Question

Timing circuits are a crucial component of VLSI chips. Here’s a simple model of such a timing circuit. Consider a complete balanced binary tree with n leaves, where n is a power of two. Each edge e of the tree has an associated length le, which is a positive number. The distance from the root to a given leaf is the sum of the lengths of all the edges on the path from the root to the leaf.

The root generates a clock signal which is propagated along the edges to the leaves. We’ll assume that the time it takes for the signal to reach a given leaf is proportional to the distance from the root to the leaf.

Now, if all leaves do not have the same distance from the root, then the signal will not reach the leaves at the same time, and this is a big problem. We want the leaves to be completely synchronized, and all to receive the signal at the same time. To make this happen, we will have to increase the lengths of certain edges, so that all root-to-leaf paths have the same length (we’re not able to shrink edge lengths). If we achieve this, then the tree (with its new edge lengths) will be said to have zero skew. Our goal is to achieve zero skew in a way that keeps the sum of all the edge lengths as small as possible.

Give an algorithm that increases the lengths of certain edges so that the resulting tree has zero skew and the total edge length is as small as possible.

Answer 1

Let L and R be the sub-trees underneath the root r.

If the height of L (where we consider height to be the maximum length of all root-leaf paths) is ∆ more than that of R, ∆ ≥ 0, then the length of the edge (r, R) is increased by ∆.

Then, we separately perform the same cycle over L and R.

Once more, by induction, argue that this greedy algorithm is efficient –

when it does not increase the length of (r, R) edge by ∆, then prove that the solution can be improved.