Integer Representations towards Efficient Counting in the Bit Probe Model

  • Gerth Stølting Brodal
  • Mark Greve
  • Vineet Pandey
  • Satti Srinivasa Rao
Conference paper

DOI: 10.1007/978-3-642-20877-5_22

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6648)
Cite this paper as:
Brodal G.S., Greve M., Pandey V., Rao S.S. (2011) Integer Representations towards Efficient Counting in the Bit Probe Model. In: Ogihara M., Tarui J. (eds) Theory and Applications of Models of Computation. TAMC 2011. Lecture Notes in Computer Science, vol 6648. Springer, Berlin, Heidelberg

Abstract

We consider the problem of representing numbers in close to optimal space and supporting increment, decrement, addition and subtraction operations efficiently. We study the problem in the bit probe model and analyse the number of bits read and written to perform the operations, both in the worst-case and in the average-case. A counter is space-optimal if it represents any number in the range [0,...,2n − 1] using exactly n bits. We provide a space-optimal counter which supports increment and decrement operations by reading at most n − 1 bits and writing at most 3 bits in the worst-case. To the best of our knowledge, this is the first such representation which supports these operations by always reading strictly less than n bits. For redundant counters where we only need to represent numbers in the range [0,...,L] for some integer L < 2n − 1 using n bits, we define the efficiency of the counter as the ratio between L + 1 and 2n. We present various representations that achieve different trade-offs between the read and write complexities and the efficiency. We also give another representation of integers that uses n + O(logn ) bits to represent integers in the range [0,...,2n − 1] that supports efficient addition and subtraction operations, improving the space complexity of an earlier representation by Munro and Rahman [Algorithmica, 2010].

Keywords

Data structure Gray code Bit probe model Binary counter Integer representation 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Gerth Stølting Brodal
    • 1
  • Mark Greve
    • 1
  • Vineet Pandey
    • 2
  • Satti Srinivasa Rao
    • 3
  1. 1.MADALGO, Department of Computer ScienceAarhus UniversityÅrhus NDenmark
  2. 2.Computer Science & Information SystemsBITS PilaniIndia
  3. 3.School of Computer Science and EngineeringSeoul National UniversityRepublic of Korea

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