Methods and apparatus for extracting capacitances exerted on diagonal interconnect lines in an integrated circuit design

[00002] This application claims the benefit of U.S. Provisional Patent Application No. 60/298,135, filed Jun. 12, 2001, entitled “Methods And Apparatus For Representing And Manipulating Polygons In A Multidimensional Space.” This application is also a continuation-in-part of United States Patent Application entitled “Method and Apparatus for Representing Multidimensional Data” filed on Mar. 15, 2000, and having the Ser. No. 09/526,266 now U.S. Pat. No. 6,625,611.

[00003] 1. Field of the Invention

[00004] The present invention is directed toward the field of computers and electronic design automation, and more particularly towards storing representations of polygons for computer and electronic design automation applications.

[00005] 2. Art Background

[00006] Many applications today analyze multidimensional data records. A multidimensional data record contains a number of data values, which are defined along a number of dimensions (also called attributes or keys) in a multidimensional space. Such records are typically stored in data files or databases.

[00007] A spatial data record is one type of multidimensional data record. Spatial data records typically describe the attributes (e.g., the position, size, shape, etc.) of geometric objects, such as points, lines, polygons, regions, surfaces, volumes, etc. Spatial records are used in many fields, including computer-aided design, computer graphics, data management systems, robotics, image processing, geographic information systems, pattern recognition, and computational geometry.

[00008] Effective data structures are needed to organize multidimensional and spatial data records efficiently, in order to optimize the time for querying these records. For instance, a sequential list of the multidimensional data records provides the simplest way of storing the records. However, the time needed for performing a query on such a list is intolerably high in most cases since each record in the list needs to be examined for each query. Numerous multidimensional data structures have been proposed for organizing multidimensional and spatial data records. Hanan Samet, The Design and Analysis of Spatial Data Structures, Addison-Wesley Publishing, 1990, includes a review of many of these data structures.

[00009] Multidimensional data structures include hierarchical data structures. Hierarchical structures are based on the principle of recursive decomposition of the data space (i.e., the object space or the image space). In other words, these data structures are created by recursively dividing the data space, and storing the data records according to their location in the divided space. Quadtrees and k-d trees are two types of hierarchical data structures.

[00010] Electronic design automation (“EDA”) applications assist engineers in designing integrated circuits (“IC's”). Specifically, these applications provide sets of computer-based tools for creating, editing, and analyzing IC design layouts. These layouts are formed by geometric shapes that represent layers of different materials and devices on IC's. Spatial data records define the spatial attributes of many of these geometric shapes. For instance, spatial data records are used to define geometric shapes that represent conductive interconnect lines. Interconnect lines route signals on the IC's. These lines are sometimes referred to as wire segments or segs. EDA applications typically characterize interconnect lines as rectangles. The rectangles depict interconnect lines deposed on the IC in a vertical and/or horizontal directions, relative to the boundaries of the integrated circuit. As described fully below, the present invention characterizes interconnect lines as polygons. The polygons depict interconnect lines deposed on the IC in diagonal directions, relative to the boundaries of the integrated circuit. Thus, it is desireable to create mechanisms to manipulate and store polygons in a multi-dimensional space.

[00011] Capacitances exerted on diagonal interconnect lines in an integrated circuit (“IC”) design are extracted. A diagonal interconnect line defines a line deposed in a direction other than zero or ninety degrees relative to the integrated circuit boundaries. A computer system stores an IC design layout and uses geometric objects to represent diagonal interconnects lines in the layout. The computer system represents each geometric object by a geometric data segment. To extract the capacitances, a plurality of regions in the multidimensional layout is defined. A plurality of hierarchical data structures for a number of said regions are created. Each hierarchical data structure corresponds to a separate region. For each geometric object in the layout, a region that the geometric object intersects and the hierarchical data structure corresponding to the identified region is identified. Furthermore, the geometric data segment of the geometric object in the identified hierarchical data structure is inserted.

[00012] In another embodiment, capacitances, exerted on diagonal interconnect lines in an IC design, are determined by adaptively defining a plurality of regions in the multidimensional layout. A data structure for each of said regions is created. The data structure stores the geometric data segments for each geometric object in its region, such that the number of regions increases as the number of geometric objects increases. The increasing region number maintains the size of each data structure relatively constant as the number of geometric objects increases.

[00046] The disclosure of U.S. Provisional Patent Application No. 60/298,135, filed Jun. 12, 2001, entitled “Methods And Apparatus For Representing And Manipulating Polygons In A Multidimensional Space” is hereby expressly incorporated herein by reference.

[00047] The present invention uses geometric objects defined in a multi-dimensional data space. The geometric objects are represented by data tuples or data segments for processing in a computer. In one embodiment, the geometric objects comprise polygons. Different embodiments of the present invention use different classes of polygons. FIG. 1 illustrates an “n” sided arbitrary polygon as an example of a geometric object. For the example geometric object of FIG. 1, the “n” sided arbitrary polygon is represented by “n” vertices. Specifically, for this embodiment, the “n” sided polygon is defined by the the vertices {X₁Y₁, X₂Y₂, X₃Y₃, X₄Y₄ and X_nY_n), relative to a two-dimensional coordinate axes. In other embodiments, the geometric objects may be defined in a coordinate system with more than two dimensions (e.g., a three-dimensional coordinate system). Although the “n” sided polygon illustrated in FIG. 1 is shown with more than five sides (e.g., “n” is greater than or equal to “6”), a geometric object may be defined by two or more vertices (e.g., “n” is greater then or equal to two).

[00048] As described more fully below, the geometric object may comprise rectangles. The rectangles may be aligned with the coordinate axes (i.e., 0 or 90 degrees relative to the xy coordinate axes) or the rectangles may be situated at angles other then 0 or 90 degrees (i.e., rectangles of arbitrary orientation in a multi-dimensional data space). The geometric objects may also include oblique rectangles with a predefined set of values that define the rectangle's orientation in a multi-dimensional space. In one embodiment, the rectangles are situated, relative to an x-y coordinate axes, in a multi-dimensional space at multiples of 30 and 45 degree angles (e.g., 30, 45, 60, 90, 120, 135, 150, and 180, etc). FIG. 2c illustrates an oblique rectangle with an orientation that is not parallel or perpendicular with the x-y coordinate axes.

[00049] An arbitrary polygon (FIG. 1) used in the present invention may comprise convex polygons. As used herein, a “convex polygon” is defined as a closed geometric object consisting of: (1) two (2) or more sides or edges; (2) the property that the interior of the polygon on the outside side of the edge, when facing forward from the edge's viewpoint, turns at least partially to the outside when the advancing around the polygon from one edge to the next edge (i.e., no collinear edge); and (3) no repetition in vertices, X_iY_i (i.e., the polygon does not have duplicate vertices). FIG. 2a illustrates an example of a convex polygon for use as a data segment. For the example convex polygon shown in FIG. 2a, the geometric object is defined by its vertices (X₀Y₀, X₁Y₁, X₂Y₂, X₃, Y₃, X₄, Y₄).

[00050] Another example of a geometric object that may be used in the present invention is an octagon. FIG. 2b illustrates an example octagon. The octagon, which consists of eight sides, is represented, in a data segment, by 8 vertices (X₀Y₀, X_1Y1 . . . X_nY_n).

[00051] As described more fully below, the geometric objects have application for defining wires in an integrated circuit design. In one embodiment, the wires are deposed at diagonal angles relative to the boundaries of the integrated circuit. As used herein, diagonal wiring geometries consist of interconnect wires of an integrated circuit or integrated circuit design situated at angles other than 0 or 90 degrees (i.e., parallel or perpendicular) to the boundaries of the integrated circuit. However, diagonal wiring geometries may also include wires situated parallel or perpendicular to the boundaries of the integrated circuit. (referred to as “Manhattan directional wiring”). U.S. patent application Ser. No. 09/733,104, entitled “Multi-Directional Wiring On A Single Metal Layer”, filed on Dec. 7, 2000, discloses embodiments for diagonal wiring geometries in integrated circuits, and is expressly incorporated herein by reference. For embodiments employing diagonal wiring geometries, the octagon may be used to represent a connector between two or more wires situated at angles other than 0 or 90 degrees. FIG. 2d illustrates an example of a wiring configuration that has two interconnect wires, represented by two oblique rectangles, joined by an octagon shaped connector.

[00052] FIG. 3 illustrates an example interconnect layer of an integrated circuit design that employs diagonal wiring. As shown in FIG. 3, the rectangles, which depict interconnect wires on an IC design, are situated at various angles relative to the boundaries of the integrated circuit. For example, interconnect wire 10 is deposed at an angle of 45 degrees, and interconnect wire 19 is deposed at an angle of 135 degrees. In one embodiment for employing diagonal wiring geometries, the wires are deposed at 30 and 45 degree angles, as well as other angles that are multiples of 30 and 45 degree angles.

[00053] FIG. 4a illustrates one embodiment for a data segment that defines a geometric object. For this embodiment, the data segment defines, in field 400, the number of sides in the geometric object. For the example arbitrary polygon shown in FIG. 1, this field stores the integer “n”, to depict that the polygon has “n” sides. The data segment of FIG. 4a also stores vertices for the geometric object. Specifically, for this example, a first vertex, X₀Y₀, is stored in field 410, a second vertex, X₁Y₁, is stored in field 415, and an “nth” vertex is stored in field 425. Although the data segment of FIG. 4a is the shown as having distinct fields for the number of sides and for each vertex, a data structure arranged in a different manner may be used without deviating from the spirit or scope of the invention.

[00054] FIG. 4b illustrates one embodiment for a data segment that defines a geometric object for an interconnect wire. Similar to the data segment of FIG. 4a, the data segment of FIG. 4b includes a field 435 that defines the number of sides in the geometric object, as well as “n” fields (440, 445, and 450) that define “n” vertices for the geometric object. In addition to defining the number of sides and vertices in a multi-dimensional space, the data segment of FIG. 4b also defines a layer and a white/gray designation, in fields 460 and 470, respectively. The layer field 460 specifies a metal wiring layer in an integrated circuit design for the interconnect wire, and the white/gray designation defines whether the segment is critical.

[00055] One embodiment of the invention generates “n” dimensional hierarchical trees (referred to herein as n-gon trees or simply “ng” trees). The ng trees are a form of a binary tree that organizes data tuples or data segments into “outside child nodes” and “inside child nodes” in accordance with a discriminator value. The discriminator value is selected for each layer or discriminator dimension in the ng tree. In one embodiment for the ng tree, one of “n” sides of a polygon is selected as the discriminator value. To create the ng tree, data segments are designated as “outside child nodes” if a data segment is outside the plane defined by the discriminator value, and data segments are selected as “inside child nodes” if the data segment is inside the plane defined by the discriminator value. This process of partitioning data segments into inside child nodes and outside child nodes is repeated recursively through each level of the ng tree.

[00056] FIG. 5 illustrates an example of an ng tree. The ng tree may be represented in a hierarchical data structure with one or more “nodes” at each level of the tree. The nodes in the ng tree of FIG. 5 are labeled based on the node's level in the tree. For this example, there are “n+1” levels in the ng tree. As described more fully below, the number of levels in the tree is dependent upon the number of data segment elements arranged in the ng tree. As described more fully below, the discriminator value for each level is determined dynamically (i.e., discriminator value=current_depth of ng tree modulo (sides of polygon in discriminating node). Thus, the ng tree cycles through the “n” sides of a discriminating node to partition nodes into outside child nodes and inside child nodes.

[00057] FIG. 6 is a flow diagram illustrating one embodiment for populating an array of ngNodes for generation of an ng tree. First, an array of ngNodes of the appropriate size is allocated to store data segments of the ng tree (block 605, FIG. 6). Pointers used in the process to generate the ng tree are initialized (S=1, N=1) (block 610, FIG. 6). Specifically, “S” is a pointer to a data segment, and “N” is a pointer to a node in the array of ngNodes. The process copies data from data segment “S” data structure into ngNode N's corresponding fields (block 615, FIG. 6). The pointers to the data segments “S” and the array of ngNodes “N” are incremented (block 620, FIG. 6). If the last data segment has not been copied into the array of ngNodes, then the process repeats steps 615 and 620 (block 625, FIG. 6). If the last data segment has been copied into the array of ngNodes, then the process determines whether there is only one node (block 630, FIG. 6). If there is only one node in the array of ngNodes, then this node is set as the root node for the ng tree (block 635, FIG. 6). Alternatively, if they are more than one node, then the process calls a “BuildOptimalNgTree” process, with the array of ngNodes as an input parameter (block 640, FIG. 6). Although the present invention is described using multiple processes for creating the ng tree and its associated hierarchical data structure, one or more software routines or processes may be implemented to perform these functions without deviating from the spirit or scope of the invention.

[00058] FIG. 7 is a flow diagram illustrating one embodiment for the BuildOptimalNgTree process. The BuildOptimalNgTree process 700 receives, as input parameters, an array of ngNodes, a low_bound node, a high_bound node, and a current_depth (block 705, FIG. 7). The low_bound parameter specifies a node with the lowest value relative to a discriminator dimension. Similarly, the high_bound parameter specifies a node with the highest value relative to a discriminator dimension. The BuildOptimalNgTree routine 700 is called recursively. The current_depth parameter specifies the level of the tree for the corresponding call to the BuildOptimalNgTree routine.

[00059] In one embodiment, the low_bound and high_bound parameters, which comprise 64-bit values, store the minimum and maximum penetration “distances” of any segments within the discriminator half planes. The distance involves the computation:
F1(coords)/sqrt(F2(coords)),
where the F2(coords) is a constant for a given half plane. In one embodiment, the process stores F1(coords) to eliminate the floating point, division, and sqrt computations.

[00062] The process sets the discriminator dimension for the current_depth (block 710, FIG. 7). In one embodiment, the process sets the discriminator dimension equal to the value of the current_depth multiplied by the modulo(number of sides of the polygon). A process, referred to herein as the “PartitionAboutMedian” is called to partition the nodes around a median (block 715, FIG. 7). This process is shown in the flow diagram of FIG. 8. The PartitionAboutMedian process includes, as input parameters, an array, a low_bound node, a high_bound node, and a current_depth. In one embodiment, the partition about median routine for the ng tree is as follows. A random element of the specified sub-array is chosen and is designated as the “median node.” In one embodiment, the median for the array of ngNodes is set from the expression:
low_bound+high_bound/2.

[00064] The new_root of the array of ngNodes is identified by the median value (block 720, FIG. 7). Based on the current_depth, one of the edges of the polygon, represented by that node, is chosen. The PartitionAboutMedian routine then goes through the rest of the nodes in the sub array partitioning the outside nodes to the left of the median node and the inside nodes to the right of the median node. This procedure does not guarantee a balanced tree (i.e., same number of nodes to the outside and to the inside of the median node). The new_root for the outside child is determined by recursively calling the BuildOptimalNgTree routine, with the array, low_bound, median −1, and current_depth+1 as parameters (block 750, FIG. 7). Similarly, to determine a new root for the inside child, the BuildOptimalNgTree routine is recursively called with array, median+1, high_bound, and current_depth+1 as input parameters (block 750, FIG. 7). The BuildOptimalNgTree routine 700 returns new_root (block 730, FIG. 7). Some embodiments of the invention use a partitioning algorithm that is disclosed in Robert Sedgewick, Algorithms in C++, Third Edition, Parts 1-4, Addison-Wesley Publishing, 1998, Chapter 7, and in Cormen, et al., Introduction to Algorithms, Section 8, 1996.

[00065] FIG. 8 is a flow diagram illustrating one embodiment for the PartitionAboutMedian routine. For this embodiment, the PartitionAboutMedian routine selects a random node in the array, or sub array, of ngNodes for designation as the median node in the array/sub array (block 810, FIG. 8). The PartitionAboutMedian routine selects the “Nth” edge of the polygon for the median node, where N is computed from the current_depth modulo(the number of sides in the polygon) (block 820, FIG. 8). The PartitionAboutMedian routine determines the outside half plane for the selected “Nth” edge (block 830, FIG. 8).

[00066] In one embodiment for determining half planes, the Nth edge of the polygon (e.g., 8^th edge of an octagon) is designated a vector from point “N−1” to point “N” of the polygon. The first edge, edge #1, is defined from point 0 (“P0”) to point 1 (“P1”), and the second edge, edge #2, is defined from point 1 (P1) to point 2 (“P2”), etc. At the last edge of the polygon, the segment wraps around so that the closing edge, N, runs from point N−1 to point 0. For example, in a 3-gon, the edges are defined by P0→P1, P1→P2, and P2→P0. A line defined by these segments splits the plane into two half planes. The half plane to the left of the line segment (looking forward from P0 to P1) is arbitrarily selected as the half plane defined by the segment. Therefore, the vector P1→P0 defines the other half plane. This embodiment for defining a half plane provides a counter-clockwise orientation to the edges of the polygon.

[00067] The equation in the form |F1(coords)|/sqrt(F2(coords)) yields the minimum distance between a point and the line that forms the border of the half plane. If the process eliminates the step of computing the absolute value, then the equation yields a negative distance for segments outside the half plane and a positive distance for segments inside the half plane. In one embodiment, if the defining line has a distance of 0 from the segment, then the segment is arbitrarily defined as inside. This permits a mathematical definition for the “inside” and “outside” of the half plane. For additional explanation on half planes, see “Pre-Calculus Mathematics”, Shanks/Brumfiel/Fleenor/Eicholz, Addison-Wesley, 1965, and “Computational Geometry in C”, O'Rourke, Cambridge University Press, 1993, reprinted in 1996 with corrections from 1994 reprint.

[00068] The process initializes an index, K, to nodes in the array/sub array (block 840, FIG. 8). A polynomial for the polygon of the array[K] node is determined (block 840, FIG. 8). The PartitionAboutMedian routine then determines whether the polynomial for array[K]'s node lies entirely in the Nth edge's outside half plane (block 860, FIG. 8). If it does, then the node array[K] is arranged as an outside node (i.e., array[outside node]), where outside node index is between low_bound and median-1 index (block 870, FIG. 8). Alternatively, if the polynomial for array[K]'s node does not lie entirely in the Nth edge's outside half plane, then the node array[K] is arranged as an inside node (i.e., array [inside node]), where inside node index is between median −1 and high_bound (block 880, FIG. 8). The PartitionAboutMedian routine then determines whether there are any nodes in the array/sub-array not yet assigned to a node of the ng tree. Specifically, the process determines whether (K−1) is equal to the number of nodes in the array of ngNodes (block 885, FIG. 8). If it is, then the PartitionAboutMedian routine ends. If (K−1) does not equal the number of nodes in the ngNode array, then the index into the ngNode array, K, is incremented (i.e., K=K+1) (block 890, FIG. 8).

[00069] Some embodiments of the invention provide a method for organizing multidimensional data tuples or segment data. A data tuple is a set of dimension values (also referred to herein as “data values” or “segment data”) that collectively represents one entity (e.g., a person, an item, a spatial object, etc.). The dimension values for each data tuple are specified along a number dimensions. These dimensions collectively define a multidimensional data space.

[00070] In some embodiments of the invention, each data tuple is formed as a data object (i.e., as an instantiation of a class in an object-oriented program). In other embodiments, however, the data tuples are not represented as data objects.

[00071] Some embodiments of the invention create a two-layered data structure to organize the multidimensional data tuples. FIG. 9 illustrates a process 700 used by some of these embodiments. As shown in this figure, the process 700 initially divides (at 705) the multidimensional data space along one or more dimension values. This division results in a number of data regions. The process 700 creates a data structure (e.g., a non-hierarchical data structure) to represent this division. This data structure represents the first data structure layer.

[00072] For each data region, the process (at 710) then creates a hierarchical data structure that represents the second data structure layer. Next, the process (at 715) inserts each data tuple into one or more hierarchical data structures. In particular, the process inserts each data tuple into the hierarchical data structure for the data region that encompasses the data tuple.

[00073] Some data tuples cross more than one data region. For some embodiments of the invention, process 700 inserts data tuples into the hierarchical data structures of each data region that they cross. For instance, in some embodiments, the process 700 divides each data tuple that crosses a data region boundary into two tuples along the data region boundary that the data tuple crosses. One of the resulting two tuples falls within one data region, while the other resulting tuple falls within the other data region. The resulting data tuples are then inserted into the hierarchical data structures of their corresponding data regions.

[00074] For some embodiments of the invention, the process 700 also inserts a data tuple into more than one hierarchical data structure if that data tuple is needed for the analysis of data tuples in more than one hierarchical data structure. For instance, a data tuple might fall outside of a particular data region but might be necessary for the analysis of one or more data tuples within that data region. In the discussion below, the term “non-source” refers to this type of data tuples. Specifically, for a particular data region, a source data tuple is a data tuple that resides in the data region, while a non-source data tuple is a data tuple that is outside of the data region but it is necessary for the analysis of one or more data tuples within the data region.

[00075] In some embodiments of the invention, the process 700 inserts a copy of some or all of each non-source data tuple for a particular region into that region's hierarchical data structure. This ensures that only one hierarchical data structure is queried for the data tuples of a particular data region. In other embodiments, the non-source data tuples for a particular data region is analyzed with the data structures of the data regions that surround the particular data region. Hence, for the data tuples in a particular data region, these embodiments not only query the data structure for that data region but also query the data structures of the surrounding data regions.

[00076] A wide range of applications are applicable to the invention to create efficient multidimensional data structures. For instance, EDA applications may use the invention's data structures to efficiently organize data relating to interconnect lines on IC's, including interconnect lines deposed in directions other than horizontal or vertical (e.g., diagonal interconnect lines). Such an organization speeds up the identification of nearby interconnect lines and hence speeds up the capacitance extraction process.

[00077] FIG. 10 illustrates one embodiment of the invention to store data relating to interconnect lines on an integrated circuit. As shown in FIG. 10, this embodiment is a two-layer data structure 900. The first layer is a two-dimensional array 905 of tile data structures 910, and the second layer is an ng tree 915 associated with each tile data structure.

[00078] Each tile data structure 910 represents a tile region on the IC layout. As further described below, some embodiments of the invention use rectangular tiles 1205 that divide the IC layout along x- and y-axes. Dividing the IC layout into tiles along the x- and y-axes divides the IC design space into rectangular buckets that are defined (1) along the x- and y-axes, by their tiles, and (2) along the z-axes by the thickness of the IC. One of ordinary skill in the art will understand that other embodiments of the invention divide the IC layout space into different shaped regions and along different sets of axes.

[00079] As shown in FIG. 10, each tile data structure 910 has an associated ng tree 915. Each tile's ng tree efficiently organizes the data relating to the source and non-source interconnect lines for that tile. For a particular tile, a source interconnect line is a line that resides in the tile, while a non-source line is a line that lies outside of the tile, but the line is necessary for the analysis of one or more lines within the tile. The tile data structure 910 will be discussed more fully below.

[00080] FIG. 11 illustrates a process 1000 that some embodiments of the invention use (1) to create the two-layer data structure 900 of FIG. 10, and (2) to query the data in this structure. As shown in FIG. 11, the process divides (at 1005) the IC layout into a number of tile regions along the x- and y-axes. The process then constructs (at 1010) a two-dimensional array of tile data structures, where each tile data structure represents a tile region. Next, the process inserts (at 1015) the data for each interconnect segment into one or more of the tile data structures. For each tile region, the process creates (at 1020) an ng tree that stores the interconnect-line data that was inserted into that region's tile data structure. Finally, the process performs (at 1025) a range query about each critical source segment that was entered into the ng trees. Each of these steps is further discussed below.

[00081] FIG. 12 presents a process 1100 for dividing an IC-data space into a number of tiles along the x and y-axes. Initially, the process gathers (at 1105) statistics about the IC layout. Some embodiments of the invention gather statistics by examining all interconnect lines in the IC's design space. In some embodiments, the statistics that are gathered include the number of interconnect segments, the dimensions of the segments, and the width and height of the chip.

[00082] Some embodiments also determine (at 1105) the maximum halo size. The halo size is the maximum distance of capacitive influence between two segments. In some embodiments, the halo size is adaptively calculated by using known prior art techniques. The halo distance may be different for each layer of the IC. Some embodiments set the halo size to the maximum halo value across all layers.

[00083] After gathering statistics, the process then specifies (at 1110) a variety of tile sizes. To do this, some embodiments specify a variety of rows and columns for dividing the IC layout. A given tile size might require the number of rows to differ from the number of columns. This tile size is beneficial in situations where the distribution of segments in the horizontal direction is significantly different from the distribution in the vertical direction.

[00084] Next, the process selects (at 1115) the first specified tile size for analysis. For this tile size, the process computes (at 1120) the number of segments N_i per tile. As further described below, some embodiments do not actually compute the number of segments N_i per tile but rather statistically determine this number.

[00085] The process then computes (at 1125) the time for querying all the records in the ng tree for that tile (i.e., compute N_ilog₂N_i for that bin). Next, the process sums (at 1130) the search times for all the tiles to obtain the total search time. Equation (4) mathematically explains the total search time. $\begin{array}{cc}\mathrm{Total}\mathrm{Search}\mathrm{time}\propto \sum _{\mathrm{Tiles}}{N}_i\log 2N_i& \left(4\right)\end{array}$

[00086] The process (at 1135) then determines whether it has examined the last tile size. If not, the process (at 1140) selects the next tile size and then repeats the process to compute the total search time for the next tile size. Otherwise, the process (at 1145) identifies the smallest calculated total search time, and selects the tile size that resulted in this search time.

[00087] Finally, the process (at 1150) divides the IC layout into a number of tiles based on the selected tile size. FIG. 13 presents an example of a division of an IC layout 1205 by a number of rectangular tiles 1210. In some embodiments of the invention, the exterior tiles (i.e., the tiles that cover the exterior sides of the IC layout) are extended to infinity, in order to protect against numerical round-off errors. FIG. 14 presents an example of such an approach.

[heading-00088] 1. Predicting the Actual Number of Segments Per Tile

[00089] Some embodiments of the invention do not actually compute (at 1120) the number of segments N_i per tile, but instead predict the average number of segments μ per tile. Some of these embodiments use the following equation (5) to compute the estimated number of segments, μ. $\begin{array}{cc}\mu =\frac{\begin{array}{c}{n}_S[a_S+w_S\left(2d+h_B\right)+h_S\left(2d+w_B\right)+\\ \left(2d+h_B\right)\left(2d+w_B\right)]\end{array}}{wh}& \left(5\right)\end{array}$
In this equation, n_S is the total number of the segments, as is the mean area of the segments, w_S is the mean width of the segments, h_S is the mean height of the the segments, W_B is the width of each tile, h_B is the height of each tile, w is the total width of the IC, and h is the total height of the IC.

[00091] Equation (5) is based on the assumption that the position of each segment in the IC layout is random. FIG. 15 illustrates how this equation was derived. This figure illustrates a particular tile 1405, the tile's halo boundary 1410, and two segments 1415 and 1420. The tile 1405 has a width of w_B and a height of h_B, while the tile's halo boundary has a width of w_B+2d and a height of h_B+2d. The two segments both have an effective width w₀ and height h₀.

[00092] As further discussed below, an interconnect segment must fall within the tile 1405 or its halo boundary 1410 for insertion into the tile's ng tree. As shown in FIG. 15, the segments 1415 and 1420 fall just outside of the halo boundary 1410 of the tile. This is because the center of segment 1415 is a distance (w₀+2d+w_B)/2 away from the center of the tile, while the center of segment 1420 is a distance (h₀+2d+h_B)/2 away from the center of the tile.

[00093] Hence, in order for a segment with a width w₀ and height h₀ to fall within the halo boundary 1310 of the tile, the center of this segment must fall within a rectangle 1425 centered at the tile and having width w₀+2d+w_B and height h₀+2d+h_B. Because the position of each segment in the IC layout is random, the probability P that such a segment overlaps the tile is equal to the area of rectangle 1425 divided by the area of the IC. Equation (6) illustrates this probability. $\begin{array}{cc}P=\frac{\left(w_0+2d+w_B\right)\left(h_0+2d+h_B\right)}{wh}& \left(6\right)\end{array}$

[00094] Equation (7) below is obtained by expanding the formula for the probability that a given segment overlaps a given tile. $\begin{array}{cc}P=\frac{\begin{array}{c}[w_O{h}_O+w_O\left(2d+h_B\right)+h_O\left(2d+w_B\right)+\\ \left(2d+h_B\right)\left(2d+w_B\right)]\end{array}}{wh}& \left(7\right)\end{array}$
The average number of segments per tile may be obtained by using equation (7) to calculate the probability for all the segments and by summing the calculated probabilities. Equation (8) explains this operation mathematically. $\begin{array}{cc}\mu =\sum _{n_S}{P}_i=\sum _{n_S}\frac{\begin{array}{c}[w_O{h}_O+w_O\left(2d+h_B\right)+h_O\left(2d+w_B\right)+\\ \left(2d+h_B\right)\left(2d+w_B\right)]\end{array}}{wh}& \left(8\right)\end{array}$
The sum of w₀h₀ becomes n_Sa_S, while the sum of w₀ becomes n_Sw_S, and the sum of h₀ becomes n_Sh_S. Hence, equation (8) can be simplified into equation (9) below: $\begin{array}{cc}\mu =\sum _{n_S}{P}_i=\frac{\begin{array}{c}[n_S{a}_S+n_S{w}_S\left(2d+h_B\right)+n_S{h}_S\left(2d+w_B\right)+\\ n_S\left(2d+h_B\right)\left(2d+w_B\right)]\end{array}}{wh}& \left(9\right)\end{array}$
By factoring out the variable n_S, Equation (5) is derived from Equation (9). As set out in equation (10) below, the formula for μ may also be given in terms of the total area A_S of the segments, the total width W_S of the segments and the total height H_S of the segments as follows: $\begin{array}{cc}\mu =\frac{\begin{array}{c}[A_S+W_S\left(2d+h_B\right)+H_S\left(2d+w_B\right)+\\ n_S\left(2d+h_B\right)\left(2d+w_B\right)]\end{array}}{wh}& \left(10\right)\end{array}$

[00098] As further described below, in some embodiments, a segment might result in multiple data-tuple entries into the ng tree of a particular tile. Whether a segment requires multiple data-tuple entries depends on: (1) the number of tile boundaries that the segment crosses; and (2) the manner that the data structure treats segments that are close to or cross tile boundaries; and (3) the particular implementation of the halo boundary surrounding the tile.

[00099] Equations (5) and (10), however, do not account for the multiple data-tuple entries into the ng tree for such segments. These equations assume that such multiple entries minimally affect the average number of segments per tile. These equations, however, may be modified by adding a constant multiplier (e.g., such as a multiplier of 1.1 or 1.2) to account for the increase in the number of segments per tile due to the segments that cross halo and tile boundaries. This multiplier can be larger for the smaller tile sizes because smaller tiles result in more segment crossings.

[00100] In one embodiment for processing geometric objects (e.g., convex polygons) to determine a tile grid, segment data for the geometric objects is replaced with the smallest axes-aligned rectangle that contains the geometric segment. FIG. 15 illustrates this technique by replacing the six-sided polygon 1415 with a smallest axes-aligned rectangle, and by replacing the oblique rectangle 1420 with a smallest axes-aligned rectangle. FIG. 16 illustrates an example of a polygon replaced with a smallest axes-aligned rectangle for purposes of computing a tiling grid. For this example, the segment is a five-sided polygon 1600. To calculate a tiling grid, an axes-aligned rectangle 1605 that encompasses the polygon 1600 is generated as shown in FIG. 16. In one embodiment, the geometric objects (e.g., polygon 1600) are represented as an order collection of vertex points. Thus, the axes-aligned rectangle may be determined by finding the minimum and maximum X and Y coordinates among the vertices of the geometric object.

[00101] The process 700 has numerous advantages. For instance, the time that this process expends on constructing its two-layered data structure increases linearly with the number of data tuples in the data space. The following example illustrates this point. As discussed above, the time for constructing a ng tree with N data tuples is proportional to Nlog₂N. However, if the data space is divided into two data regions and each data region roughly contains N/2 data tuples, the time for constructing a ng tree for each data region is proportional to ½Nlog₂(N/2). Hence, the time for constructing an ng tree for both data regions is proportional to Nlog₂(N/2), which is better than Nlog₂(N).

[00102] Similarly, if the data space is divided into R data regions with each data region containing roughly N/R data tuples, the time for constructing ng trees for all the data regions is proportional to Nlog₂(N/R). Equation (3) below explains this mathematically. $\begin{array}{cc}\mathrm{Total}\mathrm{Run}\mathrm{time}\propto \sum _R\frac{N}{R}{\log }_2\frac{N}{R}=N{\log }_2\frac{N}{R}<<N{\log }_{2}N& \left(3\right)\end{array}$
Hence, dividing the multidimensional data space into a number of data regions R reduces the total construction time. This reduction can be significant if the number of data regions R is on the same order as the number of tuples N. In fact, the construction time can be made to increase linearly with the number of data tuples N, by increasing the number of data regions R linearly with the number of data tuples N. For example, if R is selected so that it is always equal to N/1000, then the construction time will always be proportional Nlog₂(1000).

[00104] As shown in FIG. 11, the process 1000 constructs a two-dimensional array of tile data structures after the IC layout is divided into a number of tile regions along the x- and y-axes. Each particular tile data structure represents a particular tile region. In some embodiments of the invention, each tile data structure is a tile data object (i.e., an instantiation of a tile data class). Moreover, some embodiments construct the array as a two-dimensional array of tile data objects. In these embodiments, two index values identify the location of each tile data object in the two-dimensional array. Other embodiments, however, construct this array as an array of pointers, where each pointer points to a particular data object. In these embodiments, the two index values identify the location of a pointer that points to the tile data object.

[00105] FIG. 10 illustrates the tile data structure 910 that is used in some embodiments of the invention. This tile data structure 910 represents a tile by specifying one main rectangle (“MRec”), one to four halo rectangles (“HRec”), and one bounding box (“bbox”) rectangle.

[00106] As shown in FIGS. 17-19, the main rectangle 1505 is the tile rectangle that is specified by the tile-defining process of FIG. 12, while the halo rectangles 1510 correspond to halo regions that surround the tile. The halo regions are used to identify non-source interconnect lines. As described above, non-source interconnect lines are lines that fall outside of a tile (or main) rectangle 1505. Non-source interconnect lines are necessary for the analysis of interconnect lines within the tile. For example, when extracting capacitances influenced by a particular source interconnect line that is close to its tile's edge, it might be necessary to look for non-source interconnect lines outside of the tile, because such interconnect lines may lie within the halo distance of the particular interconnect line. The halo regions provide one solution for identifying non-source interconnect lines. As described below, some embodiments of the invention insert into the tile's hierarchical data structure a copy of the non-source interconnect lines that fall within the halo rectangle 1510 of the tile.

[00107] In some embodiments of the invention, a typical tile has four halo rectangles, as shown in FIG. 17. However, the tiles at the periphery of the chip have fewer halo rectangles. For example, as shown in FIG. 18, a tile in the middle of the outside edge of the IC layout has only three halo rectangles, two of which extend to infinity in the embodiments that extend the outer tiles to infinity. Also, as shown in FIG. 19, a tile in the lower-right comer of the IC layout only has two halo rectangles, both of which extend to infinity in the embodiments that extend the outer tiles to infinity. In the extreme case of a chip containing only two tiles, each would have a single halo rectangle. In addition, there is no halo rectangles in the extreme case of a chip containing only a single tile.

[00108] The bounding box rectangle encloses the tile's main and halo rectangles. As described further below, the bounding box rectangle is used to quickly identify all the interconnect lines that intersect the tile's main and halo rectangles. These identified interconnect lines can then be inserted as source or non-source interconnect lines in the hierarchical data structure of the tile.

[00109] As shown in FIG. 10, the title data structure represents the main, halo, and bounding-box rectangles by their minimum x (X_MIN), minimum y (Y_MIN), maximum x (X_MAX), and maximum y (Y_MAX) coordinates. The data structure also includes a counter for specifying the number of halo rectangles the tile has (i.e., the number of valid halo rectangles in the tile data structure). The data structure further includes a pointer 920 to the first segment inserted into the data structure. Finally, the data structure includes a pointer 925 to the root node of the tile's ng tree.

[00110] As shown in FIG. 11, the process 1000 inserts (at 1015) each interconnect segment into one or more of the tile data structures after it constructs a two-dimensional array of tile data structures. To insert an interconnect segment into a tile data structure, the process must first determine whether the interconnect segment crosses the boundaries of the main or halo rectangles of the tile data structure. If so, the process must break the interconnect segment into one or more pieces along the boundaries that it crosses. The resulting pieces are discarded if they are outside the tile's bounding box, or inserted in the tile data structure as a source or non-source segment if the portion of the segment falls within the tile's boundary box.

[00111] In one embodiment, the exact intersection of the polygon with the main or halo rectangle is determined for use as the segment data. Currently, algorithms exist to determine this intersection (see “Computational Geometry in C”, O'Rourke, Cambridge University Press, 1993, reprinted in 1996 with corrections from 1994 reprint, Section 7.4, page 243).

[00112] In another embodiment, to determine whether the interconnect segment crosses the boundaries of the main or halo rectangles, a smallest axes-aligned rectangle that contains the geometric object (e.g., polygon) is calculated. For this embodiment, the process approximates the dimensions of the geometric object. The method determines the superset of the set of tiles intersected by the polygon. In some cases, this superset is too large, thus considering more segment data than necessary. For example, if the segment data defines a narrow rectangle oblique to the tile grid, then replacing this segment data with the smallest axes-aligned rectangle results in including irrelevant tiles, and thus impedes processing. In other embodiments, the intersection of convex polygons to the tile grid is determined.

[00113] FIG. 20 pictorially illustrates the breaking up of three segments along the boundaries of a tile 1805. This tile has four halo rectangles 1810, 1815, 1820, and 1825 that surround it. As shown in FIG. 20, segment 1830 intersects the halo rectangle 1810. The segment 1830, however, does not intersect the main tile rectangle 1805. Hence, this segment is only divided into two parts, 1845 and 1850 along the boundary 1855 of the halo rectangle 1810. Part 1845 is discarded as it falls outside the tile's bounding box, while part 1850 is selected as a non-source segment to insert into the tile's data structure.

[00114] Another segment illustrated in FIG. 20 is segment 1835. This segment intersects the main rectangle 1805 and the halo rectangle 1820 of the tile. Hence, this segment is divided into three parts 1860, 1865, and 1870 along the boundaries 1875 and 1880 of the halo and main rectangles. Part 1860 is discarded as it falls outside the tile's bounding box, part 1865 is selected as a non-source segment to insert into the tile's data structure, and part 1870 is selected as a source segment to insert into the tile's data structure.

[00115] The final segment illustrated in FIG. 20 is segment 1840. This segment intersects the main rectangle 1805 and the halo rectangles 1820 and 1825 of the tile. Hence, this segment is divided into four parts: 1890, 1892, 1894, and 1896 along the boundaries 1875, 1880, and 1898 of the main and halo rectangles 1805, 1820, and 1825. Part 1890 is discarded as it falls outside the tile's bounding box. Parts 1892 and 1894 are identified as non-source segments and inserted into the tile's data structure. Part 1896 is selected as a source segment to insert into the tile's data structure.

[00116] FIG. 21 illustrates one embodiment for a process that inserts interconnect segments into one or more tile data structures. As shown in this figure, the process (at 2105) initially sets a segment number S equal to 1. Next, the process (at 2110) identifies all tiles that segment S intersects. Based on these calculations, the process retrieves one or more tile data structures from the two-dimensional array 905.

[00117] After identifying the tiles that the segment intersects, the process (at 2115) sets the tile count T_i equal to 1. The process (at 2120) then computes the intersection of the segment S with the main rectangle of the tile T_i. Some embodiments of the invention use existing algorithms to compute the intersection of a polygon with a rectangle.

[00118] Next, the process (at 2125) determines whether this intersection is empty. If so, the process transitions to step 2145. If not, the process (at 2130) creates a new segment based on the intersection of the segment, S, with the main rectangle of the tile, T_i.

[00119] In order to create a new segment for insertion into the tile, a new source segment is determined based on the intersection of the segment data with the tile's main rectangle. A new non-source segment is the intersection of the original segment with a halo rectangle. There are several embodiments for creating new segments based on the intersection of the polygon segment data with main and halo rectangles. In one embodiment, the new source segment is defined as the exact intersection of the polygon segment with the tile's main rectangle. FIG. 22 illustrates one example for determining new source segments and new non-source segments for a polygon. This example includes a halo rectangle 2240 and a main rectangle 2205. As shown in FIG. 22, a polygon includes a new source segment 2210 bounded by main rectangle 2205. A new non-source segment is created between the area of the halo rectangle 2240 and the main rectangle 2205. A third segment, labeled 2230, is not part of either the main or halo rectangle, and therefore is discarded.

[00120] In another embodiment, if the polygons are rectangles, then the new segment may be approximated by another rectangle equal to the intersection of a halo or main rectangle. FIG. 23 illustrates one embodiment for estimating a segment rectangle estimated by a bounding rectangle (e.g., halo or main rectangle). For this example, a new segment rectangle 2310 is estimated based on its intersection with bounding box 2305. The segment 2320 is discarded for purposes of this calculation. For this embodiment, the boundary of the new source rectangle is determined by extending a center line perpendicular to the end of segment 2310 at the point where the segment crosses the boundary of the bounding box.

[00121] To specify a new segment area, the process creates new segment data structure 2000, which is illustrated in FIG. 24. This data structure defines the new polygon area by its vertices (e.g., X₀,Y₀, X₁,Y₁, . . . X_n, Y_n). In one embodiment, the segment data structure for a polygon also includes a field 2010 that specifies whether the area is a source or non-source area. The process (at 2130) marks the new area as a source area since it is in the main rectangle of the tile. In another embodiment, the data structure 2000 further includes a field 2005 that specifies whether the rectangle is white or gray (i.e., critical or not), and the process (at 2030) specifies the new area as a white or gray area depending on the corresponding attribute of the segment S. The data structure 2000 also includes a pointer 2015 to link the new area to another area or segment in a list.

[00122] In one embodiment, if the critical/non-critical attribute is used, the process (at 2132) determines whether the new segment is gray (i.e., based on the information from segment S). If the new segment is gray, then the process (at 2137) inserts the new area in the list of gray segments for tile, T_i. If the new segment is not gray, then the process (at 2135) inserts the new area in the list of white segments for tile, T_i. In other embodiments, the critical/non-critical attribute is not used (e.g., graphics application), then the process transitions from block 2130 to block 2135.

[00123] In one embodiment, as shown in FIG. 25, the new rectangle is inserted by linking the pointer 920 of the tile data structure 910 to the new area, if the new area is the first area inserted into the tile data structure. Otherwise, the new area is inserted by linking the pointer 2015 of the last-inserted area to the new area.

[00124] After inserting the new area, the process (at 2145) sets the halo-rectangle number, H_{j, to} 1. The process (at 2150) then computes the intersection of the segment S with the halo rectangle H_j of the tile T_i. Embodiments for computing the intersection of a polygon with a rectangle are described above. Next, the process (at 2155) determines whether this intersection is empty. If so, the process transitions to step 2160.

[00125] If not, the process (at 2170) creates a new area based on the intersection of the segment S with the halo rectangle H_j of the tile T_i. One embodiment for creating a new area based on the intersection of a polygon and a rectangle is described above. The data structure of this new area is identical to that shown in FIG. 24. Hence, the process creates the new area by specifying its vertices (e.g., X₀, Y₀, X₁, Y₁, . . . X_n, Y_n). The process also specifies the new area as a non-source segment since it is in the halo region of the tile. The process further specifies the new area as a white or gray (i.e., critical or non-critical) area depending on the corresponding attribute of the segment S.

[00126] Next, the process (at 2175) inserts the new area in the data structure for tile T_i. As before, if another area has already been inserted in this tile data structure, the new area is inserted by linking the pointer 2015 of the last-inserted area to the new area. Otherwise, the new area is inserted by linking the pointer 920 of the tile data structure 910 to the new area.

[00127] The process (at 2160) determines whether the halo rectangle number, H_j, equals the halo counter value of the segment S. If not, the process (at 2165) increments the halo rectangle number H_j by 1, and then transitions back to step 2150 for the next halo region of tile, T_i. Otherwise, the process (at 2180) increments the tile number T_i by 1, and then determines (at 2185) whether all the identified tiles were examined. If not, the process transitions to step 2120 to repeat the process for the next intersected tile.

[00128] On the other hand, if all the identified tiles were examined, the process determines (at 2190) whether all the segments have been examined. If so, the process ends. Otherwise, the process (at 2195) increments the segment number S, and then identifies (at 2110) all tiles that the new segment S intersects. The process is then repeated for the new segment S.

[00129] As shown in FIG. 11, the process 1000 creates (at 1020) an ng tree for each tile region after it inserts the interconnect segments into the tile data structures. Each ng tree efficiently organizes the data relating to the interconnect segments in its tile. Some embodiments of the invention use a two-step process to create an ng tree for a tile. First, for this embodiment, each inserted segment is converted into an ng node. Second, the ng nodes are connected to build an ng tree. These processes are described above in conjunction with FIGS. 6, 7 and 8.

[00130] FIG. 26 illustrates one manner for constructing an ng node array. As shown in FIG. 26, some embodiments create an ng node array by creating a one-dimensional pointer array 2305 and an ng node 2310 for each pointer in the array. Each pointer in the array points to a unique ng node. Some embodiments create the ng node array by only creating the one-dimensional pointer array 2305.

[00131] In some embodiments of the invention, each ng node is an ng node object (i.e., an instantiation of an ng node class). FIG. 26 illustrates the data structure for an ng node. As shown in this figure, each ng node has two sets of fields. The first set 2315 include fields to contain data regarding the segment inserted into the node. This data includes the segment's vertices, number of sides, its source or non-source status, and its critical or non-critical status.

[00132] The second set of fields 2320 includes fields that are necessary for building and traversing the ng tree. For instance, it includes outside and inside child pointers for pointing to the outside and inside children of the node. These pointers are initialized as NIL pointers. This second set also includes the children nodes' low and high dimension values along the node's discriminator dimension. The node's discriminator dimension depends on the node's level in the tree, and hence is determined when the tree is being built. The low and high dimension values speed up the process for traversing the tree.

[00133] FIG. 27 presents a tile data structure with its associated ng tree. This figure illustrates how the tile data structure identifies the root of the ng node, and how the root ng node connects to its child nodes through its outside and inside child pointers.

[00134] FIG. 28 illustrates one manner of partitioning the IC layout 305 of FIG. 3. In this figure, the IC layout 305 is partitioned along the x- and y-axes into nine square tiles. The layout is divided into these square tiles by two row lines defined by y-dimension values 100 and 200, and two column lines defined by x-dimension values 100 and 200. FIG. 28 also shows the halo rectangles (dashed lines) that surround the main rectangle. Portions of the rectangle segments in FIG. 28 that are encompassed by the main rectangle or the halo rectangle are marked to show the new segments created as set forth in the process of FIG. 21. For example, rectangle segment 10 includes a first new segment to define the intersection with rectangle 10 with the main rectangle, and a second new segment to define the intersection of rectangle 10 with the halo rectangle.

[00135] EDA applications can use the two-layered data structure 900 to speed up capacitance extraction operations. In particular, EDA applications can use this two-layered data structure to quickly identify interconnect lines with a certain distance (i.e., a halo distance) of critical interconnect lines. These identified interconnect lines can then be used to compute the capacitances exerted on the critical interconnect lines.

[00136] FIG. 29 illustrates one process that is used in some embodiments of the invention to identify the interconnect lines within a halo distance of a critical interconnect line. This process (at 2805) initially retrieves an ng tree from the permanent storage device 825, and stores this ng tree in the system memory of the computer system. Next, the process (at 2810) selects a critical interconnect segment in the retrieved ng tree.

[00137] The process (at 2815) then calculates the halo region surrounding the selected segment. In some embodiments of the invention, this halo region is defined as a rectangle that has its centered aligned with the segment's center, and has its vertical and horizontal sides a halo distance d away from the segment's vertical and horizontal sides respectively.

[00138] Next, the process (at 2820) uses the calculated halo region as a range-query window to perform a range query on the ng tree of the selected segment. In some embodiments of the invention, the process 2800 uses known techniques for traversing the tree and performing a range query on the ng tree. For instance, the process makes branching decisions at each node by determining whether that node's discriminator value falls below, above, or within the query range along that node's discriminator dimension. The process examines: (1) the node's outside subtree when the node's discriminator value is inside the half plane for the query range, (2) the node's inside subtree when the node's discriminator value is outside the half plane for the query range, and (3) the node's outside and inside subtrees when the node's discriminator value falls within the query range.

[00139] As discussed above by reference to FIG. 27, each node in the ng tree specifies the low and high dimension values (along the node's discriminator dimension) of the nodes in its outside and inside subtree. Consequently, the process 2800 can use these low and high dimension values at each node to determine whether it would be useful to examine the subtrees originating from a node. For instance, if a node's discriminator value falls outside the plane of the query range, the process can examine the low bound of the node's outside subtree to determine whether the lowest value in that subtree is also above the query range along that node's discriminator. If so, the process does not need to examine the outside subtree since all the segments in this subtree are above the query range along that node's discriminator.

[00140] Each time the process 2800 encounters a node whose discriminator value falls within the query range, the process determines whether the remaining coordinates of the segment stored at that node fall within its range query window. In essence, the process compares the coordinates of the segment stored at that node with the coordinates of its range-query window. If the segment falls within this range-query window, the process returns this segment's data.

[00141] Some embodiments of the invention perform process 2800 for all critical (i.e., white) source segments in all the ng trees. In fact, some embodiments sequentially perform this process for all critical source segments in a retrieved ng tree, before retrieving another ng tree to analyze its segments.

[00142] One of ordinary skill in the art will understand that the invention's two-layer data structure 900 has numerous advantages. For instance, as discussed above, it speeds up the capacitance extraction operation, because it allows an EDA application to identify quickly all interconnect lines near a particular critical line by simply analyzing one small ng tree. Only one small ng tree needs to be examined for each critical segment, since each segment is stored in a small ng tree with all the interconnect lines that might capacitively couple to it.

[00143] Dividing the IC layout into smaller regions, and creating relatively smaller ng trees to store the segment data in each region, also allows the total query runtime to increase linearly with the number of interconnect segments in the layout. The runtime for performing N queries for N segments in a ng tree is proportional to Nlog₂N. However, if the IC layout is divided into two regions and each region roughly contains N/2 segments, the time for performing range queries about all the segments in each region is proportional to ½Nlog₂(N/2). Thus, the time for performing a range query about the segments in both regions is proportional to Nlog₂(N/2), which is better than Nlog₂(N).

[00144] Similarly, if the IC layout is divided into R regions with each region containing roughly N/R segments, the time for performing a range query about all the segments in all the regions is proportional to Nlog₂(N/R). Equation (1) below explains this mathematically. $\begin{array}{cc}\mathrm{Total}\mathrm{run}\mathrm{time}\propto \sum _R\frac{N}{R}{\log }_2\frac{N}{R}=N{\log }_2\frac{N}{R}<<N{\log }_{2}N& \left(1\right)\end{array}$

[00145] Hence, dividing the IC layout into smaller regions, and creating relatively smaller ng trees to store the data in each region, reduce the total query time. This reduction can be significant if the number of data regions R is on the same order as the number of segments N. In fact, the total query time can be made to increase only linearly with the number of segments, by increasing the number of data regions R linearly with the number of segments N. For example, if the number of regions R is selected so that it is always equal to N/1000, then the total query time will always be proportional Nlog2(1000).

[00146] The data structure 900 also works well with computers that have smaller system memories. This data structure does not need as much system memory because (1) an EDA application will only need to examine one tile's ng tree for each critical segment, and (2) each tile's ng tree is much smaller than a traditional ng tree that stores the data for all the segments in the IC layout.

[00147] 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 embodiments described above have only one ng tree for each tile, some embodiments of the invention have more than one ng tree for each tile. For each tile, some embodiments have one ng tree for white segments (i.e., critical segments) and one k-d tree for gray segments (i.e., non-critical segments).