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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
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
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
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
Lawler’s Algorithm
W.-S. Luk (Fudan Univ.) Lecture 3: Parametric Problems (single parameter) 2012年 8月 11日 5 / 7
Howard’s Algorithm
W.-S. Luk (Fudan Univ.) Lecture 3: Parametric Problems (single parameter) 2012年 8月 11日 6 / 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