layout: true class: typo, typo-selection --- count: false class: nord-dark, middle, center # Lecture 9: Double Patterning @luk036 👨‍💻 · 2026 📅 --- ### 🔍 Background .pull-left[ - 🏭 In the past, chips have continued to get smaller and smaller, and therefore consume less and less power. - 🚧 However, we are rapidly approaching the end of the road and optical lithography cannot take us to the next place we need to go. ] .pull-right[  ] --- ### 🖨️ Process of Lithography .pull-left[ 1. Photo-resist coating (光阻涂层) 🎨 2. Illumination (光照) 💡 3. Exposure (曝光) ☀️ 4. Etching (蚀刻) ⚒️ 5. Impurities doping (杂质掺杂) 🧪 6. Metal connection 🔗 ] .pull-right[  ] --- ### 🔬 Sub-wavelength Lithography .pull-left[ - Feature size << lithography wavelength - 45 nm vs. 193 nm 📏  ] .pull-right[ - What you see in the mask/layout is **not** what you get on the chip: - Features are distorted 🌀 - Yields are declined  ] --- ### ❓ What is Double Patterning? .pull-left[  ] .pull-right[  ] Unlike conventional optical lithography, which exposes the photoresist once under one mask, masks is exposed twice by splitting them into two, each with half its feature density. --- ### 🔑 Key technologies - *Layout fracturing* algorithm to reduce the number of rectangles and the total cut length. - *Dynamic priority search tree* for plane sweeping. 🌳 - Graph-theoretic approach: - Convert the coloring problem to a T-join problem and then solve it with Hadlock's algorithm. 🎨 - Decompose the underlying conflict graph into its tri-connected components using a data structure named *SPQR-tree*. 🌲 --- ### 📐 Polygon Fracturing Algorithm - Allow minimal overlap to reduce the number of rectangles, and thus the number of conflicts. ⚖️  --- ### ⚔️ Conflict Detection .pull-left70[ - Rule 1: If the distance between two rectangles is $\geq b$, then they are *not* in conflict. ✅ - Rule 2: Two overlapping/contacting rectangles are **NOT** conflict. 🤝 - Rule 3: Two rectangles $X$ and $Y$ are in conflict if they are $\leq b$ apart and there is a path from $X$ to $Y$ that reconstructs a "concave" polygon. ⚠️ Conflicting: $(A,C)$, $(B,D)$, but not $(A,B), (A,D)$ and $(B,C)$. ] .pull-right30[  ] --- ### 🎨 Conflict Graph  - Blue edge: positive weight (opposite color preferred) 🔵 - Green edge: negative weight (same color preferred) 🟢 --- ### 📝 Formulation of the Layout Decomposition Problem - INSTANCE: Graph $G = ({\color{salmon}V}, {\color{lime}E})$ and weight function $w:E \to Z$ - SOLUTION: Disjoint subsets of vertices $ {\color{salmon}V_0}$ and $ {\color{salmon}V_1}$ so that $ {\color{salmon}V_0} \cup {\color{salmon}V_1} = {\color{salmon}V}$ and $ {\color{salmon}V_0} \cap {\color{salmon}V_1} = \emptyset$. - MINIMIZE: total cost $\sum_{e \in {\color{lime}E}_c} w(e)$ where $E_c = \{(u,v) : u,v \in {\color{salmon}V_0}$ or $u,v \in {\color{salmon}V_1}, (u,v) \in {\color{lime}E}\}$ 💰 👉 Note: the problem is - Linear time solvable for bipartite graphs. - Polynomial time solvable for planar graphs. - But in general, NP-hard (even for tripartite graphs) 💀 --- ### Double Patterning Decomposition 🔄 .mermaid[ <pre> graph LR A[Layout] --> B[Conflict Graph] B --> C[Color Decomposition] C --> D[Mask 1] C --> E[Mask 2] style A fill:#2196f3 style B fill:#9c27b0 style C fill:#ff9800 style D fill:#4caf50 style E fill:#f44336 </pre> ] --- ### 📊 Graph-Theoretic Approach - Q: How can we produce a high-quality result when the problem is NP-hard? 🤔 - A: Observe that $G$ is a nearly planar graph: we can use Hadlock's algorithm. 🗺️ - However, the time complexity of this method is still very high. ⏳ - Solution: Graph division methods. --- class: middle, center # 🌲 SPQR-Tree --- ### 🔗 Connected Graph - Recall that a graph $G = ({\color{salmon}V}, {\color{lime}E})$ is a *connected* if every pair of vertices $u, v$ in $G$ is connected by a path. 🛤️ - A graph can be divided into its connected components in linear time. ⏱️ - Clearly, the color assignment problem can be solved independently for each connected component without affecting any QoR. 🎨 --- ### 🔗 Bi-connected Graph - A vertex is called a *cut-vertex* of a connected graph $G$ if removing it disconnects $G$. - If no cut-vertex is found in $G$, then the graph is called a bi-connected graph. 🔗 - In the following example, $a$, $b$ and $c$ are cut-vertices.  --- ### 🔗 Bi-connected Components $G'$ - A division of $G$ into its bi-connected components can be performed in linear time by using a simple depth-first search to identify cut-vertices. 🔍 - It can be easily shown that the color assignment problem can be solved for each bi-connected component separately without affecting any QoR [@chiang_fast_2005] 🎨 - Question: Is it possible to further decompose the graph? 🤔 --- ### 🔗 Tri-connected Graph - If removing a pair of vertices will disconnect $G'$, the pair is called a *separation pair* of $G'$. - If no separation pair can be found in $G'$, then it is called a *tri-connected graph*. 🔗 - In the following example, $(a,b)$, $(g,h)$, $(c,d)$, $(c,e)$ and $(c,f)$ are separation pairs.  --- ### ✂️ Tri-connected Graph Division  --- ### 🌲 SPQR-tree  - A division of $G'$ into its tri-connected components can be performed by identifying the separation pairs in linear time with the help of SPQR-tree [@gutwenger_linear_2001]. ⏱️ --- ### 🦴 Skeleton - Each tree node of SPQR-tree is associated with a tri-connected component of $G'$ called *skeleton* 💀 - A skeleton represents a contraction of $G'$ based on a set of *virtual edges*. 👻 - A skeleton was classified into four types: - Series (S): the skeleton is a cycle graph. 🔄 - Parallel (P): the skeleton contains only two vertices $s$ and $t$, and $k$ parallel edges between $s$ and $t$ where $k \ge 3$. ↔️ - Trivial (Q): the skeleton contains only two vertices $s$ and $t$, and two parallel edges between $s$ and $t$, one of which is virtual and the other is real. 👻 - Rigid (R): the skeleton is a tri-connected graph of a type other than the above. 🏗️ --- class: middle, center # ✂️ Divide-and-Conquer Method --- ### ✂️ Divide-and-conquer method Consists of three basic steps: 1. Divide a conflict graph into its tri-connected components. 2. Conquer each tri-connected component in a bottom-up manner. 3. Combine the solutions into a complete solution in a top-down manner. 🔝 We calculate two possible solutions for each component, namely $(s,t)$ of the same color and $(s,t)$ of the opposite color. 🎨 ---  --- ### Bottom-up Conquering: S Type  --- ### P Type  --- ### R Type  --- ### 🔝 Top-down Merging  --- ### ✂️ Node Splitting - Node splitting (additional rectangle splitting) for resolving conflicts. - To reduce the number of "cuts", we apply node splitting after one color assignment and then recolor. 🎨 .pull-left[ Before:  ] .pull-right[ After:  ] --- class: middle, center # 🧪 Experimental Results --- ### 45 nm SDFFRS_X2 Layer 11, 9   --- ### 45 nm Example  --- ### Random, 4K rectangles  ---  : fft_all, 320K polygons --- ### 📊 Experimental Results | \#poly | \#nodes/\#edges | w/ spqr | w/o spqr | time | cost | | ------ | --------------- | ------- | -------- | ----- | ----- | | 3631 | 31371/52060 | 13.29 | 38.25 | 65.3% | 4.58% | | 9628 | 83733/138738 | 199.94 | 2706.12 | 92.6% | 2.19% | | 18360 | 159691/265370 | 400.43 | 4635.14 | 91.4% | 1.18% | | 31261 | 284957/477273 | 1914.54 | 9964.18 | 80.7% | 1.61% | | 49833 | 438868/738759 | 3397.26 | 15300.9 | 77.8% | 1.76% | | 75620 | 627423/1057794 | 3686.07 | 17643.9 | 79.1% | 2.50% | : Experimental results of the runtime and cost reduction (with minimizing the number of stitches) --- count: false class: nord-dark, middle, center .pull-left[ # Q&A 🎤 ] .pull-right[  ]