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2015 | OriginalPaper | Chapter

Optimal Search Trees with 2-Way Comparisons

Authors : Marek Chrobak, Mordecai Golin, J. Ian Munro, Neal E. Young

Published in: Algorithms and Computation

Publisher: Springer Berlin Heidelberg

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Abstract

In 1971, Knuth gave an \(O(n^2)\)-time algorithm for the classic problem of finding an optimal binary search tree. Knuth’s algorithm works only for search trees based on 3-way comparisons, but most modern computers support only 2-way comparisons (\(<\), \(\le \), \(=\), \(\ge \), and \(>\)). Until this paper, the problem of finding an optimal search tree using 2-way comparisons remained open — poly-time algorithms were known only for restricted variants. We solve the general case, giving (i) an \(O(n^4)\)-time algorithm and (ii) an \(O(n\log n)\)-time additive-3 approximation algorithm. For finding optimal binary split trees, we (iii) obtain a linear speedup and (iv) prove some previous work incorrect.

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As defined here, a 2wcst T must determine the relation of the query \(v\) to every key in \(\mathcal{K}\). More generally, one could specify any partition \(\mathcal P\) of \(\mathcal{Q}\), and only require T to determine, if at all possible using keys in \(\mathcal{K}\), which set \(S\in \mathcal P\) contains \(v\). For example, if \(\mathcal P = \{\mathcal{K}, \mathcal{Q}\setminus \mathcal{K}\}\), then T would only need to determine whether \(v\in \mathcal{K}\). We note without proof that Theorem 1 extends to this more general formulation.

For an algorithm that works with linear (or O(1)-degree polynomial) functions of \(\beta \).

If it is possible to distinguish \(v=K_i\) from \(K_i < v < K_{i+1}\), then \(\mathcal{C}\) must have at least one operator other than \(<\), so we can add either \({ \left\langle v= K_i \right\rangle }\) or \({ \left\langle v\le K_i \right\rangle }\).

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Title: Optimal Search Trees with 2-Way Comparisons
Authors: Marek Chrobak
Mordecai Golin
J. Ian Munro
Neal E. Young
Publisher: Springer Berlin Heidelberg
Book: Algorithms and Computation
Print ISBN: 978-3-662-48970-3

Electronic ISBN: 978-3-662-48971-0

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-3-662-48971-0_7

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