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Extreme value statistics and traveling fronts: application to computer science.
Satya N Majumdar1, P L Krapivsky
1Laboratoire de Physique Quantique, UMR C5626 du CNRS, Université Paul Sabatier, 31062 Toulouse Cedex, France.
Summary
We analyze binary search tree height statistics by mapping them to polymer problems. This approach yields exact results for arbitrary entry distributions and generalizes to m-ary trees.
Area of Science:
- Computer Science
- Algorithms
- Data Structures
Background:
- Binary search trees are fundamental data structures.
- Understanding their height statistics is crucial for performance analysis.
- Existing methods often focus on random distributions.
Purpose of the Study:
- To derive exact asymptotic properties of height and balanced height in binary search trees.
- To extend these results to search trees with arbitrary entry distributions.
- To generalize findings to m-ary search trees.
Main Methods:
- Mapping the binary search tree problem to fragmentation and then to a modified directed polymer problem on a Cayley tree.
- Utilizing traveling front techniques to solve the polymer problem.
- Translating solutions back to derive properties of the original search tree problem.
Main Results:
- Derived exact asymptotic properties for binary search tree height and balanced height.
- Obtained exact results for trees with arbitrary entry arrival distributions.
- Determined the asymptotic shape of the full probability distribution of height.
- Generalized results to m-ary search trees with arbitrary distributions.
Conclusions:
- The novel mapping provides a powerful method for analyzing search tree statistics.
- The approach yields exact results beyond traditional random models.
- This work offers a comprehensive understanding of search tree height distributions.