On the cost of optimal alphabetic code trees with unequal letter costs

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.