Generalized Huffman Coding for Binary Trees with Choosable Edge Lengths

Abstract: In this paper we study binary trees with choosable edge lengths, in particular rooted binary trees with the property that the two edges leading from every non-leaf to its two children are assigned integral lengths $l_1$ and $l_2$ with $l_1+l_2 =k$ for a constant $k\in\mathbb{N}$. The depth of a leaf is the total length of the edges of the unique root-leaf-path. We present a generalization of the Huffman Coding that can decide in polynomial time if for given values $d_1,...,d_n\in\mathbb{N}_{\geq 0}$ there exists a rooted binary tree with choosable edge lengths with $n$ leaves having depths at most $d_1,..., d_n$.