Fast and efficient phase conflict detection and correction in standard ...

esults in significant speed-ups of the phase conflict detection algorithm.Experimental results on representative examples show averagespeedups of 5.9x using the proposed approach, while maintainingthe same quality of results as the method in [2].To summarize, the key features that distinguish the proposedphase conflict detection scheme from previous methods are:¯ Representation of the layout as a CCG (to be defined later) and theestablishment of its relationship to phase-assignability of its correspondinglayout.¯ Reduction of the phase conflict detection problem to a minimumweightconflict cycle problem (to be defined later) on the CCG and anoptimal polynomial-time algorithm for the problem, when the CCGis an embedded planar graph.Together, these two factors provide significant runtime advantageswithout sacrificing solution quality. The following sectionsinclude a detailed discussion of these items.3.1 CCG ConstructionThe first step of the conflict detection algorithm is to build aCCG. Given a layout Ä, theCCG ´Æ µ consists ofshifter nodes Æ, conflict edges and feature edges .1. For each shifter in Ä, create an edge shifter node Ò ¾ Æ.2. For two overlapping shifters 3 × ½ and × ¾, create a conflict edge ¾ connecting Ò ½ and Ò ¾.HereÒ ½ and Ò ¾ are the edge shifter nodes for× ½ and × ¾, respectively.3. Create a feature edge ¾ between the two shifters that are onopposite sides of a critical feature.0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 000000000000011111111111110000000000000111111111111111111111111000000000000011111111111110000000000000111111111111111111111111000000000000011111111111110000000000000111111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000011111111111111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111Edge Shifter NodeConflict Edge(a) Conflict Cycle GraphFeature EdgeOverlap Node0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 00000000000001111111111111000000000001111111111111111111111110000000000000 00000000000001111111111111000000000001111111111111111111111110000000000000 00000000000001111111111111000000000000011111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000001111111111111111111111110000000000000000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111100000000000001111111111111 0000000000000111111111111111111111111110000000000000 0000000000000111111111111111111111111110000000000000Edge Shifter Node(b) Phase Conflict GraphFigure 3. (a) CCG. (b) Phase Conflict Graph.Figure 3(a) shows an example of a CCG for the layout shownearlier in Figure 1. The CCG has 6 nodes and 6 edges. By comparison,the phase conflict graph (shown in Figure 3(b)) of thesame layout has 11 nodes and 11 edges. Experimental results ina later section show a substantial reduction in the edge count usingthe new graphs, which contributes to the runtime improvements.However, unlike the phase conflict graph, the CCG does not equatephase-assignability of its corresponding layout to bipartition. So,a CCG may be bipartite even though its corresponding layout isnot phase-assignable. For instance, even though the layout in Figure3(a) is not phase-assignable, its corresponding CCG is bipartite.Thus, a new criterion is needed for detecting phase conflictsusing the CCG.It should be further clarified that in the CCG, the feature edgesand conflict edges play different roles: nodes connected by featureedges should be assigned different phases and nodes connected byconflict edges should be assigned the same phase. Later in thissection, we discuss how the feature edges are used only to appropriatelyclassify the cycles they belong to and can be dropped3 Two shifters that are separated by less than the minimum shifter spacing arecalled overlapping shifters.151during some intermediate graph constructions, thereby producingmore speed-ups.In the rest of this section, we present the theory for phase conflictdetection using the CCG.Definition 1 A conflict cycle is a cycle which contains an oddnumber of feature edges.Theorem 1 A layout is phase-assignable if and only if the correspondingCCG has no conflict cycles.Proof: See Appendix for proof.In order to make the layout phase-assignable, it is necessary toremove all conflict cycles from the CCG by deleting edges. Thedeleted edges directly correspond to phase conflicts that have tobe corrected. Each edge in the CCG can be assigned a weight toreflect the cost of correcting the phase conflict corresponding to it.A large cost of the phase conflicts selected for correction wouldimply large changes to the layout and/or mask, which is highlyundesirable. Hence, it is essential to minimize the sum of weightsof the edges to be deleted during conflict cycle removal.The minimum-weight conflict cycle removal problem is definedas follows: Given a CCG ´Îµ, remove a minimum-weightset of edges ¼ such that the modified graph ¼´Î Ò ¼ µ doesnot have any conflict cycles.It can be easily proved that this problem is NP-hard for generalgraphs by doing a simple reduction to the minimum-weight bipartizationproblem. However, an optimal polynomial-time algorithmexists when the graph is an embedded planar graph. This optimalalgorithm is referred to as Pl CC Remove in Figure 2 and will bediscussed in detail in the next section.3.2 Optimal Minimum-Weight Conflict Cycle RemovalAlgorithmLet denote a CCG that is also an embedded planar graph. Theminimum-weight conflict cycle removal problem on is reducedto a minimum-weight perfect matching problem on a suitably constructedgraph through an intermediate reduction to an T-joinproblem [11].Definition 2 A conflict face of is a face corresponding to a conflictcycle in . Afaceof that is not a conflict face is a legal face.Definition 3 The dual graph of a CCG is constructed byrepresenting every face of with a node in .Anedge whichbelongs to faces ½ and ¾ in (whose corresponding nodes in are Ú ½ and Ú ¾ , respectively) is represented with an edge ¼Ú ½ Ú ¾ in . A node Ú ¾ corresponding to a conflict face ¾ is called a conflict node. A node that is not a conflict nodeis a legal node.Theorem 2 Removing an odd number of edges from every conflictface and an even number of edges from every legal face will generatea graph with no conflict cycles.Proof: See Appendix for proof.The problem of deleting a minimum-weight set of edges suchthat an odd number of edges are deleted from every conflict faceand an even number of edges are deleted from every legal face of translates to the following problem on its dual graph :

Find the minimum-weight set of edges Ë to be deleted in ´Îµ such that: (a) an odd number of edges in Ë are incident onevery conflict node Ù ¾ Î ; (b) an even number of edges in Ë areincident on every legal node Ú ¾ Î .This is exactly the same as the T-join problem on a graph whichcan be optimally solved. The T-join problem of a graph seeks aminimum-weight edge set Ë such that a node Ù is incident to anodd number of edges of Ë if and only if Ù belongs to the nodesubset Ì of the given graph. Our problem reduces to the T-joinproblem if the set of all conflict nodes in is denoted as the setÌ . Unlike the reductions in [2] in which Ì is the set of all nodeswith odd degrees, our set Ì may include nodes with odd or evendegrees.Next, we describe how our particular T-join problem can bereduced to a perfect matching problem on a suitably constructedgadget graph . Previous work [5] shows that the specific constructionof the gadget graph can greatly impact the runtime of theperfect matching algorithm. The authors in [2] introduce a newgadget graph construction that provides speed-ups over previouswork in this area. However, their gadget construction works onlyfor the case when Ì denotes the set of odd-degree nodes.Our gadget graph construction, henceforth referred to as ConstructGadget, consists of the following steps:1. Each node Ú in the dual graph corresponds to a set ofnodes in , which is denoted as 4 Ú . Each edge connectingÚ and Ú ¼in the dual graph is arbitrarily assigned to either Úor Ú ¼ . If is assigned to Ú, it will appear as a true node Ø Ú in Ú .If is not assigned to Ú, it will appear as ghost node Ú in Ú . The ghost nodes are not explicitly represented in but are merely pointers to their respective true nodes. Theassignment is done according to the following rules:(a) For each conflict node Ù, the parity of the number of ghostnodes in Ù is different from the parity of its node degree.(b) For each legal node Ú, the parity of the number of ghost nodesin Ú is equal to the parity of its node degree.It is easy to ensure that the conditions are always satisfiedsince it is always possible to turn a ghost node Ú into a truenode Ø Ú and add an additional edge of weight Û´µ between Ø Ú and Ø Ú¼ to meet the parity requirement at the cost of increasingthe node number in by one.2. The nodes in Ú are connected to each other by weightededges as follows:(a) An edge of 0 weight is added between Ø Ú and Ø Ú ¼.(b) An edge of weight Û´ ¼ µ is added between Ø Ú and Ú ¼.(c) An edge of weight Û´µ is added between Ú and Ø Ú ¼.(d) An edge of weight ´Û´µ ·Û´ ¼ µµ is added between Úand Ú ¼.Figure 4 illustrates the construction of the gadget graph from thedual graph. Note that the assignment of the edges (indicated byedges pointing toward the node) follows the rules specified in Step1 of the construction.Theorem 3 The T-join problem for a graph ´ÎÛ̵,where Ì denotes the set of all conflict nodes in Î , can be reducedto a minimum-weighted perfect matching on the gadget graph ´Î ¼ ¼ Û ¼ µ constructed using Construct Gadget.4 Ú is referred to as a gadget of Ú and hence which is a collection of Úsiscalled a gadget graph.1522Legal NodeConflict Node431(a) Dual GraphTrue NodeGhost Node2423314(b) Gadget graphFigure 4. Gadget Graph Construction from DualGraph.Proof: See Appendix for proof.The perfect matching problem can be optimally solved in polynomialtime. In our implementation, the code of Cook andRohe [12] is integrated.It should be noted that using the proposed CCG and our particularT-join formulation implies that the classification of a face doesnot rely on its edge number. Therefore, further simplifications canbe done on the dual graph, when it is converted to the gadget graphthat is input to the perfect matching problem. For instance, featureedges are only needed to classify the faces as conflict faces or legalfaces and can be dropped during the dual graph constructionif they cannot be picked by the phase conflict detection algorithm(in the next section we discuss why this might be the case). Thissimplification results in a further reduction in the number of nodesand edges in the gadget graph without affecting the correctnessof the above reductions. However, this simplification could not bedone with the previous bipartite formulation in [5, 6, 2], whichcontributes to the increased complexity of their gadget graphs.4 Proposed Phase Conflict Correction AlgorithmThe primary task of AAPSM-specific layout modification is tocorrect the phase conflicts selected by the conflict detection algorithm.Phase conflicts can be corrected either by adding space betweenshifters corresponding to a conflict (equivalent to correctinga conflict edge) or by widening critical features (equivalent to correctinga feature edge). However, widening critical features mayintroduce significant timing problems and hence feature edges arenot picked by the current conflict detection scheme. In this work,we assume that phase conflicts corresponding to conflict edges,that can be corrected by increasing the spacing between featuressuch that their corresponding shifters are separated by the requiredshifter spacing, are the only ones being corrected.It should be noted that merely increasing the spacing betweenthe shifters corresponding to the phase conflict may cause DRCviolations as well as introduce new phase conflicts as the relativelocations of the neighboring features may change. The work presentedin [2] solved this problem by adding end-to-end spacesthroughout the layout. The spaces are inserted such that only thelength of the polysilicon interconnect is increased. This techniquecould only be applied to standard cells and macro blocks with a lowdensity of phase conflicts. Experiments indicate that this methodwhen applied directly to standard-cell blocks can cause large increasesin area.Our layout modification algorithm exploits the fact that1

1 24356VHVVH H H H H1 2 3 4 5 6Figure 7. Hierarchical Layout and its Partition Tree.can exist between features of neighboring cells. The only differenceis that Step 2 of the layout modification algorithm may beomitted.Our proposed algorithm can also be easily extended to correctphase conflicts in hierarchical layouts. As shown in Figure 7, a hierarchicallayout can be represented using the partition tree,whereeach leaf node represent a layout region. The À (or Î ) node representsa region which is partitioned into several child regions usinghorizontal (or vertical) cut lines; there is a dashed line between anytwo neighboring child nodes which represents the cut line. Thelayout modification problem can be solved using a bottom-up algorithm.First, the phase conflicts are corrected within each leafnode by inserting end-to-end spaces. Then for each upper level,new phase conflicts are avoided by inserting gaps along the cutlines. The algorithm shown in Figure 5 is the special case whenthe input layout can be represented as a two-level tree.5 Experimental ResultsThis section presents the experiments we conducted to test thebenefits of the proposed ideas. The experiments were run on a4X400 Mhz Ultra-Sparc II with 4.0 GB of RAM. The examplesare 90 nm designs and were generated using a commercial placeand route tool. Typical values of threshold width for critical features,shifter dimensions and shifter spacing were used in the experiments.This work focuses mainly on phase conflicts that can becorrected by increasing the spacing between features in the layout.Thus, phase conflicts caused by T-shapes are not handled. Thesecan be corrected by feature widening or mask splitting [7]. Phaseconflicts caused by local line-end conflicts between neighboringfeatures are also not considered as they can be efficiently detectedand corrected during layout generation [13].5.1 Phase Conflict Detection ResultsTable 1 compares the runtime and the quality of results (numberof edges deleted, or in other words, number of phase conflictsselected for correction) of the proposed method with other state-ofthe-artapproaches. The conflict detection method presented in [2]is our main comparison point since their results are best in termsof the number of phase conflicts chosen for correction and runtimes,when compared to other state-of-the-art approaches [5, 6].Columns 1 and 2 give the design names and design statistics (likenumber of polygons and number of shifter overlaps), respectively.The results obtained after applying the method in [2] are groupedunder the sub-heading Bipartization Method in [2]. The results obtainedusing our phase conflict detection method are grouped underProposed Method. Columns Runtime(5) and Runtime(9) comparethe runtimes (in seconds) of the method in [2] and our proposedmethod, respectively 5 . The results indicate that our runtimes are5 It should be noted that only the time spent in solving the perfect matchingproblem is reported in both cases as this is the most compute-intensive portion of154significantly better than those obtained using the method in [2].An average improvement of 5.9x is obtained. This can be primarilyattributed to a significant reduction in the number of edges inthe CCG compared to the phase conflict graph [2] (the numbers areshown in Columns #edges(3) and #edges(7)) and the removal ofundeletable edges (in our case, feature edges) during intermediategraph constructions. These two factors together produced a significantreduction in the number of nodes and edges of the gadgetgraph constructed during perfect matching, when compared to theones constructed in [2] (the numbers are shown in #nodes/edgesGG(4) and #nodes/edges GG(8)). This accounts for the speedupsobtained by the proposed method. We believe these runtime advantageswill be more pronounced in larger designs.The table also shows that the quality of our results (in terms onnumber of phase conflicts chosen for correction) is better than theresults obtained using the method in [2] (Column #Conflicts(6)shows the results obtained using the method in [2] and Column#Conflicts(10) shows the results obtained with our method). Thisimprovement is primarily due to the fact that the number of edgesdeleted during planarization of the CCG (second step in Figure 2) issmaller since there are fewer edges in the CCG. Hence, the optimalalgorithm can be applied to a larger sub-graph of the original graph.5.2 Phase Conflict Correction ResultsTable 2 reports the results of using the proposed layout modificationscheme for correcting the phase conflicts chosen by thedetection step on the same layouts. Column Area reports the areaof the designs in square microns. Column Conflict specifies thenumber of phase conflicts selected for correction by the detectionalgorithm for each design (the numbers are slightly different fromthe number of conflicts in Table 1 for each design due to the use ofdifferent weighting schemes). Column Outside provides the numberof phase conflicts that are selected for correction and occur betweenfeatures of neighboring cells. The results indicate that theseconflicts account for a small fraction of the total number of phaseconflicts in any design. This strengthens our view that it might betoo conservative to leave blank space around all the cells or forcethe boundary features of each cell to have the same phase and couldcause unnecessary area increases. The fifth column (% AreaInc.)reports the percentage area increase for these layouts as a result ofthe added spaces. The area increase for these layouts ranges from3.4-9.1%, with an average increase of 6.3%. The area increasegoes up slightly with the size of the layout. For comparison, thelayout increase caused by the method in [2] is also reported inthe last column (% Area Inc. [2]). The numbers indicate the areaincreases caused by the method proposed in [2] are indeed verylarge, thereby making it unsuitable for large industrial designs.6 ConclusionsA new theory for Bright-Field phase conflict detection was presentedin this paper. The proposed method enabled the representationof the layout using a significantly leaner graph called theconflict cycle graph (CCG). The phase conflict detection methodwas reduced to a new problem called the minimum-weight conflictcycle removal problem. An optimal algorithm for solving thethe algorithm. The gadget graph construction was also sped-up by 2x using thenew graph, but the perfect matching times has a greater contribution to the totalrun-time.

Bipartization Method in [2]Proposed MethodDesign(1) # Plgns/#Shft Ovlps(2) #edges(3) #nodes/#edges GG(4) Runtime (5) # Conflicts (6) #edges(7) #nodes/#edges GG(8) Runtime(9) # Conflicts(10)Design1 10274/24580 84508 46756/153291 2.53 631 59939 15562/26943 0.38 624Design2 13630/32257 112599 62480/203971 1.90 963 79672 20910/36403 0.22 946Design3 21868/53749 182809 103912/338810 3.33 1558 128836 34806/61227 0.55 1534Design4 20425/50059 173319 98834/320499 3.18 1678 121546 33266/57705 0.48 1664Design5 25784/63760 216691 122324/397999 3.87 1854 152655 40614/70853 0.60 1839Design6 48787/157668 484999 355760/1147057 12.17 6330 325769 126112/231717 2.78 5989Design7 44121/142707 436297 339538/1084677 12.30 5010 295001 112152/207784 2.47 4865Design8 72101/237557 729980 567532/1817272 20.05 10275 489922 189986/351007 4.23 9631Design9 105882/376707 1133279 949064/2997548 41.35 18148 757581 303408/565217 7.95 17463Design10 159070/552767 1667581 1415620/4437522 66.40 27308 1115928 444082/829898 11.58 26354Table 1. Phase Conflict Detection Results.Design Area Conflict Outside %AreaInc. % Area Inc. [2]Design1 25173.96 641 148 3.4 18.1Design2 16397.82 995 197 5.4 23.1Design3 31416.21 1589 284 5.8 26.8Design4 25715.23 1724 238 6.1 28.8Design5 40409.68 1720 322 4.7 32.12Design6 61705.52 6257 770 5.8 57.47Design7 58414.06 5100 586 6.1 59.1Design8 94178.09 10141 512 7.3 80.2Design9 148231.77 18657 2672 9.1 100.0Design10 249210.41 28121 4224 9.0 100.0Table 2. Layout Modification Results for Standard-Cell Blocks.minimum-weight conflict cycle removal problem in CCGs that areembedded planar graphs was also presented. The optimal algorithmwas used as a sub-routine within the conflict detection algorithmso that a minimal solution could be obtained for non-planarCCGs. Supporting experimental results illustrated significant improvementsin the runtime (average runtime improvement of 5.9x),while maintaining the same quality of results (in terms of numberof phase conflicts chosen for correction) as the best availableprevious work in this area.A novel layout modification algorithm for making standard-cellblocks phase-assignable was also presented. Experimental resultsconfirm that the new method produced much smaller increases inarea than previous work in this area. The small area increases producedby the proposed method (average area increase of 6.3%)make it practical for use as a post-placement optimization step.It can also be applied to attain phase-assignability of standardcellblocks done with phase-assignable standard cells. The proposedlayout modification method can be extended to allow featurewidening for certain phase conflicts that cannot be corrected by increasingthe spacing between features. Another possible extensionof this work would be to integrate with a timing engine to make thelayout modifications more timing-aware.References[1] L. W. Liebmann, T. H. Newman, R. A. Ferguson, R. M. Martino,A. F. Molless, M. O. Neisser, and J. T. Weed. A ComprehensiveEvaluation of Major Phase Shift Mask Technologies for Isolated GateStructures in Logic Designs. In SPIE (2197), pages 612–623, 1994.[2] C. Chiang, A. Kahng, S. Sinha, X. Xu and A. Zelikovsky. Bright-FieldAAPSM Conflict Detection and Correction. In DATE, pages 908-913,2005.[3] A. Moniwa, T. Terasawa, N. Hasegawa and S. Okazaki. Algorithmsfor Phase-Shift Mask Design with Priority on Shifter Placement. InJpn J. of App. Phys. 34, pages 6584–6589, 1993.[4] K. Ooi, S. Hara and K. Koyama. Computer-Aided Design Softwarefor Designing Phase-Shift Masks. In Jpn J. of App. Phys. 32, pages5887–5891, 1993.155[5] P. Berman, A.B. Kahng, S. Mantik, I.L. Markov and A. Zelikovsky.Optimal Phase Conflict Removal for Layout of Dark Field AlternatingPhase Shifting Masks.. In IEEE TCAD (9), pages 1265–1278, 1999.[6] A.B. Kahng, S. Vaya and A. Zelikovsky. New Graph Bipartizationsfor Double-Exposure, Bright Field Alternating Phase-Shift Mask Layout.In ASP-DAC, pages 133–138, 2001.[7] C. Pierrat, F.A. Driessen and G. Vandenberghe. Full phase-shiftingmethodology for 65 nm node lithography. In SPIE (5040), pages 282–293, 2003.[8] K. Ooi, K. Koyama and M. Kiryu. Method of Designing Phase-Shifting Masks Utilizing a Compactor. In Jpn J. of App. Phys. 32,pages 6774–6778, 1994.[9] A. Moniwa, T. Terasawa, K. Nakajo, J. Sakemi and S. Okazaki.Heuristic Method for Phase-Conflict Minimization in AutomaticPhase- Shift Mask Design. In Jpn J. of App. Phys. 34, pages 6584–6589, 1995.[10] K. Cao, J. Hu and M. Cheng. Layout modification for library cellAlt-PSM composability. In SPIE, 2004.[11] W.J. Cook, W.H. Cunningham, W.R. Pulleyblank and A. Shrijver.Combinatorial Optimization. Willey Inter-Science, New York, 1998.[12] W. Cook and A. Rohe. Computing Minimum-Weight Perfect Matchings. August , 1998.http://www.or.unibonn.de/home/rohe/matching.html.[13] P. Ghosh, C. Kang, M. Sanie and J. Huckabay. PsmLint: BringingAltpsm benefits to the IC design stage. In SPIE (5042), pages 314–325, 2003.APPENDIXTheorem 1 A layout is phase-assignable if and only if the correspondingCCG has no conflict cycles.Proof: () Assume Ä is phase-assignable. Let all the edge shifter nodesbe colored with the same phases as the shifters in Ä. It is true that the nodecolorings of satisfy the following two conditions: nodes connected by afeature edge have different colors and nodes connected by a conflict edgehave the same color. Let us assume further that there exists a conflictcycle and let Ò ½Ò ¾Ò Ò ½ beaclosedwalkalong. By thedefinition of a conflict cycle, there are an odd number of feature edges in

. Hence, starting from Ò ½, the node phases will flip an odd number oftimes in . Therefore, the node Ò ½ will be assigned two different phases,which is impossible. Hence our assumption that , whose correspondinglayout Ä is phase-assignable, has a conflict cycle is wrong.()Assume does not contain any conflict cycles. WLOG, assume is connected (if is not connected, the proof can be done for each of connectedcomponents of ). Assume further that Ä is not phase-assignable.Then, there exists at least two shifters × ½ and × ¾ that do not satisfy theconditions for phase assignment. Let Ò ½ and Ò ¾ be the edge shifter nodescorresponding to × ½ and × ¾, respectively. Let the edge connecting them bedenoted as . Let be a cycle that contains and let Ò ¼ be an intermediatenode in (any node other than Ò ½ and Ò ¾).There are two possible cases:1. Shifters × ½ and × ¾ are on opposite sides of a critical feature and havesame color: Since the phases are only changed across feature edges,if Ò ½ has the same phase as Ò ¼, then there must be an even numberof feature edges from Ò ¼ to Ò ½. By assumption, Ò ¾ has the samephase as Ò ½, and hence as Ò ¼, Thus, there must be an even numberof feature edges from Ò ¼ to Ò ¾. Then, must contain an odd numberof feature edges and hence is a conflict cycle.2. Overlapping shifters × ½ and × ¾ have different colors: If Ò ½ and Ò ¼have the same phase, then there must be an even number of featureedges on the path from Ò ½ to Ò ¼. By assumption, Ò ¾ has an oppositephase from Ò ½ and hence an opposite phase from Ò ¼. Thus, the pathfrom Ò ¼ to Ò ¾ must have an odd number of feature edges. Hence must contain an odd number of feature edges and hence is aconflict cycle.This contradicts our initial assumption that has no conflict cycles.Hence, our assumption that Ä is not phase-assignable is wrong. ÙØDefinition 4 Two faces are neighboring faces if they share at least onecommon edge.Definition 5 Two neighboring faces can form a merged face by deletingat least one common edge.Lemma 1 The parity of the number of feature edges of the merged face oftwo neighboring faces is equal to the parity of the sum of the numbers offeature edges of two faces.Proof: Let the two faces have Ñ ½ and Ñ ¾ feature edges and they share Ñ ¿feature edges. Then the merged face has Ñ ½ · Ñ ¾ ¾Ñ ¿ feature edges,which has the same parity as Ñ ½ · Ñ ¾.ÙØLemma 2 A planar embedded graph has no conflict cycles if and onlyif all faces are legal.Proof: For a planar embedded graph, any cycle is the result of mergingÒ faces. If all faces are legal, we know that the number of feature edgesin the merged face is even from Lemma 1. Therefore, by definition, thegraph has no conflict cycles. It is obvious that if the original graph has noconflict cycles, then every face is legal (by definition).ÙØFigure 8. Deleting all common edges (in this case,only one) results in a merged face.Theorem 2 Removing an odd number of edges from every conflict faceand an even number of edges from every legal face will generate a graphwith no conflict cycles.Proof: Assume that an odd number of edges are removed from all conflictfaces and an even number of edges are removed from every legal face of156. As shown in Figure 8, the deletion of one or more common edgesresults in the creation of a merged face. Let ¼ be the graph obtained afterthe edge deletion. Any faceËin ¼must be the result of merging a set Ë offaces in . LetË Ë ½ ˾,whereË ½ is the set of all conflict faces in Ëand Ë ¾ is the set of all legal faces in Ë. The sum of the numbers of deletededges of all the faces in Ë must be even, since any deleted edge belongs totwo faces of Ë and is counted twice. The sum of the numbers of deletededges of all the faces in Ë ¾ is even since an even number of edges areremoved from any legal face. Hence, the sum of the numbers of deletededges of all the faces in Ë ½ should also be even. Since an odd number ofedges are removed from any conflict face, the number of faces in Ë ½ mustbe even. Thus, the sum of the numbers of feature edges of all the facesin Ë ½ is even. It is obvious that the sum of the numbers of feature edgesof all the faces in Ë ¾ is even since every legal face has an even number offeature edges. Therefore, the sum of the numbers of feature edges of allthe faces in Ë is even. By Lemma 1, the number of feature edges of themerged face is also even. So the merged face is legal. This is true of anyface in ¼ . Thus, ¼ has no conflict cycles (Lemma 2). ÙØTheorem 3 The T-join problem for a graph ´Î Û Ìµ,where Ì denotes the set of all conflict nodes in Î , can be reducedto a minimum-weighted perfect matching on the gadget graph ´Î ¼ ¼ Û ¼ µ, constructed using Algorithm Construct Gadget.Proof: () Mapping perfect matching solution of the gadget graph to avalid solution of the T-join problem on : For any node Ú ¾ whosegadget is Ú,1. If Ø Ú is matched within Ú, ¾ Ë;2. If Ø Ú is not matched within Ú,then ¾ Ë;3. If Ú is matched within Ú,then ¾ Ë;4. If Ú is not matched within Ú,then ¾ Ë;The set Ë thus constructed is a valid solution to the T-join problem. Assume ghost nodes are matched within Ú, ghost nodes are not matchedwith Ú, true nodes are matched within Ú and true nodes are notmatched within Ú. In any perfect matching solution, the number of nodesmatched within Ú, ´ · µ, is even. The parity of ´ · µ is the sameas the parity of ´ · · · µ, whose parity is the same as the parity of´·µ. Here, ´·µ is the number of edges ¾ Ë and ´·µ is the numberof ghost nodes in Ú. For conflict nodes, we require that the parity of thenumber of ghost nodes in Ú, ´·µ, to be different from the parity of thenode degree. Therefore, the parity of the number of edges ¾ Ë, ´ · µ,is also different from the parity of the node degree. Hence, the number ofedges ¾ Ë is odd for a conflict node. Similarly, it can be shown that forlegal node, the number of edges ¾ Ë is even. Therefore, the solution Ë isa valid solution of the T-join problem.() Mapping a solution Ë of the T-join problem to a solution of theperfect matching problem of can be done as follows: For any node Ú inthe dual graph , divide the edges connecting node Ú into four sets:¯ Ë ½ Ú ¾ Ú and ¾ ˯ Ë ¾ Ú ¾ Ú and ¾ ˯ Ë ¿ Ø Ú ¾ Ú and ¾ ˯ Ë Ø Ú ¾ Ú and ¾ ËLet the cardinality of Ë ½, Ë ¾, Ë ¿ and Ë be , , and , respectively.The ghost nodes and the true nodes are matched within Ú and theremaining nodes are matched outside Ú. Since Ë is a valid solution ofthe T-join problem, the parity of ´ · µ is the same as the parity of thenumber of ghost nodes ´·µ (by construction). Thus, ´´·µ·´·µµ ´ · µ·¾ is even. Hence, ´ · µ is even, which is the number of nodesto be matched within Ú. This is always possible since Ú is a completegraph.ÙØ