Partial alphabetic trees

In the partial alphabetic tree problem we are given a multiset of nonnegative weights W = {w₁,…, w_n}, partitioned into k≤n blocks B₁,…, B_k. We want to find a binary tree T where the elements of W resides in its leaves such that if we traverse the leaves from left to right then all leaves of B₁ precede all leaves of B_j for every i<j.Furthermore among all such trees, T has to minimize ∑^N_i=1 w_id(w_i), where d(w_i) is the depth of w_i in T. The partial alphabetic tree problem generalizes the problem of finding a Huffman tree over W (there is only one block) and the problem of finding a minimum cost alphabetic tree over W (each block consists of a single item). This fundamental problem arises when we want to find an optimal search tree over a set of items which may have equal keys and when we want to find an optimal binary code for a set of items with known frequencies, such that we have a lexicographic restriction for some of the codewords. Our main result is a pseudo-polynomial time algorithm that finds the optimal tree. Our algorithm runs in (formula presented) time where (formula presented). In particular the running time is polynomial in case the weights are bounded by a polynomial of n. To bound the running time of our algorithwe prove an upper bound of [αlog(W_sum/W_min)+1] on the depth of the optimal tree. Our algorithm relies on a solution to what we call the layered Huffman forest problem which is of independent interest. In the layered Huffman forest problem we are given an unordered multiset of weights W = {w₁,…, w_n}, and a multiset of integers D = {d₁,…, d_m}.We look for a forest F with m trees, T₁,…, T_m, where the weights in W correspond to the leaves of F, that minimizes ∑ ⁿ_i=1 w_id_F(w_i) where d_F (w_i) is the depth of w_i in its tree plus d_i w_i € T_i. Our algorithm for this problem runs in O(kn²)time.