Partial Alphabetic Trees

Abstract

In the partial alphabetic tree problem we are given a multiset of nonnegative weights W = w ₁, . . . , w _n , partitioned into k ≤ n blocks B ₁, . . . , 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 B _i precede all leaves of B _j for every i < j. Furthermore among all such trees, T has to minimize ∑ ⁱ_n =1 wid(wi), where d(wi) is the depth of wiin T. The partial alphabetic tree problem generalizes the problem of finding a Huffman tree over T (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 O(W _sum/W _min)^2α log(W _sum/W _min)n ²) time where W _sum= ∑ ⁱ_n =1 wi, W _min= mini wi, and α = 1/log ∅ 1.44 ¹. I n particular the running time is polynomial in case the weights are bounded by a polynomial of n. To bound the running time of our algorithm we prove an upper bound of ⌊α log(W _sum/W _min) + 1⌋ on the depth of the optimal tree.

Our algorithm relies on a solution to what we call the layered Hu.- man forest problem which is of independent interest. In the layered Huffman forest problem we are given an unordered multiset of weights W = w ₁, . . . , w _n, and a multiset of integers D = d ₁, . . . , d _m. We look for a forest F with m trees, T ₁, . . . , T _m, where the weights in W correspond to the leaves of F, that minimizes ∑ ⁱ_n =1 widF(wi) where dF(wi) is the depth of wiin its tree plus dj if wi ∈ Tj. Our algorithm for this problem runs in O(kn ²) time.

∅ = √5+12 1.618 is the golden ratio