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Two Skew-Binary Numeral Systems and One Application

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Abstract

We introduce two numeral systems, the magical skew system and the regular skew system, and contribute to their theory development. For both systems, increments and decrements are supported using a constant number of digit changes per operation. Moreover, for the regular skew system, the operation of adding two numbers is supported efficiently. Our basic message is that some data-structural problems are better formulated at the level of a numeral system. The relationship between number representations and data representations, as well as operations on them, can be utilized for an elegant description and a clean analysis of algorithms. In many cases, a pure mathematical treatment may also be interesting in its own right. As an application of numeral systems to data structures, we consider how to implement a priority queue as a forest of pointer-based binary heaps. Some of the number-representation features that influence the efficiency of the priority-queue operations include weighting of digits, carry-propagation and borrowing mechanisms.

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Authors and Affiliations

  1. Department of Computer Science, University of Copenhagen, Copenhagen, Denmark

    Amr Elmasry & Jyrki Katajainen

  2. The Royal Library, Copenhagen, Denmark

    Claus Jensen

Authors
  1. Amr Elmasry
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  2. Claus Jensen
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  3. Jyrki Katajainen
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Corresponding author

Correspondence to Jyrki Katajainen.

Additional information

A preliminary version of this paper entitled “The magic of a number system” [1] was presented at the 5th International Conference on Fun with Algorithms held on Ischia Island in June 2010.

A. Elmasry was supported by the Alexander von Humboldt Foundation and the VELUX Foundation.

J. Katajainen was partially supported by the Danish Natural Science Research Council under contract 09-060411 (project “Generic programming—algorithms and tools”).

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Elmasry, A., Jensen, C. & Katajainen, J. Two Skew-Binary Numeral Systems and One Application. Theory Comput Syst 50, 185–211 (2012). https://doi.org/10.1007/s00224-011-9357-0

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  • DOI: https://doi.org/10.1007/s00224-011-9357-0

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