Buffer Insertion Basics - Computer Engineering & Systems Group ...

with minimum C. In van Ginneken’s algorithm, it takes O(bn) to find one best candidate at each
buffer position.
According to convex pruning, it is easy to see that all best candidates are on the convex hull.
The following lemma says that if we sort candidates in increasing Q and C order from left to right,
then as we add wires to the candidates, we always move to the left to find the best candidates.
Lemma 1 For any T(v), let nonredundant candidates after convex pruning be α 1 ,α 2 ,...,α k , in
increasing Q and C order. Now add wire e to each candidate α j and denote it as α j + e. For any
buffer type B i , if α j gives the maximum P i (α j ) and α k gives the maximum P i (α k +e), then k ≤ j.
The following lemma says the best candidate can be found by local search, if all candidates are
convex.
Lemma 2 For any T(v), let nonredundant candidates after convex pruning be α 1 ,α 2 ,...,α k , in
increasing Q and C order. If P i (α j−1 ) ≤ P i (α j ), P i (α j ) ≥ P i (α j+1 ), then α j is the best candidate
for buffer type B i and
P i (α 1 ) ≤ · · · ≤ P i (α j−1 ) ≤ P i (α j ),
P i (α j ) ≥ P i (α j+1 ) ≥ · · · ≥ P i (α k ).
With the above two lemmas and convex pruning, best candidates are founded in amortized
O(n) time in [40] and O(b) time in [41] 1 , which are more efficient than van Ginneken’s algorithm.
1 In [40], Lemma 1 is presented differently. It says if all buffers are sorted decreasingly according to driving
resistance, then the best candidates for each buffer type in such order is from left to right.
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