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Verified Analysis of Random Binary Tree Structures

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Interactive Theorem Proving (ITP 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10895))

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Abstract

This work is a case study of the formal verification and complexity analysis of some famous probabilistic algorithms and data structures in the proof assistant Isabelle/HOL. In particular, we consider the expected number of comparisons in randomised quicksort, the relationship between randomised quicksort and average-case deterministic quicksort, the expected shape of an unbalanced random Binary Search Tree, and the expected shape of a Treap. The last two have, to our knowledge, not been analysed using a theorem prover before and the last one is of particular interest because it involves continuous distributions.

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Notes

  1. 1.

    In fact, any non-discrete probability distribution works, where by ‘non-discrete’ we mean that all singleton sets have probability 0. In the formalisation, however, we restricted ourselves to the case of a uniform distribution over a real interval.

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Acknowledgement

This work was funded by DFG grant NI 491/16-1. We thank Johannes Hölzl and Andreas Lochbihler for helpful discussions, Johannes Hölzl for his help with the construction of the tree space, and Bohua Zhan and Maximilian P. L. Haslbeck for comments on a draft. We also thank the reviewers for their suggestions.

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Authors and Affiliations

  1. Technische Universität München, 85748, Garching bei München, Germany

    Manuel Eberl, Max W. Haslbeck & Tobias Nipkow

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  1. Manuel Eberl
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  2. Max W. Haslbeck
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  3. Tobias Nipkow
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Correspondence to Manuel Eberl .

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  1. Carnegie Mellon University, Pittsburgh, PA, USA

    Jeremy Avigad

  2. Inria, Nantes, France

    Assia Mahboubi

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Eberl, M., Haslbeck, M.W., Nipkow, T. (2018). Verified Analysis of Random Binary Tree Structures. In: Avigad, J., Mahboubi, A. (eds) Interactive Theorem Proving. ITP 2018. Lecture Notes in Computer Science(), vol 10895. Springer, Cham. https://doi.org/10.1007/978-3-319-94821-8_12

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  • DOI: https://doi.org/10.1007/978-3-319-94821-8_12

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