Lec03 parametric problems

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