PATENT ABSTRACT
Disclosed herein is a method and associated apparatus for the design and manufacture of VLSI circuit which incorporates therein a method for routing connections between component tiles of the VLSI circuit being designed.

PATENT DESCRIPTION
CROSS REFERENCE TO RELATED APPLICATIONS  
       [0001]    This application claims priority based upon U.S. Provisional Patent Application Ser. No. 60/281,926 filed Apr. 6, 2001.  
         [0002]    This application is also related to U.S. patent application Ser. Nos. 10/ ______ (Atty. Docket No. 5181-89400) entitled “A Maximal Tile Generation Technique and Associated Methods of Designing and Manufacturing VLSI Circuits” and Ser. No. 10/ ______ (Atty. Docket No. 5181-89300) entitled “Active Region Management Techniques and Associated Methods of Designing and Manufacturing VLSI Circuits”, both of which were filed on______ , 2002, assigned to the Assignee of the present application and hereby incorporated by reference as if reproduced in their entirety. 
     
    
     
       BACKGROUND  
         [0003]    1. Technical Field  
           [0004]    This relates to the design and manufacture of very large scale integrated (“VLSI”) circuits and, more particularly, to a method for routing connections between component tiles of a VLSI circuit suitable for use in conjunction with the design and manufacture of VLSI circuits.  
           [0005]    2. Description of the Relevant Art  
           [0006]    A VLSI circuit is typically composed of a plurality of generally horizontal layers, each layer having a plurality of generally rectangular shaped components positioned thereon. VLSI circuit designers commonly refer to these generally rectangular shaped components as “component tiles” and to the rectangular shaped open spaces that surround the component tiles as “space tiles.” Component tiles that are to be connected on a VLSI circuit are said to form a “net”, while any component tile not connected to a particular net is considered to be an obstruction to that net. Two tiles are said to be “adjacent” if they touch along their edges and “overlapping” if there is even a single point located within the interior of both tiles. A set of tiles positioned within a routing area is said to be “maximal” if no two tiles are either overlapping or adjacent on their left or right edges.  
           [0007]    One step in the design of a VLSI circuit is to select the wire paths that extend through the space tiles to connect the electrically equivalent component tiles that form nets. A current technique used to determine these paths utilizes a tile expansion algorithm. More specifically, clear space around the component tiles forming a net is fractured into maximal space tiles. Adjoining ones of these maximal space tiles are used to define the most efficient tile path between two components. The path of the actual connection between the components, known as the wire path, is then defined as the route through the space tile path from the center of the component source tile to the center of the component destination tile.  
           [0008]    The aforementioned technique for selecting the wire paths for a VLSI circuit design suffers from two drawbacks, both of which may add to the cost of VLSI circuits manufactured in accordance with the design. First, if defined in accordance with the above-described manner, a tile path is not necessarily the optimal tile path through the clear space. Second, since the width of a tile path is typically much larger than the width of a wire path, multiple wire paths may exist through a given tile path. If the wire path located within the tile path is arbitrarily selected, the selected wire path is not necessarily the most efficient wire path potentially located within the tile path.  
         SUMMARY  
         [0009]    Disclosed herein is a method and associated apparatus for the design and manufacture of VLSI circuit which incorporates therein a method for routing connections between component tiles of the VLSI circuit being designed. In accordance with the disclosed method, maximal component tile and maximal space tile lists are constructed and, from the constructed lists, the maximal component and maximal space tiles are positioned on the routing area(s). Optimal tile path and minimum cost wire path between pins that are CTs to be connected are determined and, utilizing the determined wire path, a VLSI circuit design is generated. The minimum cost path from a starting tile S to a destination tile T is determined by creating a priority queue and a search tree ST with the starting tile S as its root. While the priority queue is not empty, a low cost tile E neighboring the starting tile S is popped and, if a tile path to destination tile T is found, the cost of the tile path is evaluated and saved as the minimum cost point path. Otherwise, for each tile F neighboring the tile E, the search tree is expanded and, for a tile path having an estimated cost greater than the current cost, the search path is pruned. Otherwise, the search is expanded by adding the tile F (with corresponding minimum cost CE)) to the priority queue Q and inserting the tile F into the search tree as a child node of the tile E. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0010]    [0010]FIG. 1 is a block diagram of a computer system in which VLSI circuit design software resides.  
         [0011]    [0011]FIG. 2 is a flow chart of a method for designing and manufacturing VLSI circuits.  
         [0012]    [0012]FIG. 3 a  is a flow chart of a method, suitable for use with the method of designing and manufacturing VLSI circuits of FIG. 2, for reconfiguring a routing area into an arrangement of maximal component tiles and maximal space tiles.  
         [0013]    [0013]FIGS. 3 b - 1  through  3   b - 2  is a flow chart of a method, suitable for use with the method of designing and manufacturing VLSI circuits of FIG. 2, of determining an optimal tile path and minimum cost wire path between pins that are CTs to be connected.  
         [0014]    [0014]FIG. 4 is a top view of first and second routing areas, each forming a respective layer of a VLSI circuit design and having at least one component tile to be connected into a net.  
         [0015]    [0015]FIG. 5 is a top view of the first routing area of FIG. 4 after reconfiguration into an arrangement of maximal component tiles and maximal space tiles.  
         [0016]    [0016]FIG. 6 is a top view of the second routing area of FIG. 4 after reconfiguration into an arrangement of maximal component tiles and maximal space tiles. 
     
    
     DETAILED DESCRIPTION  
       [0017]    Referring first to FIG. 1, a computer system  1 , for example, a personal computer (“PC”), file server or other type of computer, in which VLSI circuit design software resides will now be described in greater detail. The computer system  1  is comprised of a processor subsystem  2 , a memory subsystem  3  and an input/output (“I/O”) subsystem  4  coupled together by a bus subsystem  5 . The bus subsystem  5  encompasses the main system bus and any local or other types of busses that collectively couple the processor subsystem  2 , the memory subsystem  3  and the I/O subsystem  4  to one another. As used herein, the terms “couple” or “coupled” refer broadly to either a direct or an indirect connection between the referenced elements.  
         [0018]    The processor subsystem  2  encompasses the collective processing capability of the computer system  1 , including the central processing unit (“CPU”) as well as any secondary processing devices, for example, an arithmetic processing unit, coupled to the CPU by the bus subsystem  5 . Similarly, the memory subsystem  3  encompasses the collective storage capability of the computer system  1 , including main, auxiliary, cache and any other memory accessible by the processor subsystem  2  via the bus subsystem  5 . Finally, the I/O subsystem  4  encompasses any and all I/O devices, for example, floppy, CD-ROM or DVD drives, coupled to the bus subsystem  5 , for writing data to or reading data from the processor subsystem  2  or the memory subsystem  3 . The I/O subsystem  4  also encompasses any data communications equipment (“DCE”), for example, network interface cards or modems, which couple the computer system  1  to data terminal equipment (“DTE”), for example, a second PC, file server or web server, via a local area network (“LAN”), wide area network (“WAN”), intranet, internet or other type of network.  
         [0019]    [0019]FIG. 1 further shows plural software modules, specifically, a first software module  6  for providing the computer system  1  with VLSI circuit design functionality, a second software module  7  for providing the computer system  1  with maximal tile generation functionality, a third software module  8  for providing the computer system  1  with active region management functionality and a fourth software module  9  for providing the computer system  1  with connection routing functionality. Each of the software modules  6  through  9  is comprised of a series of instructions which are encoded in the memory subsystem  3  as computer readable program code and executable by the processor subsystem  2 . Typically, the VLSI circuit design module  6 , the maximal tile generation module  7 , the active region management module  8  and the connection routing module  9  will be stored in the auxiliary memory of the memory subsystem  3  prior to the execution thereof. A transportable computer usable medium  10 , for example, a floppy disk, CD-ROM or file transfer software, is used to copy the VLSI circuit design module  6 , the maximal tile generation module  7 , the active region management module  8  and the connection routing module  9  into the auxiliary memory of the memory subsystem  3 .  
         [0020]    As illustrated in FIG. 1, the functionality provided by the software modules  6  through  9  may be encoded in the memory subsystem  3  and/or the computer usable medium  10  as discrete computer programs, each containing computer readable program code. Alternately, the functionality provided by the software modules  6  through  9  may be encoded in the memory subsystem  3  and/or the computer usable medium  10  as separate subroutines of a single computer program containing plural computer readable program subcodes. Furthermore, while any of the software modules  6  through  9  may be executed separately, typically, the VLSI circuit design module  6  will be initially executed by the processor subsystem  3 . The VLSI circuit design module  6  will then periodically call selected ones of the maximal tile generation module  7 , the active region management module  8  and the connection routing module  9  to perform certain functions during the design of a VLSI circuit. Of course, any one of the software modules  6  through  9  may call any other one of the software modules  6  through  9  to perform certain functions on its behalf.  
         [0021]    Referring next to FIG. 2, a method for designing and manufacturing VLSI circuits will now be described in greater detail. The method commences at step  11  with the execution of the VLSI circuit design module  6  by the processor subsystem  2 . The method proceeds to step  12  where one or more routing areas are defined and a collection of component tiles are positioned on each routing area using the VLSI circuit design module  6 . Generally, a routing area comprises the surface area of a layer of an integrated circuit on which component tiles are positioned. When plural routing areas are defined by the VLSI circuit design module  6 , the integrated circuit being designed is a multi-layer integrated circuit and each routing area comprises the surface of a respective layer of the multi-layer integrated circuit.  
         [0022]    For example, FIG. 4 illustrates the components tiles (“CT”s) of a multi-layer integrated circuit design having two layers. The CTs  92 ,  94  and  96  are positioned within a first routing area  90  of a first, or upper, layer. Also shown in FIG. 4 is CT  98 , which is illustrated in phantom because it resides on a second routing area  90 ′ of a second, or lower, layer located beneath the routing area  90 . In the VLSI circuit design illustrated in FIG. 4, the CTs  92 ,  96  and  98 , also known as pins, are on the same net while the CT  94  is an obstruction. Thus, the CTs  92 ,  96  and  98  must now be connected while avoiding the CT  94 .  
         [0023]    While, in the description to follow, the routing areas  90  and  90 ′ are oftentimes described with respect to a vertical axis, hereafter termed the “S” axis, which extends from S 0  to S 8 , and a horizontal axis, hereafter termed the “D” axis, which extends from D 0  to D 9 , it should be clearly understood that the use of these terms is not intended to imply or suggest that the routing areas  90  and  90 ′ have a particular orientation, either horizontal or vertical, relative to the VLSI circuit being designed. Likewise, the description of a first span (or segment thereof) of the routing areas  90  or  90 ′ as being located above or below a second span (or segment thereof) is not intended to imply or suggest that the first and second spans are oriented in either the horizontal or vertical plane. Rather, the use of these terms is merely intended to describe their relative location within a common plane without regard to the specific orientation of that plane.  
         [0024]    Accordingly, the method proceeds to step  14  where the first routing area  90  is selected for reconfiguration into maximal component and space tiles. The method then continues on to step  16  for construction of maximal component tile and maximal space tile lists for the routing area  90 . The CTs  92 ,  94  and  96  of the routing area  90  are all maximal component tiles because there are no overlapping tiles or adjacent tiles on either the left or right sides thereof. However, the open space of the routing area  90  needs to be divided into one or more maximal space tiles. A suitable method to perform this operation is disclosed in co-pending U.S. patent application Ser. No. 10/ ______ (Atty. Docket No. 5181-89400) entitled “A Maximal Tile Generation Technique and Associated Methods of Designing and Manufacturing VLSI Circuits” and previously incorporated by reference. However, as the current example differs from that described in the above-referenced patent application in that all of the component tiles positioned on the routing area  90  are already maximal component tiles, the method set forth in the above-referenced application shall later be applied to the routing area  90  illustrated in FIG. 4.  
         [0025]    After constructing maximal component tile and maximal space tile lists for the routing area  90  in a manner to be more fully described below, the method proceeds to step  17  where it is determined if there are additional routing area for which maximal component tile and maximal space tile lists are to be constructed. As maximal component tile and maximal space tile lists must still be constructed for the routing area  90 ′, the method proceeds to step  18  for selection of the routing area  90 ′ and then returns to step  16  for construction of maximal component tile and maximal space tile lists for the routing area  90 ′. After constructing maximal component tile and maximal space tile lists for the routing area  90 ′ in a manner to be more fully described below, the method proceeds to step  17  where it is determined that maximal component tile and maximal space tile lists have been constructed for all routing areas. Accordingly, the method then proceeds to step  19  where the maximal component tile and maximal space tile lists constructed at step  16  are used to position the maximal component tiles and the maximal space tiles on the routing areas  90 ,  90 ′.  
         [0026]    After positioning the maximal component and maximal space tiles on the routing areas  90 ,  90 ′, the method proceeds to step  20  where the connection routing module  9  determines the optimal tile path and minimum cost wire path between pins that are CTs to be connected. Using the minimum cost wire path between pins that are CTs to be connected, the method continues on to step  24  where the positions of the CTs and the wire path interconnecting the connected CTs are used by the VLSI circuit design module  6  to produce a VLSI circuit design. It should be noted, however, that the foregoing description of a method of designing a VLSI circuit is highly simplified and that numerous steps in the process which are deemed as not being needed for an understanding of the disclosed techniques have been omitted for ease of description. Having completed the design of the VLSI circuit, the method then continues on to step  26  where plural VLSI circuits which conform to the design are manufactured at a facility using conventional manufacturing processes. The method then ends at step  28 .  
         [0027]    Turning now to FIG. 3 a , the method by which maximal component tile and maximal space tile lists are determined for the routing area  90  at step  16  will now be described in greater detail. The method commences at step  30  and, at step  32 , a list (AL) of active segments and a list (IL) of inactive segments of a span which extends along a bottom edge  90   a  of the routing area  90  are identified. Generally, an “active” segment of a span is a segment which passes through the interior of a component tile or along a lower edge thereof. Conversely, an “inactive” segment of a span is a segment that passes through an unoccupied portion of the routing area or along an upper edge of a component tile. Once active and inactive segments of the span have been identified, a level is then assigned to each segment thereof. The level of a segment is the point, along the S axis, generally aligned with the span of which the active or inactive segment forms a portion thereof. For example, a segment forming part of a span may be described as follows: [DX, DY, SZ] where DX is the point, along the D axis, where the active or inactive segment starts, DY is the point, along the D axis, where the active or inactive segment ends and SZ is the level, along the S axis, of the active or inactive span of which the active or inactive segment forms a portion thereof.  
         [0028]    An active segment of a first span may be said to “match” an active segment of a second span if the segments have the same start point DX and the same stop point DY along the D axis but different levels SZ along the S axis. For example, the active segments [D 1 , D 3 , S 1 ] and [D 1 , D 3 , S 2 ] are considered to be matching active segments. Conversely, the term “unmatched” active segments refer to spans which, in addition to having different levels SZ, also have different start points DX, different stop points DY or both. Likewise, an inactive segment of a first span matches an inactive segment of a second span if the segments have the same start and stop points DX and DY but different levels SZ while unmatched inactive segments also have either a different start point DX, a different stop point DY or both.  
         [0029]    A “characteristic” of a span is defined by the set of segments which comprises the span. A pair of spans may be deemed as having the same characteristic if every active and inactive segment of a first span of the span pair has a matching active or inactive segment, respectively, in a second one of the span pair. Conversely, a pair of spans may be deemed as having different characteristics if every active and inactive segment of the first span fails to have a matching active or inactive segment, respectively, in the second span.  
         [0030]    As may be clearly seen in FIG. 4, neither a lower edge nor an interior of a component tile extends along the bottom edge  90   a  of the routing area  90 . Accordingly, the bottom edge  90   a  of the routing area  90  has a span comprised of a single inactive segment which extends from DO to D 9 . As may be further seen in FIG. 4, the level of the span is SO. The level of the span is hereby designated as a first stop point SO for a set of stop points for the routing area  90  and the list of active and inactive segments of the span which extends along the bottom edge  90   a  of the routing area  90  and is generally aligned with the first stop point SO may be described as follows: 
         AL=Φ; and  
         IL={[D 0 , D 9 , SO]}.  
         [0031]    The list of ALs and the list of ILs for the span generally aligned with the first stop point S 0  are hereby designated as a current list of ALs and a current list of ILs, respectively, for the routing area  90 .  
         [0032]    Continuing on to step  34 , additional members of the set of stop points for the routing area  90  are identified. These additional stop points for the routing area  90  are those points along the S axis which are generally aligned with either a lower edge of one or more of the CTs  92 ,  94  and  96  and/or an upper edge of one or more of the CTs  92 ,  94  and  96 . Thus, from the known arrangement of the CTs  92 ,  94  and  96  illustrated in FIG. 4, the maximal tile generation module  7  identifies, in an ascending order relative to the S axis, S 1 , S 2 , S 3 , S 4 , S 5  and S 6  as additional members of the set of stop points for the routing area  90 . Finally, an upper edge  90   b  of the routing area  90  is designated as a last stop point S 8  of the set of stop points for the routing area  90 . The method then proceeds to step  36  where an empty list of maximal component tiles (CTL) and an empty list of maximal space tiles (STL) are generated.  
         [0033]    S 1 , S 2 , S 3 , S 4 , S 5  and S 6  were identified as additional stop points because they are all aligned with one or more of the lower edges of the CTs  92 ,  94  and  96  and/or the upper edges of the CTs  92 ,  94  and  96 . The edges of the CTs  92 ,  94  and  96  are used to identify additional stop points since the edges of a CT indicate transition between active and inactive regions. More specifically, a lower edge of a CT indicates the location of an inactive-to-active transition while an upper edge of a CT indicates the location of an active-to-inactive transition. The stop points are selected to coincide with either inactive-to-active or active-to-inactive transitions because, in accordance with the techniques disclosed herein and to be more fully described below, maximal component and/or space tiles are generated whenever active and/or inactive segments of a first span fails to have a matching active and/or inactive segments along a second span.  
         [0034]    At step  38 , it is determined whether there are additional stop points which require examination. If so, the method proceeds to step  40  for selection of a next stop point for examination. At step  42 , a next list of active segments and a next list of inactive segments are identified for a next span generally aligned with a next stop point of the set of stop points. The lists of active and inactive segments of the span generally aligned with the next stop point are generated using the active region management techniques disclosed in co-pending U.S. patent application Ser. No. 10/ ______ (Atty. Docket No. 5181-89300) entitled “Active Region Management Techniques and Associated Methods of Designing and Manufacturing VLSI Circuits” and previously incorporated by reference. Once the next list of active segments and the next list of inactive segments of the span generally aligned with the next stop point have been generated at step  42 , the method proceeds to step  44  where the next list of active segments is compared to the current list of active segments and, based upon that comparison, one or more maximal component tiles may be identified for inclusion in the list of maximal component tiles.  
         [0035]    More specifically, for each active segment S in the current list of active segments, the next list of active segments is examined for a matching active segment S&#39;. If there is no matching active segment S&#39; in the next list of active segments, a maximal component tile having a width generally equal to the width [DX, DY] of the active segment S and a height generally equal to the difference between the level of the active segment S and the level of the active segment S&#39; is generated. The generated maximal component tile is then inserted into the maximal component tile list CTL and the active segment S removed from the current list of active segments. After searching for a matching active segment S&#39; in the next active segment list for each active segment S in the current active segment list, any unmatched active segment S&#39; in the next active segment list is added to the current active segment list. Initially, the current active segment list will contain active segments at only one level. It should be noted, however, as the next active segment lists for various levels are examined, the current active segment list will likely contain active segments at plural levels.  
         [0036]    The method then proceeds to step  46  where the process of step  45  is repeated using the current and next list of inactive segments. By doing so, one or more maximal space tiles may be generated at step  44  and added to the maximal space tile list STL. Additionally, the current list of inactive segments will be modified by deleting the inactive segments, from the current list of inactive segments, the inactive segments having a matching inactive segment in the next list of inactive segments and by adding, to the current list of inactive segments, unmatched inactive segments from the next list of inactive segments. The method then returns to step  38  where the process described in steps  40 ,  42 ,  44  and  46  is repeated for each stop point in the set of stop points. After the last stop point in the set of stop points has been processed, the method will proceed from step  38  to step  48  where the maximal component tile list CTL and the maximal space tile list STL generated by the described method are output, typically, to the VLSI circuit design module  8  for use in connection with the design and manufacture of a VLSI circuit in accordance with the method of FIG. 2.  
         [0037]    The above-described method of generating a list of maximal component tiles CTL and a list of maximal space tiles STL for the routing area  90  shall again be described, now with respect to the example illustrated in FIGS. 4 and 5. As previously noted, FIG. 4 shows the CTs  92 ,  94  and  96  positioned in the routing area  90 . As previously set forth, the process starts at step  32  by identifying the active and inactive segments for a span extending along the bottom edge  90   a  of the routing area  90 . As there are no active segments along the bottom edge  90 a, the active and inactive segment lists for this span, which is generally aligned with the stop point SO, are initially set as follows: 
         AL=Φ; and  
         IL={[D 0 , D 9 , SO]}.  
         [0038]    At step  34 , the remaining members of the set of stop points are identified (S 1 , S 2 , S 3 , S 4 , S 5 , S 6  and S 8 ) and, at step  36 , an empty maximal component tile list CTL and an empty maximal space tile list STL are generated. As the stop points S 1 , S 2 , S 3 , S 4 , S 5 , S 6  and S 8  need to be examined, the method passes through  38  and on to step  40  where the stop point SI, the next stop point after the stop point S 0 , is selected for examination. As may be seen in FIG. 4, bottom edge  96   a  of the CT  96  is generally aligned with the stop point S 1 . The corresponding segment is, therefore, considered to be active while the remaining segments generally aligned with the stop point S 1  are considered to be inactive. Accordingly, at step  42 , the next list of active segments and the next list of inactive segments are determined to be: 
         AL=Φ; and  
         IL={[D 0 , D 9 , SO]}.  
         [0039]    Proceeding to step  44 , the current active segment list is empty. As a result, there are no matches between the current list of active segments and the next list of active segments. As a result, no maximal component tiles are generated at step  44 . Furthermore, as the entry in the next active segment list is unmatched, it is added to the current active segment list, which now becomes: 
         AL={[D 6 , D 8 , S 1 ]}. 
         [0040]    The method then proceeds to step  46  for processing of the current inactive segment list. The current inactive segment list contains a single entry [D 0 , D 9 , S 0 ]. As there is no matching span in the next inactive segment list, a space tile, hereafter referred to as ST  100  and illustrated in FIG. 5, which extends from D 0  to D 9  in the D axis along a line generally aligned with S 0  and which extends from S 0  to S 1  in the S axis is generated and added to the maximal space tile list STL. The matched entry [D 0 , D 9 , S 0 ] is deleted from the current inactive segment list and the unmatched entries [D 0 , D 6 , S 1 ] and [D 8 , D 9 , S 1 ] of the next inactive segment list are added to the inactive segment list, thereby producing the following current inactive segment list: 
         IL={[D 0 , D 6 , S 1 ], [D 8 , D 9 , S 1 ]}. 
         [0041]    The method then returns to step.  38  and, as there are additional stop points to be examined, on to step  40  where stop point S 2  is selected for examination.  
         [0042]    A span extending across the routing area  90  along a line generally aligned with the stop point S 2  passes along a lower edge  94   a  of the CT  94  and through the interior of the CT  96 . Accordingly, the next list of active and inactive segments would be as follows 
         AL={[D 3 , D 4 , S 2 ], [D 6 , D 8 , S 2 ]}; and 
         IL={[D 0 , D 3 , S 2 ], [D 4 , D 6 , S 2 ], [D 8 , D 9 , S 2 ]}. 
         [0043]    For each active segment in the current active segment list, the next active segment list is searched for matches. Here, [D 6 , D 8 , S 1 ], currently the only entry in the current active segment, matches the [D 6 , D 8 , S 2 ] entry from the next active segment list. Accordingly, no maximal component tiles are generated during step  44 . The matched entry [D 6 , D 8 , S 2 ] is deleted from the next active segment list while the unmatched entry [D 3 , D 4 , S 2 ] from the next active segment list is added to the current active segment list, thereby producing the following current active segment list: 
         AL={[D 3 , D 4 , S 2 ], [D 6 , D 8 , S 1 ]}. 
         [0044]    The method then proceeds to step  46  for processing of the current inactive segment list. The current inactive segment list contains entries [D 0 , D 6 , S 1 ] and [D 8 , D 9 , S 1 ] while the next inactive segment list contains entries [D 0 , D 3 , S 2 ], [D 4 , D 6 , S 2 ] and [D 8 , D 9 , S 2 ]. Thus, the [D 8 , D 9 , S 1 ] entry of the current inactive segment list is matched while the [D 0 , D 6 , S 1 ] entry is unmatched. Accordingly, maximal space tile ST  102 , which extends from D 0  to D 6  in the D axis along a line generally aligned with SI, extends from S 1  to S 2  in the S axis and is illustrated in FIG. 5, is generated and added to the maximal space tile list STL. The unmatched entry is then deleted from the current inactive segment list while the unmatched entries [D 0 , D 3 , S 2 ] and [D 4 , D 6 , S 2 ] of the next inactive segment list is added to the current inactive segment list, thereby producing the following current inactive segment list: 
         IL={[D 0 , D 3 , S 2 ], [D 4 , D 6 , S 2 ], [D 8 , D 9 , S 1 ]}. 
         [0045]    A span extending across the routing area  90  along a line generally aligned with the stop point S 3  passes through the interior of the CT  94  and along an upper edge  96 b of the CT  96 . Accordingly, the next list of active and inactive spans would be as follows: 
         AL={[D 3 , D 4 , S 3 ]}; and 
         IL ={[DO, D 3 , S 3 ], [D 4 , D 9 , S 3 ]}. 
         [0046]    For each active span in the current active segment list, the next active segment list is searched for matches. Here, the entry [D 3 , D 4 , S 2 ] matches the entry [D 3 , D 4 , S 3 ] while the entry [D 6 , D 8 , S 1 ] is unmatched. Accordingly, the entry [D 6 , D 8 , S 1 ] is used to generate a maximal component tile, hereafter referred to as CT  104  and illustrated in FIG. 5, which extends from D 6  to D 8  in the D axis along a line generally aligned with Si and from S 1  to S 3  in the S axis. The newly generated maximal component tile is then added to the maximal component tile list CTL. The unmatched entry [D 6 , D 8 , S 1 ] is deleted from the current active segment list and, since there are no unmatched entries from the next active segment list, the following current active segment list is produced: 
         AL={[D 3 , D 4 , S 2 ]}. 
         [0047]    The method then proceeds to step  46  for processing of the current inactive segment list. The current inactive segment list contains entries [D 0 , D 3 , S 2 ], [D 4 , D 6 , S 2 ] and [D 8 , D 9 , S 1 ] while the next inactive segment list contains entries [D 0 , D 3 , S 3 ] and [D 4 , D 9 , S 3 ]. Thus, the [D 4 , D 6 , S 2 ] and [D 8 , D 9 , S/I] entries of the current inactive segment list are unmatched while the [D 0 , D 3 , S 2 ] entry of the current inactive segment list is matched. Accordingly, the unmatched entry [D 4 , D 6 , S 2 ] is used to generate maximal space tile ST  106 , which extends from D 4  to D 6  in the D axis along a line generally aligned with S 2 , extends from S 2  to S 3  in the S axis and is illustrated in FIG. 5. Similarly, the unmatched entry [D 8 , D 9 , SI] is used to generate maximal space tile ST  108 , which extends from D 8  to D 9  in the D axis along a line generally aligned with S 2 , extends from S 1  to S 3  in the S axis and is illustrated in FIG. 5. The newly generated tiles are then added to the maximal space tile list, the unmatched entries [D 4 , D 6 , S 2 ] and [D 8 , D 9 , S 1 ] are deleted from the current inactive segment list and the unmatched entry [D 4 , D 9 , S 3 ] of the next inactive segment list is added to the current inactive segment list, thereby producing the following current inactive segment list: 
         IL={[D 0 , D 3 , S 2 ], [D 4 , D 9 , S 3 ]}. 
         [0048]    The method then returns to step  38  and, as there are additional stop points to be examined, on to step  40  where stop point S 4  is selected for examination.  
         [0049]    A span extending across the routing area  90  along a line generally aligned with the stop point S 4  along a lower edge  92   a  of the CT  92  and through the interior of the CT  94 . Accordingly, the next list of active and inactive spans generated at step  42  would be as follows: 
         AL={[D 1 , D 2 , S 4 ], [D 3 , D 4 , S 4 ]}; and 
         IL {[DO, D 1 , S 4 ], [D 2 , D 3 , S 4 ], [D 4 , D 9 , S 4 ]}. 
         [0050]    The current and next active segment lists are then processed at step  44 . The current active segment list is [D 3 , D 4 , S 2 ] while the next active segment list is [D 1 , D 2 , S 4 ], [D 3 , D 4 , S 4 ]. Thus, [D 3 , D 4 , S 2 ], the only member of the current active segment list, matches [D 3 , D 4 , S 4 ] of the next active segment list while [D 1 , D 2 , S 4 ] of the next active segment lists is unmatched. Accordingly, at step  44 , no maximal component tiles are generated and no entries are deleted while [D 1 , D 2 , S 4 ] is added to the current active segment list, thereby producing the following current active segment list: 
         AL={[D, D 2 , S 4 ], [D 3 , D 4 , S 2 ]}. 
         [0051]    The method then proceeds to step  46  for processing of the current inactive segment list. The current inactive segment list contains entries [D 0 , D 3 , S 2 ] and [D 4 , D 9 , S 3 ] while the next inactive segment list contains entries [D 0 , D 1 , S 4 ], [D 2 , D 3 , S 4 ] and [D 4 , D 9 , S 4 ]. Thus, the [D 4 , D 9 , S 3 ] entry of the current inactive segment list is matched to the entry [D 4 , D 9 , S 4 ] entry of the next inactive segment list while the [D 0 , D 3 , S 2 ] entry of the current inactive segment list and the [D 0 , D 1 , S 4 ] and [D 2 , D 3 , S 4 ] entries of the next inactive segment list are unmatched. Accordingly, the unmatched entry [D 0 , D 3 , S 2 ] of the current active span is used to generate maximal space tile ST  110 , which extends from D 0  to D 3  in the D axis along a line generally aligned with S 2 , extends from S 2  to S 4  in the S axis and is illustrated in FIG. 5. The newly generated space tile is then added to the maximal space tile list STL, the unmatched entry [D 0 , D 3 , S 2 ] is deleted from the current inactive segment list and the unmatched entries [D 0 , D 1 , S 4 ] and [D 2 , D 3 , S 4 ] of the next inactive segment list is added to the current inactive segment list, thereby producing the following current inactive segment list: 
         IL={[D 0 , D 1 , S 4 ], [D 2 , D 3 , S 4 ], [D 4 , D 9 , S 3 ]}. 
         [0052]    The method then returns to step  38  and, as there are additional stop points to be examined, on to step  40  where stop point S 5  is selected for examination.  
         [0053]    A span extending across the routing area  90  along a line generally aligned with the stop point S 5  passes through the interior of the CT  92  and along the upper edge  94   b  of the CT  94 . Accordingly, the next list of active and inactive spans generated at step  42  would be as follows: 
         AL={[D, D 2 , S 5 ]; }and  
         IL={[D 0 , D 1 , S 5 ], [D 2 , D 9 , S 5 ]}. 
         [0054]    The current and next active segment lists are then processed at step  44 . The current active segment list is [D 1 , D 2 , S 4 ] and [D 3 , D 4 , S 2 ] while the next active segment list is [D 1 , D 2 , S 5 ]. Thus, the [D 1 , D 2 , S 4 ] entry of the current active segment list matches the [D 1 , D 2 , S 5 ] entry of the next active segment list while the [D 3 , D 4 , S 2 ] entry of the current active segment list is unmatched. Accordingly, the [D 3 , D 4 , S 2 ] of the current active segment list is used to generate a maximal component tile, hereafter referred to as CT  112 , which extends from D 3  to D 5  in the D axis along a line generally aligned with the stop point S 4  and from the stop point S 2  to the stop point S 5  in the S axis, and is illustrated in FIG. 5. The newly generated maximal component tile is then added to the maximal component tile list CTL. The unmatched entry [D 3 , D 4 , S 2 ] is then deleted from the current active segment list and the matched entry [D 1 , D 2 , S 5 ] is deleted from the next active segment list, thereby resulting in the following current active segment list: 
         AL={[D 1 , D 2 , S 4 ]}. 
         [0055]    The method then proceeds to step  46  for processing of the current inactive segment list. The current inactive segment list contains entries [D 0 , D 1 , S 4 ], [D 2 , D 3 , S 4 ] and [D 4 , D 9 , S 3 ] while the next inactive segment list contains the entries [D 0 , D 1 , S 5 ] and [D 2 , D 9 , S 5 ]. Thus, the [D 2 , D 3 , S 4 ], [D 4 , D 9 , S 3 ] entries of the current inactive segment list and the [D 2 , D 9 , S 5 ] entry of the next inactive segment list are unmatched. Accordingly, the unmatched entry [D 2 , D 3 , S 4 ] of the current inactive span is used to generate maximal space tile ST  114 , which extends from D 2  to D 3  in the D axis along a line generally aligned with S 4  and which extends from S 4  to S 5  in the S axis while the unmatched entry [D 4 , D 9 , S 3 ] of the current inactive span is used to generate maximal space tile ST  116  which extends from D 4  to D 9  in the D axis along a line generally aligned with S 3  and which extends from S 3  to S 5  in the S axis. The newly generated space tiles ST  114  and ST  116 , both of which are illustrated in FIG. 5, are then added to the maximal space tile list STL, the unmatched entries [D 2 , D 3 , S 4 ] and [D 4 , D 9 , S 3 ] are deleted from the current inactive segment list and the unmatched entry [D 2 , D 9 , S 5 ] of the next inactive segment list is added to the current inactive segment list, thereby producing the following current inactive segment list: 
         IL={[D 0 , D 1 , S 4 ], [D 2 , D 9 , S 5 ]}. 
         [0056]    The method then returns to step  38  and, as there are additional stop points to be examined, on to step  40  where the stop point S 6  is selected for examination.  
         [0057]    A span extending across the routing area  90  along a line generally aligned with the stop point S 6  passes along the upper edge  92   b  of the CT  92 . Accordingly, the next list of active and inactive spans generated at step  42  would be as follows: 
         ALΦ; and 
         IL={[D 0 , D 9 , S 6 ]}. 
         [0058]    The current and next active segment lists are then processed at step  44 . The current active segment list is [D 1 , D 2 , S 4 ] while the next active segment list is empty. Thus, the [D 1 , D 2 , S 4 ] entry of the current active segment list is unmatched. Accordingly, the [D 1 , D 2 , S 4 ] of the current active segment list is used to generate a maximal component tile, hereafter referred to as CT  118 , which extends from D 1  to D 2  in the D axis along a line generally aligned with the stop point S 4  and from the stop point S 4  to the stop point S 6  in the S axis, and is illustrated in FIG. 5. The newly generated maximal component tile is then added to the maximal component tile list CTL. The unmatched entry [D 1 , D 2 , S 4 ] is then deleted from the current active segment list, thereby emptying the current active segment list and, since the next active segment list is empty as well, the current active segment list becomes the following: 
         AL=Φ. 
         [0059]    The method then proceeds to step  46  for processing of the current inactive segment list. The current inactive segment list contains entries [D 0 , D 1 , S 4 ] and [D 2 , D 9 , S 5 ] while the next inactive segment list contains the entry [D 0 , D 9 , S 6 ]. Thus, the [D 0 , D 1 , S 4 ] and [D 2 , D 9 , S 5 ] entries of the current inactive segment list and the [D 0 , D 9 , S 6 ] entry of the next inactive segment list are unmatched. Accordingly, the unmatched entry [D 0 , D 1 , S 4 ] of the current inactive span is used to generate maximal space tile ST  120 , which extends from D 0  to D 1  in the D axis along a line generally aligned with S 4  and which extends from S 4  to S 6  in the S axis while the unmatched entry [D 2 , D 9 , S 5 ] of the current inactive span is used to generate maximal space tile ST  122  which extends from D 2  to D 9  in the D axis along a line generally aligned with S 5  and which extends from S 5  to S 6  in the S axis. The newly generated space tiles ST  120  and ST  122 , both of which are illustrated in FIG. 5, are then added to the maximal space tile list STL, the unmatched entries [D 0 , D 1 , S 4 ] and [D 2 , D 9 , S 5 ] are deleted from the current inactive segment list and the unmatched entry [D 0 , D 9 , S 6 ] of the next inactive segment list is added to the current inactive segment list, thereby producing the following current inactive segment list: 
         IL={[D 0 , D 9 , S 6 ]}. 
         [0060]    The method then returns to step  38  and, as there are additional stop points to be examined, on to step  40  where the stop point S 8  is selected for examination. The stop point S 8  is generally aligned with an upper edge  90   b  of the routing area  90 . As neither a lower edge nor an interior of a component is positioned along the upper edge  90   b , the next list of active and inactive spans generated at step  42  would be as follows: 
         AL=Φ; and 
         IL={ [D 0 , D 9 , S 7 ]}. 
         [0061]    The current and next active segment lists are then processed at step  44 . The current active segment list is Φ while the next active segment list is Φ. As both the current and next active segment lists are empty, no additional maximal component tiles are generated and the current active segment list remains empty.  
         [0062]    The method then proceeds to step  46  for processing of the current inactive segment list. The current inactive segment list contains the entry [D 0 , D 9 , S 6 ] while the next inactive segment list contains the entry [D 0 , D 9 , S 8 ]. While the current and inactive segment lists contain matching entries, the stop point being examined is the last stop point in the set of stop points. Accordingly, the [D 0 , D 9 , S 6 ] entry of the current inactive segment list is used to generate a maximal space tile, hereafter referred to as maximal space tile  124 , which extends from D 0  to D 9  in the D axis along a line generally aligned with S 6 , extends from S 6  to S 8  in the S axis and is illustrated in FIG. 5. The newly generated space tile is added to the maximal space tile list STL and the entry [D 0 , D 9 , S 6 ] of the current inactive segment list is deleted therefrom, thereby emptying it. Further, as the entry [D 0 , D 9 , S 8 ] was matched to the entry [D 0 , D 9 , S 6 ], it is not added to the current inactive segment list, thereby keeping the current inactive segment list empty. The method then returns to step  38  and as all of the stop points SO through S 6  and S 8  have been examined, to step  48  for output of the generated maximal component tile and maximal space tile lists CTL and STL to the VLSI circuit design module  6 . The method then ends at step  50 .  
         [0063]    Continuing to refer to FIG. 3 a , the method of generating a list of maximal component tiles CTL and a list of maximal space tiles STL for the routing area  90 ′, shall now be described, this time, in conjunction with FIGS. 4 and 6. As previously noted, FIG. 4 shows the CT  98  positioned in the routing area  90 ′. As before, the process starts at step  32  by identifying the active and inactive segments for a span extending along a bottom edge  90   a ′ of the routing area  90 ′. As there are no active segments along the bottom edge  90   a ′, the active and inactive segment lists for this span, which is generally aligned with the stop point SO, are initially set as follows: 
         AL=Φ; and  
         IL={[D 0 , D 9 , SO]}.  
         [0064]    At step  34 , the remaining members of the set of stop points are identified (S 5  and S 7 ) and, at step  36 , an empty maximal component tile list CTL and an empty maximal space tile list STL are generated. As the stop points S 5 , S 7  and S 8  need to be examined, the method passes through  38  and on to step  40  where the stop point S 5 , the next stop point after the stop point SO, is selected for examination. As may be seen in FIG. 4, bottom edge  98   a  of the CT  98  is generally aligned with the stop point S 5 . The corresponding segment is, therefore, considered to be active while the remaining segments generally aligned with the stop point S 5  are considered to be inactive. Accordingly, at step  42 , the next list of active segments and the next list of inactive segments are determined to be: 
         AL={[D 5 , D 7 , S 5 ]}; and 
         IL={[D 0 , D 5 , S 5 ], [D 8 , D 9 , S 5 ]}. 
         [0065]    Proceeding to step  44 , the current active segment list is empty. As a result, there are no matches between the current list of active segments and the next list of active segments. As a result, no maximal component tiles are generated at step  44 . Furthermore, as the entry in the next active segment list is unmatched, it is added to the current active segment list, which now becomes: 
         AL={[D 5 ,D 7 ,S 5 ]}. 
         [0066]    The method then proceeds to step  46  for processing of the current inactive segment list. The current inactive segment list contains a single entry [D 0 , D 9 , SO]. As there is no matching span in the next inactive segment list, a space tile, hereafter referred to as ST  200  and illustrated in FIG. 6, which extends from D 0  to D 9  in the D axis along a line generally aligned with S 0  and which extends from S 0  to S 5  in the S axis is generated and added to the maximal space tile list STL. The matched entry [D 0 , D 9 , S 0 ] is deleted from the current inactive segment list and the unmatched entries [D 0 , D 5 , S 5 ] and [D 8 , D 9 , S 5 ] of the next inactive segment list are added to the inactive segment list, thereby producing the following current inactive segment list: 
         IL={[D 0 , D 5 , S 5 ], [D 8 , D 9 , S 5 ]}. 
         [0067]    The method then returns to step  38  and, as there are additional stop points to be examined, on to step  40  where stop point S 7  is selected for examination.  
         [0068]    A span extending across the routing area  90 ′ along a line generally aligned with the stop point S 7  passes along an upper edge  98   b  of the CT  98 . Accordingly, the next list of active and inactive segments would be as follows 
         AL=Φ; and  
         IL={[D 0 , D 9 , SO]}. 
         [0069]    For each active segment in the current active segment list, the next active segment list is searched for matches. Here, [D 5 , D 7 , S 5 ], currently the only entry in the current active segment, is unmatched. Accordingly, the entry [D 5 , D 7 , S 5 ] is used to generate a maximal component tile, hereafter referred to as CT  202  and illustrated in FIG. 6, which extends from D 5  to D 7  in the D axis along a line generally aligned with S 5  and from S 5  to S 7  in the S axis. The newly generated maximal component tile is then added to the maximal component tile list CTL. The unmatched entry [D 5 , D 7 , S 5 ] is deleted from the current active segment list, thereby emptying it, and, since there are no unmatched entries from the next active segment list, the current active segment list remains empty as shown below: 
         AL=Φ. 
         [0070]    The method then proceeds to step  46  for processing of the current inactive segment list. The current inactive segment list contains entries [D 0 , D 5 , S 5 ] and [D 7 , D 9 , S 5 ] while the next inactive segment list contains the entry [D 0 , D 9 , S 7 ]. Thus, the [D 0 , D 5 , S 5 ] and [D 7 , D 9 , S 5 ] entries of the current inactive segment list are unmatched. Accordingly, the unmatched entry [D 0 , D 5 , S 5 ] is used to generate maximal space tile ST  204 , which extends from D 0  to D 5  in the D axis along a line generally aligned with S 5 , extends from S 5  to S 7  in the S axis and is illustrated in FIG. 5. Similarly, the unmatched entry [D 7 , D 9 , S 5 ] is used to generate maximal space tile ST  206 , which extends from D 7  to D 9  in the D axis along a line generally aligned with S 5 , extends from S 5  to S 7  in the S axis and is illustrated in FIG. 5. The newly generated tiles are then added to the maximal space tile list, the unmatched entries [D 0 , D 5 , S 5 ] and [D 7 , D 9 , S 5 ] are deleted from the current inactive segment list and the unmatched entry [D 0 , D 9 , S 7 ] of the next inactive segment list is added to the current inactive segment list, thereby producing the following current inactive segment list: 
         IL{[D 0 , D 9 , S 7 ]}. 
         [0071]    The method then returns to step  38  and, as there are additional stop points to be examined, on to step  40  where stop point S 8  is selected for examination. The stop point S 8  is generally aligned with an upper edge  90   b ′ of the routing area  90 ′. As neither a lower edge nor an interior of a component is positioned along the upper edge  90   b ′, the next list of active and inactive spans generated at step  42  would be as follows: 
         AL=Φ; and  
         IL={[D 0 , D 9 , SO]}. 
         [0072]    The current and next active segment lists are then processed at step  44 . The current active segment list is Φ while the next active segment list is Φ. As both the current and next active segment lists are empty, no additional maximal component tiles are generated and the current active segment list remains empty.  
         [0073]    The method then proceeds to step  46  for processing of the current inactive segment list. The current inactive segment list contains the entry [D 0 , D 9 , S 7 ] while the next inactive segment list contains the entry [D 0 , D 9 , S 8 ]. While the current and inactive segment lists contain matching entries, the stop point being examined is the last stop point in the set of stop points. Accordingly, the [D 0 , D 9 , S 7 ] entry of the current inactive segment list is used to generate a maximal space tile, hereafter referred to as maximal space tile  208 , which extends from D 0  to D 9  in the D axis along a line generally aligned with S 7 , extends from S 7  to S 8  in the S axis and is illustrated in FIG. 5. The newly generated space tile is added to the maximal space tile list STL and the entry [D 0 , D 9 , S 7 ] of the current inactive segment list is deleted therefrom, thereby emptying it. Further, as the entry [D 0 , D 9 , S 8 ] was matched to the entry [D 0 , D 9 , S 7 ], it is not added to the current inactive segment list, thereby keeping the current inactive segment list empty. The method then returns to step  38  and as all of the stop points S 0 , S 5 , S 7  and S 8  have been examined, to step  48  for output of the generated maximal component tile and maximal space tile lists CTL and STL to the VLSI circuit design module  6 . The method then ends at step  50 .  
         [0074]    Referring next to FIGS. 3 b - 1  through  3   b - 2 , the method of determining an optimal tile path and minimum cost wire path between pins that are CTs to be connected will now be described in greater detail. From the application of the method of FIG. 3a to the routing areas illustrated in FIG. 4, it has been determined that the CTs  118 ,  104  and  202  are pins to be connected while the CT  112  is an obstruction to be avoided. The method commences at step  50  and, at step  52 , a starting tile S and a destination tile T are selected from amongst the CTs  118 ,  104  and  202 . The method then proceeds on to step  54  where a priority queue Q is created and to step  56  where the starting tile S is inserted into the priority queue Q. The method then continues on to step  58 , where a search tree ST having the starting tile S as its root is generated and on to step  60  where an initial path cost C between starting tile S and destination tile T is set to infinity.  
         [0075]    Proceeding on to step  62 , while the priority queue Q is not empty, steps  64  through  84  are executed as appropriate. Accordingly, the method either proceeds to step  64  or to step  86  described below. Here, as the priority queue Q currently holds the starting tile S, the method proceeds on to step  64  where the lowest cost tile E is popped from the priority queue Q. To appreciate which tile is the lowest cost tile E to be popped from the priority queue Q, a brief illustration shall be necessary. Assume, for example, that the CT  118  is selected as the starting tile S and the CT  104  is selected as the destination tile T. There are seven tiles E, specifically, the tiles ST  120 , ST  124 , ST  122 , ST  114 , ST  110 , ST  200  and ST  204 , that adjoin the tile CT  118 . The distance from each tile E to the destination tile T is evaluated and the tile E closest to the destination tile T is popped from the priority queue as the lowest cost tile E. In the foregoing example, of the seven tiles E, the ST  200  is the only tile adjoining the destination tile CT  104 . Thus, as the tile ST  200  is closest to the destination tile CT  104 , the tile ST  200  is popped at step  64  as the lowest cost tile E.  
         [0076]    Proceeding on to step  66 , if the tile E popped from the priority queue is the destination tile T, steps  68 ,  70  and  72  are executed. Otherwise, the method proceeds to step  74 , below. More specifically at step  68 , a tile path LP from the starting tile S to the destination tile T is retrieved from the search tree TS. At step  70 , the minimum cost C 2  from the starting tile S to the destination tile T is evaluated and, at step  72 , if the current cost C exceeds the evaluated minimum cost C 2 , then C is set to C 2  and the minimum cost point path is saved.  
         [0077]    Proceeding on to step  74  for each tile F neighboring the tile E, steps  76  through  84  are executed to expand the search tree. At step  76 , a minimum cost CF from the starting tile T to the tile F is evaluated and, at step  78 , a lower cost bound CT from the tile F to the destination tile T is evaluated. At step  80 , an estimated cost CE is set as the sum of CF and CT and, at step  82  if CE is greater than or equal to the current cost C, then the method continues and the search tree TS is pruned. Otherwise, as set forth in step  84 , the tile F is inserted into the priority queue Q with the estimated cost CE and the tile F is inserted into the tree TS as a child of the tile E. The method then proceeds to step  86  where, the saved minimum cost (if determined) is returned. Otherwise a routing failure is reported. The method then ends at step  88 .  
         [0078]    The input to the above-referenced algorithm is the list of space tiles created by the method of FIG. 3 a , a start tile S and a destination tile T. The output is the list of points for the minimum cost routing path. We maintain a priority queue Q and a search tree TS, where Q is initially set to S and S is set to the root of the search tree TS. When no point path has been found, the path cost is set to infinity. As illustrated herein, the search terminates only when the priority queue Q is empty. In every search, one tile E is popped from the priority queue Q. If the tile E is the destination tile T, then the minimum cost path from the root of the search tree to the destination tile T is evaluated and the cost compared with saved cost. An update is performed if a lower cost path is found and the corresponding path is saved. Steps  66  through  72  perform this task. Steps  74  through  84  expand the search and prune the branch. Process the neighbor tiles F of the tile E one by one. For each such tile F, CF, the minimum cost from S to F and CT, the lower bound cost to T are evaluated. The cost from S to T through this path cannot be lower than estimated CE=CF+CT. If this estimated cost is higher than the current cost C, then prune this search path; otherwise expand the search by adding tile F into tree TS as a child node of E.  
         [0079]    Thus, there has been described and illustrated herein, various methods suitable for use in conjunction with the design and manufacture of VLSI circuits. However, those skilled in the art should recognize that numerous modifications and variations may be made in the techniques disclosed herein without departing substantially from the spirit and scope thereof, which is defined solely by the claims appended hereto.