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 :