layout: true class: typo, typo-selection --- count: false class: nord-dark, middle, center # ⌚ Lecture 05b - Clock Skew Scheduling Under Process Variations (2) @luk036 👨‍💻 · 2026 📅 --- ### 📋 Overview .pull-left[ - A Review of CSS Issues 📝 - General Formulation 📐 - Yield-driven Clock Skew Scheduling 🎯 - Numerical Results 📊 ] .pull-right[  ] --- ### ⏱️ Minimum Clock Period Problem - 📝 Linear programming (LP) formulation $$\begin{array}{ll} \text{minimize} & {\color{coral}T_\text{CP} } \\ \text{subject to} & {\color{green}l_{ij}} \leq {\color{red}t_i} - {\color{red}t_j} \leq {\color{orange}u_{ij}} \end{array}$$ where $\text{FF}_i$ and $\text{FF}_j$ are sequentially adjacent to each other. - The above constraints are called *system of difference constraints* (see Introduction to Algorithms, MIT): - 🔑 Key: it is easy to check if a feasible solution exists by detecting negative cycles using the Bellman-Ford algorithm. --- ### 🔗 System of Difference Constraints - 🔄 In some cases, you may need to do some transformations, e.g. - ${\color{red}t_i} \leq \min_k \{ {\color{red}t_k} + {\color{coral}a_{ik}}\} \rightarrow {\color{red}t_i} - {\color{red}t_k} \leq {\color{coral}a_{ik}}, \; \forall k$ - ${\color{red}t_i} \geq \max_k \{ {\color{red}t_k} + {\color{coral}b_{ik}}\} \rightarrow {\color{coral}b_{ik}} \leq {\color{red}t_i} - {\color{red}t_k}, \; \forall k$ --- ### ⚖️ Slack Maximization (EVEN) - Slack Maximization Scheduling $$\begin{array}{ll} \text{maximum} & {\color{coral}\beta} \\ \text{subject to} & {\color{red}t_j} - {\color{red}t_i} \leq {\color{green}\mu_{ij}} - {\color{coral}\beta} \end{array}$$ (👉 Note: ${\color{green}\mu_{ij}} \text{ is not equal to } {\color{green}\mu_{ji}}$) - is equivalent to the so-called *minimum mean cycle problem* (MMC), where: - ${\color{coral}\beta}^* = \sum_{(i,j) \in {\color{lime}C}} {\color{green}\mu_{ij}} / |{\color{lime}C}|$, - ${\color{lime}C}$: critical cycle (first negative cycle) - ⚡ Can be efficiently solved by the parametric shortest path methods. --- ### ⚖️ Slack Maximization (C-PROP) - Slack Maximization Scheduling $$\begin{array}{ll} \text{maximum} & {\color{coral}\beta} \\ \text{subject to} & {\color{red}t_j} - {\color{red}t_i} \leq {\color{green}\mu_{ij}} - {\color{orange}\sigma_{ij}} {\color{coral}\beta} \end{array}$$ (we show the correctness later) - 📈 is equivalent to the *minimum cost-to-time ratio cycle problem* (MCR), where: - ${\color{coral}\beta}^* = \sum_{(i,j) \in {\color{lime}C}} {\color{green}\mu_{ij}} / \sum_{(i,j) \in {\color{lime}C}} {\color{orange}\sigma_{ij}}$, - ${\color{lime}C}$: critical cycle - Assume Gaussian distribution (Bell Shape 🔔) --- ### 📐 General Formulation $$\begin{array}{ll} \text{maximum} & {\color{darkmagenta} g({\color{coral}\beta})} \\ \text{subject to} & {\color{red}t_i} - {\color{red}t_j} \leq {\color{coral} f_{ij}({\color{coral}\beta})}, \; \forall (i,j) \in {\color{lime}E} \end{array}$$ where ${\color{coral} f_{ij}({\color{coral}\beta})}$ a linear function that represents various problems defined above. | Problem | ${\color{darkmagenta} g({\color{coral}\beta})}$ | ${\color{coral} f_{ij}({\color{coral}\beta})}$ (setup) | ${\color{coral} f_{ji}({\color{coral}\beta})}$ (hold) | | ------- | ------ | --------------------------------------------------- | ----------------------------------------- | | Min CP | $-{\color{coral}\beta}$ | ${\color{coral}\beta} - {\color{coral}D_{ij}} - {\color{coral}T_\text{s}}$ | $-{\color{coral}T_\text{h}} + {\color{coral}d_{ij}}$ | | EVEN | ${\color{coral}\beta}$ | ${\color{coral}T_\text{CP}} - {\color{coral}D_{ij}} - {\color{coral}T_\text{s}} - {\color{coral}\beta}$ | $-{\color{coral}T_\text{h}} + {\color{coral}d_{ij}} - {\color{coral}\beta}$ | | C-PROP | ${\color{coral}\beta}$ | ${\color{coral}T_\text{CP}}-{\color{coral}D_{ij}}-{\color{coral}T_\text{s}} - {\color{orange}\sigma_{ij}} {\color{coral}\beta}$ | $-{\color{coral}T_\text{h}} + {\color{coral}d_{ij}} - {\color{orange}\sigma_{ij}} {\color{coral}\beta}$ | --- ### 📐 General Formulation (cont'd) - In fact, ${\color{darkmagenta} g({\color{coral}\beta})}$ and ${\color{coral} f_{ij}({\color{coral}\beta})}$ are not necessarily linear functions. Any monotonic decreasing function will do. - Theorem: if ${\color{darkmagenta} g({\color{coral}\beta})}$ and ${\color{coral} f_{ij}({\color{coral}\beta})}$ are *monotonic decreasing* functions for all $i$ and $j$, then there is a unique solution to the problem. (prove later). - Question 1: Does this generalization have any application? 🤔 - Question 2: What if ${\color{darkmagenta} g({\color{coral}\beta})}$ and ${\color{coral} f_{ij}({\color{coral}\beta})}$ are convex but not monotone? 🤔 --- class: nord-light, middle, center ## Yield & Statistical Methods 🎯 --- ### 🔔 Non-Gaussian Distribution .pull-left[ - ~~65nm and below,~~ the path delay is likely to have a non-Gaussian distribution 🔕: 👉 Note: central limit theorem does not apply here. Why? ] .pull-right[  ] --- ### 🎯 Timing Yield Maximization - Formulation: - $\max\{\min\{\text{Pr}\{ {\color{red}t_j} - {\color{red}t_i} \leq {\color{orange}\tilde{W}_{ij}} \}\}\}$ - is not exactly timing yield but reasonable. - does not require correlation information among ${\color{orange}\tilde{W}_{ij}}$. Why? - It is equivalent to: $$\begin{array}{ll} \text{maximum} & {\color{coral}\beta} \\ \text{subject to} & {\color{red}t_i} - {\color{red}t_j} \le {\color{coral}T_\text{CP}} - F^{-1}_{ji}({\color{coral}\beta}) \\ & {\color{red}t_j} - {\color{red}t_i} \le F^{-1}_{ij}(1 - {\color{coral}\beta}) \end{array}$$ where $F_{ij}(\cdot) \text{ is CDF of } {\color{orange}\tilde{W}_{ij}}$ - Luckily, any CDF must be a monotonic increasing function. --- ### 📊 Statistical Interpretations of C-PROP - Reduce to C-PROP when ${\color{orange}\tilde{W}_{ij}}$ is Gaussian, or precisely $$F_{ij}({\color{blue}x}) = K(({\color{blue}x} - {\color{green} \mu_{ij}})/{\color{orange} \sigma_{ij}})$$ - EVEN: identical distribution up to shifting $$F_{ij}({\color{blue}x}) = H({\color{blue}x} - {\color{green} \mu_{ij}})$$ Not necessarily worse than C-PROP --- ### 📊 Comparison  --- ### 🔄 Timing Optimization Iteration Flow .mermaid[ <pre> flowchart TD style A fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px style B fill:#fff3e0,stroke:#e65100,stroke-width:3px style C fill:#e3f2fd,stroke:#1565c0,stroke-width:3px style D fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px style E fill:#ffcdd2,stroke:#c62828,stroke-width:3px style F fill:#e3f2fd,stroke:#1565c0,stroke-width:3px style G fill:#fff3e0,stroke:#e65100,stroke-width:3px style H fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px A[Start: Initial β interval] --> B{Check feasibility} B -->|Infeasible| C[Find negative cycle C] B -->|Feasible| D[Improve via shortest path] C --> E[Zero-out cycle: update β] D --> F[Update βmin] E --> G{Converged?} F --> G G -->|No| B G -->|Yes| H[Return optimal β*] </pre> ] --- class: nord-light, middle, center ## Solving Methods & Algorithms 🚀 --- ### 🛠️ Three Solving Methods in General - 🔍 Binary search based - Local convergence is slow. - 🔁 Cycle based - Idea: if a solution is infeasible, there exists a negative cycle which can always be "zero-out" with minimum effort (proof of optimality) - 🛤️ Path based - Idea: if a solution is feasible, there exists a (shortest) path from where we can always improve the solution. --- ### 🚀 Parametric Shortest Path Algorithms - 🔍 Lawler's algorithm (binary search) - 🔁 Howard's algorithm (based on cycle cancellation) - 🔀 Hybrid method - ⚙️ Improved Howard's algorithm - 📥 Input: - Interval [βmin, βmax] that includes β\* - Tol: tolerance - G({\color{salmon}V}, {\color{lime}E}): timing graph - 📤 Output: - Optimal β* and its corresponding critical cycle C --- ### ⚙️ Lawler's Algorithm .column-2.column-norule.font-sm.mb-xs[ ```uml @startuml while ((βmax - βmin) > tol) : β := (βmin + βmax) / 2; if (a neg. cycle C under β exists) then : βmax := β; else : βmin := β; endif endwhile : β* := β; @enduml ```  ] --- ### ⚙️ Howard's Algorithm .column-2.column-norule.font-sm.mb-xs[ ```uml @startuml : β := βmax; while (a neg. cycle C under β exists) : find β' such that sum{(i,j) in C | fij(β')} = 0; : β := β'; endwhile : β* := β; @enduml ```  ] --- ### 🔀 Hybrid Method .column-2.column-norule.font-sm.mb-xs[ ```uml @startuml while ((βmax - βmin) > tol) : β := (βmin + βmax) / 2; if (a neg. cycle C under β exists) then : find β' such that sum{(i,j) in C | fij(β')} = 0; : β := β'; : βmax := β; else : βmin := β; endif endwhile : β* := β; @enduml ```  ] --- ### ⚙️ Improved Howard's Algorithm .column-2.column-norule.font-sm.mb-xs[ ```uml @startuml : β := (βmin + βmax) / 2; while (no neg. cycle under β) : βmin := β; : β := (βmin + βmax) / 2; endwhile while (a neg. cycle C under β exists) : find β' such that sum{(i,j) in C | fij(β')} = 0; : β := β'; endwhile : β* := β; @enduml ```  ] --- count: false class: nord-dark, middle, center .pull-left[ ## Q&A 🎤 ] .pull-right[  ] --- ### 🔧 Backup ``` {.terminal} > pandoc -t beamer lec05b.md beamer.yaml -o slide05b.pdf ```