Abstract:
A conformal outline of a well which is to receive elements of a circuit is formed from one or more candidate rectangles which enclose input rectangles. The one or more candidate rectangles are determined based upon a cost of the candidate rectangles determined therefor based on the overlap of the candidate rectangles with one or more penalty or avoid rectangles. Each input rectangle represents an area where it is desired to place elements of the circuit and each penalty or avoid rectangle represent an area where it is desired to avoid placing elements of the circuit. To determine the candidate rectangle(s) having the most advantageous cost, a side and/or an edge of each candidate rectangle is positioned at or near plural locations where the sides and/or edges of the input rectangles reside and a cost is determined therefor. The candidate rectangle(s) having the most favorable cost are then utilized as solution rectangles for the conformal outline.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims priority from U.S. Provisional Patent Application Ser. No. 60/269,264, filed Feb. 15, 2001, entitled “Quick Creation Of Conformal Outlines For Use In Transistor Level Semiconductor Layouts”. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to automatic layout of one or more elements of an integrated circuit, especially analog elements, and, more particularly, to the creation of conformal outlines used in the layout of these elements. 
     2. Brief Description of the Prior Art 
     In order to realize the high levels of integration that are occurring with system-on-chip designs, analog and digital functionality are being included on more semiconductor chips than ever. As more of these mixed signal chips are created, automation for the layout of the analog segments of these chips is required. Currently, this is an almost completely manual process. 
     An analog cell is typically a circuit, such as an operational amplifier or comparator, that includes between ten and one hundred elements, such as transistors, resistors and capacitors. In order to automatically lay out these elements to create a circuit, a large amount of detailed geometry needs to be optimized. For example, the placement of elements needs to take maximal advantage of design rules to create a compact and correct layout. One aspect of detailed geometric creation that can have a dramatic impact on the compactness and correctness of an analog cell layout is the well and latchup of protection geometry. 
     In a typical silicon semiconductor process, mask layers are created where each N-well is created and where each P-well is created. All N-type transistors must be surrounded by a P-type well and vice versa. These wells need to be electrically connected to the correct voltages or the transistors may not perform optimally, and in the worst case will not work at all. The geometry used to electrically connect the well to the rest of the circuitry is referred to as latchup protection geometry. 
     While the problems of creating wells and latchup protection geometry also occur in the layout of digital circuits, the solutions used to create well and latchup protection geometry for digital layouts are different than the solutions required for layout of analog cells. Specifically, for most digital cells, the transistors are very similarly sized and small. These transistors tend to share active regions in addition to sharing well geometry. The methodology most often used to create digital cells is to first define the areas of the cell where the P and N wells are positioned and then to place the transistors in those areas. In contrast, in an analog layout, the devices may be much larger and may also have differing sizes. Therefore, it is preferred to first place the devices, then form the wells around the devices. 
     Another difference between digital cell layout and analog cell layout is the voltage to which each well is connected. For digital layouts, the wells are almost always tied to power and ground. In contrast, analog circuits tend to have more wells that are tied to voltages other than power and ground. 
     In general, the rules on how a well geometry is created simply specify that the well geometry must enclose certain geometry of the devices by given amounts. The rules for where latchup protection must be formed in a well geometry, however, are more complicated. Specifically, latchup protection must be formed in the well area with the devices and must be placed so that it can effectively keep the entire well at a desired voltage. In general, this latter condition is expressed as a radius from which all the devices in the well must be located from the latchup protection geometry. 
     In addition to these physical rules, there are constraints on the performance of an automated solution for the creation of well and latchup protection geometry. Namely, it must be fast enough to be used in automatic layout where a large number of candidate solutions, each having well and latchup protection geometry, may need to be evaluated. The solution must also be fast enough to be used in an interactive mode to assist the designer with manual placement. 
     The wells that are created must also satisfy aesthetic constraints in order to be acceptable to designers. These constraints can be roughly expressed to be that the well shapes should have low complexity. Shapes with rough, jagged edges are generally not acceptable to designers. Moreover, designers prefer each well group to be as large as possible. However, this latter preference is in tension with the goal of any automated well layout system to minimize the well area utilized to receive the desired circuits. 
     It is, therefore, an object of the present invention to overcome the above problems and others by providing an automated method for creating well geometries for circuits, especially analog circuits. It is also an object of the present invention to minimize the area of the generated well geometry while, at the same time, avoiding overlap between the generated well geometry and one or more areas of an integrated circuit design where it is desired to avoid placing circuits or elements of a circuit. Still other objects will become apparent to those of ordinary skill in the art upon reading and understanding the following detailed description. 
     SUMMARY OF THE INVENTION 
     Accordingly, we have invented a method of creating a conformal outline for layout of one or more devices on an integrated circuit. The method includes defining a plurality of input rectangles in a Cartesian coordinate system having a first axis and second axis, with each input rectangle having two edges parallel to the first axis and two sides parallel to the second axis. Coordinates on the first axis where the sides of each input rectangle sides reside are acquired and a sorted list is formed from the acquired coordinates. A variable I is initialized with the value of two (2) and first and second candidate rectangles having one of their respective sides positioned at the smallest and largest coordinates in the sorted list, respectively, are identified from the input rectangles. The other of the sides of the respective first and second candidate rectangles are set equal to the coordinate on the first axis stored in the I th  position in the sorted list, thereby forming a shared side. For each candidate rectangle having coordinates on the first axis which overlap coordinates of one or more input rectangles on the first axis, the lengths of the sides of the candidate rectangles are modified as necessary so that they extend from the maximum to the minimum second axis coordinates of the one or more input rectangles. If the shared side with respect to one candidate rectangle does not coincide with or overlap the shared side with respect to the other candidate rectangle, the length of the shared side with respect to the one candidate rectangle is modified as necessary so that it coincides with or overlaps the shared side with respect to the other candidate rectangle. 
     The method can also include the step of initializing a Best Solution variable that includes a cost. 
     The method can further include the steps of determining a cost of the candidate rectangles and, when the cost of the candidate rectangles is more advantageous then the cost of the Best Solution, updating the Best Solution with the position of the candidate rectangles in the Cartesian coordinate system and the cost determined therefor. 
     Next, the variable I is incremented by one (1) and a determination is made if I equals the number of coordinates in the sorted list. If not, the shared side is stepped or moved to the coordinate on the first axis stored in the I th  position in the sorted list. For each candidate rectangle having coordinates on the first axis which overlap coordinates of one or more input rectangles on the first axis, the lengths of the sides of the candidate rectangle are modified as necessary so they extend from the maximum to the minimum second axis coordinates of the one or more input rectangles. If the shared side with respect to one candidate rectangle does not coincide with or overlap the shared side with respect to the other candidate rectangle, the length of the shared side with respect to one rectangle is modified as necessary so that it coincides with or overlaps the shared side with respect to the other candidate rectangle. Next, the cost of the candidate rectangles is determined and the Best Solution is updated with the position of the candidate rectangles and the cost therefor when the cost is more advantageous than the current cost of the Best Solution. The foregoing steps are repeated until the value of the variable I equals the number of coordinates in the sorted list. 
     The cost of the candidate rectangles can be determined in numerous manners. One manner of determining the cost includes defining one or more penalty rectangles in the Cartesian coordinate system and determining an area of intersection between the candidate rectangles and the one or more penalty rectangles. The cost of the candidate rectangles can be determined based on at least two of (i) the area of the candidate rectangles; (ii) the area of intersection; and (iii) a penalty weight (PW) for the intersection. 
     Preferably, the cost is determined utilizing the following equation:        Cost   =       (     Area                 of                 Candidate                 Rectangles     )     +       ∑     k   =   0       k   =   A                         PW   k     *       ∑     i   =   0       i   =   B                       Area                 of                 Intersection                     (       Penalty                   Rects   k       ,                Candidate                   Rects   i         )     .                                      
     where A=(No. of Penalty Rects. −1); and 
     B=(No. of Candidate Rects. −1). 
     We have also invented a method of creating a conformal outline for layout of one or more devices on an integrated circuit that includes defining a plurality of input rectangles in a Cartesian coordinate system having a first axis and a second axis, where each input rectangle has two edges parallel to the first axis and two sides parallel to the second axis. Coordinates on the first axis are acquired where the sides of each input rectangle reside and a sorted list is formed from the acquired coordinates. A variable I is initialized to two (2). Next, first and second candidate rectangles having one of their respective sides positioned at the smallest and largest coordinates in the sorted list, respectively, are identified from the input rectangles and a third candidate rectangle having its sides positioned between the smallest and largest coordinates in the sorted list is also identified from the input rectangles. The side of the first candidate rectangle opposite the smallest coordinate in the sorted list and one side of the third candidate rectangle are set equal to the coordinate on the first axis stored in the I th  position in the sorted list thereby forming a shared side. A variable J is initialized to the value of the variable I plus one (1), i.e., J=I+1. Next, the side of the second rectangle opposite the largest coordinate in the sorted list and the other side of the third candidate rectangle are set equal to the coordinate on the first axis stored in the J th  position in the sorted list, thereby forming another shared side. For each candidate rectangle having coordinates on the first axis which overlap coordinates of one or more input rectangles on the first axis, the lengths of sides of the candidate rectangle are modified as necessary so they extend from the maximum to the minimum second axis coordinates of the one or more input rectangles. When the shared side with respect to one candidate rectangle does not coincide with or overlap the same shared side with respect to the other candidate rectangle, the length of the shared side with respect to the one candidate rectangle is modified as necessary so that it coincides with or overlaps the same shared side with respect to the other candidate rectangle. 
     The method can also include determining a cost of the candidate rectangles and, when the thus determined cost is more advantageous than a cost of a Best Solution variable, the Best Solution variable is updated with the coordinates of the candidate rectangles in the Cartesian coordinate system and the cost determined therefor. 
     The foregoing steps, after initialization of the variable J, are repeated until the value of J equals the number of coordinates in the sorted list. When the value of J equals the number of coordinates in the sorted list, the value of I is incremented by 1 and the foregoing steps, after identifying the first, second and third candidate rectangles from the input rectangles, are repeated until the value of I equals the number of coordinates in the sorted list minus one (1). 
     We have also invented a method of creating a conformal outline for layout of one or more devices on an integrated circuit that includes the steps of defining a plurality of input geometries in a coordinate system having a first axis and a second axis where each input geometry has first and second edges parallel to the first axis and first and second sides parallel to the second axis. Coordinates on the first axis are acquired where the sides of each input geometry reside. A sorted list is formed from these acquired coordinates. A first candidate geometry is identified having its first side residing at the smallest coordinate in the sorted list and a second candidate geometry is identified having its second side residing at the largest coordinate in the sorted list. The second side of the first candidate geometry and the first side of the second candidate geometry are positioned at a coordinate in the sorted list between the smallest and largest coordinates thereby forming a shared side. 
     For each candidate geometry which has first axis coordinates which overlap first axis coordinates of one or more input geometries, and which has at least one second axis coordinate extreme which does not coincide with a second axis coordinate extreme of the one or more input geometries, the lengths of the sides of the candidate geometry are modified so that opposite ends of each side terminate at the respective second axis coordinate extremes of the one or more input geometries. When the shared side with respect to one candidate geometry does not coincide with or completely overlap the shared side with respect to the other candidate geometry, the length of the shared side with respect to the one candidate geometry is modified so that it coincides with or completely overlaps the shared side with respect to the other candidate geometry. 
     The method can also include determining for the combination first and second candidate geometries a cost therefor. If the thus determined cost is more advantageous than a previously determined cost or no cost, the thus determined cost is stored along with the position of the first and second candidate geometries relative to the input geometries. The foregoing steps, after forming the shared side at a coordinate in the sorted list between the smallest and largest coordinates, are repeated for each position of the shared side at coordinates in the sorted list between the smallest and largest coordinates. 
     We have also invented a method of creating a conformal outline for layout of one or more devices on an integrated circuit that includes the steps of defining a plurality of input geometries in a coordinate system having a first axis and a second axis, where each input geometry has first and second edges parallel to the first axis and first and second sides parallel to the second axis. Coordinates on the first axis are acquired where the sides of each input geometry reside and a sorted list is formed of the acquired coordinates. A first candidate geometry is identified having its first side residing at the smallest coordinates in the sorted list and a second candidate geometry is identified having a second side residing at the largest coordinate in the sorted list. A third candidate geometry is also identified having its first and second sides residing between the smallest and largest coordinates in the sorted list. The second side of the first candidate geometry and the first side of the third candidate geometry are positioned at one coordinate in the sorted list between the smallest and largest coordinates in the sorted list thereby forming a first shared side. The second side of the third candidate geometry and the first side of the second candidate geometry are positioned at another coordinate in the sorted list between the one coordinate and the largest coordinate in the sorted list thereby forming a second shared side. 
     For each candidate geometry which has first axis coordinates which overlap first axis coordinates of one or more of the input geometries, and which has at least one second axis coordinate extreme which does not coincide with a second axis coordinate extreme of the one or more input geometries, the lengths of the sides of the candidate geometry are modified as necessary so that opposite ends of each side terminate at the respective second axis coordinate extremes of the one or more input geometries. When the shared side with respect to one candidate geometry does not coincide or completely overlap the shared side with respect to an adjacent candidate geometry, the length of the shared side with respect to the one candidate geometry is modified so that it coincides or completely overlaps the shared side with respect to the adjacent candidate geometry. 
     The method can also include determining a cost of the first, second and third candidate geometries and, if the thus determined cost is more advantageous than a previously determined cost or no cost, the thus determined cost is stored along with the positions of the first, second and third candidate geometries relative to the input geometries. The foregoing steps, after forming the second shared side, are repeated for each position of the second shared side at coordinates in the sorted list between the coordinate in the sorted list where the first shared side resides and the largest coordinate in the sorted list. 
     The method can further include repeating the foregoing steps, after forming the second shared side, for each position of the first shared side at coordinates in the sorted list between the one coordinate and the second from the largest coordinate in the sorted list. 
     Lastly, we have invented a method of creating a conformal outline for the layout of one or more devices on an integrated circuit. The method includes defining a plurality of adjoining candidate rectangles which encompass a plurality of input rectangles, where the outline of the adjoining rectangles define the conformal outline. 
     Preferably, each input rectangle has left and right sides, and top and bottom edges. Each candidate rectangle preferably has left and right sides, and top and bottom edges. Each pair of adjacent candidate rectangles define a shared side or edge that is coincident with the respective side or edge of at least one input rectangle. 
     The method can further include stepping each shared side or edge to at least one other side or edge, respectively, of one of the input rectangles and determining for each step of each shared side or edge a cost for the candidate rectangles. The outline of the candidate rectangles having the most advantageous cost is selected as the conformal outline. 
     The method can also include at least one of the following steps. First, for each step of each shared side, the lengths of the respective sides of at least the pair of candidate rectangles defining the shared side are adjusted whereupon the top and bottom edges of each pair of candidate rectangles are coincident with the topmost and bottommost edges of any input rectangles which overlap the candidate rectangle. Second, for each step of each shared edge, the lengths of the respective edges of at least the pair of candidate rectangles defining the shared edge are adjusted whereupon the left and right sides of each of the pair of candidate rectangles are coincident with the leftmost and rightmost sides of any input rectangles which overlap the candidate rectangle. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a top level flow diagram of the operation of a software program which implements a method in accordance with the present invention; 
     FIG. 2 is a one candidate rectangle solution for an array of input rectangles and penalty rectangles received in a Cartesian coordinate system; 
     FIGS. 3 a - 3   f  are a detailed flow diagram of the software routine shown in FIG. 1 that evaluates two and three candidate rectangle solutions for the array of input rectangles and penalty rectangles shown in FIG. 2; 
     FIGS. 4 a - 4   c  show a portion of the X-axis two-candidate rectangle solution implemented in the part of the flow diagram shown in FIGS. 3 a - 3   c;    
     FIG. 4 d  shows the starting Y-axis two-candidate rectangle solution for the part of the flow diagram shown in FIGS. 3 d - 3   f;    
     FIGS. 5 a - 5   c  show the process in step  214  of FIG. 3 a  of adjusting the top and/or bottom edges of at least one candidate rectangle so that a shared side of one candidate rectangle coincides or overlaps the same shared side with respect to another candidate rectangle; 
     FIGS. 6 a - 6   f  show part of the X-axis three-candidate rectangle solution implemented by the part of the flow diagram shown in FIGS. 3 a - 3   c ; and 
     FIGS. 6 g  and  6   h  show the starting and ending Y-axis three-candidate rectangle solution implemented by the part of the flow diagram shown in FIGS. 3 d - 3   f.   
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present invention is a method for creating one or more solution geometries, such as solution rectangles, that cover a set of input geometries, such as input rectangles, in a conformal manner. The present invention is embodied in a computer software program which is called as a sub-routine from a main program. The software program essentially receives inputs regarding (i) input rectangles to be covered, (ii) rectangles to be avoided, and (iii) the maximum number of candidate rectangles to be used for a result. Based on these three inputs, the software program returns to the main program, or routine, one or more solution rectangles that cover the set of input rectangles in a conformal manner. 
     The invention will now be described with reference to the accompanying Figures where like reference numbers correspond to like elements. In accompanying FIGS. 2,  4   a - 4   d  and  6   a - 6   h , the sides and edges of coordinate rectangles are shown offset from the sides and edges of certain input rectangles for purposes of illustration. However, it is to be appreciated that the sides and edges of the candidate rectangles overlap the sides and edges of certain input rectangles. 
     With reference to FIG. 1, a main program/routine  102  initiates a call to a sub-routine software program  104  which implements the method of the present invention. Software program  104  includes routine  106  which receives the three basic inputs, namely, the input rectangles to be covered, the rectangles to be avoided, and the maximum number of candidate rectangles to be used for a result. Program flow then advances to routine  108  which initializes a “Best Solution” variable and sets a variable N equal to 1. Next, program flow advances to routine  110  which finds the N candidate rectangle solution and its cost. Since at this time N equals 1, routine  110  finds a one candidate rectangle solution and the cost of this solution. 
     With reference to FIG.  2  and with continuing reference to FIG. 1, routine  110  finds the one rectangle solution by superimposing a plurality of input rectangles  12 ,  14 ,  16 ,  18 ,  20  and  22 , i.e., the input rectangles to be covered, in a Cartesian coordinate system  24  having a first X axis  26  and a second Y axis  28 . Each input rectangle  12 - 22  has top and bottom edges parallel to X axis  26  and left and right sides parallel to Y axis  28 . Routine  110  also superimposes one or more penalty rectangles  30 ,  32 , and  34 , i.e., the rectangles to be avoided, in Cartesian coordinate system  24 . The positions of input rectangles  12 - 22  relative to the positions of penalty rectangles  30 - 34  are received in routine  106 . Each penalty rectangle  30 - 34  can have an overlapping or nonoverlapping relationship with each other or with one or more input rectangles  12 - 22  and vice versa. 
     Next, a minimum X axis coordinate x 1  where a side of one of the input rectangles, e.g., input rectangle  12 , resides; a maximum X axis coordinate x 12  where a side of one of the input rectangles, e.g., input rectangle  20 , resides; the minimum Y axis coordinate y 1  where an edge of one of the input rectangles, e.g., input rectangle  22 , resides; and the maximum Y axis coordinate y 10  where an edge of one of the input rectangles, e.g., input rectangle  16 , resides are determined. Next, a candidate rectangle  36  is defined in Cartesian coordinate system  24  with its left side  38  and its right side  40  positioned at the smallest X axis coordinate x 1  and the largest X axis coordinate x 12 , respectively, of input rectangles  12 - 22 . Candidate rectangle  36  also has its bottom edge  42  and its top edge  44  positioned at the smallest Y axis coordinate y 1  and the largest Y axis coordinate y 10 , respectively, of input rectangles  12 - 22 . 
     Next, the cost of this one candidate rectangle solution, i.e., the cost of candidate rectangle  36 , is determined. Preferably, this cost is determined utilizing the following equation (1):        Cost   =       (     Area                 of                 Candidate                   Rectangle        (   s   )         )     +       ∑     k   =   0       k   =   A                         PW   k     *       ∑     i   =   0       i   =   B                       Area                 of                 Intersection                     (       Penalty                     Rect        (   s   )       k       ,                Candidate                     Rect        (   s   )       i         )     .                                      
     where A=(No. of Penalty Rects. −1); 
     B=(No. of Candidate Rects. −1); and 
     PW=(Penalty Weight). 
     The invention, however, is not to be construed as limited to utilizing equation (1) to determine the cost of candidate rectangle solution(s). 
     As can be seen from equation (1), the cost of an N-candidate rectangle solution is determined as a function of the area of the candidate rectangle(s), and the area of intersection between the candidate rectangle(s) and the penalty rectangle(s). In addition, a penalty weight (PW) is associated with the area of intersection so that the significance of the area of intersection to the determination of the cost of the solution can be regulated as desired. 
     Once the cost of candidate rectangle  36  is determined, software program  104  advances to routine  112  which updates the Best Solution variable with the position of candidate rectangle  36  relative to input rectangles  12 - 22  and the cost of this solution. Next, in routine  114 , the current value of the variable N is compared to the maximum number of candidate rectangles received in routine  106 . For the purpose of describing the invention, it will be assumed that the maximum number of candidate rectangles that can be received in routine  106  is three (3). However, this is not to be construed as limiting the invention. 
     If the current value of the variable N equals the maximum number of candidate rectangles, program flow advances to routine  116  which returns the Best Solution variable and, more particularly, the position of the candidate rectangle(s) relative to the input rectangles to main program/routine  102  for further processing. However, if the current value of the variable N does not equal the maximum number of candidate rectangles, program flow advances to routine  118  where the value of the variable N is incremented by one (1). 
     Next, program flow advances to routine  120  where all possible N-candidate rectangle solutions and their costs are evaluated. For each N-candidate rectangle solution, the position of the candidate rectangles relative to the input rectangles and the cost of the solution are saved in the Best Solution variable if the cost of the solution is less than the cost of the solution currently saved in the Best Solution variable. In other words, the current value of the Best Solution variable is replaced by the N-candidate rectangle solution having a more advantageous, e.g., lower, cost. 
     More specifically, and with reference to FIGS. 3 a - 3   f , and  4   a - 4   d , routine  120  includes step  201  which acquires the coordinates on the X axis  26  where the sides of each input rectangle  12 - 22  reside. Program flow then advances to step  202  which forms a sorted list of the acquired X axis coordinates. For example, in FIG. 4 a , input rectangle  12  has its sides residing at X axis coordinates x 1  and x 3 ; input rectangle  14  has its sides positioned at X axis coordinates x 2  and x 5 ; input rectangle  16  has its sides positioned at X axis coordinates x 4  and x 7 ; input rectangle  18  has its sides positioned at X axis coordinates x 6  and x 9 ; input rectangle  20  has its sides positioned at X axis coordinates x 8  and x 12 ; and input rectangle  22  has its sides positioned at X axis coordinates x 10  and x 11 . Once X axis coordinates x 1 -x 12  are acquired, they are sorted, preferably in order from smallest to largest, i.e., from x 1  to x 12 , to form a sorted list of X axis coordinates. 
     Next, in step  204 , the left edge of a first candidate rectangle  46  is assigned to the first X axis coordinate x 1  in the sorted list. Then, in step  206 , the right edge of a second candidate rectangle  48  is assigned to the last coordinate x 12  in the sorted list. 
     Program flow then advances to step  208  where a variable I is assigned a value of two (2). Next, in step  209 , a determination is made whether the current value of the variable N equals to the maximum number of candidate rectangles. If not, program flow advances to step  210  which assigns the right side of candidate rectangle  46  and the left side of candidate rectangle  48  to the coordinate stored in the I th  position in the sorted list thereby forming a shared or adjoining side  49 . In this case, since I is equal to two (2), shared side  49  is positioned at X axis coordinate x 2 . Program flow then advances to step  212 . 
     In step  212 , the top and bottom edges of each candidate rectangle  46  and  48  are set equal to the maximum and minimum Y axis values, i.e., the Y axis coordinate extremes, of any input rectangle  12 - 22  having X axis coordinates which overlap X axis coordinates of the candidate rectangle. For example, as shown in FIG. 4 b  in contrast to FIG. 4 a , the lengths of the left and right sides of candidate rectangle  46  are adjusted so that the bottom and top edges of candidate rectangle  46  are positioned at Y axis coordinates y 3  and y 8  where the bottom and top edges, respectively, of input rectangle  12  reside. Similarly, the lengths of the sides of candidate rectangle  48  are adjusted so that the bottom and top edges of candidate rectangle  48  are positioned at Y axis coordinates y 1  and y 10  where the bottom edge of input rectangle  22  and the top edge of input rectangle  16 , respectively, reside. Program flow then advances to step  214 . 
     In step  214 , the position of the top and/or bottom edges of one or more of candidate rectangles  46  and  48  are modified as necessary so that shared side  49  with respect to one candidate rectangle, e.g., candidate rectangle  48 , coincides with or completely overlaps shared side  49  with respect to the other candidate rectangle, e.g., candidate rectangle  46 . Candidate rectangles  46  and  48  shown in FIG. 4 b  meet all the conditions set forth in step  214  and therefore require no modification of the position of their top and/or bottom edges. Examples where the top and/or bottom edges of a pair of candidate rectangles  50  and  52  require repositioning are shown in FIGS. 5 a - 5   c.    
     In FIG. 5 a , candidate rectangles  50  and  52  have a shared side  54 . As can be seen, shared side  54  with respect to candidate rectangle  50  does not coincide with or completely overlap shared side  54  with respect to candidate rectangle  52 , and vice versa. To meet this requirement in step  214 , the lengths of the sides of candidate rectangle  50  can be extended upward, as shown in FIG. 5 b , so that the top edges of candidate rectangles  50  and  52  are aligned. Alternatively, as shown in FIG. 5 c , the lengths of the sides of candidate rectangle  52  can be extended downward so that the bottom edges of candidate rectangles  50  and  52  are aligned. The decision to reposition the top edge of candidate rectangle  50  or the bottom edge of candidate rectangle  52  can be based on, among other things, the avoidance of overlap with a penalty rectangle, the avoidance of overlap with another candidate rectangle, and a minimization of the additional area to be added by such repositioning. However, other criteria can also be utilized. It can be seen that the modified candidate rectangles  50  and  52  shown in FIGS. 5 b  and  5   c , respectively, meet all the conditions set forth in step  214 , namely, that adjacent candidate rectangles  50  and  52  share a side  54  and that the shared side  54  of one rectangle  50  or  52  coincides with or overlaps the shared side  54  of the other rectangle  52  or  50 . 
     Referring back to FIGS. 3 a - 3   f  and  4   a - 4   d , program flow then advances to step  216  where the cost of candidate rectangles  46  and  48  is determined utilizing equation (1) above. If the cost of candidate rectangles  46  and  48  is less than the current cost saved in the Best Solution variable, the positions of candidate rectangles  46  and  48  relative to input rectangles  12 - 22  and the cost of this solution are saved in the Best Solution variable. 
     Program flow then advances to step  217  where a determination is made if the value of variable N equals the maximum number of candidate rectangles. If not, program flow advances to step  218  where the variable I is incremented by one (1). Program flow then advances to step  220  where a determination is made whether the current value of the variable I equals the number of X axis coordinates in the sorted list. If not, program flow returns to step  210 . Steps  210 - 220  are then repeated until the variable I equals the number of X axis coordinates in the sorted list. 
     Repeating steps  210 - 220  for each value of the variable I has the effect of sequentially stepping shared side  49  to each X axis coordinate in the sorted list from X axis coordinate x 2 , shown in FIG. 4 b , to X axis coordinate x 11 , shown in FIG. 4 c . For each position of shared side  49  at an X axis coordinate in the sorted list, adjustment of the position of the top and/or bottom edges of candidate rectangles  46  or  48  in accordance with the requirements of steps  212  and  214  is repeated and the cost of candidate rectangles  46  and  48  is determined. If any thus determined cost is more advantageous, e.g., less than, the current cost saved in the Best Solution variable, the positions of candidate rectangles  46  and  48  relative to input rectangles  12 - 22  and the cost of this solution are saved in the Best Solution variable. 
     When shared side  49  has been stepped to each X axis coordinate between the first and last X axis coordinates, the same process is repeated for a shared edge stepped to each Y axis coordinate between the first and last Y axis coordinates. More specifically, when step  220  determines that the value of the variable I equals the number of X axis coordinates in the sorted list, program flow advances to step  221  in FIGS. 3 d - 3   f . Steps  221 - 240  are essentially the same steps as steps  201 - 220 , respectively, except that steps  221 - 240  are for two candidate rectangles having a shared edge positioned at Y axis coordinates between the first and last Y coordinates in a sorted list of Y coordinates of input rectangles  12 - 22 . An example of candidate rectangles  56  and  58  having a shared or adjoining edge  60  positioned at Y axis coordinate y 2  is shown in FIG. 4 d . Steps  230 - 240  sequentially step shared edge  60  to each Y axis coordinate in the sorted list of Y axis coordinates between Y axis coordinates y 1  and y 10  in the same manner as steps  210 - 220  step shared edge  49  to each X axis coordinate in the sorted list of X axis coordinates between X axis coordinates x 1  and x 12 . For each position of shared edge  60  between Y axis coordinates y 1  and y 10 , steps  232 - 236  are repeated. 
     In step  232 , the left and right sides of each candidate rectangle are set equal to the maximum and minimum X axis values, i.e., the X axis coordinate extremes, of any input rectangle having its Y axis coordinates overlapping the Y axis coordinates of the candidate rectangle. 
     In step  234 , the position of the right and/or left sides of at least one candidate rectangle are modified as necessary so that shared edge  60  with respect to one candidate rectangle, e.g.,  56 , coincides with or overlaps shared edge  60  with respect to the other candidate rectangle, e.g.,  58 . 
     In step  236 , the cost of first and second candidate rectangles  56  and  58  for each position of shared side  60  at a Y axis coordinate between Y axis coordinates y 1  and y 10  is determined and saved, along with the corresponding position of first and second candidate rectangles  56  and  58  relative to input rectangles  12 - 22 , in the Best Solution variable if the thus determined cost is less than the cost currently saved in the Best Solution variable. 
     Referring back to FIG.  1  and with continuing reference to FIGS. 3 d - 3   f , when step  240  determines that the value of the variable I equals the number of Y coordinates in the sorted list of Y coordinates, all of the steps associated with routine  120  for all two candidate rectangle solutions are complete and program flow returns to routine  114  which determines if the variable N equals the maximum number of candidate rectangles. If not, program flow advances to routine  118  where the value of the variable N is incremented by one (1), i.e., N=3. Program flow then advances to routine  120  for evaluation of all three candidate rectangle solutions. 
     In this iteration, routine  120  executes steps  201 - 209 ,  212 - 220 ,  221 - 228 , and  232 - 240  discussed above. In addition, since the variable N equals the maximum number of candidate rectangles, i.e., 3, routine  120  further executes steps  250 - 260  shown in FIGS. 3 a - 3   c  and steps  270 - 280  shown in FIGS. 3 d - 3   f . These steps will now be described with reference to FIGS. 6 a - 6   h  for all 3-candidate rectangle solutions. 
     With specific reference to FIGS. 3 a - 3   f  and  6   a - 6   h , routine  120  initially performs step  201  which acquires the coordinates on X axis  26  where the sides of each input rectangle  12 - 22  reside. Program flow then advances to step  202  which forms a sorted list of the X axis coordinates. Program flow then advances to step  204  where the left edge of a first candidate rectangle  62  is assigned to the first X axis coordinate x 1  in the sorted list. Next, in step  206  the right edge of a second candidate rectangle  64  is assigned to the last coordinate x 12  in the sorted list. In step  208  the variable I is set equal to 2 and in step  209  a determination is made whether the variable N equals the maximum number of candidate rectangles. Since, in this iteration of routine  120 , the variable N equals the maximum number of candidate rectangles, i.e., 3, program flow advances to step  250  where a variable J is set equal to the current value of the variable I plus one (1), i.e., J=I+1. Program flow then advances to step  252 . 
     In step  252 , the right side of first candidate rectangle  62  and the left side of a third candidate rectangle  66  are set equal to the coordinate stored in the I th  position in the sorted list to form a shared or adjoining side  68 . In addition, the right side of third candidate rectangle  66  and the left side of second candidate rectangle  64  are set equal to the coordinates stored in the J th  position in the sorted list thereby forming a shared or adjoining side  70 . In this iteration of step  252 , I is equal to two (2) and J is equal to three (3). Thus, shared sides  68  and  70  are positioned at X axis coordinates x 2  and x 3 , respectively. Program flow then advances to step  212 , bypassing step  210 . 
     In step  212 , the top and bottom edges of each candidate rectangle  62 ,  64  and  66  are set equal to the maximum and the minimum Y axis values, i.e., the Y axis coordinate extremes, of each input rectangle  12 - 22  having its X axis coordinates overlapping the X axis coordinates of candidate rectangle  62 ,  64  or  66 . For example, as shown in FIG. 6 b  in contrast to FIG. 6 a , the lengths of the sides of candidate rectangle  62  are adjusted so that the bottom and top edges of candidate rectangle  62  are positioned at Y axis coordinates y 3  and y 8 , respectively, where the bottom and top edges of input rectangle  12  reside. Similarly, the lengths of the sides of candidate rectangle  66  are adjusted so that the bottom and top edges of candidate rectangle  66  are positioned at Y axis coordinates y 2  and y 8  where the bottom edge of input rectangle  14  and the top edge of input rectangle  12 , respectively, reside. Furthermore, the lengths of the sides of candidate rectangle  64  are adjusted so that the bottom and top edges of candidate rectangle  64  are positioned at Y axis coordinates y 1  and y 10  where the bottom edge of input rectangle  22  and the top edge of input rectangle  16 , respectively, reside. 
     In step  214 , the position of the top and/or bottom edges of at least one candidate rectangle  62 ,  64  or  66  are modified as necessary so that each shared side  68  and  70  with respect to one candidate rectangle coincides or completely overlaps the same shared side  68  and  70  with respect to the other candidate rectangle. In FIG. 6 b , shared side  68  with respect to candidate rectangle  66  completely overlaps shared side  68  with respect to candidate rectangle  62 , and shared side  70  with respect to candidate rectangle  64  completely overlaps shared side  70  with respect to candidate rectangle  66 . Therefore, no modification of the position of the top and/or bottom edges of candidate rectangle  62 ,  64  or  66  is required. 
     Program flow then advances to step  216  where the cost of candidate rectangles  62 ,  64  and  66  is determined utilizing equation (1) above. If the cost of candidate rectangles  62 ,  64  and  66  is less than the current cost saved in the Best Solution variable, the position of candidate rectangles  62 ,  64  and  66  relative to input rectangles  12 - 22  and the cost of this solution are saved in the Best Solution variable. 
     Program flow then advances to step  217  which determines if the variable N equals the maximum number of candidate rectangles, i.e., 3. In this iteration of step  217 , the value of the variable N equals the maximum number of candidate rectangles and program flow advances to step  254  where the variable J is incremented by one (1). Program flow then advances to step  256  which determines if the value of variable J equals the number of X coordinates in the sorted list. If not, program flow advances to step  252 . 
     For this iteration of step  252 , the variable I is equal to two (2) and the variable J is equal to four ( 4 ). In step  252 , shared side  68  remains positioned at the I th  position in the sorted list. In this case, since the value of the variable I has not changed since the prior iteration of step  252 , shared side  68  remains at X axis coordinate x 2 . However, in this iteration of step  252 , the position of shared side  70  is set equal to the J th  position in the sorted list. Since the value of the variable J equals four (4), shared side  70  steps from X axis coordinate x 3  in the previous iteration of step  252 , shown in FIG. 6 b , to X axis coordinate x 4  for the current iteration of step  252 , shown in FIG. 6 c.    
     Program flow then advances to step  212  where the top and bottom edges of each candidate rectangle  62 ,  64  and  66  are set equal to the maximum and minimum Y axis values of any input rectangle  12 - 22  having its X axis coordinates overlapping the X axis coordinates of the candidate rectangle. 
     Program flow then advances to step  214  where the position of the top and/or bottom edges of one or more of candidate rectangles  62 ,  64  and  66  are modified as necessary so that each shared side  68  and  70  with respect to one candidate rectangle coincides or completely overlaps the same shared side  68  or  70  with respect to the other candidate rectangle. Candidate rectangles  62 ,  64  and  66  shown in FIG. 6c meet all of the conditions set forth in steps  212  and  214  and therefore require no modification of the positions of their top and/or bottom edges. 
     Program flow then advances to step  216  where the cost of candidate rectangles  62 ,  64  and  66  is determined utilizing equation (1) above. If the thus determined cost is less than the current cost saved in the Best Solution variable, the position of candidate rectangles  62 ,  64  and  66  relative to input rectangles  12 - 22  and the cost of this solution are saved in the Best Solution variable. Program flow then advances to step  217  where a determination is made whether the value of the variable N equals the maximum number of candidate rectangles. In this iteration of step  217 , the variable N equals the maximum number of candidate rectangles. Thus, program flow advances to step  254  where the value of the variable J is incremented by one (1). In step  256  a determination is made if the value of the variable J equals the number of X coordinates in the sorted list. In this iteration, the variable J equals five, which is less than the number of X coordinates in the sorted list, e.g., 12. Since the variable J does not equal the number of X coordinates in the sorted list, program flow advances to step  252  where the position of shared side  70  is stepped to the next X axis coordinate, i.e., X axis coordinate x 5 , in the sorted list. 
     Sequentially repeating steps  252 ,  212 ,  214 ,  216 ,  217 ,  254  and  256  for each value of the variable J has the effect of sequentially stepping shared side  70  from X axis coordinate x 3  to X axis coordinate x 11 , while, at the same time, shared side  68  remains positioned at X axis coordinate x 2 . For each position of shared side  70  at an X axis coordinate in the sorted list, the process of adjusting the top and bottom edges of candidate rectangles  62 ,  64  or  66  in accordance with the requirements of steps  212  and  214  is repeated. In addition, the cost of candidate rectangles  62 ,  64  and  66  for each position of shared side  70  at an X axis coordinate in the sorted list is determined and stored, along with the position of candidate rectangles  62 ,  64  and  66  relative to input rectangles  12 - 22 , in the Best Solution variable if the thus determined cost is less than the current cost stored in the Best Solution variable. 
     With reference to FIG. 6 d  and with continuing reference to FIGS. 3 a - 3   c , when step  256  determines that the variable J equals the number of X coordinates in the sorted list, e.g., 12, program flow advances to step  258  where the variable I is incremented by one (1). During the first iteration of step  258 , the variable I is set equal to 3. Program flow then advances to step  260  where a determination is made whether the value of the variable I equals the number of X coordinates in the sorted list minus one, i.e., 11. Stated differently, in step  260 , a determination is made if the value of the variable I is equal to one less than the number of X coordinates in the sorted list. If not, program flow advances to step  250  where the variable J is set equal to the current value of the variable I plus one, i.e., J=I+1. In this iteration, since the variable I equals three (3), the variable J equals four (4). Program flow then 
     step  251 , shared side  68  is positioned at the X axis coordinate stored in the I th  position in the sorted list. In this iteration of step  251 , since the value of I is equal to three (3), shared side  68  is stepped from X axis coordinate x 2  to X axis coordinate x 3 . 
     Program flow then advances to step  252  where shared side  70  is positioned at the X axis coordinate stored in the J th  position in the sorted list. In this iteration of step  252 , since J equals four (4), shared side  70  is stepped to X axis coordinate x 4 . Steps  252 ,  212 ,  214 ,  216 ,  217 ,  254  and  256  are then sequentially repeated until the variable J equals the number of coordinates in the sorted list. Sequentially repeating steps  252 ,  212 ,  214 ,  216 ,  217 ,  254  and  256  for each value of the variable J has the effect of stepping shared side  70  from X axis coordinate x 4  to X axis coordinate x 11  while shared side  68  remains positioned at X axis coordinate x 3 . 
     With reference to FIG. 6 e  and with continuing reference to FIG. 6 d , each time step  256  determines that the variable J equals the number of X coordinates in the sorted list, steps  258 ,  260 ,  250  and  251  are executed. Each repetition of steps  258 ,  260 ,  250  and  251  has the effect of stepping shared side  68  from its current X axis coordinate, e.g., x 3 , to each X axis coordinate in the sorted list of X axis coordinates, up to and including X axis coordinate x 10 . 
     For each position of shared side  68  at an X axis coordinate from its current X axis coordinate to X axis coordinate x 10 , steps  252 ,  212 ,  214 ,  216 ,  217 ,  254  and  256  are sequentially repeated until the variable of J equals the number of coordinates in the sorted list. Since the value of the variable I will always be less than the value of the variable J, each repetition of steps  252 ,  212 ,  214 ,  216 ,  217 ,  254  and  256  has the effect of stepping shared side  70  to each X axis position in the sorted list between the current X axis position of shared side  68  and X axis coordinate x 12 . Thus, when shared side  68  is positioned at X axis coordinate x 4 , shared side  70  is stepped from X axis coordinate x 5  to X axis coordinate x 11 . For each position of shared side  70  at an X axis coordinate in the sorted list between its current X axis coordinate and X axis coordinate x 11 , the process of adjusting the top and/or bottom edges of candidate rectangle  62 ,  64  and  66  in accordance with steps  212  and  214  and the determination of the cost thereof in accordance with step  216  is made. If, for any position of shared sides  68  and  70  at X axis coordinates in the sorted list, step  216  determines that the cost thereof is more advantageous than the current cost stored in the Best Solution variable, the position of candidate rectangles  62 ,  64  and  66  relative to input rectangles  12 - 22  and the cost of this solution are saved in the Best Solution variable. 
     The process of sequentially stepping shared side  68  from X axis coordinate x 2  to X axis coordinate x 10 , wherein for each position of shared side  68  the position of shared side  70  is stepped through each X axis coordinate between the current X axis coordinate position of shared side  68  and X axis coordinate x 12 , continues until shared side  68  is positioned at X axis coordinate x 10  and shared side  70  is positioned at X axis coordinate x 11 . As shown in FIG. 6 f , when shared sides  68  and  70  are positioned at X axis coordinates x 10  and x 11 , respectively, no further stepping of shared sides  68  or  70  to another next X axis coordinate in the sorted list is possible while at the same time maintaining the integrity of candidate rectangles  62 ,  64  and  66 . 
     As discussed above, for each position of shared side  68  and  70 , the position of the top and/or bottom edges of one or more candidate rectangles  62 ,  64  and  66  are modified as necessary in accordance with steps  212  and  214 . Thereafter, the cost of candidate rectangles  62 ,  64  and  66  for each position of shared sides  68  and  70  are determined and stored, along with the position of candidate rectangles  62 ,  64  and  66  relative to input rectangles  12 - 22 , in the Best Solution variable if the thus determined cost is more advantageous, e.g., less than, the current cost stored in the Best Solution variable. 
     When shared sides  68  and  70  have been stepped to the positions shown in FIG. 6 f,  the same process is repeated for two shared edges stepped through the Y axis coordinates between the first and last Y axis coordinates. More specifically, with reference to FIGS. 3 d - 3   f  and with continuing reference to FIGS. 3 a - 3   c,  when step  260  determines that the value of the variable I equals one less than the number of X coordinates in the sorted list, program flow advances to step  221 . As discussed above, steps  221 - 240  are essentially the same as steps  201 - 220 , respectively. In addition, steps  270 - 280  are essentially the same as steps  250 - 260 , respectively, except that three candidate rectangles  72 ,  74  and  76  have shared or adjoining edges  78  and  80  that are stepped to Y axis coordinates of input rectangles  12 - 22  along the Y axis. More specifically, candidate rectangles  72 ,  74  and  76  have shared edges  78  and  80  initially positioned at Y axis coordinates y 2  and y 3 , respectively, as shown in FIG. 6 g.  Executing steps  222 - 237  and  270 - 280  in the order shown FIGS. 3 d - 3   f  has the effect of stepping shared side  78  from Y axis coordinate y 2  to Y axis coordinate y 8 , where for each position of shared side  78  at one of these Y axis coordinates, shared side  80  is stepped through each Y axis coordinate between the current position of shared side  78  and Y axis coordinate y 9 . 
     For each position of shared sides  78  and  80  at Y axis coordinates in the sorted list, the left and right sides of candidate rectangles  72 ,  74  and  76  are adjusted in accordance with steps  232  and  234 . Step  232  is similar to step  212  except that the position of the left and right sides of each candidate rectangle  72 ,  74  and  76  are set equal to the minimum and maximum X axis values, i.e., the X axis extremes, of any input rectangle  12 - 22  having its Y axis values overlapping the Y axis values of the candidate rectangle. Likewise, step  234  is similar to step  214  except that the position of the left and right sides of each candidate rectangle  72 ,  74  and  76  are modified as necessary so that shared side  78  or  80  with respect to one candidate rectangle coincides or completely overlaps the same shared side  78  or  80  with respect to the other candidate rectangle. Step  236  is similar to step  216  which stores the position of candidate rectangles  72 ,  74  and  76  relative to input rectangles  12 - 22  and the cost of candidate rectangles  72 ,  74  and  76  in the Best Solution variable if the thus determined cost is more advantageous, e.g., less than, the current cost stored in the Best Solution variable. 
     With reference to FIG. 6 h,  when the value of the variable N equals the maximum number of solution rectangles, e.g., 3, steps  221 - 237  and  270 - 280  are executed in the order shown in FIGS. 3 d - 3   f  until shared edges  78  and  80  are positioned at Y axis coordinates y 8  and y 9 , respectively. When step  280  determines that the value of I equals one less than the number of Y coordinates in the sorted list, program flow exits routine  120  and advances to routine  114  in FIG.  1 . When routine  114  determines that the variable N equals the maxumum number of solution rectangles, in this case 3, program flow advances to routine  116  which returns the positions(s) of the candidate rectangle(s) relative to input rectangles  12 - 22  having the most advantageous, e.g., least, cost to main program/routine  102  for processing. The candidate rectangle(s) returned to main program/routine  102  are also referred to as the solution rectangle(s). In the foregoing example, the most advantageous cost is the least cost. However, this is not to be construed as limiting the invention. 
     Main program/routine  102  utilizes the outline of the solution rectangle(s) as the conformal outline for the layout of one or more analog or digital devices, such as transistors, on an integrated circuit in a manner known in the art. 
     The invention has been described with reference to the preferred embodiment. Obvious modifications and alterations will occur to others upon reading and understanding the preceding detailed description. For example, the maximum number of candidate rectangle solutions can be increased to four or more provided that the above described method is adapted to account for this increase in the maximum number of candidate rectangles to be processed. To this end, the number of shared sides for four or more candidate rectangles solutions and the movement of these shared sides needs to be coordinated in the manner described above for the shared sides of the three (3) candidate rectangle solution. It is intended that the invention be construed as including all such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.