Partial Alphabetic Trees

• Authors: Arye Barkan;Haim Kaplan
• Venue: ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
• Year: 2002

In the partial alphabetic tree problem we are given a multiset of nonnegative weights W = {w1, . . . ,wn}, partitioned into k 驴 n blocks B1, . . . , Bk. 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 Bi precede all leaves of Bj for every i j. Furthermore among all such trees, T has to minimize 驴i=1n wid(wi), where d(wi) is the depth of wi 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 O((Wsum/Wmin)2驴 log(Wsum/Wmin)n2) time where Wsum = &Sigmai=1n wi, Wmin = mini wi, and 驴 = 1/log 驴 驴 1.44. 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 algorithm we prove an upper bound of 驴驴 log(Wsum/Wmim) + 1驴 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, . . . , wn}, and a multiset of integers D = {d1, . . . , dm}. We look for a forest F with m trees, T1, . . . , Tm, where the weights in W correspond to the leaves of F, that minimizes 驴i=1n widF (wi) where dF (wi) is the depth of wi in its tree plus dj if wi 驴 Tj. Our algorithm for this problem runs in O(kn2) time.