The Compression Optimality of Asymmetric Numeral Systems

Source coding has a rich and long history. However, a recent explosion of multimedia Internet applications (such as teleconferencing and video streaming, for instance) renews interest in fast compression that also squeezes out as much redundancy as possible. In 2009 Jarek Duda invented his asymmetri...

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Published inEntropy (Basel, Switzerland) Vol. 25; no. 4; p. 672
Main Authors Pieprzyk, Josef, Duda, Jarek, Pawłowski, Marcin, Camtepe, Seyit, Mahboubi, Arash, Morawiecki, Paweł
Format Journal Article
LanguageEnglish
Published Switzerland MDPI AG 17.04.2023
MDPI
Subjects
Algorithms
ANS
Approximation
Asymmetry
Coding
Complexity
Compression ratio
Entropy
entropy coding
Internet
Internet software
lossless compression
Markov analysis
Markov chains
Markov processes
Mathematical analysis
Multimedia
Multimedia communications
Multiplication & division
Optimization
Probability
Probability distribution
Quantum computers
Quantum computing
Redundancy
source coding
Statistical analysis
Streaming media
Teleconferencing
Video compression
Video transmission
Germany
source coding
entropy coding
ANS
lossless compression
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Abstract Source coding has a rich and long history. However, a recent explosion of multimedia Internet applications (such as teleconferencing and video streaming, for instance) renews interest in fast compression that also squeezes out as much redundancy as possible. In 2009 Jarek Duda invented his asymmetric numeral system (ANS). Apart from having a beautiful mathematical structure, it is very efficient and offers compression with a very low coding redundancy. ANS works well for any symbol source statistics, and it has become a preferred compression algorithm in the IT industry. However, designing an ANS instance requires a random selection of its symbol spread function. Consequently, each ANS instance offers compression with a slightly different compression ratio. The paper investigates the compression optimality of ANS. It shows that ANS is optimal for any symbol sources whose probability distribution is described by natural powers of 1/2. We use Markov chains to calculate ANS state probabilities. This allows us to precisely determine the ANS compression rate. We present two algorithms for finding ANS instances with a high compression ratio. The first explores state probability approximations in order to choose ANS instances with better compression ratios. The second algorithm is a probabilistic one. It finds ANS instances whose compression ratios can be made as close to the best ratio as required. This is done at the expense of the number θ of internal random “coin” tosses. The algorithm complexity is O(θL3), where L is the number of ANS states. The complexity can be reduced to O(θLlog2L) if we use a fast matrix inversion. If the algorithm is implemented on a quantum computer, its complexity becomes O(θ(log2L)3).
AbstractList Source coding has a rich and long history. However, a recent explosion of multimedia Internet applications (such as teleconferencing and video streaming, for instance) renews interest in fast compression that also squeezes out as much redundancy as possible. In 2009 Jarek Duda invented his asymmetric numeral system (ANS). Apart from having a beautiful mathematical structure, it is very efficient and offers compression with a very low coding redundancy. ANS works well for any symbol source statistics, and it has become a preferred compression algorithm in the IT industry. However, designing an ANS instance requires a random selection of its symbol spread function. Consequently, each ANS instance offers compression with a slightly different compression ratio. The paper investigates the compression optimality of ANS. It shows that ANS is optimal for any symbol sources whose probability distribution is described by natural powers of 1/2. We use Markov chains to calculate ANS state probabilities. This allows us to precisely determine the ANS compression rate. We present two algorithms for finding ANS instances with a high compression ratio. The first explores state probability approximations in order to choose ANS instances with better compression ratios. The second algorithm is a probabilistic one. It finds ANS instances whose compression ratios can be made as close to the best ratio as required. This is done at the expense of the number θ of internal random "coin" tosses. The algorithm complexity is O(θL3), where is the number of ANS states. The complexity can be reduced to O(θLlog2L) if we use a fast matrix inversion. If the algorithm is implemented on a quantum computer, its complexity becomes O(θ(log2L)3).
Source coding has a rich and long history. However, a recent explosion of multimedia Internet applications (such as teleconferencing and video streaming, for instance) renews interest in fast compression that also squeezes out as much redundancy as possible. In 2009 Jarek Duda invented his asymmetric numeral system (ANS). Apart from having a beautiful mathematical structure, it is very efficient and offers compression with a very low coding redundancy. ANS works well for any symbol source statistics, and it has become a preferred compression algorithm in the IT industry. However, designing an ANS instance requires a random selection of its symbol spread function. Consequently, each ANS instance offers compression with a slightly different compression ratio. The paper investigates the compression optimality of ANS. It shows that ANS is optimal for any symbol sources whose probability distribution is described by natural powers of 1/2. We use Markov chains to calculate ANS state probabilities. This allows us to precisely determine the ANS compression rate. We present two algorithms for finding ANS instances with a high compression ratio. The first explores state probability approximations in order to choose ANS instances with better compression ratios. The second algorithm is a probabilistic one. It finds ANS instances whose compression ratios can be made as close to the best ratio as required. This is done at the expense of the number θ of internal random “coin” tosses. The algorithm complexity is O ( θ L 3 ) , where L is the number of ANS states. The complexity can be reduced to O ( θ L log 2 L ) if we use a fast matrix inversion. If the algorithm is implemented on a quantum computer, its complexity becomes O ( θ ( log 2 L ) 3 ) .
Source coding has a rich and long history. However, a recent explosion of multimedia Internet applications (such as teleconferencing and video streaming, for instance) renews interest in fast compression that also squeezes out as much redundancy as possible. In 2009 Jarek Duda invented his asymmetric numeral system (ANS). Apart from having a beautiful mathematical structure, it is very efficient and offers compression with a very low coding redundancy. ANS works well for any symbol source statistics, and it has become a preferred compression algorithm in the IT industry. However, designing an ANS instance requires a random selection of its symbol spread function. Consequently, each ANS instance offers compression with a slightly different compression ratio. The paper investigates the compression optimality of ANS. It shows that ANS is optimal for any symbol sources whose probability distribution is described by natural powers of 1/2. We use Markov chains to calculate ANS state probabilities. This allows us to precisely determine the ANS compression rate. We present two algorithms for finding ANS instances with a high compression ratio. The first explores state probability approximations in order to choose ANS instances with better compression ratios. The second algorithm is a probabilistic one. It finds ANS instances whose compression ratios can be made as close to the best ratio as required. This is done at the expense of the number θ of internal random “coin” tosses. The algorithm complexity is O(θL[sup.3]), where L is the number of ANS states. The complexity can be reduced to O(θLlog[sub.2]L) if we use a fast matrix inversion. If the algorithm is implemented on a quantum computer, its complexity becomes O(θ(log2L)[sup.3]).
Source coding has a rich and long history. However, a recent explosion of multimedia Internet applications (such as teleconferencing and video streaming, for instance) renews interest in fast compression that also squeezes out as much redundancy as possible. In 2009 Jarek Duda invented his asymmetric numeral system (ANS). Apart from having a beautiful mathematical structure, it is very efficient and offers compression with a very low coding redundancy. ANS works well for any symbol source statistics, and it has become a preferred compression algorithm in the IT industry. However, designing an ANS instance requires a random selection of its symbol spread function. Consequently, each ANS instance offers compression with a slightly different compression ratio. The paper investigates the compression optimality of ANS. It shows that ANS is optimal for any symbol sources whose probability distribution is described by natural powers of 1/2. We use Markov chains to calculate ANS state probabilities. This allows us to precisely determine the ANS compression rate. We present two algorithms for finding ANS instances with a high compression ratio. The first explores state probability approximations in order to choose ANS instances with better compression ratios. The second algorithm is a probabilistic one. It finds ANS instances whose compression ratios can be made as close to the best ratio as required. This is done at the expense of the number θ of internal random "coin" tosses. The algorithm complexity is O(θL3), where L is the number of ANS states. The complexity can be reduced to O(θLlog2L) if we use a fast matrix inversion. If the algorithm is implemented on a quantum computer, its complexity becomes O(θ(log2L)3).Source coding has a rich and long history. However, a recent explosion of multimedia Internet applications (such as teleconferencing and video streaming, for instance) renews interest in fast compression that also squeezes out as much redundancy as possible. In 2009 Jarek Duda invented his asymmetric numeral system (ANS). Apart from having a beautiful mathematical structure, it is very efficient and offers compression with a very low coding redundancy. ANS works well for any symbol source statistics, and it has become a preferred compression algorithm in the IT industry. However, designing an ANS instance requires a random selection of its symbol spread function. Consequently, each ANS instance offers compression with a slightly different compression ratio. The paper investigates the compression optimality of ANS. It shows that ANS is optimal for any symbol sources whose probability distribution is described by natural powers of 1/2. We use Markov chains to calculate ANS state probabilities. This allows us to precisely determine the ANS compression rate. We present two algorithms for finding ANS instances with a high compression ratio. The first explores state probability approximations in order to choose ANS instances with better compression ratios. The second algorithm is a probabilistic one. It finds ANS instances whose compression ratios can be made as close to the best ratio as required. This is done at the expense of the number θ of internal random "coin" tosses. The algorithm complexity is O(θL3), where L is the number of ANS states. The complexity can be reduced to O(θLlog2L) if we use a fast matrix inversion. If the algorithm is implemented on a quantum computer, its complexity becomes O(θ(log2L)3).
Source coding has a rich and long history. However, a recent explosion of multimedia Internet applications (such as teleconferencing and video streaming, for instance) renews interest in fast compression that also squeezes out as much redundancy as possible. In 2009 Jarek Duda invented his asymmetric numeral system (ANS). Apart from having a beautiful mathematical structure, it is very efficient and offers compression with a very low coding redundancy. ANS works well for any symbol source statistics, and it has become a preferred compression algorithm in the IT industry. However, designing an ANS instance requires a random selection of its symbol spread function. Consequently, each ANS instance offers compression with a slightly different compression ratio. The paper investigates the compression optimality of ANS. It shows that ANS is optimal for any symbol sources whose probability distribution is described by natural powers of 1/2. We use Markov chains to calculate ANS state probabilities. This allows us to precisely determine the ANS compression rate. We present two algorithms for finding ANS instances with a high compression ratio. The first explores state probability approximations in order to choose ANS instances with better compression ratios. The second algorithm is a probabilistic one. It finds ANS instances whose compression ratios can be made as close to the best ratio as required. This is done at the expense of the number θ of internal random “coin” tosses. The algorithm complexity is O(θL3), where L is the number of ANS states. The complexity can be reduced to O(θLlog2L) if we use a fast matrix inversion. If the algorithm is implemented on a quantum computer, its complexity becomes O(θ(log2L)3).
Audience Academic
Author Pieprzyk, Josef
Pawłowski, Marcin
Duda, Jarek
Mahboubi, Arash
Morawiecki, Paweł
Camtepe, Seyit
AuthorAffiliation 1 Institute of Computer Science, Polish Academy of Sciences, 01-248 Warsaw, Poland
4 School of Computing and Mathematics, Charles Sturt University, Port Macquarie, NSW 2444, Australia
3 Institute of Computer Science and Computer Mathematics, Jagiellonian University, 30-348 Cracow, Poland
2 Data61, CSIRO, Sydney, NSW 2122, Australia
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Issue 4
Keywords source coding
entropy coding
ANS
lossless compression
Language English
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Snippet Source coding has a rich and long history. However, a recent explosion of multimedia Internet applications (such as teleconferencing and video streaming, for...
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StartPage 672
SubjectTerms Algorithms
ANS
Approximation
Asymmetry
Coding
Complexity
Compression ratio
Entropy
entropy coding
Internet
Internet software
lossless compression
Markov analysis
Markov chains
Markov processes
Mathematical analysis
Multimedia
Multimedia communications
Multiplication & division
Optimization
Probability
Probability distribution
Quantum computers
Quantum computing
Redundancy
source coding
Statistical analysis
Streaming media
Teleconferencing
Video compression
Video transmission
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Title The Compression Optimality of Asymmetric Numeral Systems
URI https://www.ncbi.nlm.nih.gov/pubmed/37190460
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Volume 25
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