Lec03 parametric problems
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This document summarizes a lecture on parametric problems with a single parameter given by W.-S. Luk at Fudan University in 2012. It defines parametric potential problems where the goal is to maximize a parameter subject to a constraint involving a monotonic decreasing function of the parameter. It provides examples where the function is linear or nonlinear. It also outlines several algorithms for solving these problems, including Lawler's algorithm based on binary search and Howard's algorithm based on cycle cancellation.
![Examples
d(β) is linear (m − βs):
Optimal clock period scheduling problem [?]
Slack maximization problem [?]
Yield-driven clock skew scheduling (Gaussian) [?]
d(β) is non-linear:
Yield-driven clock skew scheduling (non-Gaussian) [?]
Multi-domain clock skew scheduling [?]
W.-S. Luk (Fudan Univ.) Lecture 3: Parametric Problems (single parameter) 2012 年 8 月 11 日 3/7](https://image.slidesharecdn.com/lec03parametricproblems-120813015454-phpapp02/85/Lec03-parametric-problems-3-320.jpg)
![Algorithms
Lawler’s algorithm (binary search based)
Howard’s algorithm (cycle cancellation)
Young’s algorithm (path based) [?]
Burns’ algorithm (path based) [?]
for clock period optimization problem (all elements of s are either 0 or
1)
Several hybrid methods are also proposed
W.-S. Luk (Fudan Univ.) Lecture 3: Parametric Problems (single parameter) 2012 年 8 月 11 日 4/7](https://image.slidesharecdn.com/lec03parametricproblems-120813015454-phpapp02/85/Lec03-parametric-problems-4-320.jpg)
Lec03 parametric problems
- 1. Lecture 3: Parametric Problems (single parameter) Wai-Shing Luk (陆伟成) Fudan University 2012 年 8 月 11 日 W.-S. Luk (Fudan Univ.) Lecture 3: Parametric Problems (single parameter) 2012 年 8 月 11 日 1/7
- 2. Parametric Potential Problem (PPP) Consider a parameter potential problem: maximize β subject to y ≤ d(β), A·u =y where d(β) is a monotonic decreasing function. If d(β) is linear (m − βs) and s is non-negative, the problem reduces to the well-known minimum cost-to-time ratio problem. If s = constant, it further reduces to the minimum mean cycle problem. Parametric flow problem can be defined similarly. W.-S. Luk (Fudan Univ.) Lecture 3: Parametric Problems (single parameter) 2012 年 8 月 11 日 2/7
- 3. Examples d(β) is linear (m − βs): Optimal clock period scheduling problem [?] Slack maximization problem [?] Yield-driven clock skew scheduling (Gaussian) [?] d(β) is non-linear: Yield-driven clock skew scheduling (non-Gaussian) [?] Multi-domain clock skew scheduling [?] W.-S. Luk (Fudan Univ.) Lecture 3: Parametric Problems (single parameter) 2012 年 8 月 11 日 3/7
- 4. Algorithms Lawler’s algorithm (binary search based) Howard’s algorithm (cycle cancellation) Young’s algorithm (path based) [?] Burns’ algorithm (path based) [?] for clock period optimization problem (all elements of s are either 0 or 1) Several hybrid methods are also proposed W.-S. Luk (Fudan Univ.) Lecture 3: Parametric Problems (single parameter) 2012 年 8 月 11 日 4/7
- 5. Lawler’s Algorithm W.-S. Luk (Fudan Univ.) Lecture 3: Parametric Problems (single parameter) 2012 年 8 月 11 日 5/7
- 6. Howard’s Algorithm W.-S. Luk (Fudan Univ.) Lecture 3: Parametric Problems (single parameter) 2012 年 8 月 11 日 6/7
- 7. Remarks To solve the feasibility problem, c.f. Lecture 2 Need to solve feasibility problems many times. Therefore, data structures, such as Precede graph, Fibonacci heap or Spanning tree/forest, may be used for efficiency W.-S. Luk (Fudan Univ.) Lecture 3: Parametric Problems (single parameter) 2012 年 8 月 11 日 7/7