Patent Publication Number: US-8112733-B2

Title: Method and apparatus for routing with independent goals on different layers

Description:
CLAIM OF BENEFIT TO PRIOR APPLICATIONS 
     This application is a divisional application of U.S. patent application Ser. No. 10/335,180, filed on Dec. 31, 2002, now issued as U.S. Pat. No. 7,480,885. U.S. patent application Ser. No. 10/335,180 claims the benefit of U.S. Provisional Patent Application 60/427,131, filed on Nov. 18, 2002. U.S. Pat. No. 7,480,885 is incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     The invention is directed towards a method and apparatus for routing with independent goals on different layers. 
     BACKGROUND OF THE INVENTION 
     An integrated circuit (“IC”) is a device (e.g., a semiconductor device) that includes many electronic components, such as transistors, resistors, diodes, etc. These components are often interconnected to form multiple circuit components, such as gates, cells, memory units, arithmetic units, controllers, decoders, etc. An IC includes multiple layers of wiring that interconnect its electronic and circuit components. Traditionally, IC&#39;s use preferred direction (“PD”) wiring models, which specify a preferred wiring direction for each of their wiring layers. In preferred direction wiring models, the preferred direction typically alternates between successive wiring layers. One example of a PD wiring model is the PD Manhattan wiring model, which specifies alternating layers of preferred-direction horizontal and vertical wiring. 
     Design engineers design IC&#39;s by transforming logical or circuit descriptions of the IC&#39;s into geometric descriptions, called layouts. IC layouts typically include (1) circuit modules (i.e., geometric representations of electronic or circuit IC components) with pins, and (2) interconnect lines (i.e., geometric representations of wiring) that connect the pins of the circuit modules. A net is typically defined as a collection of pins that need to be connected. A list of all or some of the nets in a layout is referred to as a net list. 
     To create layouts, design engineers typically use electronic design automation (“EDA”) applications. These applications provide sets of computer-based tools for creating, editing, and analyzing IC design layouts. One EDA tool is a router that defines routes for interconnect lines that connect the pins of nets. Routing is generally divided into two phases: global routing and detailed routing. For each net, global routing generates a “loose” route for the interconnect lines that are to connect the pins of the net. The “looseness” of a global route depends on the particular global router used. After global routes have been created, the detailed routing creates specific individual routes for each net. 
     While some commercial global routers today might allow an occasional diagonal jog, these routers do not typically explore diagonal routing directions consistently when they are specifying the routing geometries of the interconnect lines. This, in turn, increases the total wirelength (i.e., total length of interconnect lines) needed to connect the nets in the layout. Therefore, there is a need for a routing method and apparatus that considers diagonal routing directions. There is also a need for a new way of identifying and costing routes. 
     SUMMARY OF THE INVENTION 
     Some embodiments of the invention provide a method of routing. The method selects a net with a set of routable elements in a multi-layer layout region. In some embodiments, the method identifies a route for the net based on different congestion goals on different layers. In other embodiments, the method identifies a route for the net based on different congestion goals between different layer pairs. In some embodiments, the method identifies a route for the net based on both the different congestion goals on different layers and between different layer pairs. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The novel features of the invention are set forth in the appended claims. However, for purpose of explanation, several embodiments of the invention are set forth in the following figures. 
         FIG. 1  illustrates a 4×4 section of a congestion grid. 
         FIG. 2  illustrates a section of a length grid that divides each Gcell created by the congestion grid into four nodes. 
         FIG. 3  illustrates the four nodes in each Gcell on a particular layer. 
         FIGS. 4-7  illustrate the directions of edges on interconnect layers  2 - 5  in some embodiments of the invention. 
         FIG. 8  illustrates edges that cross the Gcells created by the congestion grid. 
         FIGS. 9-12  illustrate four examples of internal zigs between the four nodes of a Gcell. 
         FIG. 13  presents a three-dimensional side view of the example illustrated in  FIG. 9 . 
         FIGS. 14-21  illustrate eight examples of external zigs between the four nodes of a particular Gcell and the eight nodes in four Gcells that are adjacent to the particular Gcell. 
         FIG. 22  presents a three-dimensional side view of the example illustrated in  FIG. 14 . 
         FIG. 23  illustrates a via location in a Gcell. 
         FIG. 24  illustrates a via location in a Gcell. 
         FIG. 25  illustrates a process that conceptually represents the overall flow of the router in some embodiments of the invention. 
         FIGS. 26A ,  26 B, and  26 C present three examples that illustrate how some embodiments compute the capacity of a congestion edge between two Gcells on a given layer. 
         FIG. 27  illustrates a route-generation process that the routing process uses to generate a route for a particular net in some embodiments of the invention. 
         FIG. 28  presents one example of shadow nodes. 
         FIG. 29  illustrates a path-generation process that the route-generation process uses in some embodiments. 
         FIG. 30  illustrates an example of a back trace operation used by the path-generation process of  FIG. 29 . 
         FIG. 31  conceptually illustrates a computer system with which one embodiment of the invention is implemented. 
         FIG. 32  illustrates an example of a multi-layer global route that is produced by a router of some embodiments of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In the following description, numerous details are set forth for purpose of explanation. However, one of ordinary skill in the art will realize that the invention may be practiced without the use of these specific details. In other instances, well-known structures and devices are shown in block diagram form in order not to obscure the description of the invention with unnecessary detail. 
     Several embodiments of the invention provide a router that routes a set of nets in a region of an integrated circuit (“IC”) layout. Each routed net includes a set of routable elements in the IC-layout region. The routable elements are pins in the embodiments described below, although they might be other elements in other embodiments. 
     In the embodiments described below, the router uses a five-layer wiring model that has horizontal wiring on wiring layer  1 , vertical wiring on wiring layer  2 , horizontal wiring on wiring layer  3 , +45° diagonal wiring on wiring layer  4 , and −45° diagonal wiring on wiring layer  5 . One of ordinary skill will realize that the router can use other wiring models in other embodiments. In some embodiments, a line is “diagonal” if it forms an angle other than 0° or 90° with respect to the layout&#39;s Cartesian coordinate axes, which are typically parallel with the layout&#39;s boundary and/or the boundary of the layout&#39;s expected IC. On the other hand, an interconnect line is “horizontal” or “vertical” if it forms an angle of 0° or 90° with respect to one of the coordinate axes of the layout. 
     In the embodiments described below, the router partitions an IC-layout region into several square sub-regions. For each net being routed, the router then identifies a global route that connects the set of sub-regions that contain at least one pin of the net. Each net&#39;s global route is a set of edges (i.e., interconnect lines) that connects the set of sub-regions that contain the net&#39;s pins. The identified routes might have horizontal, vertical, and ±45° diagonal edges in the embodiments described below. 
     In these embodiments, the edges that are used to define each route are part of a routing graph used by the router. Section I provides an overview of this routing graph. Next, Section II provides the overall flow of the router. Section III then describes route-generation and path-generation processes used by the router. Section IV describes a computer system that can be used to implement some embodiments of the invention. 
     I. Routing Graph, Congestion Grid, and Length Grid 
     In some embodiments, the router uses two grids to create a routing graph. The first grid is a coarser grid that divides the IC layout into a number of sub-regions, called Gcells. The second grid is a finer grid that divides each Gcell into four sub-regions. In the embodiments described below, the Gcells are square. This shape well supports ±45° routing, as any set of ±45° wiring tracks that cut through a square Gcell will fill its horizontal and vertical boundaries consistently. One of ordinary skill will realize that other embodiments might use different shaped Gcells. 
     On each wiring layer, each of the four sub-regions in each Gcell is represented by a node at the center of the sub-region. The embodiments described below use the coarser grid to measure route congestion in the layout region, and use the finer grid to measure route lengths. Accordingly, below, the coarser grid is referred to as the congestion grid, while the finer grid is referred to as the length grid. 
       FIGS. 1 and 2  illustrate small sections of the congestion and length grids. As shown in these figures, intersecting horizontal and vertical lines form both these grids.  FIG. 1  illustrates a 4×4 section of the congestion grid  100 . This section divides a portion of an IC region into 16 Gcells  105 . In the embodiments described below, the congestion grid divides the IC region into many more Gcells (e.g., tens or hundreds of thousands). 
       FIG. 2  illustrates a section of the length grid  200  that corresponds to the section of the congestion grid  100  illustrated in  FIG. 1 . As shown in this figure, the length grid divides each Gcell  105  into four nodes  205  on each wiring layer.  FIG. 3  illustrates the four nodes in each Gcell on a particular layer. There are a number of planar and non-planar edges between the nodes defined by the length grid  200 . These edges are referred to as “node edges” in the discussion below. 
     A. Planar Edges 
     A planar node edge connects two adjacent routing-graph nodes. Each such edge represents a set of wiring tracks along the edge&#39;s particular direction that connect the two sub-regions represented by the edge&#39;s two nodes. Planar node edges have different directions on different wiring layers.  FIGS. 4 through 7  illustrate the directions of these edges on layers  2 - 5  in some embodiments. Some embodiments assume that there are no planar node edges between routing-graph nodes on layer  1 , as this layer is often quite congested. Some of these embodiments promote all the pins on layer  1  to layer  2 . Other embodiments, however, specify planar node edges on layer  1 . In some of these embodiments, the planar node edges on layer  1  are in the same direction as node edges on layer  3 . 
       FIG. 4  illustrates that on layer  2  a vertical node edge  405  exists between each pair of vertically adjacent nodes, while  FIG. 5  illustrates that on layer  3  a horizontal node edge  505  exists between each pair of horizontally adjacent nodes.  FIGS. 6 and 7  illustrate that on layers  4  and  5 , ±45° diagonal node edges exist only between certain pairs of diagonally adjacent nodes. Specifically,  FIG. 6  illustrates that 45° diagonal node edges exist between northwest nodes  605  and southeast nodes  610  of different Gcells. As shown in this figure, no 45° diagonal node edges are incident on northeast nodes  615  and southwest nodes  620 .  FIG. 7  illustrates that −45° diagonal node edges exist between northeast node  615  and southwest nodes  620  of different Gcells. As shown in this figure, no −45° diagonal node edges are incident on northwest nodes  605  and southeast nodes  610 . 
     In the embodiments described below, each Manhattan node edge on layer  2  or  3  has a unit length cost (L). In these embodiments, each diagonal node edge on layer  4  or  5  has a length cost that equals the unit length cost times the square root of two (L*√{square root over (2)}). Also, the use of a node edge across a Gcell boundary reduces the capacity of the boundary, and is thereby assessed a wire congestion cost. 
     The router examines wire congestion at Gcell boundaries on each layer available for routing. Specifically, on each available-routing layer, the router computes capacities at Gcell boundaries for wiring along the particular layer&#39;s direction. On a particular layer, the wiring resources (i.e., wiring tracks) across a Gcell boundary can be conceptually represented as a planar “congestion edge” across that boundary on the particular layer in the layer&#39;s wiring direction. 
       FIG. 8  presents a two-dimensional diagram that illustrates the congestion edges on layers  2 - 5  for the routing directions illustrated in  FIGS. 4-7 .  FIG. 8  illustrates one horizontal congestion edge across each vertical boundary between horizontally adjacent Gcells, one vertical congestion edge across each horizontal boundary between vertically adjacent Gcells, and two ±45° diagonal congestion edges across each boundary between each pair of adjacent Gcells. In this example, each vertical congestion edge is on layer  2 , each horizontal congestion edge is on layer  3 , each 45° congestion edge is on layer  4 , and each −45° congestion edge is on layer  5 . 
     The router keeps track of one congestion-grid capacity on each layer at each boundary between adjacent Gcells. Accordingly, each congestion edge is associated with all node edges that cross the same Gcell boundary on the same layer as the congestion edge. As illustrated in  FIGS. 4-7 , certain planar node edges cross the Gcell boundaries. In the embodiments described below, certain non-planar edges between layers  4  and  5  cross Gcell boundaries. These non-planar edges are further described in Section I.B.3. 
     In some embodiments that use the wiring model illustrated in  FIGS. 4-7 , the association between the congestion edges and the node edges is as follows. Each horizontal congestion edge on layer  3  is associated with the pair of horizontal node edges that cross the same Gcell boundary as the horizontal congestion edge on the layer  3 . Each vertical congestion edge on layer  2  is associated with the pair of vertical node edges that cross the same Gcell boundary as the vertical congestion edge on layer  2 . 
     Each 45° diagonal congestion edge on layer  4  (1) is associated with a 45° diagonal node edge that crosses the same Gcell boundary as the 45° diagonal congestion edge on layer  4 , and (2) can be associated with two non-planar node edges between layers  4  and  5  that cross the same Gcell boundary as the 45° congestion edge. Each −45° diagonal congestion edge on layer  5  (1) is associated with a −45° diagonal node edge that crosses the same Gcell boundary as the −45° diagonal congestion edge on layer  5 , and (2) can be associated with two non-planar node edges between layers  4  and  5  that cross the same Gcell boundary as the −45° congestion edge. The association between ±45° congestion edges and non-planar node edges will be described below in Section I.B.3. 
     Node edges start and terminate on nodes. Congestion edges, on the other hand, do not have explicit start and end points in some embodiments. This is because unlike node edges that are used to define routes, congestion edges function only to evaluate usage versus capacity. The router&#39;s use of node and congestion edges is further described below. 
     B. Non-Planar Edges: Vias. 
     In the embodiments described below, the router can define routes that use non-planar node edges. In these embodiments, non-planar node edges exist (1) between each pair of nodes that are overlapping and that are in two adjacent routing layers (e.g., are in layers  2  and  3 ), (2) between certain pairs of non-overlapping nodes that are within the same Gcell and that are on adjacent diagonal layers  4  and  5 , and (3) between certain pairs of non-overlapping nodes that are within adjacent Gcells and that are on adjacent diagonal layers  4  and  5 . Each non-planar node edge represents a via between the two layers traversed by the edge. A non-planar edge that is between non-overlapping nodes in layers  4  and  5  also represents wiring to and from the edge&#39;s via. Each of the non-planar edge types will now be described further. 
     1. Non-Planar Edge Between Overlapping Nodes. 
     The routing graph includes a non-planar node edge between each pair of overlapping nodes that are on two adjacent routing layers. Each such non-planar edge represents a via between the edge&#39;s two nodes. Each such edge is assessed a wirelength cost and a via congestion cost. The wirelength cost equals a via-scalar factor (X) times the unit length cost (L) (i.e., is assessed a wirelength cost X*L). The via-scalar factor is 1 in some embodiments, while it is greater or less than one in other embodiments. The use of any non-planar edge also incurs a via congestion cost that represents the potential difficulty in placing too many vias between the two layers traversed by the non-planar edge in the Gcell associated with the non-planar edge&#39;s via. For a non-planar edge between two overlapping nodes, the Gcell associated with the edge&#39;s vias is the Gcell containing the two nodes. 
     2. Non-Planar Edges Between Non-Overlapping Nodes in the Same Gcell: Internal Zigs 
     Non-planar node edges exist between certain pairs of non-overlapping nodes that are within the same Gcell and that are on adjacent diagonal layers  4  and  5 . Such non-overlapping nodes are called internal zigs.  FIGS. 9 through 12  illustrate four internal zigs that some embodiments define between layers  4  and  5  in a Gcell. Each of these figures presents a two-dimensional top view of the routing graph.  FIG. 13  presents a three-dimensional side view of the example illustrated in  FIG. 9 . 
     In  FIGS. 9 and 13 , an internal zig  900  goes from a northwest node  905  on layer  4  to a northeast node  910  on layer  5  in a Gcell  920 . On layer  4 , 45° node edges run through northwest nodes (such as node  905 ) but not through northeast nodes (such as node  910 ). Conversely, on layer  5 , −45° node edges run through northeast nodes (such as node  910 ) but not through northwest nodes (such as node  905 ). Accordingly, the internal zig  900  allows a route running through node  905  or  910  to change layers and directions. As shown in  FIGS. 9 and 13 , this zig has three components. Two of its components are planar segments, where one segment is a 45° edge that runs northerly from the sub-region represented by node  905  on layer  4 , while the other segment is a −45° edge that runs southerly to the sub-region represented by node  910  on layer  5 . The third component is a non-planar component that is at the location where the two planar components overlap. The non-planar component represents a via, while the planar components represent wiring to and from the via.  FIGS. 9 and 13  show the location of this intersection (i.e., the via location) to be on the Gcell boundary. However, in a detailed route representation of the internal zig  900 , this intersection might occur anywhere within the sub-region  935  illustrated in  FIG. 9 . 
     The embodiments described below assess three costs for the internal zig  900 . First, an internal zig is assessed a wirelength cost that equals a via-scalar factor (X) times the unit length cost (L) (i.e., is assessed a wirelength cost X*L). Second, an internal zig is assessed an additional wirelength cost, which is the unit length cost times the square root of two (i.e., it is L*√{square root over (2)}). This additional wirelength cost represents the approximate wirelength necessary to traverse to and from the actual via location. Third, there is a via congestion cost associated with the internal zig. This via congestion cost represents the potential difficulty in placing too many vias between the two layers traversed by the internal zig in the Gcell associated with this zig&#39;s via. The Gcell associated with an internal zig&#39;s via is the Gcell containing the two nodes of the zig. 
     As mentioned above, the internal zig might not result in a via in Gcell  920  but might result in a via in Gcell  925  above it. Accordingly, unlike the embodiments described below, other embodiments might assess a via congestion cost for the Gcell  925  and/or assess a wire congestion cost to account for the congestion that the wiring associated with the internal zig might cause across the boundary between Gcells  920  and  925 . As further described below for external zigs, via and wire congestion costs should be accounted for together, as the location of the via will determine the layer on which wires cross the congestion grid. Via congestion and wire congestion costs are further described below. 
     The internal zigs  1000 ,  1100 , and  1200  that are illustrated in  FIGS. 10 ,  11 , and  12  are analogous to the internal zig  900 , except that they connect different pairs of nodes in the Gcell  920 . These three zigs are costed in the same manner as the zig  900 . 
     3. Non-Planar Edges Between Non-Overlapping Nodes in the Adjacent Gcells: External Zigs 
     Non-planar node edges exist between certain pairs of non-overlapping nodes that are within adjacent Gcells and that are on adjacent diagonal layers  4  and  5 . Such non-overlapping nodes are called external zigs.  FIGS. 14 through 21  illustrate eight external zigs that some embodiments define between one of four nodes of a particular Gcell ( 920 ) and one of eight nodes in the four Gcells ( 1410 ,  1415 ,  1420 , and  1425 ) that are adjacent to the particular Gcell ( 920 ). Each of these figures presents a two-dimensional top view of the routing graph.  FIG. 22  presents a three-dimensional side view of the example illustrated in  FIG. 14 . 
       FIGS. 14 and 22  illustrate an external zig  1400  between the node  930  on layer  4  of the Gcell  920  and the node  1405  on layer  5  of the Gcell  1410 , which is adjacent to Gcell  920 . On layer  4 , 45° node edges run through southeast nodes (such as node  930 ) but not through southwest nodes (such as node  1405 ). Conversely, on layer  5 , −45° node edges run through southwest nodes (such as node  1405 ) but not through southeast nodes (such as node  930 ). Accordingly, the external zig  1400  allows a route running through node  930  or  1405  to change layers and directions. 
     This zig traverses has three different components. Two of its components are planar segments, where one segment is a 45° edge that runs northerly from the sub-region represented by node  930  on layer  4 , while the other segment is a −45° edge that runs southerly to the sub-region represented by node  1405  on layer  5 . The third component is a non-planar component that is at the location where the two planar components overlap. The non-planar component represents a via, while the planar components represent wiring to and from the via. 
     There are four costs associated with the external zig  1400 . First, an external zig is assessed a wirelength cost that equals a via-scalar factor (X) times the unit length cost (L) (i.e., is assessed a wirelength cost X*L). Second, the external zig is assessed an additional wirelength cost, which is the unit length cost times square root of two (i.e., it is L*√{square root over (2)}). This extra wirelength cost represents the approximate wirelength necessary to traverse to and from the actual via location. 
     The third and fourth cost components are the via congestion cost and the wire congestion cost. As mentioned above, the use of any non-planar edge incurs a via congestion cost that represents the potential difficulty in placing too many vias between the two layers traversed by the non-planar edge in the Gcell associated with the non-planar edge&#39;s via. The wire congestion cost, on the other hand, represents the congestion that the wiring associated with the external zig causes across the Gcell boundary crossed by the external zig. 
     The via and wire congestion costs of an external zig depend on the actual location of the via represented by the external zig. However, an external zig specifies only that a via between layers  4  and  5  is placed close to the boundary between two Gcells (e.g., Gcells  920  and  1410 ), and does not specify an actual location of the via between layers  4  and  5 . In other words, an external zig can be associated with a via location in either of the two Gcells that it traverses, and can be associated with either of the two diagonal congestion edges that are defined across the boundary between the two Gcells. Consequently, in some embodiments, the router associates the external zig with one of the Gcells and one of the diagonal congestion edges, in order to assign the via and wire congestion costs for using the external zig. To do this, the router first computes two sets of via and wire congestion costs, where (1) the first set is based on a via location in one Gcell and on a Gcell boundary-crossing along a particular congestion edge, and (2) the second set is based on a via location in the other Gcell and on a Gcell-boundary crossing along the other congestion edge. The router then identifies the set with the smaller aggregate via and wire congestion costs. It then specifies the external zig&#39;s via location and congestion edge as the identified set&#39;s via location and congestion edge. 
     For instance, in  FIGS. 14 and 22 , the via for the external zig  1400  can be located in Gcell  920  or Gcell  1410 . Accordingly, a first set of via and wire congestion costs V 1  and W 1  is computed based on an assumption that a detail route generated from the external zig  1400  would result in a via location in Gcell  920 .  FIG. 23  illustrates such a via location. For such a location, the via congestion cost V 1  is computed. The cost V 1  represents the increase in the via congestion between layers  4  and  5  in the Gcell  920 . Section III will describe how via congestion costs are computed in some embodiments. 
     As shown in  FIG. 23 , the via location in Gcell  920  will require a −45° edge to cross the congestion-grid boundary  1430  between Gcells  920  and  1410  on layer  5 . Hence, for this via location, the wire congestion cost W 1  is computed. The cost W 1  represents the increase in the congestion in the −45° direction on layer  5  across the Gcell boundary  1430 . This wire congestion cost is computed by reference to the capacity and usage of congestion edge  2305 , which represents the wiring tracks in the −45° direction across the Gcell boundary  1430 . Section III will describe how wire congestion costs are computed in some embodiments. 
     For the external zig  1400 ,  FIG. 24  illustrates a via location in Gcell  1410 . A second set of via and wire congestion costs V 2  and W 2  is computed for this via location. The via congestion cost V 2  represents the increase in the via congestion between layers  4  and  5  in the Gcell  1410 . Also, as shown in  FIG. 24 , this via location will require a 45° edge to cross the boundary  1430  on layer  4 . Hence, for this via location, the wire congestion cost W 2  represents the increase in congestion cost in the 45° direction on layer  4  across the Gcell boundary  1430 . This wire congestion cost is computed by reference to the capacity and usage of congestion edge  2310 , which represents the wiring resources in the 45° direction across the Gcell boundary  1430 . 
     Once the two sets of costs are computed for the example in  FIGS. 14 and 22 , two aggregate values A 1  and A 2  are obtained by using a linear equation to sum the via and wire costs in each set. For instance, in some embodiments, A 1  equals a*V 1 +b*W 1 , while A 2  equals a*V 2 +b*W 2 , where a and b are scalar values. The scalar values a and b are equal to 1 in some embodiments, while, in other embodiments, they differ from each other and/or are greater or less than 1. 
     After computing the aggregate values, the external zig is associated with the set that results in the smaller aggregate value. For instance, if the aggregate value A 1  of the first set is smaller than the second set&#39;s aggregate value A 2  in the example illustrated in  FIGS. 23 and 24 , the via location, edge crossing, and via and wire congestion costs of the first set are selected as the via location, edge crossing, and via and wire congestion costs of the external zig  1400 . In other words, the router specifies the Gcell  920  as the Gcell that contains the via of the external zig  1400 . It associates this external zig with the congestion edge  2305  (i.e., with a −45° edge crossing on layer  5 ). The router also specifies this external zig&#39;s incremental via and wire congestion costs as the values V 1  and W 1  (i.e., as the first-set incremental via and wire congestion costs). 
     The via and wire congestion costs in and across Gcells are values that continually evolve as the router embeds more routes. Hence, each time the router explores using a non-planar edge, the router uses the above-described approach to select the optimal via location and edge crossing for an external zig at that time. 
     The external zigs  1500 - 2100  that are illustrated in  FIGS. 15-21  are analogous to the external zig  1400 , except that they connect different node pairs. These seven external zigs  1500 - 2100  are costed in exactly the same manner as the zig  1400 . 
     C. Route Representation with Respect to the Length and Congestion Edges 
     As described below, the router identifies the global route for a net by performing one or more path searches that identify one or more route segments that connect one or more pairs of pins/Steiner points of the net. Each path search tries to identify a path between two sets of nodes associated with the net along the node edges. If the path search identifies a path between the two sets, it embeds the identified path by reference to the node edges that the path traversed to go from one node set to the other. Hence, the router ends up defining each net&#39;s route in terms of the node edges. 
     However, in several instances, the discussion below refers to congestion edges used by a route or a path, where a path is an actual or potential portion of a route. A route or path is said to use a particular congestion edge if it is defined by reference to a planar or non-planar node edge that crosses the same boundary as the congestion edge on the same layer as the congestion edge. In other words, a route or path is said to use a particular congestion edge when it uses (1) a planar node edge associated with the particular congestion edge, or (2) a non-planar node edge that the router has associated with the particular congestion edge for the route&#39;s or path&#39;s use of the non-planar node edge. 
     Even though the embodiments described below define global routes by reference to the node edges, one of ordinary skill will realize that other embodiments might define a global route differently. For instance, some embodiments might define a global route in terms of the congestion edges. 
     II. Overall Flow of Router 
       FIG. 25  illustrates a process  2500  that conceptually represents the overall flow of the router in some embodiments of the invention. As shown in this figure, the process  2500  initially uses (at  2505 ) the congestion and length grids  100  and  200  to partition the IC layout region into numerous Gcells, with four nodes on each routing layer in each Gcell. As described above, these Gcells and nodes define a routing graph in which the router defines and embeds routes. 
     Next, the process computes (at  2510 ) the capacities of congestion edges between adjacent Gcells. These edges were described above by reference to  FIG. 8 . The capacity of a congestion edge is typically determined by a variety of factors, such as the size of the sub-regions, the pitch (width and spacing) of the wiring tracks represented by the edge, and the obstructions near edge. 
       FIGS. 26A and 26B  present two examples that illustrate how the process  2500  computes the capacity of a congestion edge in some embodiments. In the description of these examples, “pitch” refers to the sum of the default wire width and spacing for a given layer, projected in the wiring direction for that layer onto a Gcell boundary that it crosses. For a ±45-degree wire, this projection increases width and spacing by a factor of √{square root over (2)} compared to their ordinary values. 
     To compute the capacity of a congestion edge that crosses a Gcell boundary on a given layer, the process  2500  (1) defines a parallelogram about the Gcell boundary, (2) identifies potential obstacles in the parallelogram, (3) identifies the intersection of the projection in the layer&#39;s wiring direction of the identified obstacle with the boundary, (4) specifies any identified intersection as a blocked portion of the boundary, and (5) derives the congestion-edge capacity from the unblocked portion of the boundary. 
     In some embodiments, the process identifies a parallelogram about a Gcell boundary in the following manner. It identifies a first pair of parallel sides of the parallelogram by translating the Gcell boundary onto the midpoint of each of the two Gcells in the direction of the layer&#39;s wiring direction. These two sides will be parallel to the boundary between the Gcells and will traverse through the Gcell midpoints. The second pair of the parallelogram&#39;s parallel sides are in the layer&#39;s wiring direction and connect to the parallelogram&#39;s first pair of sides (i.e., each side in the second pair terminates at one end of each side of the first pair). 
       FIG. 26A  illustrates a parallelogram  2615  that is defined for a congestion edge  2632  that crosses a Gcell boundary  2630  on layer  4 , while  FIG. 26B  illustrates a parallelogram  2665  that is defined for a congestion edge  2634  that crosses a Gcell boundary  2680  on layer  3 . The boundary  2630  is between Gcells  2605  and  2610 , while the boundary  2680  is between Gcells  2655  and  2660 . The diagonal congestion edge  2632  is associated with one 45° node edge that crosses the boundary  2630  on layer  4 , while the horizontal congestion edge is associated with two horizontal node edges that cross the boundary  2680  on layer  3 . 
     In  FIG. 26A , the wiring direction on layer  4  is the 45° diagonal direction. Accordingly, the boundary  2630  is translated onto the center of the Gcells  2605  and  2610  in the 45° diagonal direction. This translation defines two parallel vertical sides  2620  and  2625  of the parallelogram  2615 . These two sides are parallel to the boundary  2630  and respectively run through the center of Gcells  2605  and  2610 . The other two sides of the parallelogram are sides  2635  and  2640 , which are in the layer&#39;s wiring direction (which is the 45° direction) and connect to sides  2620  and  2625 . 
     In  FIG. 26B , the wiring direction is horizontal. Accordingly, the boundary  2680  is translated onto the center of the Gcells  2655  and  2660  in the horizontal direction. This translation defined two parallel vertical sides  2670  and  2675  of the parallelogram  2665 . These two sides are parallel to the boundary  2680  and respectively run through the center of Gcells  2655  and  2660 . The other two sides of the parallelogram  2665  are sides  2685  and  2690 , which are in the layer&#39;s wiring direction (which is the horizontal direction) and connect to the sides  2670  and  2675 . 
     After identifying the parallelogram about a congestion-edge&#39;s Gcell boundary, the process then identifies each potential obstacle (e.g., each piece of pin, obstruction, or pre-route metal) that falls in the parallelogram. For each potential obstacle identified in the parallelogram, the process then identifies the portion of the boundary that the obstacle would intersect if the obstacle were moved across the boundary in the layer&#39;s wiring direction. For instance,  FIG. 26A  illustrates a pin  2642  on layer  4  that falls within the parallelogram  2615 . As shown in this figure, this pin would intersect portion  2644  of the boundary if it were moved across the boundary  2630  in the 45° direction.  FIG. 26B  illustrates a pin  2692  on layer  3  that falls within the parallelogram  2665 . As shown in this figure, the pin  2692  would intersect portion  2694  of the boundary  2680  if it were moved across this boundary in the horizontal direction. 
     The process treats all identified intersected portions of the boundary as blocked segments of the boundary. The process then estimates the capacity of a congestion edge to be the total length (T U ) of all unblocked intervals on the congestion edge&#39;s boundary that are at least one pitch long, divided by pitch (P), i.e., the capacity of the congestion edge equals 
     
       
         
           
             
               
                 T 
                 U 
               
               P 
             
             . 
           
         
       
     
     Some embodiments might not treat each piece of pin, obstruction, or pre-route metal as a blockage on the interval of boundary onto which it translates in the routing direction. Also, some embodiments might differently define the region to examine near a boundary crossed by a congestion edge. For instance, some embodiments might define different parallelograms on the diagonal layers. Instead of the parallelogram  2615  in  FIG. 26A , some embodiments might define a parallelogram  2652  illustrated in  FIG. 26C . This parallelogram  2652  has the Gcell centers and the Gcell boundary endpoints as its four vertices. 
     After  2510 , the process computes (at  2515 ) the via capacity between each two adjacent layers in each Gcell. In a given Gcell, the via capacity between two adjacent layers is computed as a user-adjustable constant times the maximum of all capacities of planar congestion edges into the Gcell on either of the layers. The constant is typically less than 2. In some embodiments, it is 1.7. 
     After  2515 , the process then identifies (at  2518 ) a set of potential Steiner points for each net that it is routing. Steiner points for a net can be found by (1) representing each pin of the net as a unique (x,y) position given by the centroid of its pin geometry, (2) assigning an edge cost between any two points in the plane equal to the octilinear distance between them, and (3) invoking a procedure given in “A fast and simple Steiner routing heuristic”, by Manjit Borah, Robert Owens, and Mary Jane Irwin, Discrete Applied Mathematics 90 (1999), pp. 51-67. One manner of computing the octilinear distance between two points is described in U.S. patent application Ser. No. 10/174,662, entitled “Method and Apparatus for Estimating Distances in a Region,” and filed on Jun. 19, 2002, now issued as U.S. Pat. No. 6,879,934. The U.S. patent application Ser. No. 10/174,662 is incorporated herein by reference. One skilled in the art will recognize that a variety of other heuristics may be used to generate Steiner points. Also, the Steiner set for a net might be an empty set in certain situations. 
     Next, for each net that it is being routed, the process identifies (at  2520 ) a congestion-unaware route that does not account for via congestion within the Gcells or wire congestion at Gcell boundaries. The generation of a congestion-unaware route for a net will be further described below in Section III. 
     After  2520 , the process performs two nested loops. The inner loop identifies one set of routes for each net being routed, while the outer loop causes the inner loop to run several (e.g., 8) times to generate several (e.g., 8) sets of routes. The generated sets of routes typically differ. These sets often differ because, in the embodiments described below, the inner loop uses a route-generation process that employs a costing function that accounts for resources used by the routes previously identified by the inner loop. The outer loop runs from  2525  to  2545 , while the inner loop runs from  2530  to  2540 . 
     At  2525 , the process sorts the nets. In some embodiments, the first time the process reaches  2525  it sorts the nets in an ascending order of the lengths of their congestion-unaware routes, which were identified at  2520 . The process then selects (at  2530 ) a net according to the order specified at  2525 . It then identifies (at  2535 ) a route for the selected net. To identify this route, the process typically uses a route generation process that employs a costing function that accounts for resources used by the routes previously identified at  2535 . No previously identified route exists for the first net in the first pass of the process  2500  through  2535 . However, one or more such routes exist in every subsequent pass through  2535 . The route identification at  2535  will be further described below in Section III. As mentioned above, the process  2500  typically uses a route generation process at  2535  to identify a route for a net. However, in some cases, the process might not identify a new route at  2535  for a net, but rather might identify a previous route (e.g., the most recent route) that it previously identified for the net. 
     After identifying a route for the selected net, the process determines (at  2540 ) whether it has generated a route for all the nets in the current pass through  2525 - 2545  (i.e., whether the selected net is the last net in the order specified in the last pass through  2525 ). If not, the process selects (at  2530 ) the next net in the order specified in the last pass through  2525 , identifies (at  2535 ) a route for this net, and then determines (at  2540 ) whether this net is the last net in the order specified in the last pass through  2525 . 
     Once the process determines (at  2540 ) that it has generated a route for all the nets in its current pass through  2525 - 2545 , the process determines (at  2545 ) whether it has generated the desired number (S) of route sets. If not, the process returns to  2525  to initiate another pass through the outer loop (i.e., through  2525  to  2545 ) so that it can generate another set of routes. For this pass through, the process can specify (at  2525 ) the same net order as, or a different net order than, the previous pass through the outer loop. Some embodiments specify a different net order for each pass through the outer loop in an attempt to increase the differences between the sets of generated routes. 
     When the process determines (at  2545 ) that it has generated the desired number of route sets, the process then identifies (at  2550 ) one set of routes from all the generated routes. Different embodiments use different techniques to select (at  2550 ) one combination of routes from the set of identified routes. One suitable technique is randomized rounding, which is described in Randomized Algorithm, by Rajeev Motwani and Prabhakar Raghavan, Cambridge University Press (1995, 1997). 
     Several other suitable techniques are described in U.S. patent application Ser. No. 10/335,239, entitled “Method and Apparatus for Solving an Optimization Problem in an Integrated Circuit Layout,” filed on Dec. 31, 2002, now issued as U.S. Pat. No. 7,216,308. This application is incorporated herein by reference. One technique described in this incorporated application identifies one set of routes by first specifying a set that has one identified route for each net. It then iteratively examines all the nets. During the examination of each particular net, the process iteratively examines all the identified routes for the particular net. During the examination of each particular route for each particular net, the process replaces the current route for the particular net in the solution set with the particular route if the replacement would improve the solution set. Under this approach, the set that remains after all the identified routes of all the nets have been examined is the set identified at  2550 . 
     After  2550 , the process ends. 
     III. Route Generation 
       FIG. 27  illustrates a route-generation process  2700  that the routing process  2500  can use (at  2535 ) to generate a route for a particular net in some embodiments. The process  2700  starts (at  2705 ) by initializing a variable, Route_Length, to zero. The process uses this variable to specify the length of the route that it tries to construct for the net. 
     The process then identifies (at  2710 ) the nodes (i.e., the length-grid sub-regions on each layer) that contain the particular net&#39;s set of pins and Steiner points. These nodes will be referred to as the configuration nodes of the particular net. After identifying the configuration nodes once for a net, some embodiments store the configuration nodes for the net, so that they can be retrieved the next time that they are needed. In some cases, a pin or a Steiner point can be in more than one length-grid sub-region (i.e., more than one node). Hence, each pin or Steiner point is associated with a set of nodes. Also, in some embodiments, each Steiner point is specified only by an x- and y-coordinate. Hence, it can be on any layer in the routing graph. Accordingly, in some embodiments, the node on each layer that includes the x- and y-coordinates of a net&#39;s Steiner point is added to the net&#39;s configuration nodes. 
     After identifying the configuration nodes for the particular net, the process specifies (at  2715 ) source and target node sets for a first path search. In some embodiments, the process specifies the target set as a node set that is associated with a particular pin of the net. It then specifies the nodes of all other pins and Steiner points in the net&#39;s configuration that are within a certain distance of the target set as source nodes. In some embodiments, this distance is a certain percentage greater than the distance between the target set and the node in the net&#39;s configuration closest to the target set. 
     On layers  4  and  5 , the net pins might be in nodes that do not have planar edges running through them. However, such nodes are target and source nodes that need to be expanded to and from during a path search. It would be inefficient to reach these nodes only through non-planar edges. Accordingly, to address this situation, some embodiments specify one or more nodes that are adjacent to such nodes on the same layer as “shadow nodes.” A shadow node of a particular node in effect augments the representation of the particular node&#39;s pin in a path search. A particular node&#39;s shadow is a source node when the particular node is a source node, and is a target node when the particular node is a target node. In other words, a path can expand from a particular node&#39;s shadow node when the particular node serves as a source node. When the particular node is a target of a path search, an expansion to the particular node&#39;s shadow node is treated as an expansion to the target node. 
       FIG. 28  presents one example of shadow nodes. This figure illustrates a pin  2805  in a node  2810  on layer  4 . Layer  4  has 45° node edges that connect the northwest and southeast nodes on this layer. Node  2810 , however, is a northeast node that does not have a planar node edge running through it. This node could be a target or source node. However, it would be difficult to reach this node since it has no incident planar node edge. Consequently, to address this situation, some embodiments specify node  2815  and/or node  2820  as shadow nodes of node  2805 . As shadow nodes, node  2815  and  2820  can be treated as source nodes of a path search when node  2810  is a source node, and they can be treated as target nodes of a path search when node  2810  is a target node. 
     Some embodiments would specify both node  2815  and node  2820  as shadow nodes of node  2810 . Other embodiments would specify only one of these two nodes as a shadow node. For instance, some of these embodiments would specify node  2815  as the shadow node since pin  2805  is closer to the node edge  2825  running through node  2815  than the node edge  2830  running through node  2820 . Other embodiments might take this approach (i.e., might expand only from the shadow node that is closest to the actual pin) only when one or more node edges connected to the shadow node is not blocked (e.g., by an obstacle). When a node edge connecting to the shadow node that is closest to the actual pin is blocked, some of these embodiments might select the other adjacent node as an additional shadow node. 
     After specifying (at  2715 ) the source and target sets for a path search, the process directs (at  2720 ) a path-generation process to identify and embed the lowest-cost path between the specified source and target sets. If the path-search process embeds the lowest-cost path, the path-generation process increments the Route_Length by the length of the embedded path. The path-generation process is further described below by reference to  FIG. 29 . 
     At  2725 , the route-generation process determines whether the path-generation process was able to identify and embed a path between the specified source and target sets. If not, the process  2700  has failed to find a route for the net. Accordingly, it returns (at  2730 ) a notification specifying its failure and then ends. 
     The router&#39;s response to this notification was not illustrated in  FIG. 25 , in order not to obscure the description of the router&#39;s flow with unnecessary details. However, it should be noted that the router responds differently to this notification in different embodiments. For instance, in some embodiments, the router can remove from the routing problem a net that the process  2700  fails to route. In other embodiments, the router removes the net only if it repeatedly fails to find a route for the net after re-adjusting the net order and trying to find a complete routing solution. 
     If the process determines (at  2725 ) that the path-generation process identified and embedded a path, it determines (at  2735 ) whether it has routed all the pins of the net. If so, the process  2700  notifies (at  2740 ) the process  2700  that it has embedded a route for the net and provides this route and its associated Route_Length. 
     If the process  2700  determines (at  2735 ) that it has not routed all the pins of the net, the process specifies (at  2745 ) new source and target sets for another path search. In some embodiments, the process specifies (at  2745 ) as the target node set (1) all the nodes that are associated with the routed pins and Steiner points, and (2) all nodes that are currently on the one or more paths that the path-generation process has embedded for the net during the current route generation. In some embodiments, the process specifies (at  2745 ) as the source node set all nodes associated with any unrouted pin and Steiner point in the net&#39;s configuration that are within a certain distance of the target set. In some embodiments, this distance is a certain percentage greater than the distance between the target set and a node in the net&#39;s configuration that is closest to the target set and that is associated with a pin or Steiner that has not yet been routed. After specifying the source and target sets at  2745 , the process  2700  returns to  2720  to direct the path-generation process to identify and embed the lowest-cost path between the specified source and target sets. The operation of the process  2700  from  2720  was described above. 
     A. Path Generation. 
     At  2720 , the route-generation process  2700  calls a path-generation process to identify and embed a path between source and target node sets in the routing graph. In some embodiments, the router uses an A* path-generation process  2900  that is illustrated in  FIG. 29 . This process has two phases: (1) a path exploration phase, during which the process identifies a path between the specified source and target node sets, and (2) a path-embedding phase, during which the process embeds the identified path. 
     The process  2900  is an iterative best-first search that at each iteration tries to extend a partial solution with the best estimated cost. Specifically, during its path exploration phase, the process  2900  starts its path search by specifying the start of one or more paths from one or more source nodes. It then iteratively identifies one or more path expansions about the lowest cost path, until it identifies a path that connects a source node and a target node. Each identified expansion about a path is from a “current node” (also called “start node”) reached by the path being extended to a “destination node” that neighbors the current node. 
     For each expansion, the process computes an {circumflex over (F)} cost, which is the estimated cost of the path from a source node through the expansion&#39;s destination node to a target node. This cost can be expressed as:
 
 {circumflex over (F)}=G+Ĥ.   (1)
 
In this equation, G represents the cost of the path that has reached the expansion&#39;s destination node, while Ĥ represents an estimated cost of a path from the expansion&#39;s destination node to the set of target nodes. In the embodiments described below, the Ĥ cost expresses the lower-bound estimate of the shortest path from the expansion&#39;s destination node to the target set. Accordingly, in these embodiments, the {circumflex over (F)} cost expresses the estimated cost of a lowest-cost path from a source node through the expansion&#39;s destination node to a target node. Also, in these embodiments, the G and hence the {circumflex over (F)} account for several different types of costs, such as a wirelength, wire congestion, and via congestion. Each of these costs is further described below.
 
     As shown in  FIG. 29 , the process  2900  initially (at  2905 ) identifies and sets the Ĥ of each source node that the process  2700  specified for the current path search. Each node&#39;s Ĥ expresses the estimated distance between the node and the target set in the current path search. During each search, the process  2900  stores the Ĥ cost for each node after computing this cost, so that it only has to compute it once for each node reached in each search. Different embodiments compute a node&#39;s Ĥ differently. Some embodiments use a rectilinear bounding box technique that is used in conventional A* path searches. Other embodiments, however, use the novel techniques described in the above-incorporated U.S. patent application Ser. No. 10/174,662. One technique described in this application identifies two bounding boxes that each enclose the target set. One bounding box has sides that are parallel to one of the layout&#39;s coordinate axes. The other bounding has sides that are rotated by 45° with respect to the layout&#39;s coordinate axes. This technique then identifies the distance between the node and each of the two bounding boxes. It then identifies the node&#39;s Ĥ as the longer of the two identified distances. 
     For each source node, the process  2900  also specifies (at  2905 ) a “drop,” which is path identifier that represents a path expansion. Specifically, a drop represents an expansion from a start node to a destination node by referring to the destination node as its node and referring back to the drop of the start node. Drops allow the process  2900  to keep track of the paths that it explores. For each drop, the process also stores (1) a G cost, which is the cost of a path from a source node to the drop&#39;s node through the sequence of expansions that led to the drop, and (2) an {circumflex over (F)} cost, which is the drop&#39;s G cost plus the Ĥ cost of the drop&#39;s node (i.e., the Ĥ cost of the destination node of the expansion for which the drop was specified). One of ordinary skill will realize that other embodiments might not use drops or might implement drops differently. 
     At  2905 , a drop that is defined for a source node refers to the source node as its node and defines the drop&#39;s prior drop as null. The process sets the G cost of each drop defined at  2905  to zero, and sets the drop&#39;s {circumflex over (F)} cost equal to Ĥ cost of the drop&#39;s node. At  2905 , the process stores the specified drops in a storage structure, which, in some embodiments, is a priority queue (e.g., a heap) that is ordered based on the {circumflex over (F)} costs of the drops. 
     Next, at  2910 , the process retrieves from the priority queue a drop with the smallest {circumflex over (F)} cost, and specifies this drop as the Current_Drop. The process then “closes” (at  2915 ) the drop&#39;s node. A closed node is a node to which the process can no longer expand during the path search. Consequently, the closing of node at  2915  prevents the process  2900  from expanding to this node during the current path search. 
     The process then determines (at  2920 ) whether the Current_Drop&#39;s node is a node in the target set for the current path search. If not, the process performs a series of operations in a loop from  2930  to  2965 , in order to explore all possible expansions about the Current_Drop. Specifically, at  2930 , the process identify one of the possible expansions about the Current_Drop. Table 1 below lists all the possible expansions from the Current_Drop&#39;s node for a wiring model that allows routing only on layers  2 - 5 . 
     
       
         
           
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 Drop&#39;s 
                   
                   
               
               
                 Layer 
                 Planar Expansion 
                 Non-Planar Expansions 
               
               
                   
               
             
            
               
                 2 nd  Layer 
                 If drop&#39;s node is not on the north or south 
                 Expansion to the node directly above 
               
               
                   
                 boundary of the layout, two planar 
                 the drop&#39;s node. 
               
               
                   
                 expansions are possible to the two nodes 
               
               
                   
                 that are adjacent to the drop&#39;s node in the 
               
               
                   
                 ±90° directions. If the drop&#39;s node is on 
               
               
                   
                 the south or north layout boundary, then 
               
               
                   
                 one planar expansion is available to the 
               
               
                   
                 one node adjacent to the drop&#39;s node in the 
               
               
                   
                 +90° or −90° direction. 
               
               
                 3 rd  Layer 
                 If drop&#39;s node is not on the east or west 
                 Expansion to nodes directly above and 
               
               
                   
                 boundary of the layout, two planar 
                 below the drop&#39;s node. 
               
               
                   
                 expansions are possible to the two nodes 
               
               
                   
                 that are adjacent to the drop&#39;s node in the 
               
               
                   
                 0° and 180° directions. If the drop&#39;s node 
               
               
                   
                 is on the east or west boundary of the 
               
               
                   
                 layout, then one planar expansion is 
               
               
                   
                 available to the one node adjacent to the 
               
               
                   
                 drop&#39;s node in the +180° or 0° direction. 
               
               
                 4 th  Layer 
                 If drop&#39;s node is not on periphery of the 
                 Expansion to nodes directly above and 
               
               
                   
                 layout and it is either the northwest or 
                 below the drop&#39;s node. 
               
               
                   
                 southeast node of a Gcell, two planar 
                 Also, two internal zig expansions, 
               
               
                   
                 expansions are possible to the two nodes 
                 where each expansion is to a node that 
               
               
                   
                 that are adjacent to the drop&#39;s node in the 
                 is in layer 5 in the same Gcell as the 
               
               
                   
                 +45° and −135° directions. If the drop&#39;s 
                 drop&#39;s node. 
               
               
                   
                 node is on the periphery of the layout and 
                 Up to two external zig expansions, 
               
               
                   
                 it is either the northwest or southeast node 
                 where each such expansion is to a 
               
               
                   
                 of a Gcell, then zero planar expansion is 
                 node that is in layer 5 in another Gcell 
               
               
                   
                 possible or one planar expansion is 
                 which is adjacent to the Gcell 
               
               
                   
                 available to the one node adjacent to the 
                 containing the drop&#39;s node. 
               
               
                   
                 drop&#39;s node in the +45° or −135° 
               
               
                   
                 direction. There are no planar expansions 
               
               
                   
                 from a drop&#39;s node if this node is the 
               
               
                   
                 northeast or southwest node of a Gcell. 
               
               
                 5 th  Layer 
                 If drop&#39;s node is not on periphery of the 
                 Expansion to nodes directly above and 
               
               
                   
                 layout and it is either the southwest or 
                 below the drop&#39;s node. 
               
               
                   
                 northeast node of a Gcell, two planar 
                 Also, two internal zig expansions, 
               
               
                   
                 expansions are possible to the two nodes 
                 where each expansion is to a node that 
               
               
                   
                 that are adjacent to the drop&#39;s node in the 
                 is in layer 4 in the same Gcell as the 
               
               
                   
                 −45° and +135° directions. If the drop&#39;s 
                 drop&#39;s node. 
               
               
                   
                 node is on the periphery of the layout and 
                 Up to two external zig expansions, 
               
               
                   
                 it is either the southwest or northeast node 
                 where each such expansion is to a 
               
               
                   
                 of a Gcell, then zero planar expansion is 
                 node that is in layer 4 in another Gcell 
               
               
                   
                 possible or one planar expansion is 
                 which is adjacent to the Gcell 
               
               
                   
                 available to the one node adjacent to the 
                 containing the drop&#39;s node. 
               
               
                   
                 drop&#39;s node in the −45° or +135° direction. 
               
               
                   
                 There are no planar expansions from a 
               
               
                   
                 drop node if this node is the southeast or 
               
               
                   
                 northwest node of a Gcell. 
               
               
                   
               
            
           
         
       
     
     In some embodiments that use a wiring model that allows routing on layer  1 , the expansion possibilities on layer  1  are similar to the expansion possibilities on layer  3 , except that there are no non-planar expansions to a layer below. Also, in these embodiments, the process can expand from a node on layer  2  to a node directly below on layer  1 . 
     After selecting an expansion at  2930 , the process determines (at  2935 ) whether the destination node of the expansion is a closed node. If so, the process transitions to  2965 , which is further described below. Otherwise, the process computes (at  2940 ) a G cost for the expansion. The computation of this cost is described further below. 
     After  2940 , the process determines (at  2945 ) whether the destination node of the expansion specified at  2930  has been previously reached in the current path search. If not, the process (at  2960 ) computes and stores the destination node&#39;s Ĥ cost. The computation of this cost was described above at  2905 . At  2960 , the process also specifies a drop for the expansion specified at  2930 . The process associates the specified drop with the expansion&#39;s destination node, and sets the drop&#39;s previous drop to the Current_Drop. The process also (1) sets this specified drop&#39;s G cost to the G cost computed at  2940  for the expansion, and (2) sets the specified drop&#39;s {circumflex over (F)} cost to the sum of the drop&#39;s G cost and the Ĥ cost of the drop&#39;s node (i.e., the expansion&#39;s destination node). The process then stores (at  2960 ) the drop specified at  2960  in the priority queue based on its {circumflex over (F)} cost. From  2960 , the process transitions to  2965 , which will be described below. 
     If the process determines (at  2945 ) that the specified expansion&#39;s destination node has been previously reached in the current path search, the process determines (at  2950 ) whether the identified expansion&#39;s G cost (computed at  2940 ) is less than the G cost of the drop in the priority queue that is associated with the expansion&#39;s destination node. If not, the expansion specified at  2930  is not the cheapest expansion to its destination node. Hence, in this situation, the process stops its examination of the expansion identified at  2930 , and transitions to  2965 , which is further described below. 
     On the other hand, if the process determines (at  2950 ) that the identified expansion&#39;s G cost (computed at  2940 ) is less than the G cost of the drop in the priority queue that is associated with the expansion&#39;s destination node, the process removes the drop associated with the destination node from the priority queue, and specifies a new drop for this node. The process associates the new drop with the identified expansion&#39;s destination node, and sets the drop&#39;s previous drop to the Current_Drop. The process also (1) sets this newly specified drop&#39;s G cost equal to the identified expansion&#39;s G cost (computed at  2940 ) for the expansion, and (2) sets the specified drop&#39;s {circumflex over (F)} cost to the sum of the drop&#39;s G cost and the Ĥ cost of the drop&#39;s node (i.e., the Ĥ cost of the expansion&#39;s destination node). The process then stores (at  2955 ) the newly specified drop in the priority queue based on its {circumflex over (F)} cost. From  2955 , the process transitions to  2965 . 
     At  2965 , the process determines whether there is any expansion about the Current_Drop&#39;s node that it has not yet examined. If so, the process transitions back to  2930  to identify another expansion, and then performs the subsequent operations to determine whether to specify a drop for this newly identified expansion. 
     When the process determines (at  2965 ) that it has examined all expansions about the Current_Drop&#39;s node, the process determines (at  2970 ) whether the priority queue that stores the drops is empty. If so, the process has failed to find a path between the specified source and target sets. Accordingly, it returns (at  2975 ) a notification specifying its failure and then ends. On the other hand, when the process determines (at  2970 ) that the priority queue is not empty, the process transitions back to  2910  to retrieve the drop with the smallest {circumflex over (F)} cost from the priority queue and then to perform the above-described operations for this drop. 
     The process has found a path between the source and target sets when it determines (at  2920 ) that the Current_Drop&#39;s node is a target. In this situation, the process transitions from  2920  to  2925 . At  2925 , the process also embeds the identified path between the source and target sets. Starting at the Current_Drop on the target, the embedding “back traces” the sequence of drops that reached the target and generates an ordered list of (1) nodes associated with the drops and (2) node edges between these nodes. 
       FIG. 30  illustrates an example of a back trace operation. In this example, the path search has identified a path between a source node  3005  and a target node  3010 . This path has traversed through layers  3 - 5 . To identify this path, the path search has identified a series of drops  3015   a - 3015   h . With the exception of the first source-node drop  3015   a  that has a null back reference, each drop has a reference to a previous drop in the path. The back trace would start at the drop on the target and follow each drop&#39;s back reference to identify all of the path&#39;s drops, the nodes associated with these drops, and the edges between the nodes of successive drops. In this manner, the back trace would identify an ordered list of nodes  3020  between the source and target nodes  3005  and  3010 , and an ordered list of edges  3025   a - 3025   g  between these nodes. 
     Some embodiments then define the global route path by reference to the ordered list of node edges produced through the back trace. In the example illustrated in  FIG. 30 , this ordered list would include edges  3025   a - 3025   g . Other embodiments would identify the global route path by reference to the ordered list(s) of nodes and node edges identified in the back trace. In the example illustrated in  FIG. 30 , this ordered list(s) would include edges  3025   a - 3025   g  and node  3005 , nodes in the set  30020 , and node  3010 . 
     At  2925 , the process  2900  increments the Route_Length by the length of the identified global route path. This length is simply the sum of the length of the node edges identified through the back trace, where the length of a planar horizontal or vertical node edge is L, the length of a diagonal node edge is L*√{square root over (2)}, the length of a non-planar nodes edge between overlapping nodes is X*L, and the length of a non-planar edge between non-overlapping nodes is (X+√{square root over (2)})*L. 
     At  2925 , the process also determines if the source node that it reached through its back trace at  2925 , includes one or more pins that are marked as unrouted. If the process identifies one or more such pins at  2925 , it then selects one of these pins and marks it as routed. When there are no such pins in the source node reached through the back trace, the source node contains at least one Steiner point that is marked unrouted. Hence, when the source node does not contain an unrouted pin, the process selects (at  2925 ) a previously unrouted Steiner point and marks it as routed. 
     In some cases, the source and target node sets partially or completely overlap. In such cases, the path search transitions to  2925  the first time it reaches  2920 , and hence it will not identify any expansions. At  2925 , the back trace then simply identifies a previously unrouted pin or Steiner in a node that is in both the source and target node sets as a routed pin or Steiner. In other words, this back trace results in an empty set of global-route node edges as a pin or Steiner is reached without requiring any node edges. In this situation, some embodiments define a special “node-internal” edge between a previously unrouted “point” and a newly routed “point” in the source/target node, where a point in this case can be a pin or a Steiner point. Some of these embodiments use such an edge purely for internal bookkeeping, while others augment the notion of the global route to include this node-internal edge. When all of a net&#39;s pins are in the same node, the router only identifies node-internal edges. In such a situation, the resulting global route can be specified as either null (e.g., an empty set of node edges), or as a collection of node-internal edges. 
     One of ordinary skill will realize that the path-generation process might be implemented differently in other embodiments. For instance, in some embodiments, the Ĥ cost might not specify a lower bound on the shortest path between a drop&#39;s node and a target set. In addition, some embodiments might compute the {circumflex over (F)} cost slightly differently. For instance, some embodiments might express the {circumflex over (F)} cost as:
 
 {circumflex over (F)}=G+ 2 *Ĥ.  
 
Such a cost would bias the search process to expand about the drops that are closer to the target set. Alternative embodiments might express the {circumflex over (F)} cost as:
 
 {circumflex over (F)}=G+Ĥ+Ĵ.  
 
where Ĵ represents the estimated computational effort needed to complete the path from the current drop. The embodiments that use alternative {circumflex over (F)} cost might not satisfy the admissibility requirement. Also, instead of an A* search, other embodiments might perform other types of path searches.
 
     B. Costing an Expansion 
     As described above, the path-generation process  2900  computes the G cost of an expansion at  2940 . The embodiments described below use a cost function with several exponential components that depend on wirelength, wire-congestion, and via-congestion costs. In costing an expansion, the congestion components account for all the routes that the router has identified at  2535  before the current path search on the edges being considered. In this manner, these components bias the path search to spread the routes to favor evenly distributed congestion profiles over those with sharp peaks. Even though exponential cost terms are described below, one of ordinary skill will realize that other embodiments might use other types of costing functions to evaluate the routes. 
     1. General Exponential Costing Expression 
     For some embodiments, Equation (A) below provides a cost function that represents the G cost of an expansion to a destination node d. 
                   G   =       (       Y   L     *     ⅇ       ɛ   L     *           F   ^     L     ⁡     (   d   )           H   ^     MIN             )     +     (       ∑     j   =   1     m     ⁢       Y   j     *     (       ⅇ       ɛ   C     *       new   ⁢           ⁢   usage   ⁢           ⁢     (   j   )           goal   ⁡     (     layer   ⁡     (   j   )       )       *     capacity   ⁡     (   j   )               -     ⅇ       ɛ   C     *       old   ⁢           ⁢   usage   ⁢           ⁢     (   j   )           goal   ⁡     (     layer   ⁡     (   j   )       )       *     capacity   ⁡     (   j   )                 )         )     +     (       ∑     t   =   1     K     ⁢       Y   t     (           ⁢       ⅇ       ɛ   V     *       new   ⁢           ⁢   usage   ⁢           ⁢     (   t   )           goal   ⁡     (     layer   ⁢           ⁢     pairs   ⁡     (   t   )         )       *   capacity   ⁢           ⁢     (   t   )             -     ⅇ       ɛ   V     *       old   ⁢           ⁢   usage   ⁢     (   t   )           goal   ⁡     (     layer   ⁢           ⁢     pairs   ⁡     (   t   )         )       *     capacity   ⁡     (   t   )                 )                   (   A   )               
In this equation, the first exponential component represents a wirelength cost, the second exponential component represents a wire congestion cost, and the third exponential component represents a via congestion cost. Table 2 provides a definition for the variables in these components.
 
     
       
         
           
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 Term 
                 Definition 
               
               
                   
               
             
            
               
                 e 
                 The base of the natural logarithm. 
               
               
                 ε L , ε C , ε V   
                 User-adjustable exponential-multiplier parameters. 
               
               
                 Y L , Y i , and Y t   
                 Normalizing variables. 
               
               
                 {circumflex over (F)} L  (d) 
                 The estimated length of the path from a source node through the 
               
               
                   
                 expansion&#39;s destination node d to a target node. This estimated length 
               
               
                   
                 equals the length of the path that has reached the expansion&#39;s destination 
               
               
                   
                 node d, plus the destination node&#39;s Ĥ. 
               
               
                 Ĥ MIN   
                 The minimum Ĥ of the set of sources of the current path search. 
               
               
                 j 
                 One of m congestion edges that are used by the path that has reached the 
               
               
                   
                 expansion&#39;s destination node d. If the path has not used any congestion 
               
               
                   
                 edge, then m equals 0 and the wire congestion component of 
               
               
                   
                 Equation (A) is zero. 
               
               
                 old usage (j) 
                 The cumulative number of times that all the routes previously identified 
               
               
                   
                 at 2535 use the congestion edge j. In some embodiments, the router 
               
               
                   
                 computes and updates the old usage(j) value after each iteration of 2535 
               
               
                   
                 of the process 2500. 
               
               
                 new usage(j) 
                 The cumulative number of times that the congestion edge j is used by 
               
               
                   
                 (1) all the routes previously identified at 2535, and (2) the path that has 
               
               
                   
                 reached the expansion&#39;s destination node d. New usage (j) equals 
               
               
                   
                 old usage (j) plus 1. 
               
               
                 capacity(j) 
                 The estimated number of available tracks of the congestion edge j. In 
               
               
                   
                 some embodiments, this capacity is a value from 0 to 16. This number 
               
               
                   
                 was computed at 2510 of the process 2500. 
               
               
                 goal(layer(j)) 
                 A target upper bound on the congestion ratio on the layer that contains 
               
               
                   
                 edge j. 
               
               
                 t 
                 One of K non-planar edges that are on the path that has reached the 
               
               
                   
                 expansion&#39;s destination node d. If the path has not used any non-planar 
               
               
                   
                 edge, then t equals 0 and the via congestion component of Equation (A) 
               
               
                   
                 is zero. 
               
               
                 old usage(t) 
                 In all the routes previously identified at 2535, the number of non-planar 
               
               
                   
                 edges that via in the same Gcell as edge t between the same two layers as 
               
               
                   
                 edge t. With the exception of external zigs, the Gcell that contains the via 
               
               
                   
                 represented by edge t is the Gcell that contains the two nodes connected 
               
               
                   
                 by edge t. If edge t is an external zig, the Gcell of the via associated with 
               
               
                   
                 the external zig edge t is the Gcell that results in the smaller aggregate via 
               
               
                   
                 and wire congestion costs for this instance of use of the external zig t, as 
               
               
                   
                 described in Section I.B.3. After each iteration of 2535 of the process 
               
               
                   
                 2500, the router in some embodiments computes and updates the via 
               
               
                   
                 usage values for each pair of adjacent routing layers in each Gcell. 
               
               
                 new usage(t) 
                 In all the routes previously identified at 2535, and in the path that has 
               
               
                   
                 reached the expansion&#39;s destination node d, the number of non-planar 
               
               
                   
                 edges that via in the same Gcell as edge t between the same two layers as 
               
               
                   
                 edge t. See discussion in the definition of old usage (t) regarding the 
               
               
                   
                 Gcell that contains edge t&#39;s via. New usage(t) equals old usage(t) plus 1. 
               
               
                 capacity(t) 
                 The estimated number of vias that can traverse the same two layers as 
               
               
                   
                 edge t in this edge&#39;s Gcell. In some embodiments, this capacity is a value 
               
               
                   
                 from 0 to 27. This capacity was computed at 2515 of the process 2500. 
               
               
                 goal(layer pairs(t)) 
                 A target upper bound on the via congestion ratio between the two 
               
               
                   
                 adjacent layers traversed by the non-planar edge t. 
               
               
                   
               
            
           
         
       
     
     Each of the three components in Equation (A) includes an exponential expression. The exponential expressions are normalized to the same scale by multiplying them by the normalizing factors Y L , Y j , and Y t . In some embodiments, the multiplier Y L  is expressed by the following equation: 
                 Y   L     =       Y   LI     *     ⅇ       ɛ   L     *         ∑     N   =   1     R     ⁢     Length   ⁡     (   N   )           T   .   E   .   L               ,         
where (1) Y LI  is a constant, (2) N is one of the R routes identified thus far at  2535 , (3) length(N) is the length of the route N, and (4) T.E.L stands for total estimated length and equals the sum of the estimated length of each net&#39;s route. In some embodiments, the total estimated length is the sum of the lengths of the congestion-unaware routes (identified at  2520 ) of all nets; in other embodiments, it is the sum of a lower_bound on the route lengths of all the nets (e.g., it is the sum of the bounding box of each net).
 
     In some embodiments, Y LI  equals 1, while it equals another value 
             (     such   ⁢             ⁢             ⁢   as   ⁢           ⁢     1     T   .   E   .   L   .         )         
in other embodiments. The multiplier Y L  is a value that is re-computed after the identification of each route at  2535 . Hence, each time that the router is identifying (at  2535 ) a route for a net, the multiplier Y L  is based on all the routes that the router has identified before this iteration of  2535 .
 
     The multiplier Y j  is also different in different embodiments. For instance, in some embodiments, this multiplier equals 1 or some other constant. Some embodiments might use different constants for different edges. In other embodiments, this multiplier equals 
               1     goal   (       (     layer   ⁡     (   j   )       )     *     capacity   ⁡     (   j   )             ,         
where goal(layer(j)) and capacity(j) are defined in Table 2 above. In still other embodiments, this multiplier is represented by the following equation:
 
                 Y   j     =     1       C   _     L         ,         
where  C   L  is the average initial-capacity constant on each layer and is represented by:
 
                 C   _     L     =       1     Number   ⁢           ⁢   of   ⁢           ⁢   Layers       ⁢       ∑     Layers   ⁢           ⁢   L       ⁢           ⁢       (       ⅇ     (       ɛ   C         goal   ⁡     (   L   )       *     (     Average   ⁢           ⁢   Planar   ⁢           ⁢   Edge   ⁢           ⁢   Capacity   ⁢           ⁢   on   ⁢           ⁢   L     )         )       -   1     )     .               
Specifying the multiplier Y j  based on the average initial-capacity cost  C   L  centers the initial capacity costs of the edges about 1.
 
     The multiplier Y t  is also different in different embodiments. For instance, in some embodiments, this multiplier equals to 1 or some other constant. Some embodiments might use different constants for different non-planar edges. In other embodiments, this multiplier equals 
               1     goal   (       (     layer   ⁢           ⁢     pairs   ⁡     (   t   )         )     *     capacity   ⁡     (   t   )             ,         
where goal(layer pairs(t)) and capacity(t) are defined in Table 2 above. In still other embodiments, this multiplier is represented by the following equation:
 
                 Y   j     =     1       C   _     LP         ,         
where  C   LP  is the average initial-capacity constant for each layer pair and is represented by:
 
                 C   _     LP     =       1     Number   ⁢           ⁢   of   ⁢           ⁢   Adjacent   ⁢           ⁢   Layer   ⁢           ⁢   Pairs       ⁢       ∑     Layers   ⁢           ⁢   Pairs   ⁢           ⁢   LP       ⁢           ⁢       (       ⅇ     (       ɛ   V         goal   ⁡     (   LP   )       *     (     Average   ⁢           ⁢   Non   ⁢     -     ⁢   Planar   ⁢           ⁢   Edge   ⁢           ⁢   Capacity   ⁢           ⁢   Between   ⁢           ⁢   LP     )         )       -   1     )     .               
Specifying the multiplier Y t  based on the average initial-capacity constant  C   LP  centers the initial capacity costs of the non-planar edges about 1. One of ordinary skill will realize that other embodiments might specify their normalizing constants differently.
 
     In all three components of Equation (A), the base of the exponential expression is the natural-logarithm base e. Other embodiments, however, might use a different base. In addition, other embodiments might formulate differently the exponent of each exponential expression in Equation (A). In some embodiments, the exponential multipliers ε L , ε C , ε V  are real numbers between 2 to 16. For instance, in some embodiments, all three multipliers equal 9, even though these multipliers do not need to be the same value. As mentioned above, each of these multipliers is adjustable by the user in some embodiments. 
     Each route or path includes a set of node edges, which can be planar or non-planar. The cost of each planar and non-planar node edge was described above in Section I. Table 3 reiterates each of these costs for a route or a path, and describes how each of these costs is factored in each of the components of Equation (A). 
     
       
         
           
               
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
               
                   
                 Wirelength 
                   
                 Via Congestion 
               
               
                 Node Edge 
                 Component 
                 Wire Congestion Component 
                 Component 
               
               
                   
               
             
            
               
                 Planar Node 
                 Increments 
                 If edge f goes from one Gcell 
                 No effect. 
               
               
                 Edge f on 
                 length of route 
                 to another (i.e., crosses a 
               
               
                 Layers 2 and 3 
                 or path by unit 
                 Gcell boundary), then edge f 
               
               
                   
                 length cost L. 
                 is associated with a 
               
               
                   
                   
                 congestion edge j, and 
               
               
                   
                   
                 therefore a new usage(j) is 
               
               
                   
                   
                 defined by incrementing old 
               
               
                   
                   
                 usage(j) by one. Otherwise, no 
               
               
                   
                   
                 effect. 
               
               
                 Planar Node 
                 Increments 
                 Define new usage(j) by 
                 No effect. 
               
               
                 Edge f on 
                 length of route 
                 incrementing old usage(j) by 
               
               
                 Layers 4 and 5 
                 or path by 
                 one, where node edge f is 
               
               
                   
                 length cost 
                 associated with congestion 
               
               
                   
                 L * {square root over (2)}. 
                 edge j. 
               
               
                 Non-Planar 
                 Increments 
                 No effect. 
                 Define new usage(t) by 
               
               
                 Node Edge t 
                 length of route 
                   
                 incrementing old usage(t) 
               
               
                 between directly 
                 or path by 
                   
                 (associated with vias 
               
               
                 overlapping 
                 length cost 
                   
                 between layers a and b in 
               
               
                 nodes on layers 
                 X * L, where X 
                   
                 Gcell g) by one. 
               
               
                 a and b in Gcell 
                 is a via-scalar 
               
               
                 g. 
                 factor. 
               
               
                 Internal Zig t 
                 Increments 
                 No effect. 
                 Define new usage(t) by 
               
               
                 between non- 
                 length of route 
                   
                 incrementing old usage(t) 
               
               
                 overlapping 
                 or path by 
                   
                 (associated with vias 
               
               
                 nodes on layers 
                 length cost X * L 
                   
                 between layers a and bin 
               
               
                 a and b in Gcell 
                 plus L * {square root over (2)}. 
                   
                 Gcell g) by one. 
               
               
                 g. 
               
               
                 External Zig t 
                 Increments 
                 Use the approach described in 
                 Define new usage(t) by 
               
               
                 between non- 
                 length of route 
                 Section I.B.3 to associate the 
                 increment old usage(t) by 
               
               
                 overlapping 
                 or path by 
                 instance of the external zig t&#39;s 
                 one, where old usage (t) is 
               
               
                 nodes on layers 
                 length cost X * L 
                 use with a congestion edge j. 
                 the via usage between 
               
               
                 a and b in 
                 plus L * {square root over (2)}. 
                 Define new usage(j) by 
                 layers a and b in the Gcell 
               
               
                 adjacent Gcells 
                   
                 incrementing old usage(j) by 
                 that is assigned to contain 
               
               
                 g1 and g2. 
                   
                 one. 
                 via for this instance of 
               
               
                   
                   
                   
                 edge t&#39;s use, per the 
               
               
                   
                   
                   
                 approach described above 
               
               
                   
                   
                   
                 in Section I.B.3. 
               
               
                   
               
            
           
         
       
     
     One of ordinary skill will realize that other embodiments might use different exponential cost functions than the one illustrated in Equation (A). For instance, some embodiments might use an equation that has the same wire and via congestion components as Equation (A), but the following wirelength component instead of Equation (A)&#39;s wirelength component. 
               Wirelength   ⁢           ⁢   Component     =       Y   LI     *     ⅇ       ɛ   L     *         ∑     N   =   1     R     ⁢     Length   ⁢           ⁢     (   N   )           T   .   E   .   L           *       Length   ⁡     (   d   )       .             
In this equation, Length(d) is the length of the path p that has reached the expansion&#39;s destination node d. Other embodiments might use an equation that uses both this wirelength component and the wirelength component of Equation (A).
 
     2. Deriving Expansion Cost from the G Cost of the Expansion&#39;s Start Node 
     Equation (A) provides a general expression of the costing function that represents the G cost of an expansion in some embodiment. However, in some embodiments, the process  2900  does not actually use Equation (A) to compute the G cost of an expansion. Instead, it derives this G cost from the G cost of the Current_Drop in the following manner. 
     Assume that the Current_Drop specifies (i.e., is the last drop of) a path p′, while the expansion from the Current_Drop specifies a current path p, which is an extension of the path p′ to the expansion&#39;s destination node. The Current_Drop&#39;s G cost, G(Current_Drop), is the cost of the path p′ that has reached the Current_Drop. 
     The process  2900  first computes a G 1  cost that is illustrated in Equation (B) below. 
                     G   1     =       G   ⁡     (   Current_Drop   )       +       Y   L     *     (       ⅇ       ɛ   L     *           F   ^     L     ⁡     (   p   )           H   ^     MIN           -     ⅇ       ɛ   L     *           F   ^     L     ⁡     (     p   ′     )           H   ^     MIN             )                 (   B   )               
In this equation, {circumflex over (F)} L  (p) equals the length of a path p plus the destination node&#39;s Ĥ. It represents an estimated length of a path from a source node through the expansion&#39;s destination node d to a target node. It is equivalent to {circumflex over (F)} L (d), which was described above. {circumflex over (F)} L (p′) equals the length of a path p′ plus the Ĥ of the expansion&#39;s start node. It represents an estimated length of a path from a source node through the expansion&#39;s start node to a target node. The remaining terms of Equation (B) are as described above for Equation (A). The exponential expression
 
               Y   L     *     ⅇ       ɛ   L     *           F   ^     L     ⁡     (     p   ′     )           H   ^     MIN                 
represents the wirelength cost of path p′, while the exponential expression
 
               Y   L     *     ⅇ       ɛ   L     *           F   ^     L     ⁡     (   p   )           H   ^     MIN                 
represents the wirelength cost of path p. Hence, Equation (B) illustrates that the G 1  cost can be obtained by adding the incremental wirelength cost for the expansion to the G cost of the Current_Drop, since path p is an extension of path p′ by the current expansion.
 
     The length of the path p can be obtained from the length of the path p′, as illustrated in Table 4 below. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 4 
               
               
                   
                   
               
               
                   
                 Expansion to 
                   
               
               
                   
                 Destination Node 
                 Length of Path P 
               
               
                   
                   
               
             
            
               
                   
                 Planar expansion 
                 Length of p′ plus L, where L is the 
               
               
                   
                 on layer 2 or 3 
                 unit length cost. 
               
               
                   
                 Planar expansion 
                 Length of p′ plus L * {square root over (2)}. 
               
               
                   
                 on layer 4 or 5 
               
               
                   
                 Non-planar expansion 
                 Length of p′ plus X * L, where X is a 
               
               
                   
                 between overlapping 
                 via-scalar factor. 
               
               
                   
                 nodes 
               
               
                   
                 Internal or External Zigs 
                 Length of p′ plus (X + {square root over (2)}) * L. 
               
               
                   
                   
               
            
           
         
       
     
     The G cost of the expansion equals the G 1  cost expressed in Equation (B) if the expansion is a planar expansion that does not cross a Gcell boundary. However, if the expansion is a planar expansion along a node edge f that crosses a Gcell boundary, and the node edge f is associated with a congestion edge j, then the expansion&#39;s G cost is a G 2  cost expressed in Equation (C). 
                     G   2     =       G   1     +       Y   j     *       (       ⅇ       ɛ   C     *       new   ⁢           ⁢     usage   ⁡     (   j   )             goal   ⁡     (     layer   ⁡     (   j   )       )       *     capacity   ⁡     (   j   )               -     ⅇ       ɛ   C     *       old   ⁢           ⁢     usage   ⁡     (   j   )             goal   ⁡     (     layer   ⁡     (   j   )       )       *     capacity   ⁡     (   j   )                 )     .                 (   C   )               
As illustrated in this equation, the G 2  cost equals the G 1  cost expressed in Equation (B) plus an exponential wire congestion cost for crossing the Gcell boundary. In Equation (C), the terms are as defined above. The two exponential terms in Equation (C) represent costs after and before the expansion. Hence, Equation (C) illustrates that the G 2  cost can be obtained by adding the incremental wire congestion cost for the expansion to the G 1  cost.
 
     If the expansion is along a non-planar edge t between two directly overlapping nodes or is along an internal zig expansion t between two non-overlapping nodes in a Gcell, the expansion&#39;s G cost is a G 3  cost expressed in Equation (D) below. 
                     G   3     =       G   1     +       Y   t     *       (       ⅇ       ɛ   V     *       new   ⁢           ⁢     usage   ⁡     (   t   )             goal   ⁡     (     layer   ⁢           ⁢     pairs   ⁡     (   t   )         )       *     capacity   ⁡     (   t   )               -     ⅇ       ɛ   V     *       old   ⁢           ⁢     usage   ⁡     (   t   )             goal   ⁡     (     layer   ⁢           ⁢     pairs   ⁡     (   t   )         )       *     capacity   ⁡     (   t   )                 )     .                 (   D   )               
As illustrated in this equation, the G 3  cost equals the G 1  cost expressed in Equation (B) plus exponential via congestion cost due to the via expansion. In Equation (D), the terms are as defined above. The two exponential terms in Equation (D) represent costs after and before the expansion. Hence, Equation (D) illustrates that the G 3  cost can be obtained by adding the incremental via congestion cost for the expansion to the G 1  cost.
 
     If the expansion is an external zig t that connects two non-overlapping nodes in two adjacent Gcells, the expansion&#39;s G cost is a G 4  cost expressed in Equation (E) below. 
                     G   4     =       G   1     +       Y   j     *     (       ⅇ       ɛ   C     *       new   ⁢           ⁢     usage   ⁡     (   j   )             goal   ⁡     (     layer   ⁡     (   j   )       )       *     capacity   ⁡     (   j   )               -     ⅇ       ɛ   C     *       old   ⁢           ⁢     usage   ⁡     (   j   )             goal   ⁡     (     layer   ⁡     (   j   )       )       *     capacity   ⁡     (   j   )                 )       +       Y   t     *       (       ⅇ       ɛ   V     *       new   ⁢           ⁢     usage   ⁡     (   t   )             goal   ⁡     (     layer   ⁢           ⁢     pairs   ⁡     (   t   )         )       *     capacity   ⁡     (   t   )               *     ⅇ       ɛ   V     *       old   ⁢           ⁢     usage   ⁡     (   t   )             goal   ⁡     (     layer   ⁢           ⁢     pairs   ⁡     (   t   )         )       *     capacity   ⁡     (   t   )                 )     .                 (   E   )               
As illustrated in this equation, the G 4  cost equals the G 1  cost expressed in Equation (B) plus exponential via and wire congestion costs for the via expansion. In Equation (E), the terms are as described above. The two positive exponential terms represent costs after the expansion, while the two negative exponential terms represent costs before the expansion. Hence, Equation (E) illustrates that the G 4  cost can be obtained by adding the incremental wire and via congestion costs for the expansion to the G 1  cost.
 
     To compute the incremental via and wire congestion costs, the path search process needs to associate this instance of the external zig t&#39;s use with a via location and congestion edge, pursuant to the approach described in Section I.B.3. Specifically, the process examines two different via locations and congestion edges for this use of the external zig t, and associates the external zig with the via location and edge crossing that results in the smaller aggregate via and wire congestion costs. The process then uses the incremental via and wire congestion costs of the associated via location and edge crossing in Equation (E) to express the cost of the path p. 
     For instance, assume that the external zig t is zig  1400  that was described above by reference to FIGS.  14  and  22 - 24 . Assume further that for this instance of external zig t, the smaller aggregate via and wire congestion cost can be obtained by placing the external zig&#39;s via in Gcell  1410 . Hence, for this use of the external zig, the path search process associates this use of the external zig  1400  with the congestion edge  2310 , and specifies the Gcell for the external zig&#39;s via as the Gcell  1410 . In this situation, the incremental via congestion cost is based on the old and new usage values for vias between layers  4  and  5  in Gcell  1410 . The incremental wire congestion cost is based on the old and new usage values for the congestion edge  2310 . 
     C. Generating a Congestion-Unaware Route for a Net 
     As described above, the process  2500  identifies (at  2520 ) the congestion-unaware route for each net. To generate the congestion-unaware route for a net, the process can use route-generation and path-generation processes that are similar to the above-described route-generation and path-generation processes  2700  and  2900 , except for the costing of expansions at  2940 . To generate the congestion-unaware route for a net, some embodiments cost the expansions at  2940  in a non-exponential manner that disregards the via and wire congestion costs and focuses solely on the wirelength cost. For instance, in these embodiments, a planar expansion in layer  2  or  3  has a G cost that equals the Current_Drop&#39;s G cost plus a unit length cost L. A planar expansion in layer  4  or  5  has a G cost that equals the Current_Drop&#39;s G cost plus L*√{square root over (2)}. A non-planar expansion between two overlapping nodes has a G cost that equals the Current_Drop&#39;s G cost plus X*L, where X is the via-scaling factor. An internal zig or an external zig expansion has a G cost that equals the Current_Drop&#39;s G cost plus (X+√{square root over (2)})*L. Some embodiments disallow expansion on a node edge for which the associated congestion edge has a capacity less than 1. 
     IV. Computer System 
       FIG. 31  conceptually illustrates a computer system with which one embodiment of the invention is implemented. Computer system  3100  includes a bus  3105 , a processor  3110 , a system memory  3115 , a read-only memory  3120 , a permanent storage device  3125 , input devices  3130 , and output devices  3035 . 
     The bus  3105  collectively represents all system, peripheral, and chipset buses that support communication among internal devices of the computer system  3100 . For instance, the bus  3105  communicatively connects the processor  3110  with the read-only memory  3120 , the system memory  3115 , and the permanent storage device  3125 . 
     From these various memory units, the processor  3110  retrieves instructions to execute and data to process in order to execute the processes of the invention. The read-only-memory (ROM)  3120  stores static data and instructions that are needed by the processor  3110  and other modules of the computer system. The permanent storage device  3125 , on the other hand, is a read-and-write memory device. This device is a non-volatile memory unit that stores instruction and data even when the computer system  3100  is off. Some embodiments of the invention use a mass-storage device (such as a magnetic or optical disk and its corresponding disk drive) as the permanent storage device  3125 . Other embodiments use a removable storage device (such as a floppy disk or Zip® disk, and its corresponding disk drive) as the permanent storage device. 
     Like the permanent storage device  3125 , the system memory  3115  is a read-and-write memory device. However, unlike storage device  3125 , the system memory is a volatile read-and-write memory, such as a random access memory. The system memory stores some of the instructions and data that the processor needs at runtime. In some embodiments, the invention&#39;s processes are stored in the system memory  3115 , the permanent storage device  3125 , and/or the read-only memory  3120 . 
     The bus  3105  also connects to the input and output devices  3130  and  3135 . The input devices enable the user to communicate information and select commands to the computer system. The input devices  3130  include alphanumeric keyboards and cursor-controllers. The output devices  3135  display images generated by the computer system. For instance, these devices display IC design layouts. The output devices include printers and display devices, such as cathode ray tubes (CRT) or liquid crystal displays (LCD). 
     Finally, as shown in  FIG. 31 , bus  3105  also couples computer  3100  to a network  3165  through a network adapter (not shown). In this manner, the computer can be a part of a network of computers (such as a local area network (“LAN”), a wide area network (“WAN”), or an Intranet) or a network of networks (such as the Internet). Any or all of the components of computer system  3100  may be used in conjunction with the invention. However, one of ordinary skill in the art will appreciate that any other system configuration may also be used in conjunction with the invention. 
     The above-described router can produce multi-layer global routes that have horizontal, vertical, and diagonal edges.  FIG. 32  provides an example of one such global route. This route  3200  traverses layers  3 ,  4 , and  5 . As shown in this figure, this route includes a via between layers  3  and  4 , and an external zig via between layers  4  and  5 . 
     While the invention has been described with reference to numerous specific details, one of ordinary skill in the art will recognize that the invention can be embodied in other specific forms without departing from the spirit of the invention. For instance, even though the router described above is a flat global router, one of ordinary skill will realize that the invention can be practiced with hierarchical routers, such as a router described in U.S. patent application Ser. No. 10/013,819, filed on Dec. 7, 2001, now issued as U.S. Pat. No. 7,003,754. 
     Also, the routes and paths described above are defined with respect to the node edges illustrated in  FIGS. 4 through 7 . These routes and paths, however, can be defined differently. For instance, they can be defined with respect to a set of edges that are orthogonal to the edges illustrated in  FIGS. 4 through 7 . 
     In addition, many aspects of the invention can be practiced without the congestion and length grids illustrated in  FIGS. 1 and 2 . Alternatively, they can be practiced with different congestion and length grids, or different structures for these grids. For instance, in some embodiments, intersecting horizontal and vertical lines might not form one or both these grids. Also, nodes might not be defined in the center of the length-grid sub-regions. 
     Although the router described above uses horizontal, vertical, and ±45 diagonal wiring, many aspects of the invention can be practiced with a different set of interconnect lines. Also, some embodiments might use costing equations that are different than those described in Equations (A)-(E) above. For instance, some embodiments described above express the wirelength cost of a path p as 
               Y   L     *       ⅇ       ɛ   L     *           F   ^     L     ⁡     (   p   )           H   ^     MIN           .           
Some embodiments might express such a path cost as
 
                 Y   L     *     ⅇ       ɛ   L     *           F   ^     L     ⁡     (   p   )         B   *       H   ^     MIN               ,         
where B is a factor related to the importance of the net. This factor B is smaller (e.g., it is 1) for important time-critical nets that need shorter routes, while it is larger (e.g., it is 3) for non-critical nets that can have longer routes. Accordingly, this factor causes the path search to try to identify shorter paths for critical nets, by increasing the wirelength cost of these nets much faster during path searches than the wirelength cost of non-critical nets. Other embodiments might use other exponential and non-exponential expressions in their cost functions. Thus, one of ordinary skill in the art would understand that the invention is not to be limited by the foregoing illustrative details, but rather is to be defined by the appended claims.