Journal of the ACM (JACM)
Combinatorial theory (2nd ed.)
Combinatorial theory (2nd ed.)
An asymptotic theory for recurrence relations based on minimization and maximization
Theoretical Computer Science
Delay optimization of linear depth boolean circuits with prescribed input arrival times
Journal of Discrete Algorithms
The delay of circuits whose inputs have specified arrival times
Discrete Applied Mathematics
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For some k=2 let d=(d"1,d"2,...,d"k)@?R""0^k. We denote the concatenation of k vectors a"1,a"2,...,a"k@?@?"n"="0R^n by a"1a"2...a"k and use @e to denote the empty vector. We consider a recursively defined function D"d:@?"n"="0R^n-R@?{-~} with D"d(@e)=-~, D"d((a))=a for a@?R and D"d(a)=min{max{D"d(a"i)+d"i|1@?i@?k}|a=a"1a"2...a"k with a"i@?@?m=0n-1R^m for 1@?i@?k} for a@?R^n with n=2. The function D"d equals the cost of an optimal alphabetic code tree with unequal letter costs and the above recursion naturally generalizes a recursion studied by Kapoor and Reingold [S. Kapoor, E.M. Reingold, Optimum lopsided binary trees, J. Assoc. Comput. Mach. 36 (1989) 573-590]. If z(n) denotes the vector consisting of n=0 zeros, then let f(@a)=max{i@?N"0|D"d(z(i))@?@a} for @a@?R. Let d=min{d"1,d"2,...,d"k} and D=max{d"1,d"2,...,d"k}. Our main result is that D"d(z(@?i=1nf(a"i)))@?D"d(a)@?D"d(z(@?i=1nf(a"i)))+6D-2d for a=(a"1,a"2,...,a"n)@?R"="0^n. This result is useful for the analysis of the asymptotic growth of D"d.