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arXiv · 1505.00357

Optimal Search Trees with 2-Way Comparisons

Marek Chrobak·Mordecai Golin·J. Ian Munro·Neal E. Young
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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, while most modern computers support only 2-way comparisons (e.g., $<, \le, =, \ge$, 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. Also, for finding optimal binary split trees, we (iii) obtain a linear speedup and (iv) prove some previous work incorrect.

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Marek Chrobak, Mordecai Golin, J. Ian Munro, Neal E. Young. 2015-05-02. Optimal Search Trees with 2-Way Comparisons. https://doi.org/10.1007/978-3-662-48971-0_7

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