By M. Gasteier, S. Liao, X. Song on the crossing distribution problem, J.-Y. Jou on two-level logic minimization for low power, F. Vahid on procedure cloning, Q. Wang, G. Yeap on power reduction and power delay trade-offs, others M. Glesner

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This term is always less than 1. Therefore, k−1 k−1+ i=1 w(Pi ) < k. w(Pi ) + 1 n i=1 On the other hand, if we take any k points (strictly one from each x-coordinate obeying dominance condition; assuming at least one chain of length k exists, ACM Transactions on Design Automation of Electronic Systems, Vol. 10, No. 1, January 2005. 44 • S. O. Memik et al. we should be able to do that) the sum of the weights of these k, points will be k k·1+ i=1 w(Pi ) , w(Pi ) + 1 n i=1 which is definitely larger than the sum of k−1 points.

W(Pi ) + 1 n i=1 On the other hand, if we take any k points (strictly one from each x-coordinate obeying dominance condition; assuming at least one chain of length k exists, ACM Transactions on Design Automation of Electronic Systems, Vol. 10, No. 1, January 2005. 44 • S. O. Memik et al. we should be able to do that) the sum of the weights of these k, points will be k k·1+ i=1 w(Pi ) , w(Pi ) + 1 n i=1 which is definitely larger than the sum of k−1 points. This contradicts the initial assumption of having a (k − 1)-chain with a maximum-weight sum.

Once we guarantee finding a chain of length k, combined with the dominance property from Definition 1, each point in the resulting k-chain must correspond to a matching for a distinct operation. Therefore, this k-chain corresponds to the schedule of the path. Assuming that we find a chain of length k, no two points can have the same x-coordinate, since the k-chain would not possess the dominance property from Definition 1 in that case. Hence we guarantee that each operation is included in the solution with a valid matching.

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