Integer Representations towards Efficient Counting in the Bit Probe Model
 Gerth Stølting Brodal,
 Mark Greve,
 Vineet Pandey,
 Satti Srinivasa Rao
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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 worstcase and in the averagecase. A counter is spaceoptimal if it represents any number in the range [0,...,2^{ n } − 1] using exactly n bits. We provide a spaceoptimal counter which supports increment and decrement operations by reading at most n − 1 bits and writing at most 3 bits in the worstcase. 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 < 2^{ n } − 1 using n bits, we define the efficiency of the counter as the ratio between L + 1 and 2^{ n }. We present various representations that achieve different tradeoffs 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,...,2^{ n } − 1] that supports efficient addition and subtraction operations, improving the space complexity of an earlier representation by Munro and Rahman [Algorithmica, 2010].
 Bose, P., Carmi, P., Jansens, D., Maheshwari, A., Morin, P., Smid, M.H.M. Improved methods for generating quasigray codes. In: Kaplan, H. eds. (2010) Algorithm Theory  SWAT 2010. Springer, Heidelberg, pp. 224235 CrossRef
 Fredman, M.L. (1978) Observations on the complexity of generating quasigray codes. SIAM Journal on Computing 7: pp. 134146 CrossRef
 Gray, F.: Pulse code communications. U.S. Patent (2632058) (1953)
 Ziaur Rahman, M., Ian Munro, J. (2010) Integer representation and counting in the bit probe model. Algorithmica 56: pp. 105127 CrossRef
 Title
 Integer Representations towards Efficient Counting in the Bit Probe Model
 Book Title
 Theory and Applications of Models of Computation
 Book Subtitle
 8th Annual Conference, TAMC 2011, Tokyo, Japan, May 2325, 2011. Proceedings
 Pages
 pp 206217
 Copyright
 2011
 DOI
 10.1007/9783642208775_22
 Print ISBN
 9783642208768
 Online ISBN
 9783642208775
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 6648
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 Data structure
 Gray code
 Bit probe model
 Binary counter
 Integer representation
 Industry Sectors
 eBook Packages
 Editors

 Mitsunori Ogihara ^{(16)}
 Jun Tarui ^{(17)}
 Editor Affiliations

 16. Department of Computer Science, University of Miami
 17. Department of Information and Communication Engineering, University of ElectroComm
 Authors

 Gerth Stølting Brodal ^{(18)}
 Mark Greve ^{(18)}
 Vineet Pandey ^{(19)}
 Satti Srinivasa Rao ^{(20)}
 Author Affiliations

 18. MADALGO, Department of Computer Science, Aarhus University, IT Parken, Åbogade 34, DK8200, Århus N, Denmark
 19. Computer Science & Information Systems, BITS Pilani, 333031, India
 20. School of Computer Science and Engineering, Seoul National University, Republic of Korea
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