A combinatorial approach to Golomb forests
Optimal binary prefix-free codes for infinite sources with geometrically distributed frequencies, e.g., P={p i(1−p)} i=0 ∞, 0<p<1 , were first (implicitly) suggested by Golomb over 30 years ago in the context of run-length encodings. Ten years later Gallager and Van Voorhis exhibited such opti...
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| Vydáno v: | Theoretical computer science Ročník 263; číslo 1; s. 283 - 304 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
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Elsevier B.V
2001
Elsevier |
| Témata: | |
| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | Optimal binary prefix-free codes for infinite sources with geometrically distributed frequencies, e.g.,
P={p
i(1−p)}
i=0
∞,
0<p<1
, were first (implicitly) suggested by Golomb over 30 years ago in the context of run-length encodings. Ten years later Gallager and Van Voorhis exhibited such optimal codes for all values of
p. These codes were derived by using the Huffman encoding algorithm to build optimal codes for
finite sources and then showing that the finite codes converge in a very specific sense to the infinite one. In this note, we present a new combinatorial approach to solve the same problem, one that does not use the Huffman algorithm, but instead treats a coding tree as an infinite sequence of integers and derives properties of the sequence. One consequence of this new approach is a complete characterization of
all of the optimal codes; in particular, it shows that for all
p,0<p<1, except for an easily describable countable set, there is a unique optimal code, but for each
p in this countable set there are an
uncountable number of optimal codes. Another consequence is a derivation of infinite codes for geometric sources when the encoding alphabet is no longer restricted to be the binary one. A final consequence is the extension of the results to optimal forests instead of being restricted to optimal trees. |
|---|---|
| AbstractList | Optimal binary prefix-free codes for infinite sources with geometrically distributed frequencies, e.g.,
P={p
i(1−p)}
i=0
∞,
0<p<1
, were first (implicitly) suggested by Golomb over 30 years ago in the context of run-length encodings. Ten years later Gallager and Van Voorhis exhibited such optimal codes for all values of
p. These codes were derived by using the Huffman encoding algorithm to build optimal codes for
finite sources and then showing that the finite codes converge in a very specific sense to the infinite one. In this note, we present a new combinatorial approach to solve the same problem, one that does not use the Huffman algorithm, but instead treats a coding tree as an infinite sequence of integers and derives properties of the sequence. One consequence of this new approach is a complete characterization of
all of the optimal codes; in particular, it shows that for all
p,0<p<1, except for an easily describable countable set, there is a unique optimal code, but for each
p in this countable set there are an
uncountable number of optimal codes. Another consequence is a derivation of infinite codes for geometric sources when the encoding alphabet is no longer restricted to be the binary one. A final consequence is the extension of the results to optimal forests instead of being restricted to optimal trees. Optimal binary prefix-free codes for infinite sources with geometrically distributed frequencies, e.g., P = {p super(i)(1 - p)} super( arrow down )b sub(i) super(!) sub(=) sub(0), 0 < p < 1, were first (implicitly) suggested by Golomb over 30 years ago in the context of run-length encodings. Ten years later Gallager and Van Voorhis exhibited such optimal codes for all values of p. These codes were derived by using the Huffman encoding algorithm to build optimal codes for finite sources and then showing that the finite codes converge in a very specific sense to the infinite one. In this note, we present a new combinatorial approach to solve the same problem, one that does not use the Huffman algorithm, but instead treats a coding tree as an infinite sequence of integers and derives properties of the sequence. One consequence of this new approach is a complete characterization of all of the optimal codes; in particular, it shows that for all p, 0 < p < 1, except for an easily describable countable set, there is a unique optimal code, but for each p in this countable set there are an uncountable number of optimal codes. Another consequence is a derivation of infinite codes for geometric sources when the encoding alphabet is no longer restricted to be the binary one. A final consequence is the extension of the results to optimal forests instead of being restricted to optimal trees. copyright 2001 Elsevier Science B.V. All rights reserved. |
| Author | Golin, Mordecai J. |
| Author_xml | – sequence: 1 givenname: Mordecai J. surname: Golin fullname: Golin, Mordecai J. email: golin@cs.ust.hk organization: Department of Computer Science, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, People's Republic of China |
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| Keywords | Combinatorial method Optimal forest Tree(graph) Golomb forest Coding Probability distribution Graph theory Hoffman encoding problem Convergence |
| Language | English |
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| References | R.G. Gallager, D.C. Van Voorhis, Optimal source codes for geometrically distributed integer alphabets, IEEE Trans. Inform. Theory March 1975 228–230. Abrahams (BIB1) 1994; 331B Humblet (BIB7) 1978; IT-24 S.W. Golomb Run length encodings IEEE Tran. Informat. Theory IT-12 (1966) 399–401. N. Merhav, G. Seroussi, M.J. Weinberger, Optimal prefix codes for two-sided geometric distributions (Abstract), Proc. Internat. Sympos. on Information Theory, 1997, p. 71. N. Merhav, G. Seroussi, M.J. Weinberger, Universal probability assignment in the class of two-sided geometric distributions (Abstract), Proc. Internat. Symp. on Information Theory, 1997, p. 70. J. Abrahams, Code and parse trees for lossless source encoding, Sequences 1997 (1997). Hwang (BIB8) 1974; 69 Kato, Sun Han H. Nagaoka (BIB10) 1996; 42 Yao, Hwang (BIB15) 1990; 24 Huffman (BIB6) 1952; 40 A. Kato, Huffman-like optimal prefix codes and search codes for infinite alphabets, Manuscript, January 20, 1997. Sedgewick (BIB14) 1983 Hassan (BIB5) 1984; 32 Linder, Tarokh, Zeger (BIB11) 1997; 43 10.1016/S0304-3975(00)00250-4_BIB9 Hwang (10.1016/S0304-3975(00)00250-4_BIB8) 1974; 69 10.1016/S0304-3975(00)00250-4_BIB4 10.1016/S0304-3975(00)00250-4_BIB2 10.1016/S0304-3975(00)00250-4_BIB3 Abrahams (10.1016/S0304-3975(00)00250-4_BIB1) 1994; 331B Humblet (10.1016/S0304-3975(00)00250-4_BIB7) 1978; IT-24 Sedgewick (10.1016/S0304-3975(00)00250-4_BIB14) 1983 10.1016/S0304-3975(00)00250-4_BIB12 10.1016/S0304-3975(00)00250-4_BIB13 Linder (10.1016/S0304-3975(00)00250-4_BIB11) 1997; 43 Hassan (10.1016/S0304-3975(00)00250-4_BIB5) 1984; 32 Kato (10.1016/S0304-3975(00)00250-4_BIB10) 1996; 42 Yao (10.1016/S0304-3975(00)00250-4_BIB15) 1990; 24 Huffman (10.1016/S0304-3975(00)00250-4_BIB6) 1952; 40 |
| References_xml | – volume: 331B start-page: 265 year: 1994 end-page: 271 ident: BIB1 article-title: Huffman-type codes for infinite source distributions publication-title: J. Franklin Inst. – reference: R.G. Gallager, D.C. Van Voorhis, Optimal source codes for geometrically distributed integer alphabets, IEEE Trans. Inform. Theory March 1975 228–230. – volume: 69 start-page: 146 year: 1974 end-page: 150 ident: BIB8 article-title: On finding a single defective in binomial group testing publication-title: J. Amer. Statist. Assoc. – reference: S.W. Golomb Run length encodings IEEE Tran. Informat. Theory IT-12 (1966) 399–401. – volume: 24 start-page: 167 year: 1990 end-page: 175 ident: BIB15 article-title: On optimal nested group testing algorithms publication-title: J. Statist. Plann. Inference – volume: 32 start-page: 423 year: 1984 end-page: 439 ident: BIB5 article-title: A dichotomous search for a geometric random variable publication-title: Oper. Res. – reference: A. Kato, Huffman-like optimal prefix codes and search codes for infinite alphabets, Manuscript, January 20, 1997. – reference: N. Merhav, G. Seroussi, M.J. Weinberger, Universal probability assignment in the class of two-sided geometric distributions (Abstract), Proc. Internat. Symp. on Information Theory, 1997, p. 70. – year: 1983 ident: BIB14 publication-title: Algorithms – reference: N. Merhav, G. Seroussi, M.J. Weinberger, Optimal prefix codes for two-sided geometric distributions (Abstract), Proc. Internat. Sympos. on Information Theory, 1997, p. 71. – reference: J. Abrahams, Code and parse trees for lossless source encoding, Sequences 1997 (1997). – volume: 43 start-page: 2026 year: 1997 end-page: 2028 ident: BIB11 article-title: Existence of optimal prefix codes for infinite source alphabets publication-title: IEEE Trans. Inform. Theory – volume: IT-24 start-page: 110 year: 1978 end-page: 112 ident: BIB7 article-title: Optimal source coding for a class of integer alphabets publication-title: IEEE Trans. Inform. Theory – volume: 42 start-page: 977 year: 1996 end-page: 984 ident: BIB10 article-title: Huffman coding with an infinite alphabet publication-title: IEEE Trans. Inform. Theory – volume: 40 start-page: 1098 year: 1952 end-page: 1101 ident: BIB6 article-title: A method for the construction of minimum-redundancy codes publication-title: Proc. IRE – year: 1983 ident: 10.1016/S0304-3975(00)00250-4_BIB14 – ident: 10.1016/S0304-3975(00)00250-4_BIB2 – ident: 10.1016/S0304-3975(00)00250-4_BIB12 doi: 10.1109/ISIT.1997.612986 – ident: 10.1016/S0304-3975(00)00250-4_BIB9 – ident: 10.1016/S0304-3975(00)00250-4_BIB3 doi: 10.1109/TIT.1975.1055357 – ident: 10.1016/S0304-3975(00)00250-4_BIB4 doi: 10.1109/TIT.1966.1053907 – volume: 32 start-page: 423 issue: 2 year: 1984 ident: 10.1016/S0304-3975(00)00250-4_BIB5 article-title: A dichotomous search for a geometric random variable publication-title: Oper. Res. doi: 10.1287/opre.32.2.423 – volume: IT-24 start-page: 110 issue: 1 year: 1978 ident: 10.1016/S0304-3975(00)00250-4_BIB7 article-title: Optimal source coding for a class of integer alphabets publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.1978.1055813 – volume: 24 start-page: 167 year: 1990 ident: 10.1016/S0304-3975(00)00250-4_BIB15 article-title: On optimal nested group testing algorithms publication-title: J. Statist. Plann. Inference doi: 10.1016/0378-3758(90)90039-W – ident: 10.1016/S0304-3975(00)00250-4_BIB13 doi: 10.1109/ISIT.1997.612985 – volume: 331B start-page: 265 issue: 3 year: 1994 ident: 10.1016/S0304-3975(00)00250-4_BIB1 article-title: Huffman-type codes for infinite source distributions publication-title: J. Franklin Inst. doi: 10.1016/0016-0032(94)90099-X – volume: 40 start-page: 1098 year: 1952 ident: 10.1016/S0304-3975(00)00250-4_BIB6 article-title: A method for the construction of minimum-redundancy codes publication-title: Proc. IRE doi: 10.1109/JRPROC.1952.273898 – volume: 43 start-page: 2026 issue: 6 year: 1997 ident: 10.1016/S0304-3975(00)00250-4_BIB11 article-title: Existence of optimal prefix codes for infinite source alphabets publication-title: IEEE Trans. Inform. Theory doi: 10.1109/18.641571 – volume: 42 start-page: 977 issue: 3 year: 1996 ident: 10.1016/S0304-3975(00)00250-4_BIB10 article-title: Huffman coding with an infinite alphabet publication-title: IEEE Trans. Inform. Theory doi: 10.1109/18.490559 – volume: 69 start-page: 146 issue: 345 year: 1974 ident: 10.1016/S0304-3975(00)00250-4_BIB8 article-title: On finding a single defective in binomial group testing publication-title: J. Amer. Statist. Assoc. doi: 10.1080/01621459.1974.10480141 |
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| Snippet | Optimal binary prefix-free codes for infinite sources with geometrically distributed frequencies, e.g.,
P={p
i(1−p)}
i=0
∞,
0<p<1
, were first (implicitly)... Optimal binary prefix-free codes for infinite sources with geometrically distributed frequencies, e.g., P = {p super(i)(1 - p)} super( arrow down )b sub(i)... |
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| SubjectTerms | Combinatorics Combinatorics. Ordered structures Exact sciences and technology Graph theory Mathematics Sciences and techniques of general use |
| Title | A combinatorial approach to Golomb forests |
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