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 $$\begin{array}{ll} \text{minimize} & {\color{coral}T_\text{CP} } \\ \text{subject to} & l_{ij} \leq {\color{red}t_i} - {\color{red}t_j} \leq u_{ij} \end{array}$$ 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. - ${\color{red}t_i} \leq \min_k \{ {\color{red}t_k} + a_{ik}\} \rightarrow {\color{red}t_i} - {\color{red}t_k} \leq a_{ik}, \; \forall k$ - ${\color{red}t_i} \geq \max_k \{ {\color{red}t_k} + b_{ik}\} \rightarrow b_{ik} \leq {\color{red}t_i} - {\color{red}t_k}, \; \forall k$ --- Slack Maximization (EVEN) ------------------------- - Slack Maximization Scheduling $$\begin{array}{ll} \text{maximum} & {\color{coral}\beta} \\ \text{subject to} & {\color{red}t_j} - {\color{red}t_i} \leq \mu_{ij} - {\color{coral}\beta} \end{array}$$ (👉 Note: $\mu_{ij} \text{ is not equal to } \mu_{ji}$) - is equivalent to the so-called *minimum mean cycle problem* (MMC), where: - ${\color{coral}\beta}^* = \sum_{(i,j) \in C} \mu_{ij} / |C|$, - $C$: critical cycle (first negative cycle) - Can be efficiently solved by the parametric shortest path methods. --- Slack Maximization (C-PROP) --------------------------- - Slack Maximization Scheduling $$\begin{array}{ll} \text{maximum} & {\color{coral}\beta} \\ \text{subject to} & {\color{red}t_j} - {\color{red}t_i} \leq \mu_{ij} - \sigma_{ij} {\color{coral}\beta} \end{array}$$ (we show the correctness later) - is equivalent to the *minimum cost-to-time ratio cycle problem* (MCR), where: - ${\color{coral}\beta}^* = \sum_{(i,j) \in C} \mu_{ij} / \sum_{(i,j) \in C} \sigma_{ij}$, - $C$: critical cycle - Assume Gaussian distribution (Bell Shape 🔔) --- General Formulation ------------------- $$\begin{array}{ll} \text{maximum} & g(\beta) \\ \text{subject to} & {\color{red}t_i} - {\color{red}t_j} \leq f_{ij}(\beta), \; \forall (i,j) \in {\color{lime}E} \end{array}$$ where $f_{ij}(\beta)$ a linear function that represents various problems defined above. | Problem | $g(\beta)$ | $f_{ij}(\beta)$ (setup) | $f_{ji}(\beta)$ (hold) | | ------- | ------ | --------------------------------------------------- | ----------------------------------------- | | Min CP | $-\beta$ | $\beta - D_{ij} - T_\text{s}$ | $-T_\text{h} + d_{ij}$ | | EVEN | $\beta$ | $T_\text{CP} - D_{ij} - T_\text{s} - \beta$ | $-T_\text{h} + d_{ij} - \beta$ | | C-PROP | $\beta$ | $T_\text{CP}-D_{ij}-T_\text{s} - \sigma_{ij} \beta$ | $-T_\text{h} + d_{ij} - \sigma_{ij} \beta$ | --- General Formulation (cont'd) ---------------------------- - In fact, $g(\beta)$ and $f_{ij}(\beta)$ are not necessarily linear functions. Any monotonic decreasing function will do. - Theorem: if $g(\beta)$ and $f_{ij}(\beta)$ 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 $g(\beta)$ and $f_{ij}(\beta)$ are convex but not monotone? --- 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: - $\max\{\min\{\text{Pr}\{ {\color{red}t_j} - {\color{red}t_i} \leq \tilde{W}_{ij} \}\}\}$ - is not exactly timing yield but reasonable. - does not require correlation information among $\tilde{W}_{ij}$. Why? - It is equivalent to: $$\begin{array}{ll} \text{maximum} & {\color{coral}\beta} \\ \text{subject to} & {\color{red}t_i} - {\color{red}t_j} \le T_\text{CP} - F^{-1}_{ji}({\color{coral}\beta}) \\ & {\color{red}t_j} - {\color{red}t_i} \le F^{-1}_{ij}(1 - {\color{coral}\beta}) \end{array}$$ where $F_{ij}(\cdot) \text{ is CDF of } \tilde{W}_{ij}$ - Luckily, any CDF must be a monotonic increasing function. --- Statistical Interpretations of C-PROP ------------------------------------- - Reduce to C-PROP when $\tilde{W}_{ij}$ is Gaussian, or precisely $$F_{ij}(x) = K((x - \mu_{ij})/\sigma_{ij})$$ - EVEN: identical distribution up to shifting $$F_{ij}(x) = H(x - \mu_{ij})$$ Not necessarily worse than C-PROP --- Comparison ----------  --- ### Timing Optimization Iteration Flow 🔄 .mermaid[ <pre> flowchart TD style A fill:#4caf50,color:#fff style B fill:#ff9800,color:#fff style C fill:#2196f3,color:#fff style D fill:#9c27b0,color:#fff style E fill:#f44336,color:#fff style F fill:#2196f3,color:#fff style G fill:#ff9800,color:#fff style H fill:#4caf50,color:#fff 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> ] --- 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 ```