Modified Golomb Code For Integer Representation

IJSRD - International Journal for Scientific Research & Development| Vol. 3, Issue 10, 2015 | ISSN (online): 2321-0613
All rights reserved by www.ijsrd.com 1100
Modified Golomb Code for Integer Representation
Nelson Raja Joseph1 Jaganathan P2 Domnic Sandanam3
1
Department of Computer Science 2,3
Department of Computer Applications
1
Bharathiyar University, Coimbatore, India 2
PSNA College of Engineering & Technology Dindigul,
India 3
National Institute of Technology, Tiruchirappalli India
Abstract— In this computer age, all the computer applications
handle data in the form of text, numbers, symbols and
combination of all of them. The primary objective of data
compression is to reduce the size of data while data needs to
be stored and transmitted in the digital devices. Hence, the
data compression plays a vital role in the areas of data storage
and data transmission. Golomb code, which is a variable-
length integer code, has been used for text compression,
image compression, video compression and audio
compression. The drawback of Golomb code is that it
requires more bits to represent large integers if the divisor is
small. Alternatively, Golomb code needs more bits to
represent small integers if the divisor is large. This paper
proposes Modified Golomb Code based on Golomb Code,
Extended Golomb Code to represent small as well as large
integers compactly for the chosen divisor. In this work, as an
application of Modified Golomb Code, Modified Golomb
Code is used with Burrows-Wheeler transform for text
compression. The performances of Golomb Code and
Modified Golomb Code are evaluated on Calcary corpus
dataset. The experimental results show that the proposed code
provides better compression rate than Golomb code on an
average. The performance of the proposed code is also
compared with Extended Golomb Codes (EGC). The
comparison results show that the proposed code achieves
significant improvement for the binary files of Calgary
corpus comparing to EGC.
Key words: Variable Length Code, Golomb Code, Modified
Golomb Code, Burrows-Wheeler Compression
I. INTRODUCTION
The main aim of data compression is to store data with the
minimum number of bits in storage devices and transmit in
low band width communication networks. Data-compression
methods can be generally classified into two types i.e lossy
and lossless. In lossless compression, data can be compressed
and decompressed as exactly identical with the source data
without any loss of data. Lossless compression technique is
used, in which the decompressed data must be identical to the
source data such as financial data, executable programs, text
documents, and source code. Lossless data compression is
used in many applications such as zip tools and wireless
sensor networks. Lossy data compression has a certain loss
of information and decompressed data is not 100% identical
to the source data. Lossy data compression technique is used
to compress video, audio and images. Various codes have
been applied for data compression [1].
In contrast with the fixed-length codes, statistical
coding methods achieve compression by assigning short-
length codes to the more frequent occurring symbols and
long-length codes to rarely occurring symbols of the source
file which needs to be compressed. The statistical methods
require the probabilities of the input symbols to generate
variable-length codes. Huffman coding [2] and Shannon-
Fano [3] methods are examples for statistical methods which
use symbol tables while decoding the compressed data. There
are other coding methods such as Elias Gamma codes, Elias
Delta code, Golomb code, Fibonacci codes [4] and Extended
Golomb Code (EGC) [5], which do not require the probability
values of the input data to produce variable-length codes and
these methods are called as variable-length integer coding
methods or variable-length integer codes. Since variable-
length integer codes do not require symbol table and
probability values, these are more preferable in the
applications which require fast encoding and storage.
In this paper, we propose a new code, Modified
Golomb Code (MGC), to produce variable-length codes by
representing non-negative integers. Alternatively, MGC can
encode or represent non-negative integers very compactly.
Golomb Code (GC) [6] has been used in several applications
such as lossless image codecs, audio codecs and search
engines [7 - 8]. But, the disadvantage of GC is that it requires
more bits to represent large integers if the divisor (d) is small.
Alternatively, GC needs more bits to represent small integers
if the divisor is large. Hence, GC could not be the best choice
for the applications which have the distribution of small and
large integers. To overcome the drawback of GC, we propose
Modified Golomb Code based on GC.
II. GOLOMB CODE (GC)
Golomb Code was proposed by Solomon Golomb in 1966 for
lossless data compression. In GC, the compact representation
of non-negative integers ni depends on the selection of the
divisor d. In the first step of GC, the given number n (>0) is
first divided by a divisor d. The quotient (q) and the remainder
(r) of the given n are then used to generate codes. The
formula given in Equation (1) is used to calculate the quotient
(q) and the remainder (r) for the given n.








1
1
qdnr
d
n
q (1)
GC contains two parts. The first part is the quotient
value of (q +1) which is coded in unary code (i.e q zeros
followed by single one or q ones followed by single zero) [1]
and the second part is binary code of remainder (r). For
example, when divisor d = 3, it produces three remainders, 0,
1, 2, and are coded as 0, 10 and 11 respectively (See Table 1).
Table 2 shows the GC for divisors d =2, 3 and 4. The bit
lengths of GC (d=2, 3 and 4) to represent the integers in the
range 0 - 255 are calculated and are given in Table 4. It is
observed from Table 4 that GC (d=2) offers compact
representation for small range (1-5) and provides poor
representation for middle (32-63) and large (64-255) range of
integers. Also, for other divisors (d = 4, 8), GC does not give
better representation for middle and large range of integers.
In order to improve the integer representation of GC, a new
method based on GC is proposed in this paper.
Remainders
Binary codes
d=2 d=3 d=4

(IJSRD/Vol. 3/Issue 10/2015/254)
0 0 0 00
1 1 10 01
2 - 11 10
3 - - 11
Table 1: Codes for remainders divisor d = 2, 3 and 4
Integer n
GC
d=2 d=3 d=4
1 0 | 0 0 | 0 0 | 00
2 10 | 0 0 | 10 0 | 01
3 10 | 1 0 | 11 0 | 10
4 110 | 0 10 | 0 0 | 11
5 110 | 1 10 | 10 10 | 00
6 1110 | 0 10 | 11 10 | 01
7 1110 | 1 110 | 0 10 | 10
8 11110 | 0 110 | 10 10 | 11
9 11110 | 1 110 | 11 110 | 00
10 111110 | 0 1110 | 0 110 | 01
Table 2: GC for the integers 1 to 10
III. MODIFIED GOLOMB CODE (MGC)
In this section, a new variable-length integer code, Modified
Golomb Code (MGC), is proposed to represent non-negative
integers compactly. The proposed MGC is designed based on
GC and EGC. In GC, the given number n (>0) is first divided
by a divisor d to obtain the quotient (q) and the remainder (r).
Then, the q and r of given n are used to generate codes. But,
the number of bits required by unary in GC is more for large
range of integers. Hence, GC has the drawback of requiring
long-bit length to represent middle, large range of integers. In
EGC, the given integer n (>0) is divided by a divisor d
recursively until the last quotient becomes zero. The
remainders (ri) obtained in each division and the number of
divisions (C) are used to generate codes. The drawback of
EGC is that the divisions are made successively until the last
quotient becomes zero whether the successive division gives
better representation ( less bits) or not. Hence, in MGC, if the
number of bits needed to represent current quotient is less
than the number bits required after the division (i.e bits
requirements to represent next quotient, remainder and
count), then the division will be stopped. Due to this
condition, MGC can overcome the drawback of GC and to
achieve better representation for large integers than EGC.
In MGC, the given integer n is divided by a divisor
d (2m
≤ d<2m+1
) successively until either the condition qc
becomes zero or
C C log d ) < ( 1) ( 1) log d
2 2
(q (q 1 C C )c c 1
        

.
Alternatively, In MGC, successive division will be
stopped when qc = 0 or < log d
2
(q (q 1 )c c 1
 

. Here, qc is
the quotient obtained in Cth
division. In MGC, all the
remainders of n obtained by the divisor d are preserved ri (i=1,
2...C). MGC has three parts to represent an given integer n:
the quotient (qc) , count (C) and remainders (ri). The quotient
(qc) and the count (C) are encoded using binary code and
unary code (described in section 2), respectively. The
remainders ri are coded using binary code. The format of
MGC is given as: Binary Code (qc) | Unary Code (C) | Binary
Code (rc rc-1…r1).
A. Algorithm for MGC Integer Encoding
1) The non-negative integer n is divided by the divisor
d (2m
≤ d<2m+1
) repeatedly C times until any one of
the following conditions is satisfied.
 ( log d
2
q q 1+ )c c 1
 

 qc = 0
2) Count the number of divisions made as C and
preserve the remainders produced in each division
as r1, r2….rc.
3) Encode the last quotient (qc) obtained in step-1 and
the count (C) obtained in step-2 using log2(m+1)bits
and unary code, respectively. The remainder ri is
coded in  2log (2 1)
m
d   bits when qc =0 && C≥2, in
log2(d-1) bits when qc =0 && C =1 and in log2d bits
for all other cases. Then, the MGC for n is generated
by combining the codes for qc, C and ri in the coding
format given below:
Binary Code (qc) | Unary Code (C) | Binary Code (rC
, rC-1…r1)
Repeat steps 1- 3 for all the integers to be coded.
It is shown in the Table 3 that the possible last
quotients and remainders for the divisors d=3&4 if the
proposed method is applied to represent integers from 1 to
255. It is observed from the Table 3 that the number of
possible last quotients for d=3 is two (0,1), for d=4 it is three
(0,1,2). Also, the last remainder is only 2 when qc=0 and C ≥
2 for d=3 and for d=4, it is only 3. These are the unique pattern
occurred due the condition given in the algorithm. The same
trend happens for other devisors also. According to this, the
remainders, the last quotient and the last remainder are coded
as given in the encoding algorithm.
n
d=3 d=4
qc rc c qc rc c
1 0 1 1 0 1 1
2 0 2 1 0 2 1
3 1 0 1 0 3 1
4 1 1 1 1 0 1
5 1 2 1 1 1 1
10 1 1 2 2 2 1
15 1 2 2 0 3 2
25 0 2 2 1 2 2
50 1 2 3 0 3 3
100 1 0 4 1 2 3
200 0 2 5 0 3 4
255 1 0 5 0 3 4
Table 3: The last quotients and remainders of MGC for the
integers 1 to 255
n
MGC
d=3 d=4
1 0|1|0 0| 1 | 0
2 0|1|1 0| 1| 10
3 1|1|0 0 | 1| 11

4 1|1|10 10 | 1 | 00
5 1|1|11 10 |1 | 01
6 0|01|0 10 | 1 | 10
7 0|01|10 10 | 1 | 11
8 0|01|11 11 | 1 | 00
9 1|01|00 11 |1 | 01
10 1|01|010 11 | 1 | 10
Table 4: MGC for the integers 1 to 10
Illustration: n = 50, d = 3
Here, Since q3 < q3/3 +1+ Code (remainder of q3/3),
dividing further is stopped.
MGC(50) = Binary Code 3
( )q | Unary Code (C) |
Code 3 2 1
( , , )r r r
= Binary Code (1) | Unary Code (3) | Code ( 2, 1, 2)
= 1| 001| 111011 Use Algorithm. (for d =3, remainders 0 (0),
1(10), 2(11))
Table 4 shows the MGC for integers 1 to 10 for d = 3 and 4
B. Algorithm for MGC Integer Decoding
The following steps are used to decode the compressed data.
1) Read log2m bits and decode the bits into respective
last quotient and assign into qc.
2) Read the C bits until bit '1' is encountered, which is
used to read the C number of remainders.
3) Then, read  2log (2 1)
m
d   bits if qc =0 and C≥2 (else)
log2(d-1) bits if qc =0 and C =1 (else) log2d bits for
all other cases and decode the first remainder. Then,
read ((C-1) × log2d) number of bits further to decode
(C-1) remainders.
Repeat steps 1- 3 for all the integers to be decoded.
Decode: 1| 001| 111011 ; d = 3 , C = 3
3q = 1; C = 3; Codes: 11,10, 11 denote the remainders 3
r =
2, 2
r = 1, 1
r = 2, respectively.
2q = 3q  d + 3r (C = 3) (the value of 3q = 1 & 3r = 2)
2q = 1  3 + 2  2q = 5
1
q = 2
q  d + 2
r (C = 2) (the value of 3
q = 5 & 2
r = 1)
1
q = 5  3 + 1  1
q = 16
n = 1
q  d + 1
r (C = 1) (the value of 1
q = 16 & 1
r =2)
n = 16  3 + 2 = 50.
In general, the given integer n is decoded using eq.(2) .
n =
1
( )i i
i c
q d r

  (2)
IV. BIT-LENGTH COMPARISON
The bit lengths of MGC, GC and EGC for divisor (d=3 and
4) have been calculated and are given in Table 5. It is
observed from Table 5 that GC gives compact representation
for small range of integers (i.e 1 - 10) and gives poor
representation for other range of integers.
n d=3 d=4
MGC GC EGC MGC GC EGC
1 3 3 2 3 3 2
2 3 3 2 4 3 3
3 3 4 4 4 3 3
4 4 4 4 5 3 5
5 4 4 5 5 4 5
10 6 5 6 5 5 6
15 6 7 7 5 6 6
25 8 10 8 8 8 8
50 10 19 10 8 15 9
100 11 35 12 11 27 11
200 13 69 13 11 52 12
255 13 87 14 11 66 12
Table 5: Bit length of comparison of MGC, GC and EGC
MGC offers significantly better representation for
small to large range of integers. For small values, MGC is one
bit longer than GC. But, GC requires more bits than MGC for
mid-range values and large values. It is also observed from
Table 5 that MGC achieves better representation significantly
than EGC for large integers.
V. EXPERIMENTAL RESULTS AND DISCUSSION
Variable length integer codes (VLC) have been used to
compress text data [11], medical data [12] and remote sensing
data [13]. In this section, as an application of MGC, MGC is
used as the final stage coder of BWT compressor for text data
compression as shown in Figure 1. BWT compressor has four
stages as shown in Figure 1. In first stage of BWT
compressor, BWT computes the permutation of the given
input. Then, move-to-front (MTF) coder encodes the output
of first stage of BWT. After this, the output of MTF will be
encoded by run-length encoding(RLE). In the final stage, the
output of RLE will be encoded by the VLC coders. In the
experiment, Calgary corpus dataset [9] is used to test the
performance of MGC. The calgary corpus dataset contains
both text files (bib, book1, book2, news, paper1, paper2,
paper3, paper6, progc, progl, progp, trans and binary files
(geo, obj1, obj2, pic). Compression rate given in equation (3)
is used as a metric for performance evaluation. The
compression results of MGC are compared with the results of
GC and EGC as given in Table 6.
Fig. 1: Stages of Burrows-Wheeler Compressor
fileinputin thesymbolsofNumber
fileCompressedtheofSize
ratenCompressio 
(3)
It is observed from Table 6 that MGC achieves low
compression rate on an average than GC. GC provides better
compression rate for text files (bib, book1, book2, news,
paper1-paper6) of calgary corpus and gives poor compression
rate when d is increased. But, it achieves better results for
binary files when d is large. When MGC is compared with
GC, MGC gives better results for both text and binary files
than GC (d=8,16 ) with large divisor. For small divisor (d=4),
GC performs better than MGC for some of the text files and
gives poor performance for binary files. However, when a

large divisor is selected, GC may achieve better result for
binary files, but it cannot achieve better results for text files
compared to MGC. The reason is that GC needs more bits for
small integers when d is large and require more bits for large
integer when d is small. Since the output of BWT compressor
contains more small range of integers and less middle range
of integers for text files; and contains all range of integers
(small, middle and large integers) for binary files including
rand file, GC with small divisor can perform well at some
extent for text files and might not perform well for binary files
when compared to MGC. The distributions of the integers for
sample files (bib, geo, rand) are shown in the Fig.2. It is
observed from Fig.2 that bib file shows the significant
distribution of small range of integers and other files (rand,
geo) show the significant distribution of small, middle and
large range of integers. It is concluded that GC could not
perform well for the files which contain significant
distribution of small, middle and large range of integers. Both
MGC and EGC could perform well for both text files and
binary files. Since MGC can obtain better representation of
middle and large range of integers as shown in Table 5 than
EGC, MGC could perform well than EGC when the
collections contain more middle and large range integers. It
is observed from Table 6 that MGC obtains better
compression performance than EGC for geo, obj1, pic and
rand files. But, it is inferior to EGC for text files. Finally, it is
concluded that the performances of the codes depend on the
distributions of integers. The proposed code could perform
well for the collections which contain significant distributions
of middle and large range integers comparing to GC and
EGC.
Cor
pus
GC MGC
EG
C
d=4 d=8 d=16
d=
3
d=
4
d=2 d=3
bib
2.36
2
2.57
6
2.999
2.3
54
2.5
33
2.21
9
2.21
6
book
1
3.13
0
3.60
0
4.301
3.2
56
3.5
86
3.07
7
3.07
4
book
2
2.75
1
3.09
7
3.659
2.7
75
3.0
32
2.57
3
2.58
8
geo
11.1
20
7.35
0
5.873
5.4
11
5.3
48
6.15
2
5.56
3
news
3.26
8
3.41
2
3.889
3.1
20
3.3
40
2.99
7
2.97
2
obj1
7.72
6
5.52
9
4.978
4.2
79
4.2
87
4.62
8
4.29
4
obj2
4.41
6
3.60
5
3.512
2.8
98
2.9
93
2.87
8
2.78
5
pape
r1
2.98
5
3.26
6
3.807
2.9
51
3.1
88
2.75
2
2.76
7
pape
r2
2.88
1
3.24
6
3.839
2.9
32
3.1
97
2.74
5
2.75
5
pape
r3
3.16
0
3.53
0
4.158
3.2
20
3.4
93
3.04
2
3.04
5
pape
r5
3.09
2
3.23
1
4.525
3.6
46
3.8
88
3.49
1
3.46
6
pape
r6
3.11
1
3.38
5
3.934
3.0
48
3.2
89
2.82
8
2.85
0
pic
0.94
2
0.86
1
0.911
0.8
27
0.8
62
0.88
4
0.83
4
pogc
3.09
2
3.23
1
3.681
2.9
28
3.1
34
2.76
3
2.75
7
prog
l
2.14
5
2.35
2
2.744
2.1
00
2.2
66
1.90
1
1.94
0
prog
p
2.22
2
2.37
9
2.740
2.1
01
2.2
48
1.87
3
1.91
9
trans
1.98
3
2.12
7
2.453
1.8
81
2.0
07
1.66
9
1.72
3
rand
23.5
86
14.0
45
9.789
9.9
40
9.5
86
12.0
69
10.4
90
Com
p.
Rate
*
3.55
2
3.34
0
3.670
2.9
25
3.0
99
2.85
1
2.79
7
Table 6: Compression performance (bits per symbol) of GC
and MGC
* rand file is not included in avg. calculation.
(a)
(b)
(c)
Fig. 2: The distribution of integers in (a) rand, (b) geo (c)
bib

VI. CONCLUSION
In this paper, a new variable-length integer code is proposed.
It has been designed based on GC and EGC. GC provides
better representation for small range of integers. But, the
proposed MGC offers competitive and compact
representation for small, mid-range and large integers when
compared to GC and EGC. The overall performance of MGC
is better than GC. The performance of MGC for text
compression on calgary corpus dataset is tested and the
experimental results show that MGC performs better than GC
and EGC when the collections which contain significant
distributions of middle and large range integers.
REFERENCES
[1] D. Salomon. Variable-length Codes for Data
Compression. Springer-Verlag, London, pp. 69-100,
2007.
[2] Huffman D.A, 1952: “A method for the construction of
minimum-redundancy codes", Proceedings of the
Institute of Radio Engineers, Cambridge Vol.40,
pp.1098-1101.
[3] [Shannon C.E, 1948: A Mathematical Theory of
Communication, Bell System Technical Journal,
Vol.27, pp.379-423, 623-656.
[4] S. David, “Data Compression Book”, 2nd ed. New
York: Springer-Verlag, 2004, pp. 41–11.
[5] K. Somasundaram and S. Domnic, “Extended golomb
code for integer representation”, IEEE Transactions on
Multimedia, vol. 9, no. 2, pp239–246, 2007.
[6] Golomb S.W, 1966: “Run-length encodings”, IEEE
Transactions on Information Theory, Vol.12 (3),
pp.399-401.
[7] S. Buttcher, C. L. A. Clarke, and G. V. Cormack.
Information Retrieval: Implementing and Evaluating
Search Engines. MIT Press, Cambridge MA, 2010.
[8] Witten, Ian Moffat, Alistair Bell, Timothy. "Managing
Gigabytes: Compressing and Indexing Documents and
Images." Second Edition. Morgan Kaufmann
Publishers, San Francisco CA. 1999.
[9] Burrows M, Wheeler D, 1994: “A block sorting lossless
data compression algorithm”, Technical Report 124,
Digital Equipment Corporation.
[10] Witten I.H, Bell T, 1990: “The Calgary / Canterbury
Text Compression Corpus”
[11] Peter Fenwick, “Burrows–Wheeler compression with
variable length integer codes,” Softw—Pract. Exper.,
vol. 32, no. 13, pp. 1307–1316, Nov. 2002.
[12] SM Basha, BC Jinaga , " A Novel Optimized Golomb-
Rice Technique for the reconstruction in Lossless
Compression of Digital Images", ISRN Signal
Processing, Vol.2013, 2013.
[13] Jing-Jing Zheng et.al, Fast algorithm for remote sensing
image progressive compression", IEEE IGARSS,
Honolulu, HI, 2010.

Modified Golomb Code For Integer Representation

Recommended

More Related Content

What's hot (20)

Similar to Modified Golomb Code For Integer Representation (20)

More from IJSRD (20)

Recently uploaded (20)

Modified Golomb Code For Integer Representation