A simpler analysis of Burrows–Wheeler-based compression
In this paper, we present a new technique for worst-case analysis of compression algorithms which are based on the Burrows–Wheeler Transform. We mainly deal with the algorithm proposed by Burrows and Wheeler in their first paper on the subject [M. Burrows, D.J. Wheeler, A block sorting lossless data...
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| Vydáno v: | Theoretical computer science Ročník 387; číslo 3; s. 220 - 235 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier B.V
22.11.2007
|
| Témata: | |
| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | In this paper, we present a new technique for worst-case analysis of compression algorithms which are based on the Burrows–Wheeler Transform. We mainly deal with the algorithm proposed by Burrows and Wheeler in their first paper on the subject [M. Burrows, D.J. Wheeler, A block sorting lossless data compression algorithm, Technical Report 124, Digital Equipment Corporation, Palo Alto, California, 1994], called
bw0. This algorithm consists of the following three essential steps: (1) Obtain the Burrows–Wheeler Transform of the text, (2) Convert the transform into a sequence of integers using the move-to-front algorithm, (3) Encode the integers using Arithmetic code or any order-0 encoding (possibly with run-length encoding).
We achieve a strong upper bound on the worst-case compression ratio of this algorithm. This bound is significantly better than bounds known to date and is obtained via simple analytical techniques. Specifically, we show that for any input string
s
, and
μ
>
1
, the length of the compressed string is bounded by
μ
⋅
|
s
|
H
k
(
s
)
+
log
(
ζ
(
μ
)
)
⋅
|
s
|
+
μ
g
k
+
O
(
log
n
)
where
H
k
is the
k
th order empirical entropy,
g
k
is a constant depending only on
k
and on the size of the alphabet, and
ζ
(
μ
)
=
1
1
μ
+
1
2
μ
+
⋯
is the standard zeta function. As part of the analysis, we prove a result on the compressibility of integer sequences, which is of independent interest.
Finally, we apply our techniques to prove a worst-case bound on the compression ratio of a compression algorithm based on the Burrows–Wheeler Transform followed by distance coding, for which worst-case guarantees have never been given. We prove that the length of the compressed string is bounded by
1.7286
⋅
|
s
|
H
k
(
s
)
+
g
k
+
O
(
log
n
)
. This bound is
better than the bound we give for
bw0. |
|---|---|
| AbstractList | In this paper, we present a new technique for worst-case analysis of compression algorithms which are based on the Burrows–Wheeler Transform. We mainly deal with the algorithm proposed by Burrows and Wheeler in their first paper on the subject [M. Burrows, D.J. Wheeler, A block sorting lossless data compression algorithm, Technical Report 124, Digital Equipment Corporation, Palo Alto, California, 1994], called
bw0. This algorithm consists of the following three essential steps: (1) Obtain the Burrows–Wheeler Transform of the text, (2) Convert the transform into a sequence of integers using the move-to-front algorithm, (3) Encode the integers using Arithmetic code or any order-0 encoding (possibly with run-length encoding).
We achieve a strong upper bound on the worst-case compression ratio of this algorithm. This bound is significantly better than bounds known to date and is obtained via simple analytical techniques. Specifically, we show that for any input string
s
, and
μ
>
1
, the length of the compressed string is bounded by
μ
⋅
|
s
|
H
k
(
s
)
+
log
(
ζ
(
μ
)
)
⋅
|
s
|
+
μ
g
k
+
O
(
log
n
)
where
H
k
is the
k
th order empirical entropy,
g
k
is a constant depending only on
k
and on the size of the alphabet, and
ζ
(
μ
)
=
1
1
μ
+
1
2
μ
+
⋯
is the standard zeta function. As part of the analysis, we prove a result on the compressibility of integer sequences, which is of independent interest.
Finally, we apply our techniques to prove a worst-case bound on the compression ratio of a compression algorithm based on the Burrows–Wheeler Transform followed by distance coding, for which worst-case guarantees have never been given. We prove that the length of the compressed string is bounded by
1.7286
⋅
|
s
|
H
k
(
s
)
+
g
k
+
O
(
log
n
)
. This bound is
better than the bound we give for
bw0. |
| Author | Landau, Shir Kaplan, Haim Verbin, Elad |
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| Cites_doi | 10.1145/1082036.1082039 10.1145/214762.214771 10.1007/s00453-004-1094-1 10.1016/0166-218X(93)00116-H 10.1007/11841036_67 10.1109/TIT.1987.1057284 10.1109/TIT.1975.1055349 10.1145/5684.5688 10.1145/290159.290162 10.1145/45072.45074 10.1145/382780.382782 10.1002/spe.426 10.1145/1082036.1082043 10.1007/978-3-540-30213-1_23 |
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| Keywords | Text compression Worst-case analysis Burrows–Wheeler Transform Distance coding |
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| References_xml | – ident: b25 article-title: bzip2, a program and library for data compression – volume: 32 start-page: 99 year: 2002 end-page: 111 ident: b8 article-title: Second step algorithms in the Burrows–Wheeler compression algorithm publication-title: Software - Practice and Experience – volume: 29 start-page: 320 year: 1986 end-page: 330 ident: b4 article-title: A locally adaptive data compression scheme publication-title: Communications of the ACM – volume: 40 start-page: 1098 year: 1952 end-page: 1101 ident: b19 article-title: A method for the construction of minimum-redundancy codes publication-title: Proceedings of the Institute of Radio Engineers – ident: b1 article-title: The Canterbury Corpus – volume: 30 start-page: 520 year: 1987 end-page: 540 ident: b26 article-title: Arithmetic coding for data compression publication-title: Communications of the ACM – reference: E. Binder, Distance coder, Usenet group comp.compression, 2000 – reference: M. Burrows, D.J. Wheeler, A block sorting lossless data compression algorithm, Technical Report 124, Digital Equipment Corporation, Palo Alto, California, 1994 – reference: R. Grossi, A. Gupta, J.S. 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Rao, Succinct indexable dictionaries with applications to encoding k-ary trees and multisets, in: Proc. 13th ACM-SIAM Symposium on Discrete Algorithms, SODA ’02, 2002, pp. 233–242 – reference: P. Ferragina, R. Giancarlo, G. 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Navarro, An alphabet friendly FM-index, in: Proc. 11th Symposium on String Processing and Information Retrieval, SPIRE ’04, 2004, pp. 150–160 – reference: R. Grossi, A. Gupta, J.S. Vitter, When indexing equals compression: Experiments with compressing suffix arrays and applications, in: Proc. 15th ACM-SIAM Symposium on Discrete Algorithms, SODA ’04, 2004, pp. 636–645 – volume: 52 start-page: 552 year: 2005 end-page: 581 ident: b14 article-title: Indexing compressed text publication-title: Journal of the ACM – reference: P. Ferragina, R. Giancarlo, G. 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