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Optimized Selection Sort Algorithm for Two Dimensional Array
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AbstractThe main idea of Optimized Selection Sort Algorithm
(OSSA) is based on the already existing selection sort algorithm,
with a difference that old selection sort; sorts one element either
smallest or largest in a single iteration while optimized selection
sort, sorts both the elements at the same time i.e smallest and
largest in a single iteration. In this study we have developed a
variation of OSSA for two-dimensional array and called it Optimized
Selection Sort Algorithms for Two-Dimensional arrays OSSA2D. The
hypothetical and experimental analysis revealed that the
implementation of the proposed algorithm is easy. The comparison
shows that the performance of OSSA2D is better than OSSA by four
times and when compared with old Selection Sort algorithm the
performance is improved by eight times (i.e if OSSA can sort an
array in 100 seconds, OSSA2D can sort it in 24.55 Seconds, and
similarly if Selection Sort takes 100 Seconds then OSSA2D take only
12.22 Seconds). This performance is remarkable when the array size
is very large. The experiential results also demonstrate that the
proposed algorithm has much lower computational complexity than the
one dimensional sorting algorithm when the array size is very
large. Index Terms Algorithm analysis, Sorting algorithms, Two
Dimensional Arrays, Computational Complexity.Introduction All
Sorting and Searching are the tasks that are repeatedly encountered
in various computer applications. Since they reveal basic tasks
that must be deal with quite frequently, researchers have tried in
the past to develop algorithms efficient in terms of least memory
requirement and least time requirement. Searching together with
sorting is perhaps the most common operation used in Computing. It
was always a matter of great attraction for the researchers that to
reduce the time a computer takes in sorting and searching data
[1][2].
That's why, the researcher tried to develop speedy,
well-organized and economical algorithms for sorting and ordering
lists and arrays. This was one of the fundamental fields of
research in computing for decades. The development of optimized
sorting algorithms will leads to efficient computing as whole. It
is observed that, the time to complete the task for most sorting
algorithms depends on the amount of data. Consequently to analyze
an algorithm, it is required to find a relationship showing how the
time needed for the algorithm depends on the amount of data. It is
observed that, for an algorithm, when the amount of data is
doubled, the time needed is also doubled. Similarly from the
analysis of another algorithm it was cleared that when the amount
of data is increased the time to complete the task of sorting is
increased by a factor of four.
It is also a matter of great concern that, the nature of the
data also has an effect on the efficiency of the algorithm. A
sorting algorithm may behave differently if the original data set
is almost sorted rather than if it is random or it may be in the
reverse order.
The creation of spatial data structures that are important in
computer graphics and geographic information systems is necessarily
a sorting procedure. A well-organized sort routine is also a useful
unit in implementing algorithms like sparse matrix multiplication
and parallel programming patterns like Map Reduce [3][ 4].
The idea of optimized selection is based on the old selection
sort [5]. Unlike the old selection sort which sort data item one at
a time, the optimized selection sort algorithm (OSSA) sort two data
elements (smallest as well largest) in a single iteration. It
reduced the execution time of the selection sort by almost fifty
percent, although this algorithm also has a time complexity of
[5].
Enormous data in many fields is normally represented by the use
of Matrices. It is felt by the computer professional to have
efficient sorting algorithms for sorting the two-dimensional
arrays. There is plenty of algorithms which mainly focuses on
one-dimensional array sorting, but a few algorithms have been
proposed for two-dimensional array sorting [6][7]. In [7], the
result is extended from one-dimensional array to two-dimensional by
the use of interpolation and filtering algorithms which results in
a complexity of
Two-dimensional array is treated as one dimensional array by
most of the earlier sorting algorithms to sort these elements [8].
Simple selection sorting is the widely sorting algorithm used due
to its simplicity and less implementation complexity. Although it
has a time complexity of [9] on sorting one dimensional array with
elements as comparison operations are required. The time required
for sorting large arrays (e.g. arrays with 1000000*1000000
elements) is practically much time consuming.In this article, we
propose an optimized sort algorithm for two dimensional arrays, to
solve the above mentioned problem of sorting. The hypothetical and
experimental results proved that this proposed algorithm has a high
efficiency level.
The organization of the paper is as follows. The algorithm is
proposed in Section 2. In Section 3 We analyze the proposed
algorithm. In Section 4 we present the approach to sorting
one-dimensional arrays by the proposed algorithm. Section 5
presents Experimental results. Section 6 contained
conclusion.Proposed Optimized selection sort algorithm for two
dimensional arraysThe main idea of Optimized Selection sort is
based on already existing selection sort algorithm, with a
difference that selection sort, sorts one element either smallest
or largest in a single iteration while optimized selection sort,
sorts the two elements at a time i.e smallest and largest in a
single iteration. We have developed a variation of the already
existing selection sort for two-dimensional array and called it
Optimized Selection Sort Algorithms for Two-Dimensional array.
We have a matrix in the form of as follows
Before going to the stepwise details of the algorithm we will
create a new array which will store sorted elements. The working of
our proposed optimized selection algorithm for two-dimensional
array is depicted in the following steps:
STEP 1: Each row of the array should be sorted using the
Optimized Selection Sorting Algorithm (Let assume that the elements
are sorted in descending order in each row).STEP2:Initialize the
index values of the target array i.e Y such that STEP3:Find the
Largest element from the first column and placed it into the
successor the largest will now be the first elementof the
row.STEP4:Update the index values for STEP5:Goto step (3) if AR is
not empty, else goto Exit.STEP6:EXIT.Analysis of OSSA2DThe Most of
the time in sorting process the algorithms major cost is on the
comparisons of the data items to be sorted. Therefore we analyzed
the time complexity of by computing the number of comparison which
is performed. As we know for sorting one dimension array the number
of comparison required by the selection sort is but we have
introduced the OSSA which performs less number of comparisons and
it becomes [1]. Let assume that there are rows in a two-dimensional
array. So times comparison will be required to sort all the rows of
the array.
Now after sorting each row of the array in descending order by
the above mechanism we will now search the maximum element in the
first column. For this purpose the number of comparison needed is
and there are rows and columns. So for all the comparison in the
array it needs. After finding the element in the first column it
will be transferred to the target array.
Therefore the total number of comparison needed for the
two-dimensional array to be sorted using OSSA for two dimensional
array will be:
The time complexity for the algorithm after ignoring the
constant and lower terms will be:
In [6] the efficiency of old selection sort algorithm for
one-dimensional array having the same number of elements as that
with the two-dimensional array is found as follows.
For one dimensional array the number of comparison needed by
selection sort;
(1)/2 (2)
For two-dimensional array with equal number of elements as that
of one-dimensional array the number of comparisons are;
(3)The efficiency Calculated for efficient two dimensional
sorting algorithms in [9] is;
Now calculating the efficiency of optimized selection sort
algorithm for two-dimensional array, against the efficiency of
optimized selection sort for one dimension array;
Again calculating the efficiency of optimized selection sort for
two-dimensional array against old selection sort for the same
number of data items;
Similarly calculating the efficiency of optimized selection sort
of two-dimensional against the efficient selection sort for
two-dimensional array; from equation and we have
Now consider if we have the values of and the value of After
putting the values in equation (5) and we =
The above result shows that our algorithm is 400 times efficient
than the ordinary selection sort for one dimensional array having
same amount of data.
if we put the values in the equation (7)
The above result shows that our algorithm is still more
efficient than the already existing algorithm for two-dimensional
array having same amount of data. Conversion of Single Dimensional
Array to Two Dimensional ArrayIt has been observed in the previous
section that the sorting of two-dimensional array is far more
efficient than the one-dimensional array for the same number of
elements. In this section we have tried to convert a
one-dimensional array with constant elements to a two-dimensional
array having data elements. According to equation (5) we have
And ( )
Since is a constant and > = 1 so according to geometric
inequality
SS Algorithm
OSSA
EfSSA
OSSA for 2 Dimensional Array
1000499500249750123991.161995.55
50001249750062487501401714700856.8
10000499950002499750039750001987500
150001.12E+085624625073109693655485
200002E+0899995000112637085631854
When then the values of is the minimum value.
(8)
From the above equation we can calculate the number of
comparison of two-dimensional arrays and found that it has the
lowest value. As the number of comparison of one-dimensional array
is given by , so if the number is increased by 100 times so we need
, almost 2500 times more comparisons as that of the two dimensional
array which is , 1875 times which is much more less than the
previous one OSSA for one dimensional array. We can calculate the
values of by the following method.
Some time it might happen that the calculated value of so we
need to fill the empty cells of the array with imaginary values
depending upon the data to be sorted. If the data is sorted in the
descending then we will fill the empty cells with infinitesimal
values and with infinite values otherwise.
ResultsAfter The following table shows the number of comparison
of one-dimensional array and two-dimensional sorting algorithm.
Table 2. Table showing time consuming of different
algorithms
When we plot these results we had;
Figure 1: The plot of Selection Sort and Optimized Selection
Sort Algorithm
Figure 2: The plot of Efficient Selection Sort and Optimized
Selection Sort Algorithms for 2 Dimensional Algorithms
Figure 3: The plot of OSSA and Optimized Selection Sort
Algorithms for 2 Dimensional Algorithms
Figure 4: The plot of Selection Sort and Optimized Selection
Sort Algorithms for 2 Dimensional AlgorithmsFrom above graph, one
can easily observed that with the increase in number of data
elements, the line graph plotted by OSSA is taking less time as
compared to graph plotted by Old SS and it will be true for any
number of data.Conclusions
It is evident from the above results that optimized selection
algorithm for two-dimensional arrays is more efficient than the
simple selection sorting algorithm for the same amount of data, and
hence proved our claim to be an efficient algorithm of the order
of.
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