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Clean Reversible Simulations of Ranking Binary Trees

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Reversibility and Universality

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 30))

Abstract

We propose clean reversible simulations of ranking binary trees and unranking as reversible algorithms for reversible computing systems, which are useful for enumerating and randomly generating binary trees. Algorithms for ranking binary trees and their inverses have been studied since the 1970s. Each of these algorithms can be converted into a reversible simulation by saving all data and control information otherwise lost, and each pair of ranking and unranking reversible programs can be combined to realize a clean reversible simulation by using the Bennett method. However, such a clean reversible simulation requires multiple traversal of the given data and/or intermediate data as well as additional storage proportional to the length of the computation. We show that for Knott’s ranking and unranking algorithms, additional storage usage can be reduced by using the proper assertions of reversible loops in embedded reversible simulations. We also show a clean reversible simulation that involves only one traversal. The running time and memory usage of the proposed clean reversible simulations are asymptotically equivalent to those of the original programs by Knott with intermediate garbage of constant size. In general, the derivation strategy of efficient reversible programs from irreversible ones has not yet been established, and this study can be seen as one of the case studies. All the reversible programs presented in this paper can be run on an interpreter of the reversible programming language Janus.

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Notes

  1. 1.

    A video compression standard.

  2. 2.

    A variable is zero-cleared if it is set to zero, and an array is zero-cleared if all its elements set to zero.

  3. 3.

    In this paper, reversible procedures generated by embedding have the suffix E, e.g., catalanE.

  4. 4.

    The indices of the tree permutations (see Sect. 4.4) in [21] lie in the range one or above, but in this paper, they lie in the range zero or above for simplicity.

  5. 5.

    \(\# X\) is the number of elements in the set X.

  6. 6.

    The suffix A indicates that the procedures are implemented in Algol.

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Acknowledgements

The authors would like to thank Kota Kimura for insightful comments on an early draft of this paper, and the anonymous reviewers for their helpful and constructive comments. Preliminary versions of this paper were presented at the 33rd Workshop of the Japan Society for Software Science and Technology and the 19th JSSST Workshop on Programming and Programming Languages. This work was supported by JSPS KAKENHI Grant Number JP25730049 and Nanzan University Pache Research Subsidy I-A-2 for the 2017 academic year.

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  1. Department of Software Engineering, Nanzan University, Nagoya, Japan

    Yuhi Ohkubo, Tetsuo Yokoyama & Chishun Kanayama

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  1. Yuhi Ohkubo
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Correspondence to Tetsuo Yokoyama .

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  1. Unconventional Computing Centre, University of the West of England, Bristol, United Kingdom

    Andrew Adamatzky

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Ohkubo, Y., Yokoyama, T., Kanayama, C. (2018). Clean Reversible Simulations of Ranking Binary Trees. In: Adamatzky, A. (eds) Reversibility and Universality. Emergence, Complexity and Computation, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-319-73216-9_11

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