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

For an efficient pruning implementation and thus an efficient buffering algorithm, a sorted list
is used to maintain the solution set. The solution set Γ is increasingly sorted according to C, and
thus Q is also increasingly sorted if Γ does not contain any inferior solutions.
By a straightforward implementation, when adding a wire, the number of candidate solutions
will not change; when inserting a buffer, only one new candidate solution will be introduced. More
efforts are needed to merge two branches T l and T r at v. For each partial solution in Γ l , find the
first solution with larger Q value in Γ r . If such a solution does not exist, the last solution in Γ r will
be taken. Since Γ l and Γ r are sorted, we only need to traverse them once. Partial solutions in Γ r are
similarly treated. It is easy to see that after merging, the number of solutions is at most |Γ l | + |Γ r |.
As such, given n buffer positions, at most n solutions can be generated at any time. Consequently,
the pruning procedure at any vertex in T runs in O(n) time.
3.4 Pseudo-code
In van Ginneken’s algorithm, a set of candidate solutions are propagated from sinks to driver.
Along a branch, after a candidate buffer location v is processed, all solutions are propagated to
its upstream buffer location u through wire insertion. A buffer is then inserted to each solution to
obtain a new solution. Meanwhile, inferior solutions are pruned. At a branching point, solution
sets from all branches are merged by merging process. In this way, the algorithm proceeds in
the bottom-up fashion and the solution with maximum required arrival time at driver is returned.
Given n buffer positions in T , van Ginneken’s algorithm can compute a buffer assignment with
maximum slack at driver in O(n 2 ) time since any operation at any node can be performed in O(n)
time. Refer to Figure 4 for the pseudo-code of van Ginneken’s algorithm.
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