Compressing the dependent elements of multiset

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 01 | Jan -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1630
Compressing the Dependent Elements of Multiset
Yogesh Shankar Landge1, Prof. Dr.Girish K. Patnaik2
1, Student, Department of Computer Engineering, SSBT’s COET, Bambhori, Jalgaon (M.S.), India
2, Prof. and Head, Department of Computer Engineering, SSBT’s COET, Bambhori, Jalgaon (M.S.), India
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Abstract - A multiset is an unordered collection of
mathematical objects with repetitions allowed. Many
authors showed that there is only one way to represent the
multiset, i.e. the representation of multiset as sequences. But
no one showed that the multiset can also be represented in
multiplitive form. By performing operations like (union,
intersection, sum and difference) on multisets having
multiplitive form, the dependent elements of multiset are
obtained. And can be compressed and decompressed in the
similar manner as the independent elements of the multiset,
because both the elements are related to each other. A
Lossless compression and decompression algorithms for
multisets; taking advantage of the multisets representation
structure is described. The encoding technique that
transforms the dependent elements of multiset into an
order-invariant tree representation, and derive an
arithmetic code that optimally compresses the tree. The
proposed lossless compression algorithm achieves different
arithmetic codes for dependent elements and independent
elements of multiset and generating their lengths
respectively on the basis on certain operations performed on
multisets. With the help of the proposed system both the
elements independents and dependents of multiset can be
compressed and decompressed simultaneously.
Key Words: Lossless Data Compression, Dependent
Elements of Multiset, Multiplicative Form, Union
Operation, Intersection Operation, Sum Operation,
Difference Operation…
1. INTRODUCTION:
The theory of sets is absolutely necessary to the world
of mathematics. But in set theory where repetitions of
objects are not allowed it often become difficult to
complex systems. If one considers those complex systems
where repetitions of objects become certainly inevitable,
the set theoretical concepts fails and thus one need more
sophisticated tools to handle such situations. This leads to
the initiation of multiset (M-set) theory as a generalization
of set theory.
A multiset is a set that each item in the set has a
multiplicity which specifies how many times the item
repeats. The multisets are represented in different types
of forms.an multiset containing one occurrence of p, two
occurrences of q, and three occurrences of r is notationally
written as [[p; q; q; r; r; r]] or [p; q; q; r; r; r] or [p; q; r]1;2;3
or [1.p; 2.q; 3.r] or [p.1; q.2; r.3], depending on one's taste
and convenience.
An information pressure calculation makes an
interpretation of a data article to a compacted grouping of
yield images, from which the first data can be recouped
with a coordinating decompression calculation.
Compressors (and their coordinating decompressors) are
planned with the objective that the compacted yield is, by
and large, less expensive to store or transmit than the first
information.
For example if one wants to store a large data file, it
may be preferable to first compress it to a smaller size to
save the storage space. Compressed files are much more
easily exchanged over the internet since they upload and
download much faster. We require the capacity to
reconstitute the first record from the compacted rendition
whenever. Information pressure is a system for encoding
decides that permits considerable decrease in the
aggregate number of bits to store or transmit a document.
The more data being managed, the more it expenses
regarding stockpiling and transmission costs. To put it
plainly, Data Compression is the procedure of encoding
information to less bits than the first representation so it
consumes less storage space and less transmission time
while conveying more than a system. Data compression
algorithms are classified in two ways i.e. lossy and lossless
data compression algorithm. A compression algorithm is
utilized to changeover information from a simple to-utilize
arrangement to one advanced for smallness. In like
manner, an uncompressing system gives back the data to
its unique structure.
The motivation behind the present study is the
knowledge of the representation of multisets in different
forms and the certain operations performed on multisets.
With the help of the two concepts, the new idea can be
generated, i.e. the detailed knowledge could be obtained in
the field of the dependent elements of multiset.
The contribution of the present study is to extend the
concept of multiset theory proposed by Steinruecken [11].
The author does not give any method to generate a
mutiset having the dependent elements. The main
contribution of the present study is to how to obtain the
dependent elements of multiset. After that, applying a
lossless compression algorithm on the dependent
elements of multiset to compress them into a tree based
arithmetic codes and vice-versa.
In this paper, Section 2 describes Literature Survey based
on multiset used in various domains, the proposed solution
based on analysis of lossless multiset compression algorithm
is described in Section 3, Section 4 gives Result and

Discussion, and the conclusion implicating benefits of
proposed solution is described in Section 5.
2. LITERATURE SURVEY:
Singh and Isah, in [1], proposed a note on category of
multisets (Mul).The authors overcome the problem where
the consistency criterion for the existence of mono, epi,
and iso-morphisms in Multiset is not considered. The
authors provided the consistency criterion for the
existence of mono, epi, and iso-morphisms by using
mapping concept between multi-sets. The authors have
given the criterion that objects of the same sort cannot be
mapped to objects of different sorts. (i) f: X→ Y is a
monomorphism in Multiset iff: X→ Y is one-to-one, sort-
preserving. For example, f: [a, a, a] → [b, b, b] can be a
monomorphism. (ii) f: X→ Y is an epimorphism in Mul iff:
X→Y is onto, sort-preserving. For example, f: [a, a, a] → [b,
b] can be an epimorphism. (iii) f: X→ Y is an isomorphism
in Mul iff: X→ Y is a bijection, sort-preserving. The
advantage of the proposed solution is that it makes easier
to identify the type of morphism between multisets.
Ghilezan, in [2], proposed binary relations and algebras
on multisets. Contrary to the notion of a set or a tuple, a
multiset is an unordered collection of elements which do
not need to be different. As multisets are already widely
used in combinatorics and computer science. The author
applied algebras on multiset theory. The author has
considered generalizations of known results that hold for
equivalence and order relations on sets and get several
properties that are specific to multisets. Furthermore, the
author has also exemplified the novelty that brings this
concept by showing that multisets are suitable to
represent partial orders. The advantage of the proposed
concept is that it proves two algebras on multisets cannot
be isomorphic even if their root algebras are isomorphic.
Isah and Tella, in [3], proposed the concept on
categories of multiset (Mul) and topological spaces
(Top).The authors have illustrated the concept of an
isomorphism in categorical context and shows that a
bimorphism is not necessarily an isomorphism in the
categories multiset and topological spaces. Multisets
(considered as objects) and multiset functions (considered
as morphisms) together determine the category of
multisets, denoted Mul. For example, f: [a, b, c] → [d, e, e],
defined by f(a) = d, f(b) = e, f(c) = e, is an mset morphism
which is a monomorphism and an epimorphism, and
therefore a bimorphism, but not an isomorphism: since
f(b) and f(c) are of the same sort but b and c are of
different sorts. The multiset morphism f: [a, a, b] → [c, c,
d], defined by f(a) = c, f(b) = d, f(a) = c, is an isomorphism.
Moreover, the morphism g: [a, b] → [c, d], defined by g(a) =
c, g(b) = d, is also an isomorphism. The advantage of the
proposed concept is that both Mul and Top are not
balanced thus; a bimorphism in Mul, Top is not necessarily
an isomorphism. And, whereas Set, Abelian group, Ring
and Group are balanced since every bimorphism is an
isomorphism.
Girish and John, in [4], proposed the idea on multiset
topologies. General topology is defined as a set of sets but
multiset topology is defined as a set of multisets. The
concept of topological structures and their generalizations
is one of the most powerful notions in branches of science
such as chemistry, physics and information systems. In
most applications the topology is used to handle the
qualitative information. In any information system, some
situations may occur, where the respective counts of
objects in the universe of discourse are not single. In such
situations where dealing with collections of information in
which duplicates are allowed. In such cases multisets play
an important role in processing the information. The
information system dealing with multisets is said to be an
information multisystem. Thus, information multisystems
are more compact when compared to the original
information system. In fact, topological structures on
multisets are generalized methods for measuring the
similarity and dissimilarity between the objects in
multisets as universes. The theoretical study of general
topology on general sets in the context of multisets can be
a very useful theory for analyzing an information
multisystem. Most of the theoretical concepts of multisets
are originated from combinatorics. Combinatorial
topology is the branch of topology that deals with the
properties of geometric figures by considering the figures
as being composed of elementary geometric figures. The
combinatorial method is used not only to construct
complicated figures from simple ones but also to deduce
the properties of the complicated from the simple. In
combinatorial topology it is remarkable that the only
machinery to make deductions is the elementary process
of counting. In such situations, elements can occur more
than once. The advantage of the proposed work is that the
theory of M-topology may be useful for studying
combinatorial topology with collections of elements with
duplicates.
Tella and Daniel, in [5], proposed the concept of the
symmetric groups under multiset perspective. Research
on the multiset theory has not yet gained ground and is
still in its infant stages. According to the authors, the
research carried out so far shows a strong analogy in the
behaviour of multisets and sets and it is possible to extend
some of the main notions and results of sets to that of
multisets for instance, the theoretical aspects of multisets
By extending the notions of relations, functions,
composition and partition have been explored. The
advantage of the proposed concept is that it defines a
symmetric multigroup and derives the analogous Cayleys
theorem.
Tella and Daniel, in [6], proposed a study of group
theory in the context of multiset theory. Unlike set, a
multiset is an unordered collection of elements where
elements can occur more than once .the problem was that
the study of classical group theory in the context of sets
was not being presented, because sets do not allow
elements to occur more than once. The authors have
presented the study of classical group theory in the
context of multisets because multisets allow repetition of

elements. But the disadvantage of the proposed study is
that it does not address the concepts of Lagranges
theorem, homomorphism of groups and symmetric groups
in the context of multigroup theory.
Girish and John, in [7], proposed the concept of Rough
Multisets and Information Multisystems. The most
essential property of multisets is the multiplicity of the
elements that allows distinguishing it from a set and
considering it as a qualitatively new mathematical
concept. In any information system, some situations may
occur, where the respective counts of objects in the
universe of discourse are not single. There are many
situations where dealing with collections of information in
which duplicates are significant. In such cases multisets
play an important role in processing the information. The
information system dealing with multisets is said to be an
information multisystem. Thus, information multisystems
are more compact when compared to the original
information system. The authors have given a new
dimension to Pawlaks rough set theory, replacing its
universe by multisets. This is called a rough multiset and is
a useful structure in modelling information multisystem.
The processes involved in the intermediate stages of
reactions in Chemical systems are a typical example of a
situation which gives rise to multiset relations.
Information multisystem is represented using rough
multisets and is more convenient than ordinary rough
sets. The authors have proposed the idea with Yagers
theory of multisets. After presenting the theoretical study,
the concepts mset relations, equivalence mset relations,
partitions, and knowledge mset base have been
established. Finally the concepts of rough multisets and
related properties with the help of lower mset
approximation and upper mset approximations have been
introduced. The advantage of the proposed concept shows
is that rough multisets are important frameworks for
certain types of information multisystems.
Kuo, in [8], proposed the concept of Multiset
Permutations in Lexicographic Order. The author
proposed a simple and flexible method for generating
multiset permutations in lexicographic order. Multiset
permutation can be applied in combinations generation,
since a combination of p items out of q items set is a
special case of multiset permutations that contain q 1s and
(q-p) 0s. For example, a combination of 5 items out of 10
items can be described as 1010011001 to stand the first
item, the 4th item, the 5th item, 8th item and the last item
are picked up. Obviously, this is a permutation of a
multiset 5 0, 5 1. In other words, the advantage of the
proposed technique is that the multiset permutations can
be generated directly without any help of remapping. The
new method is conceptually easy to understand and
implement and is well-suited to a wide variety of
permutation problems.
Singh et al., in [9], proposed the concept of
Complementation in Multiset Theory. The paper
delineates the problem related to difference and
complementation in Multiset Theory. It is shown that none
of the existing approaches succeeds in resolving the
attendant difficulties without assuming some contrived
stipulations. In this paper, the authors have endeavoured
to revisit the problem related to difference and
complementation in multiset theory. It is shown that the
attendant difficulties do remain unresolved unless some
contrived stipulations are assumed. It has been
demonstrated that the technique of introducing a
cardinality bounded multiset space proves advantageous
in many ways especially in Database systems. However, as
investigated above none of the existing approaches
succeeds if the goal is to augment the theory of multiset
endowed with a Boolean algebraic structure without
assuming some contrived situations. In fact, the obvious
alternatives seem to be exhausted. However, some further
adept explications cannot be ruled out.
Singh et al., in [10], focused on Some Applications of -
Cuts in Fuzzy Multiset Theory. The authors briefly
describe fuzzy set, multiset, fuzzy multiset, -cuts and
related results. In particular, it is shown how -cuts turn
out to be an intertwining thread between multiset and
fuzzy multisets. It has been shown that either of the
aforesaid representations allows us to extend various
properties of crisp multisets and operations on crisp
multisets to their fuzzy multiset counterparts. These
extensions are cut worthy since in each representation, a
given crisp property or operation is required to be valid
for each crisp multiset involved in the representation.
Steinruecken, in [11], proposed the concept of
compressing sets and multisets of sequences. The problem
was that the information was being wasted for encoding
multisets by simply storing its elements in some
sequential order. The author has given the solution to the
problem, by proposing a technique that transforms the
multiset into an order-invariant tree representation, and
derives an arithmetic code that optimally compresses the
tree. The advantage of the technique is that the
information is saved while encoding the multisets. And on
the other hand, the disadvantage of the technique is that it
does not compress the dependent elements of multisets.
The past woks on the concept of the multiset theory
have discussed here. And it concludes that the research on
the multiset theory is still in at infant stage. The past
works presented are covered different areas in multisets,
but the idea on compressing the dependent elements of
multisets are not explored before. In [11], the author has
provided the concept of compressing the independent
elements of multiset, where the author represented
multiset as sequences. But the author does not provide the
method to generate the dependent elements of multiset
and compressing the dependent elements of multiset into
tree-based arithmetic codes. The proposed system gives
the method to generate the dependent multiset from
independent multisets and compresses the dependents
elements of multiset into arithmetic codes similar to the
independent elements of mutiset.

3. PROPOSED SOLUTION:
In tree based encoding and decoding, multisets can be
encoded and decoded using various techniques. Tree
based encoding needs to ensure that the compression
should be lossless compression. The compression and
decompression of independent multisets has been done
with the help of encoding and vice-versa. But in the
existing system, when the scenario is multisets having the
dependent elements, there is no way to obtain the
dependent elements of multiset and after that, compress
and decompress like the independent elements of multiset
to convert into arithmetic codes.
In the proposed system it is assure that encoding is
done with lossless compression and it is done on
dependent multisets. Different operations are performed
on multisets in order to generate dependent multisets on
which encoding is done. The Proposed system takes into
considerations two things that are required to provide the
solution to the problem.
I. The representation of the multisets as
Multiplicative form instead of Sequences Form.
and
II. The different types of operations performed on
multisets of Multiplicative form.
There are certain types of operations like union,
intersection, sum and difference can be applied on
multisets, after performing the respective operations on
multiset A and multiset B (assume), the resulting multiset
(assume multiset c) is obtained.
a. If the union operation is performed on multiset A
and multiset B, the resulting multiset C is
obtained, where the multiset A and multiset B
having the independent element while the
multiset C contains the dependents elements.
b. If the intersection operation is performed on
multiset A and multiset B, the resulting multiset C
is obtained, where the multiset A and multiset B
c. If the sum operation is performed on multiset A
and multiset B, the resulting multiset C is
obtained, where the multiset A and multiset B
d. If the difference operation is performed on
multiset A and multiset B, the resulting multiset C
is obtained, where the multiset A and multiset B
The existing system only compresses and decompresses
the independent elements of multiset (taking one multiset
at a time) and generated the respective arithmetic codes
for the multiset. It is known that there are two parts of
multisets, one part is multiset having the independent
elements and second part is multiset having the
dependent element. To obtain the second part of multiset
that is multiset having the dependent element, firstly the
multiset is represented in the multiplicative form. And
secondly the certain operations like union, intersection,
sum and difference are perform on multisets having the
independent elements, so the resulting multiset is a
multiset having the dependents element. This is the
process to get the second part of multiset (multiset having
the dependents element). Now, performing similar the
lossless compression and decompression algorithms, on
the dependent elements of multisets, the resulting
arithmetics codes are generated based on operations
performed. With the help of arithmetics codes, the
difference between independent elements of multisets and
dependent element of multiset can be made.
There are certain advantages using the proposed
system:
I. At the same time both the part of multiset
(independent and dependent) can be com-
pressed and decompressed with the help of the
proposed system.
II. It minimizes transmission time by simultaneously
executing both the part of multisets.
3.1 Assumptions:
In the proposed system following assumptions are
made.
 The multisets are being taken in the form of
alphabets (a, b, c, d and e).
 a, b, c, d and e are the elements of multisets where
Consider a=1 b=2 c=3 d=4 e=5.
 1, 2, 3, 4, and 5 are the decimal value for the
elements of multisets a, b, c, d, and e respectively.
3.2 Proposed System Architecture:
Architecture is a system that unifies its components into
a coherent and functional block. Figure 3.1 shows
architecture of the proposed system.
Fig -3.1: Architecture of the Proposed System
The Figure 3.1 shows the architecture of proposed
algorithm. The proposed algorithm is capable of
generating the dependent multisets. The dependent
multisets contain the dependent elements of multisets.
and then the proposed algorithm generates arithmetic
code for the dependent elements of multisets.

3.3 Design:
To generate both the dependent and independent
elements of multisets, the modification in the past scheme
can be done by performing the operations on multisets. To
design the proposed system the above consideration is
kept in mind that ensures the generation of both the
elements and producing the arithmetic codes respectively.
Encoding Algorithm:
Require: Input (Multiset 1, Multiset 2), Assume (a=1, b=2,
c=3, d=4, e=5).
1: Take Two Multisets as input.
2: Perform operations over multisets: Union, Intersection,
Sum, and Difference.
3: Operatios Result over multisets: Dependent Multisets
that contain total elements N, and length of string of
elements L.
4: Encode N, L using Binomial Probability Distribution in
tree form.
5: Let (t0) and t1 denote children of tree T, and n0 and n1
the children’s count.
6: Encode n using a binomial code, n=Binomial (n0+n1, Ө)
7: if (n0 >0) then
8: Call encode node (t0)
9: end if
10: if (n1 >0) then
11: Call encode node (t1)
12: end if
Decoding Algorithm:
Require: Input(Arithmetic codes),
Assume(a=1,b=2,c=3,d=4,e=5)
1: Decode N, using the same code over positive integers.
2: Return T, Decode node (N, L)
3: if (l >0) then
4: Decode n1 using binomial code
5: end if
6: if (n0 >0) then
7: t0 ← Decode node (n0, l-1)
8: end if
9: if (n1 >0) then
10: t0 ← Decode node (n1, l-1)
11: end if
12: Return a new tree node with count n and children t0
and t1.
Fig -3.2: Flowchart of the Proposed System
4. RESULT AND DISCUSSION:
Result and discussion section is a primary part of
research work. Evaluation of the proposed approach
versus existing approach is carried out in the result and
discussion section. Result section represents the
experimental results of the proposed approach as well as
the existing approach. Evaluation of the both the
approaches are carried out in the discussion section on the
basis of obtained results. In some cases various
parameters i.e. performance metrics like tables are used to
evaluate the system, in order to decide which one is the
best.
4.1 Experimental Results:
Experimental results present the effectiveness of
proposed system, in which involvement of representing
multiset in multiplicative form is proved better by
carrying out experiments. Experimental result presents
the effectiveness of proposed system, in which

involvement of proposed lossless compression algorithm
is proved better by carrying out experiments.
Results are carried out using java. In proposed system,
arithmetic codes are generated both for dependent as well
as independent elements of multiset at the same time
instead of generating arithmetic code for only the
independent elements of multiset. The results are taken by
executing the algorithm with all combinations and average
of 5 multisets are taken into consideration for each and
every operation. Below Results show the arithmetic codes
for dependent and independent multiset and the length
respectively. All the evaluation is done on i3 processor
with 4GB RAM and may vary as and when the hardware is
changed.
Table -4.1: shows the comparison of arithmetic codes
generated for the dependent multiset and independent
multiset using union operation.
Multis
et 1
Multi
set 2
Depen
dent
Multis
et
Arith
metic
Codes
Independent
Multiset
Arith
metic
Codes
2.a,1.b
,1.c
3.a,2.
c
3.a,1.b
,2.c
3 1 2 2.a,1.b,1.c,3.a
,2.c
5 1 3
1.a,2.b
,1.c
2.a,4.
c
2.a,2.b
,4.c
2 2 4 1.a,2.b,1.c,2.a
,4.c
3 2 5
5.a,3.b
,4.d
1.a,2.
b,5.d
5.a,3.b
,5.d
5 3 5 5.a,3.b,4.d,
1.a,2.b,5.d
6 5 9
6.a,5.b
,1.c
9.a,1.
b,2.c
9.a,5.b
,2.c
9 5 2 6.a,5.b,1.c,
9.a,1.b,2.c
15 6 3
10.a,2.
b,4.d
9.a,3.
b,1.d
10.a,3.
b,4.d
10 3 4 10.a,2.b,4.d,9
.a,3.b,1.d
19 5 5
multiset using intersection operation.
Multis
et 1
Multi
set 2
Depen
dent
Multis
et
Arith
metic
Codes
Independent
Multiset
Arith
metic
Codes
2.a,1.b
,1.c
3.a,2.
c
2.a,1.c 2 1 2.a,1.b,1.c,3.a
,2.c
5 1 3
1.a,2.b
,1.c
2.a,4.
c
1.a,1.c 1 1 1.a,2.b,1.c,2.a
,4.c
3 2 5
5.a,3.b
,4.d
1.a,2.
b,5.d
1.a,2.b
,4.d
1 2 4 5.a,3.b,4.d,
1.a,2.b,5.d
6 5 9
6.a,5.b
,1.c
9.a,1.
b,2.c
6.a,1.b
,1.c
6 1 1 6.a,5.b,1.c,
9.a,1.b,2.c
15 6 3
10.a,2.
b,4.d
9.a,3.
b,1.d
9.a,2.b
,1.d
9 2 1 10.a,2.b,4.d,9
.a,3.b,1.d
19 5 5
multiset using sum operation.
Multis
et 1
Multi
set 2
Depen
dent
Multis
et
Arith
metic
Codes
Independent
Multiset
Arith
metic
Codes
2.a,1.b
,1.c
3.a,2.
c
5.a,1.b
,3.c
5 1 3 2.a,1.b,1.c,3.a
,2.c
5 1 3
1.a,2.b
,1.c
2.a,4.
c
3.a,2.b
,5.c
3 2 5 1.a,2.b,1.c,2.a
,4.c
3 2 5
5.a,3.b
,4.d
1.a,2.
b,5.d
6.a,5.b
,9.d
6 5 9 5.a,3.b,4.d,
1.a,2.b,5.d
6 5 9
6.a,5.b
,1.c
9.a,1.
b,2.c
15.a,6.
b,3.c
15 6 3 6.a,5.b,1.c,
9.a,1.b,2.c
15 6 3
10.a,2.
b,4.d
9.a,3.
b,1.d
19.a,5.
b,5.d
19 5 5 10.a,2.b,4.d,9
.a,3.b,1.d
19 5 5
multiset using difference operation.
Multis
et 1
Multi
set 2
Depen
dent
Multis
et
Arith
metic
Codes
Independent
Multiset
Arith
metic
Codes
2.a,1.b
,1.c
3.a,2.
c
1.a,1.c 1 1 2.a,1.b,1.c,3.a
,2.c
5 1 3
1.a,2.b
,1.c
2.a,4.
c
1.a,3.c 1 3 1.a,2.b,1.c,2.a
,4.c
3 2 5
5.a,3.b
,5.d
1.a,2.
b,4.d
4.a,1.b
,1.d
4 1 1 5.a,3.b,5.d,
1.a,2.b,4.d
6 5 9
9.a,5.b
,2.c
6.a,1.
b,1.c
3.a,4.b
,1.c
3 4 1 9.a,5.b,2.c,
6.a,1.b,1.c
15 6 3
10.a,3.
b,4.d
9.a,2.
b,1.d
1.a,1.b
,3.d
1 1 3 10.a,3.b,4.d,9
.a,2.b,1.d
19 5 5
4.2 Discussion:
The purpose of the discussion section is to state
interpretations and opinions, explain the implications of
findings in the experimental evaluation. Main function of
discussion section is to answer the questions posed in the
Introduction, explain how the results support the answers
and, how the answers fit in with existing knowledge on the
topic. The discussion section is important to know the
detail advantage and need about proposed solution.
Results of experiment evaluate performance of
proposed system at various levels and steps. Performance
varies as the operation performed varies. The proposed
algorithm has the capacity to compress both the elements
dependents as well as independents into the arithmetic
codes at the same time by using the representation of
multiset in a multiplicative form. Hence, reducing the
amount of bandwidth required to store the text data over
the communication channels. It also reduces the
transmission time.

5. CONCLUSION AND FUTURE WORK:
The proposed system is developed to solve the problem
of not compressing both the independent as well as the
dependent elements of multisets simultaneously. To solve
the problem, it is ensuresed that the multisets can also be
represented in various forms. The multiplicative form is
the one of them. The advantage of representing multiset in
multiplicative form is that the dependent multiset can be
obtained by performing operations on multisets having
multiplicative form. After that the second part of multiset
(multiset having dependent element can be compressed in
the form of arithmetic codes and vice versa). The proposed
system not only compresses the independent elements of
multisets but also the dependent elements of multiset by
performing operations on multisets.
In the future, it would be a point of research to
compress and decompress the special type of Multiset i.e.,
Fuzzy Multiset into arithmetic codes.
REFERENCES
[1] D. Singh and A. I. Isah, "A note on category of multisets
(Mul)," International Journal of Algebra, vol. 7, no. 2, pp.
73-78, 2013.
[2] S. G. and Jovanka Pantovi, “Binary relations and
algebras on multisets," Publications De LInstitute
Mathematique, vol. 9, no. 5, pp. 111-117, 2014.
[3] A. I. Isah and Y. Tella, “On categories of multisets (Mul)
and topological spaces (top),"International Journal of
Science and Technology, vol. 2, no. 6, pp. 442-445, 2013.
[4] K. P. Girish and S. J. John, “On multiset topologies,"
Theory and Applications of Mathematics Computer
Science, vol. 2, no. 1, pp. 37-52, 2012.
[5] Y. Tella and S. Daniel, “Symmetric groups under
multiset perspective," IOSR Journal of Mathematics, vol. 7,
no. 5, pp. 47-52, 2013.
[6] Y.Tella and S. Daniel, “A study of group theory in the
context of multiset theory, "International Journal of
Science and Technology, vol. 2, no. 8, pp. 609-615, 2013.
[7] K. P. Girish and S. J. John, “Rough multisets and
information multisystems," Hindawi Publishing
Corporation Advances in Decision Science, vol. 5, no. 6, pp.
1-17, 2011.
[8] T. Kuo, “Multiset permutations in lexicographic order,"
International Journal of Emerging Technology and
Advanced Engineering, vol. 4, no. 1, pp. 442-448, 2014.
[9] D. Singh, A. M. Ibrahim, and T. Yohana,
“Complementation in multiset theory," International
Mathematical Forum, vol. 6, no. 38, pp. 1877-1884, 2011.
[10] D. Singh, A. J. Alkali, and A. I. Isah, “Some applications
of -cuts in fuzzy multiset theory," Journal of Emerging
Trends in Computing and Information Sciences, vol. 5, no.
4, pp. 328-335, 2014.
[11] C. Steinruecken, “Compressing sets and multisets of
sequences," Ieee Transactions on Information Theory, vol.
61, no. 3, pp. 1485-1490, 2015.
BIOGRAPHIES
Yogesh Shankar Landge is a
Student in Department of
Computer Engineering, SSBT’s
COET, Bambhori, Jalgaon (M.S.),
India.
Prof. Dr. Girish K. Patnaik is a
Professor and Head in Department
of Computer Engineering, SSBT’s
COET, Bambhori, Jalgaon (M.S.),
India.

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