Approximately Optimum Search Trees in External Memory Models
Abstract
We examine optimal and near optimal solutions to the classic binary search tree problem of Knuth. We are given a set of n keys (originally known as words), B_1, B_2, ..., B_n and 2n+1 frequencies. {p_1, p_2, ..., p_n} represent the probabilities of searching for each given key, and {q_0, q_1, ..., q_n} represent the probabilities of searching in the gaps between and outside of these keys. We have that Σ_{i=0}^n q_i + Σ_{i=1}^n p_i = 1. We also assume without loss of generality that q_{i1}+p_i+q_i != 0 for any i ϵ {1,...,n}. The keys must make up the internal nodes of the tree while the gaps make up the leaves. Our goal is to construct a binary search tree such that expected cost of search is minimized. First, we reexamine an approximate solution of Guttler, Mehlhorn and Schneider which was shown to have a worst case bound of c * H + 2 where c >= 1/(H(1/3,2/3)) ~ 1.08, and H = Σ_{i=1}^{n} p_i * log_2(1/p_i) + Σ_{j=0}^{n} q_i * log_2(1/q_j) is the entropy of the distribution. We give an improved worst case bound on the heuristic of H+4. Next, we examine the optimum binary search tree problem under a model of external memory. We use the Hierarchical Memory Model of Aggarwal et al. The model has an unlimited number of registers, R_1, R_2, ... each with its own location in memory (a positive integer). We have a set of memory sizes m_1, m_2, ..., m_l which are monotonically increasing. Each memory level has a finite size except m_l which we assume has infinite size. Each memory level has an associated cost of access c_1, c_2, ..., c_l. We assume that c_1 < c_2 < ... < c_l. We propose two approximate solutions which run in O(n) time where n is the number of words in our data set. Using these methods, we improve upon a bound given in Thite's 2001 thesis under the related HMM_2 model in the approximate setting. We also examine the related problem of binary trees on multisets of probabilities where keys are unordered and we do not differentiate between which probabilities must be leaves, and which must be internal nodes. We provide a simple O(n log_2(n)) algorithm that is within an additive (n+1)(2n) of optimal on a multiset of n keys.
Cite this work
Oliver David Lester Grant
(2016).
Approximately Optimum Search Trees in External Memory Models. UWSpace.
http://hdl.handle.net/10012/10479
Other formats
Related items
Showing items related by title, author, creator and subject.

Characterizing User Search Intent and Behavior for Click Analysis in Sponsored Search
Ashkan, Azin (University of Waterloo, 20130524)Interpreting user actions to better understand their needs provides an important tool for improving information access services. In the context of organic Web search, considerable effort has been made to model user behavior ... 
An Analysis of Peer Activities to Inform Foreign Language Learning: Word Searches, Voice, and the Use of NonTarget Languages
Reichert, Tetyana (University of Waterloo, 20101004)This empirical study investigates language use and collaborative learning in informal nonclassroom settings by learners of German as a Foreign Language (GFL). I examine learner interactions resulting from a language course ... 
Web Search, Web Tutorials & Software Applications: Characterizing and Supporting the Coordinated Use of Online Resources for Performing Work in FeatureRich Software
Fourney, Adam (University of Waterloo, 20150806)Web search and other online resources serve an integral role in how people learn and use featurerich software (e.g., Adobe Photoshop) on a daily basis. Users depend on web resources both as a first line of technical ...