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

Algorithm: MiLa(T u )/MiLa(T u,v )
Input: Subtree rooted at node u or edge (u, v)
Output: A set of candidate solutions Γ u
Global: Routing tree T and buffer library B
1. if u is a leaf, Γ u = (C u , q u , 0, 0) // q is required arrival time
2. else if u has one child node v or the input is T u,v
2.1 Γ v = MiLa(v)
2.2 Γ u = ∪ γ∈Γv (addWire((u, v), γ))
2.3 Γ b = ∅
2.4 for each b in B
2.4.1 Γ = ∪ γ∈Γu (addBuffer(γ, b))
2.4.2 prune Γ
2.4.3 Γ b = Γ b ∪ Γ
2.5 Γ u = Γ u ∪ Γ b
3. else if u has two child edges (u, v) and (u, z)
3.1 Γ u,v = MiLa(T u,v ), Γ u,z = MiLa(T u,z )
3.2 Γ u = Γ u ∪ merge(Γ u,v , Γ u,z )
4. prune Γ u
5. return Γ u
Figure 12: The MiLa algorithm.
algorithm for buffer insertion with one buffer type, where n is the number of possible buffer positions.
His algorithm finds a buffer insertion solution that maximizes the slack at the source. In
1996, Lillis, Cheng and Lin [12] extended van Ginneken’s algorithm to allow b buffer types in time
O(b 2 n 2 ).
Recently, many efforts are taken to speedup the van Ginneken’s algorithm and its extension.
Shi and Li [39] improved the time complexity of van Ginneken’s algorithm to O(b 2 n log n) for
2-pin nets, and O(b 2 n log 2 n) for multi-pin nets. The speedup is achieved by four novel techniques:
predictive pruning, candidate tree, fast redundancy check, and fast merging. To reduce the
quadratic effect of b, Li and Shi [40] proposed an algorithm with time complexity O(bn 2 ). The
speedup is achieved by the observation that the best candidate to be associated with any buffer
must lie on the convex hull of the (Q,C) plane and convex pruning. To utilize the fact that in real
applications most nets have small numbers of pins and large number of buffer positions, Li and
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