, the resulting values of x(Ni, j) are given below:
[0172] $\begin{array}{c}x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,1\right)=0,x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,2\right)=0,x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e1,3\right)=1\\ x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,1\right)=0,x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e2,2\right)=1\\ x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e3,1\right)=1,x\ue8a0\left(N\ue89e\text{\hspace{1em}}\ue89e3,2\right)=0\end{array}$

These solutions correspond to path t[0173] _{13 }for net N1, path t_{22 }for net N2, and path t_{31 }for net N3.
Provide Out as Input for Detailed Router

The optimized paths through each partition are provided as input to the detailed router ,as show in Detailed Routing step [0174] 130 of FIG. 1. The detailed router uses information about the number of tracks for each metal layer to assign a track for each node through which the net must be routed. The output of the detailed router is representative of the routed circuit design, as indicated by Output Representative of Routed Circuit Design step 140 of FIG. 1.

High performance designs require extensive resources for routing. The interconnection line length and congestion constraints provided in routing tools by current EDA vendors require multiple iterations to finalize a design and may not take into account constraints such as timing constraints. The present invention introduces a new constraintdriven global routing algorithm which takes design constraints into account while performing the global routing step. Initial testing results show that this global router helps achieve better results than are achieved by current commercially available routers. The global router of the present invention can be used as a standalone tool before performing the detailed routing step, or the global router can be used to route constraintdriven nets together with other global routers that do not handle constraintdriven nets. [0175]
Other Embodiments

The present invention is well adapted to attain the advantages mentioned as well as others inherent therein. While the present invention has been depicted, described, and is defined by reference to particular embodiments of the invention, such references do not imply a limitation on the invention, and no such limitation is to be inferred. The invention is capable of considerable modification, alteration, and equivalents in form and function, as will occur to those ordinarily skilled in the pertinent arts. The depicted and described embodiments are examples only, and are not exhaustive of the scope of the invention. [0176]

The foregoing described embodiments show different components contained within other components (i.e., the floor planning component may be a component of the global routing tool or the floor planning component may be a standalone tool). It is to be understood that such described architectures are merely examples, and that in fact many other architectures can be implemented which achieve the same functionality. In an abstract but still definite sense, any arrangement of components to achieve the same functionality is effectively “associated” such that the desired functionality is achieved. Hence, any two components herein combined to achieve a particular functionality can be seen as “associated with” each other such that the desired functionality is achieved, irrespective of architectures or intermediate components. Likewise, any two components so associated can also be viewed as being “operably connected,” or “operably coupled,” to each other to achieve the desired functionality. [0177]

The foregoing detailed description has set forth various embodiments of the present invention via the use of block diagrams, flowcharts, and examples. It will be understood by those within the art that each block diagram component, flowchart step, operation and/or component illustrated by the use of examples can be implemented, individually and/or collectively, by a wide range of hardware, software, firmware, or any combination thereof. [0178]

The present invention has been described in the context of software running on fully functional computer systems; however, those skilled in the art will appreciate that the present invention is capable of being distributed as a program product in a variety of forms, and that the present invention applies equally regardless of the particular type of signal bearing media used to actually carry out the distribution. Examples of signal bearing media include recordable media such as floppy disks and CDROM, transmission type media such as digital and analog communications links, as well as media storage and distribution systems developed in the future. [0179]

The abovediscussed embodiments may be implemented by software modules that perform certain tasks. The software modules discussed herein may include script, batch, or other executable files. The software modules may be stored on a machinereadable or computerreadable storage medium such as a disk drive. Storage devices used for storing software modules in accordance with an embodiment of the invention may be magnetic floppy disks, hard disks, or optical discs such as CDROMs or CDRs, for example. A storage device used for storing firmware or hardware modules in accordance with an embodiment of the invention may also include a semiconductorbased memory, which may be permanently, removably or remotely coupled to a microprocessor/memory system. Thus, the modules may be stored within a computer system memory to configure the computer system to perform the functions of the module. Other new and various types of computer readable storage media may be used to store the modules discussed herein. [0180]

In the example described herein, path delay is used for pruning the set of solutions for each net. Other constraints such as noise avoidance can be used to prune the solutions. Nets whose solutions appear in the same gNode as a net to be avoided can be dropped. This criteria will slow the algorithm down, since pruning of solutions might be expensive. [0181]

A cell insertion constraint can also be implemented for this algorithm. Solutions which have the highest chance of connecting to the preplaced cells to satisfy the delay or transition time requirements are the only ones considered for each net. [0182]

Shielding requirements can be considered by the global router of the present invention while performing global routing. The solutions for a net that would use the shielding tracks required by the net can be eliminated from the solution set. Shielding constraints may lead to pruning solutions that may have originally appeared to be viable routing solutions. [0183]

The above description is intended to be illustrative of the invention and should not be taken to be limiting. Other embodiments within the scope of the present invention are possible. Those skilled in the art will readily implement the steps necessary to provide the structures and the methods disclosed herein, and will understand that the process parameters and sequence of steps are given by way of example only and can be varied to achieve the desired structure as well as modifications that are within the scope of the invention. Variations and modifications of the embodiments disclosed herein can be made based on the description set forth herein, without departing from the scope of the invention. [0184]

Consequently, the invention is intended to be limited only by the scope of the appended claims, giving full cognizance to equivalents in all respects. [0185]