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 ≥ 0 R n by a 1 a 2 ⋯ a k and use ϵ to denote the empty vector. sider a recursively defined function D d : ⋃ n ≥ 0 R n → R ∪ { − ∞ } with D d ( ϵ ) = − ∞ , D d ( ( a ) ) = a for a ∈ R and D d ( a ) = min { max { D d ( a i ) + d i ∣ 1 ≤ i ≤ k } ∣ a = a 1 a 2 ⋯ a k  with  a i ∈ ⋃ m = 0 n − 1 R m  for  1 ≤ i ≤ k } for a ∈ R n with n ≥ 2 . nction 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 ( α ) = max { i ∈ N 0 ∣ D d ( z ( i ) ) ≤ α } for α ∈ 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 = 1 n f ( a i ) ) ) ≤ D d ( a ) ≤ D d ( z ( ∑ i = 1 n f ( a i ) ) ) + 6 D − 2 d 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 .