Non-Uniform Gap Distribution Library Sort

Non-Uniform Gap Distribution Library Sort
Neetu Faujdar, Rakhi Mittal
Amity University Uttar Pradesh, Department of CSE, Noida, India
Email: neetu.faujdar@gamil.com, rakhimittal1108@gmail.com
Abstract—The current study is focused towards a sorting
algorithm referred as Library Sort. Previous scientific
limitations have revealed many utilities of sorting algorithms.
Each sorting algorithm varies according to their strength and
weaknesses. The Bender et al proposed the library sort with
uniform gap distribution (LUGD) i.e. there is equal number of
gap provided after each element. But what happens if there are
many elements that belong to the same place in the array and
there is only one gap after that element. To overcome the
problems of LUGD, this paper focuses on the implementation of
library sort algorithm along with non-uniform gap distribution
(LNGD). The proposed algorithm is investigated using the
concept of mean and median. The LNGD is tested using the
four types of test cases which are random, reverse sorted, nearly
sorted and sorted dataset. The experimental result of proposed
algorithm is compared to the LUGD and LNGD proved to
provide better results in all the aspects of execution time like re-
balancing and insertion. Improvement has been achieved that
ranges from 8% to 90%.
Index Terms— Sorting, Insertion Sort, Library Sort, LNGD.
I. INTRODUCTION
Michael A. Bender proposed the library sort which can be
defined as follows [2], it is a sorting algorithm that uses
insertion sort, but with gaps in the array to accelerate
subsequent insertions [3]. Library sort is also called the
gapped insertion sort. Suppose, if a librarian wants to
keep his books alphabetically on a long shelf starting with
the alphabet „A‟ and continuing to the right till the
alphabet „Z‟ and there is no gap between the books, and if
the librarian has some more books that belong to section
„B‟, then he has to search the correct place in section „B‟.
Once he finds the correct place in section „B‟, he will
have to move every book from the middle of section „B‟
to the end of the section „Z‟ in order to make space for the
new book. This is called insertion sort. However, if the
librarian leaves a gap after every letter book, then there
will be a space after section „B‟ letter books and then he
will only have to move a few books to make space for the
new books, this is the primary principle of library sort.
Thus the library sort is an extension of insertion sort. The
author achieves the O(logn) insertions with high
probability using the evenly distributed gap, and the
algorithm runs O(nlogn) time complexity with high
probability [7]. The time complexity of library sort
O(nlogn) is better than the insertion sort, which is O(n2
)
[3]. In library sort author used the uniform gap
distribution after each element and gap is denoted by the
„ε‟. The detailed analysis of the library sort has been done
in the paper [1]. By execution time analysis, author found
that, execution time is inversely proportional to the value
of epsilon in most of the cases. At some point the value of
epsilon (ε) reached at the saturation point due to the
presence of extra spaces. Later, these extra spaces are
used to insert the data. By space complexity analysis,
found that the space complexity of the library sort
algorithm increases linearly that is, when the value of
epsilon increases, the memory consumption also
increased in the same proportion. By execution time
analysis of re-balancing, found that increase the re-
balancing factor „a‟ from 2 to 4 then the execution time
of library sort algorithm will also increase as it moves
towards the traditional insertion sort. So, to find out the
best result of library sort algorithm, the value of epsilon
should be optimized and re-balancing factor should be
minimized or ideally equal to 2. This is what about the
library sort using the uniform gap distribution. The
application of leaving gaps for insertions in a data
structure is used by [8]. This idea has found recent
application in external memory and cache-oblivious
algorithms in the packed memory structure of Bender,
Demaine and Farach-Colton and later is used in [9-10].
In this paper, a new approach proposed that is library sort
with non-uniform gap distribution (LNGD). And new
approach results are analyzed using the four types of test
cases which are random, nearly sorted, reverse sorted and
sorted test cases. The paper also compares the LUGD
with the LNGD in all the aspects of execution time like
re-balancing and insertion.
The remaining section of this paper is organized as
follows: Section 2 contains the detailed description of
LNGD algorithm. The execution time comparison of
LUGD and LNGD has been shown in section 3. The
testing and comparison based on re-balancing is
described in section 4. The performance analysis in all
aspects has been shown in section 5. In section 6
concluded the presented work along with its future scope.
II. RELATED WORK
In this section, briefly surveyed related work based on
insertion and library sort. Tarundeep et al presented a
new sorting algorithm EIS (Enhanced Insertion Sort). It is
based on the hybrid sort. The suggested algorithm
achieved O(n) time complexity which is compared O(n2
)
of insertion sort. In this the effectiveness of the algorithm
is also proved. The hybrid based sort is analyzed,
implemented, tested and compared with other major
sorting algorithms [14].
Partha et al shows that how to improve the
performance of insertion sort. He suggested a new
approach which compared to the original version of
International Journal of Computer Science and Information Security (IJCSIS),
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49 https://sites.google.com/site/ijcsis/
ISSN 1947-5500

insertion sort. The new approach is also compared with
bubble sort. The experimental results have shown that
proposed approach performs better in worst case [15].
Michael et al proposed the library sort algorithm which
is also called gapped insertion sort. It is the enhanced of
the insertion sort. The authors showed that library sort
has insertion time O(logn) with high probability. The
total running time is O(nlogn) with high probability [2].
Franky et al investigated the improvement of worst
case running time of insertion sort. So the author
presented the rotated library sort. The suggested approach
has O(√nlogn) operation per insertion and the worst case
running time is O (n1.5
logn) [16].
Neetu et al overcomes some issues of library sort. The
detailed experimental analysis of library sort is done by
the author. The performance of library sort is measured
using the dataset [1].
III. PROBLEM STATEMENT
Library sort is an improvement of insertion sort with
O(nlogn) time complexity. Library sort is also called
gapped insertion sort or say that insertion sort with gaps.
The Bender et al proposed the library sort with uniform
gap distribution i.e. there is equal number of gap provided
after each element. But what happens if there are many
elements that belong to the same place in the array and
there is only one gap after that element.
So to overcome this problem in this paper, library sort
with non-uniform gap distribution is proposed i.e.
element can have non-uniform gap. The suggested
algorithm is investigated using the concept of mean and
median.
IV. PROPOSED LIBRARY SORT ALGORITHM WITH NON-
UNIFORM DISTRIBUTION (LNGD)
The LNGD algorithm consists of three steps. The first
two steps will be the same as the LUGD algorithm [1],
but the third step will be different.
Step1. Binary Search with blanks: In library sort insert
a number in the space where it belongs and to find that
binary search is used. The array „S‟ is sorted but with
gaps. As in computer, gaps of memory will hold some
value and this value is fixed to sentinel value that is „-1‟.
Due to this reason, directly cannot use the binary search
for sorting. So, modify the binary search. After finding
the mid element, if it comes out to be „-1‟ then move
linearly left and right until get a non-zero value. These
values are named as m1 and m2. Based on these values,
define values for new low, high and mid used in a binary
search. Another difference in the binary search is that, it
is not only searches the element in the list, but also
reports the correct position where we have to insert the
number [1]. Working of step 1 is illustrated with the help
of an example.
Example: In the following array „-1‟ shows the gaps in
the array. The array position is starts from 0 up to 9. Now
let search an element say 5.
low=0
high=9
mid=(0+9)/2 = 4 = S[4]
Here S[4] = 5 got the element and terminate the search.
In this array, do not have element 5 but we are going to
search it.
Here also low =0
High=9
Mid =(0+9) /2 =4 = S[4]
S[4]= S[mid]= -1
In this case, find m1 and m2 as a mid which are
represented by S[m1] and S[m2] greater than „-1‟ in both
the direction limiting to low and high respectively. Here
the value of m1=S[2]=3 and the value of
m2=S[6]=7. According to m1 and m2 values, update the
low and high to perform binary search.
Step2. Insertion: Library sort is also known by the name
gapped insertion sort. The presence of gap for inserting
any new number at a particular position removes the task
of shifting the elements up to the next gap [1]. Working
of step 2 is explained with the help of an example.
Example: Insert the elements in the manner of 2i
in the
array. i.e. in the power of 2. This is stored in S[i]. S[i]
=pow (2, i) where „i‟ is the pass number i.e i= 0, 1, 2, 3...
if i=0 then S1= 20
=1. Now search the position of the
insertion element in the array and add the element at
position returned by the search function. Next time i=1
then S1=21
=2, and S[i] = pow(2, i-1) to pow(2,
i) i.e. the value of S1 is 1 to 2 and so on for all values of
„i‟.
Step3. Re-balancing: Re-balancing is done after
inserting 2i
elements where i=1, 2, 3, 4... and the spaces
are added when re-balancing is called. In the previous
approach, the gaps were uniform in nature. In the
proposed technique, non-uniform gap distribution is
given based on the property of insertion sort. This
property tells that more updates should be done in the
beginning of an array for generating more gaps. Gaps are
generated using the equation (1).
2/)/(* nRatio (1)
Here µ is mean and  is standard deviation.
rationee /*2 (2)
1 -1 3 -1 5 -1 7 -1 9
1 -1 3 -1 -1 -1 7 -1 9 -1
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Initially have e+ee gaps, but „ee‟ is decreased when
parsed the number equal to the ratio.
ALGORITHM: LNGD Re-balancing
Input: List of elements and re-balancing factor e.
Output: List with non uniform gaps.
Compute µ and 
Ratio  n*( µ/)/2
ee  2*n/ratio
while(l < n) do
if(j% ratio ==0 && j>0 && e+ee>0
)then
ee--
endif
if(S[j] ! = -1) then
reba[i] = S[j]
i++
j++
l++
for k=0 to ee+e do
reba[i] = -1
i++
endfor
else
j++
endif
for k = 0 to i do
S[k] = reba[k]
endfor
endwhile
end
V. EXECUTION TIME COMPARISON OF LUGD AND
LNGD
LUGD and LNGD algorithms have been tested on a
dataset [T10I4D100K (.gz)] [11-13] by increasing the
value of the gap   . The dataset contains the 1010228
items. Data set contains four cases [26-30].
(1) Random with repeated data (Random data)
(2) Reverse sorted with repeated Data (Reverse sorted
data)
(3) Sorted with repeated data (Sorted data)
(4) Nearly sorted with repeated data (Nearly sorted data)
Table 1 shows the execution time of LUGD and LNGD
algorithms in microseconds using the above mentioned
cases.
The performance of the LUGD and LNGD are compared
with random data, nearly sorted data, reverse sorted and
sorted data. The execution time in micro-seconds is
presented in Table 1. The Results are presented for
different value of „ε‟. Epsilon (ε) is the minimum number
of gaps between the two elements. The execution time
comparison of LUGD and LNGD algorithms has also
been shown in Fig. 1 to 4. In all Fig. 1 to 4, the X-axis
represents the different value of gap and the Y-axis
represents the execution time in microseconds.
0
20000000
40000000
60000000
80000000
1E+09
1.2E+09
ε =1 ε =2 ε =3 ε =4
ExecutionTimeinMicrosec
Value of Gap
Random Data
LUGD LNGD
Fig. 1. Execution time comparison of LUGD and LNGD
Time in Microseconds
Data Set Random Nearly Sorted Reverse Sorted Sorted
Value of ε LUGD LNGD LUGD LNGD LUGD LNGD LUGD LNGD
ε =1 981267433 862909204 864558882 306063385 1.451E+09 1.329E+09 861929937 313078205
ε =2 729981576 708580455 620115904 230939335 1.065E+09 1.022E+09 609647355 234697961
ε =3 119727535 101921406 358670053 185759986 278810310 125152235 356489846 195120953
ε =4 23003046 10557332 117188830 107729204 263693774 116417058 116590140 106897060
TABLE 1. Execution Time of LUGD and LNGD Algorithms in microsecond based on Gap Values
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Fig. 1 shows the comparison of LUGD and LNGD for
different values of the gap. It can be seen from the graph
that the LNGD has outperformed LUGD. The maximum
improvement in execution time by LNGD is 36.7% for
the value of ε=4.
0
10000000
20000000
30000000
40000000
50000000
60000000
70000000
80000000
90000000
1E+09
ε =1 ε =2 ε =3 ε =4
TimeinMicrosec
Value of Gap
Nearly Sorted Data
LUGD LNGD
Fig. 2 describes the execution time of the two algorithms
LUGD and LNGD on the nearly sorted data. Major
improvement found in the case of ε=1. Observed that the
improvement in execution time by LNGD is 64.59% at ε
=1. With observations, found that execution time is
inversely proportional to the value of „ε‟. The execution
time is calculated as 8% in the case of ε=4.
Fig. 3. The execution time comparison between LUGD
and LNGD using reverse sorted data. In the case of
reverse sorted data the trend for execution time is
reversed. It is nearly 8% for the ε =1, and it further
decreases for ε =2, ε =3 and ε =4 upto 55%. The same has
been shown in Fig. 3.
Fig. 4 describes the execution time of both the algorithms
on the sorted data, the improvement can be seen from the
ε=1 to ε =4. It is maximum at ε =1 that is 63.67% and
minimum at ε =4 that is 8%.
0
20000000
40000000
60000000
80000000
1E+09
1.2E+09
1.4E+09
1.6E+09
ε =1 ε =2 ε =3 ε =4
TimeinMicrosec
Value of Gap
Reverse Sorted data
LUGD LNGD
0
10000000
20000000
30000000
40000000
50000000
60000000
70000000
80000000
90000000
1E+09
ε =1 ε =2 ε =3 ε =4
TimeinMicrosec
Value of Gap
Sorted Data
LUGD LNGD
VI. REBALANCING BASED COMPARISON OF LUGD
AND LNGD
Re-balancing is used after inserting ai
element, this
increases the size of the array. The size of the array will
depend on „ε‟. To do this process, require an auxiliary
array of the same size so as to make a duplicate copy with
gaps. Re-balancing is necessary after inserting ai
element,
and re-balancing calculated till ai
where „a‟= 2, 3, 4 and i
= 0, 1, 2, 3, 4….. with the value of gaps „ε‟ = 1, 2, 3, 4.
Now showing that how to calculate the re-balancing of
LUGD and LNGD with the help of an example.
1. Example of re-balancing using LUGD
algorithm
(A) For example, when ε=1, then how re-balancing
will be performed if a=2.
2i
= 20
, 21
, 22
, 23
, 24
……………
=1, 2, 4, 8, 16 ……………
1.1 Re-balance for 20
=1
=2
1 -1
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After re-balancing this array is
=4
1 2 3 4
After re-balancing this array is as follows:
In this manner, re-balance can done of the array in the
power of 2i
.
(B) Example when ε=1, then how re-balancing will
be performed if a=3.
3i
= 30
, 31
, 32
, 33
, 34
……………
= 1, 3, 9, 27…………………
=1
=3
In the above array only one space is empty. This shows
that only one element can be inserted. On the other hand,
according to re-balancing factor 31
=3 require two spaces
in the array. In this situation, need to shift the data to
make space for the new element. In this way the
performance of the algorithm degrades as having the
larger number of swapping to generate the spaces which
is same as that in the case of traditional insertion
sort.
2. Example of re-balancing using LNGD
algorithm
(A) For example, how re-balancing will be
performed if a=2.
2i
= 20
, 21
, 22
, 23
, 24
……………
=1, 2, 4, 8, 16 ……………
In the proposed algorithm used two parameters „ee‟ and
„ratio‟ which is defined prior in the algorithm along with
the value of gaps. To understand this concept, consider an
example; say the list to be sorted is 1, 2, 3, and 4. The
average and standard deviation are calculated first. The
mean and standard deviation are calculated to 2.5 and 1.2
respectively. The ratio and „ee‟ is calculated using
equation (1) and (2).
Ratio = 3.
ee = 8/3=2 as integer
The total gaps are 1+2 = 3
=1
=2
After re-balancing the array is described below.
In this case initially j=1 that means have e+ee gaps that is
equal to 3. At second iteration „j‟ is equal to 2, now have
the condition that is j=ratio so we decrement the value of
„ee‟ by 1. Initially we have 3 gaps, then 2 gaps.
After re-balancing, the array is:
=4
1 2 3 4
After re-balancing this array is as follows:
1 -1 -1 -1 2 -1 -1 3 -1 -1 4 -1
Initially have 3 gaps for j=1.
 For j=2, j%ratio is equal to zero, therefore „ee‟ is
decremented by 1.
 For j=3, the value remains unchanged to 2 gaps
 For j=4, again the value is decremented by 1 so
there is only single gap.
(B) Example for how re-balancing will be performed
if a=3.
3i
= 30
, 31
, 32
, 33
, 34
……………
= 1, 3, 9, 27…………………
=1
After re-balancing, the array is described as:
1 -1 -1 -1
=3
1 2 3 4
After re-balancing, the array is as follows:
1 -1 -1 -1 2 -1 -1 3 -1 -1
The spaces are calculated using the equation (1). In the
similar manner, re-balancing of the array for the
remaining value of the „a‟ is held till the re-balancing is
not possible. The reason for re-balancing is not being
possible in the array according to requirement spaces is
not possible. In this way performance of algorithm
degrades as require the larger number of swaps to
generate the spaces which is same as that in the case of
traditional insertion sort.
Table 2 describes the execution time of the LUGD and
LNGD algorithm using different type of data set that are
1 2
1 -1 2 -1
1 -1 2 -1 3 -1 4 -1
1 -1
1 -1 -1 -1
1 -1 -1 -1 2 -1 -1
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random data, nearly sorted data, reverse sorted data and
sorted data. Along with the different dataset
value, the table also describes the value of „ε‟ and re-
balancing factor „a‟. The re-balancing comparison of
LUGD and LNGD algorithms is shown in Fig. 5 to 8.
From fig. 5 to 8, the X-axis represents the value of gap (ε)
and re balancing factor (a) and Y-axis represents the
execution time in microseconds.
0
50000000
1E+09
1.5E+09
2E+09
2.5E+09
3E+09
3.5E+09
ε =1ε =2ε =3ε =4ε =1ε =2ε =3ε =4ε =1ε =2ε =3ε =4
a=2 a=3 a=4
Executiontimeinmicrosec
Value of gap and re-balancing factor
Random data
LUGD
LNGD
Fig. 5. Rebalancing time comparison of random data
Fig. 5 describes the plot at random data for the different
values of „ε‟ and re-balancing factor „a‟. It is observed
from Fig. 5, as if increase the re-balance factor in the case
of LUGD the execution time also increases significantly,
but in the case of LNGD the improvement of execution
time achieved upto 94% in comparison to LUGD.
0
50000000
1E+09
1.5E+09
2E+09
2.5E+09
3E+09
ε =1ε =2ε =3ε =4ε =1ε =2ε =3ε =4ε =1ε =2ε =3ε =4
a=2 a=3 a=4
Nearly sorted data
LUGD
LNGD
Fig. 6. Rebalancing time comparison of nearly sorted data
Fig. 6 shows the comparison of LUGD with LNGD at
different gaps and re-balancing factors. Again the
improvement is upto 92% at a=4 and ε=4.
0
50000000
1E+09
1.5E+09
2E+09
2.5E+09
3E+09
3.5E+09
ε =1ε =2ε =3ε =4ε =1ε =2ε =3ε =4ε =1ε =2ε =3ε =4
a=2 a=3 a=4
Reverse sorted data
LUGD
LNGD
Fig.7. Rebalancing time comparison of reverse sorted
data
Fig. 7 shows the comparison of execution time of reverse
sorted data at the different values of „ε‟ and re-balancing
factor „a‟. Initially, in this case results are improved by
8%, but maximum upto 57% at ε=4 and a=4.
0
50000000
1E+09
1.5E+09
2E+09
2.5E+09
3E+09
3.5E+09
ε =1ε =2ε =3ε =4ε =1ε =2ε =3ε =4ε =1ε =2ε =3ε =4
a=2 a=3 a=4
Value of gap and rebalancing factor
Sorted data
LUGD
LNGD
Fig.8. Rebalancing time comparison of sorted data
Fig. 8 represents the result on the sorted data with the
different value of gaps and re-balancing factor. The result
shows that maximum improvement achieved is upto 91%
in comparison to that of LUGD.
TABLE 2. TIME TAKEN BY LUGD AND LNGD ALGORITHM IN MICROSECOND FOR DIFFERENT VALUE OF RE-BALANCING AND
GAP
Type of Dataset
Re-
balancing
Value
of ε
Random Nearly Reverse Sorted
LUGD LNGD LUGD LNGD LUGD LNGD LUGD LNGD
2
ε =1 981267433 862909204 864558882 306063385 1450636163 1328993502 861929937 313078205
ε =2 729981576 708580455 620115904 230939335 1065247938 1022310950 609647355 234697961
ε =3 119727535 106921406 358670053 185759986 278810310 125152235 356489846 195120953
ε =4 23003046 14557332 117188830 107729204 263693774 116417058 116590140 106897060
3 ε =1 2622591059 869209660 2214715182 308010395 2832112301 1556949795 3011802732 339671533
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ε =2 2103580421 709709631 1964645906 231895871 2585747568 1280383725 2651992181 249815851
ε =3 2043974421 666999939 1728175857 185620741 2195021514 1239442185 1962122927 206018101
ε =4 1620914312 657130080 1600879365 155075390 2130261056 1263332585 1620374625 170410859
4
ε =1 2942693856 912631839 2467933298 300698051 3239333534 1656255915 3281368964 367329723
ε =2 2705332601 484314092 2510103530 227562280 3154811065 1545638611 2923182920 266428823
ε =3 2676681610 327850683 2613423098 183893005 3013676930 1366501241 2378347887 210265375
ε =4 2611656774 146342570 2157740458 153786055 2993363707 1270043884 2222906193 181557846
VII. CONCLUSION AND FUTURE WORK
The proposed approach of LNGD proved to be a better
algorithm in comparison to that of LUGD. Improvement
has been achieved that ranges from 8% to 90%. The
improvement of 90% has been found in the cases where
the LUGD was performing poorer. The performance of
LNGD is better for different values of re balancing factor
which was not achieved in the case LUGD. The LNGD
and LUGD both algorithms are implemented in C
language. The program of both algorithms is designed at
Borland C++ 5.02 compiler and executed on the Intel
core I5 processor-3230 M CPU @ 2.60 GHz machine,
and the programs runs at 2.2 GHz clock speed.
In the future, investigation can be done on the locality of
data in more details. This will help not only in allocating
the spaces accurately, but may also reduce the extra
spaces which have been allocated and will act as an
overhead both on the space and execution time of the
program.
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Vol. 16, No. 3, March 2018
ISSN 1947-5500

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