A STATISTICAL COMPARATIVE STUDY OF SOME SORTING ALGORITHMS

International Journal in Foundations of Computer Science & Technology (IJFCST), Vol.5, No.4, July 2015
DOI:10.5121/ijfcst.2015.5403 21
A STATISTICAL COMPARATIVE STUDY OF
SOME SORTING ALGORITHMS
Anchala Kumari1
, Niraj Kumar Singh2*
and Soubhik Chakraborty3
1
Department of Statistics, Patna University, Patna
2
Department of Computer Science & Engineering, BIT Mesra, Ranchi
3
Department of Mathematics, BIT Mesra, Ranchi
ABSTRACT
This research paper is a statistical comparative study of a few average case asymptotically optimal sorting
algorithms namely, Quick sort, Heap sort and K- sort. The three sorting algorithms all with the same
average case complexity have been compared by obtaining the corresponding statistical bounds while
subjecting these procedures over the randomly generated data from some standard discrete and continuous
probability distributions such as Binomial distribution, Uniform discrete and continuous distribution and
Poisson distribution. The statistical analysis is well supplemented by the parameterized complexity
analysis.
KEYWORDS
Parameterized complexity, Statistical bound, Empirical-O, Computer experiment.
1. INTRODUCTION
Quick sort and Heap sort are the two standard sorting techniques used in literature for sorting
large data sets. These two methods exhibit the very same average case complexity of O(Nlog2N),
where N is the size of input to be sorted. Quick sort, possibly, is the best choice for sorting
random data. The sequential access nature of this algorithm keeps the associated constant terms
small and hence resulting in an efficient choice among the algorithms with similar asymptotic
class. The worst case complexity of quick sort is that of O(N2
) while the heap sort in worst case
exhibit the same (Nlog2N) complexity. The ϴ(Nlog2N) tight complexity of heap sort gives it an
edge over much used quick sort for use in stringent real time systems. New-sort, an improved
version of Quick sort which introduced by Sundararajan and Chakarborty [1] also confirms to
the same average and worst case complexity as that of Quick sort ,i.e., O(Nlog2N) and O(N2
)
respectively. But this New sort technique uses an auxiliary array, increasing the space complexity
thereby. A further improvement over the New sort was made by removing the concept of
auxiliary array from it and this sorting algorithm was named as K-sort [2]. Interestingly, for
typical inputs, on average the K-sort also consumes O(Nlog2N) time complexity. In a recent
research paper Singh et al. [3] explores the quick sort algorithm in detail, where it discusses
various interesting patterns obtained for runtime complexity data.
In this paper the three sorting algorithms all with the same average case complexity have been
compared by obtaining the corresponding statistical bounds while subjecting these procedures
over the randomly generated data from some standard discrete and continuous probability
distributions such as Binomial distribution, Uniform discrete and continuous distribution and
Poisson distribution.

22
ASYMPTOTIC ANALYSIS
Asymptotic analysis of Quick sort: The average and the best case recurrence of quick sort is T(N)
= 2T(N/2) + ϴ(N), T(1)=0. This upon solving yields a running time of O(Nlog2N). The worst
case recurrence is T(N) = T(N-1) + ϴ(N), T(1)=0, which results in O(N2
) complexity of quick sort
algorithm.
Asymptotic analysis of Heap sort: The best and worst case complexity of heap sort belong to
ϴ(Nlog2N) complexity.
Asymptotic analysis of K-sort: Due to its peculiar similarity the asymptotic time complexities of
K-sort is similar to that of quick sort.
EMPERICAL-O ANALYSIS
Empirical-O is an empirical estimate of the statistical bound over a finite range, obtained by
supplying numerical values to the weights which emerge from computer experiment. A computer
experiment being defined as a series of runs of a code for various inputs and is called
deterministic if it gives identical outputs if the code is re run for the same input.
Statistical bound (non probabilistic): If Wijis the weight of (a computing) operation of type i in
the jth
repetition (generally time is taken as a weight) and y is a “stochastic realization of the
deterministic T= ∑1.Wij where we count one for each operation repetition irrespective of the type,
the statistical bound of the algorithm is the asymptotic least upper bound of y expressed as a
function of N, N being the algorithm’s input size. T is realised for a fixed input while y is
expressed for a fixed size of the input. It is important to know y becomes stochastic for those
algorithms where fixing size does not fix all the computing operations. Sorting algorithm fall in
this category
Now we perform the empirical analysis of the results obtained by applying the specified
algorithms over the input data generated from the probability distributions mentioned earlier. The
codes were written in Dev C++ 5.8.2 and analysis was performed using Minitab statistical
Package.
The response (CPU time to run the code), the mean time in seconds is given in the tables 1-4 and
relative performance plots are presented in figures 1-4. Average case analysis is performed
directly on program run time to estimate the weight based statistical bound over a finite range by
running computer experiments [4][5]. This estimate is called empirical-O[6]][7]. Time of an
operation is taken as weight. Weighing permits collective consideration of all operations into a
conceptual bound which we call a statistical bound in order to distinguish it from the count based
mathematical bounds that are operation specific.
Sample size: The various inputs vary from a size of 1-10 lac which may be considered as
reasonably large for practical data set.
2. RELATIVE PERFORMANCE ANALYSIS OF DIFFERENT
ALGORITHMS
2.1 Discrete Uniform Distribution
Discrete Uniform distribution is characterised by single parameter k. In this experiment k has
been fixed at 1000.

23
Table 1. Data for discrete uniform distribution
N Heap sort
(HS)
K- Sort
(KS)
Quick Sort
(QS)
100000 .0528 .0526 .03040
200000 .0998 .1842 .0966
300000 .1778 .4032 .1904
400000 .2436 .6340 .3096
500000 .2878 .9424 .4750
600000 .3472 1.3376 .6594
700000 .4028 1.775 .860
800000 .4622 2.886 1.092
900000 .5216 2.9168 1.301
1000000 .5828 3.5912 1.744
Some interesting results are seemed to emerge from the above table. For N< 200000, quick sort
gives better performance as heap sort compared to other two algorithms but for N>=30000, the
heap sort outperforms the quick sort. The reason for ill performance of Quick sort may is due to
increase in number of ties in data set as N increases. As for as the K-sort is considered it gives
the worst performance at this particular value of k.
1000000
800000
600000
400000
200000
0
4
3
2
1
0
N
Y-Data
H S
K S
Q S
V ariab le
R e la tiv e pe rform a nce for dis cre te uniform dis tribution K= 1 0 0 0
Figure 1. Relative plots of quick heap and K-sort algorithms (k=1000)
For Discrete Uniform inputs in average case Quick sort, and K-Sort exhibit a time complexity of
Oemp(N2
) whereas that of heap sort it is Oemp(Nlog2N) [2].
2.2 Binomial Distribution
The Binomial distribution has two parameters m and p, m being the number of independent
Bernoulli trials and p the probability of success in each trial. The mean time given in table 2 was
obtained by varying N between 100000 to 1000000 and fixing m and p at 1000 and 0.5
respectively and making several runs for the same input.

24
Table 2. Data for Binomial distribution
N Heap sort
(HS)
K-sort
(KS)
Quick
sort
(QS)
100000 .4998 .6096 .2872
200000 1.0450 4.0198 1.1166
300000 1.6344 5.3716 2.46360
400000 2.24000 9.9072 4.3850
500000 2.8534 15.02379 6.8992
600000 3.4602 22.8832 9.8526
700000 4.0792 30.51688 13.5338
800000 4.74800 41.0869 17.66879
900000 5.3766 49.628720 22.0784
1000000 5.9798 62.4471 22.2676
1 0 0 0 0 0
8 0 0 0 0
6 0 0 0 0
4 0 0 0 0
2 0 0 0 0
0
7 0
6 0
5 0
4 0
3 0
2 0
1 0
0
N
Y-Data
H S
K S
Q S
V ar ia b le
R e la ti v e p e r f o r m a n c e p l o t f o r B in o m i a l in p u t
Figure 2. Relative plots of quick heap and K-sort algorithms (m=1000 and p=0.5)
The input data table (2) and relative performance plot (figure 2) supports the fact that on the
average K-sort consumes more time as compared to other two algorithms for sorting array of
same size. However for Binomial inputs quick sort and K-sort both confirms to Oemp(N2
), whereas
heap sort has O(Nlog2N) complexity.
2.3 Poisson Distribution
The Poisson distribution depends on the parameter λ. Lambda (which is both the mean and the
variance) should not be large as this is the distribution of rare events. While performing the
empirical analysis with Poisson inputs, its parameter, λ is fixed at 5.0

25
Table 3. Data for Poisson Input
N Heap sort
(HS)
K-sort
(KS)
Quick
sort
(QS)
20000 .015 .1778 .0964
40000 .019 .7158 .3190
60000 .0342 1.7450 .7250
80000 .047 2.8082 1.2696
100000 .0636 4.5356 1.9696
120000 .0716 6.320 2.8502
140000 .0818 8.5972 3.8226
160000 .094 11.2604 4.979
18000 .110 14.736 6.3406
200000 .125 18.0549 7.8044
2 0 0 0 0 0
1 5 0 0 0 0
1 0 0 0 0 0
5 0 0 0 0
0
2 0
1 5
1 0
5
0
N
Y-Data
H S
K S
Q S
V a r ia b le
R e l a t i v e P e r f o r m a n c e p l o t f o r P o i s s o n i n p u t
Figure 3. Relative plots of quick heap and K-sort algorithms (λ=5.0)
For Poisson inputs, heap sort gives the good performance as compared to other two algorithms.
Quick sort and K-sort both have Oemp(N2
) complexity whereas if we have a look on the three
graph (figure 3), it is obvious that heap sort again confirms to Oemp(Nlog2N) complexity,
2.4 Uniform Continuous Distribution
Table 4 gives the mean execution time for the data simulated from uniform continuous
distribution [0, θ ].
Table 4. Data for Uniform Continuous Distribution
N Heap sort
(HS)
K-sort
(KS)
Quick
sort
(QS)
100000 .055 .0310 .0340
200000 .11277 .0748 .062
300000 .1780 .1092 .0872

26
400000 .2400 .1468 .1064
500000 .3150 .2002 .1312
600000 .3842 .2624 .159
700000 .4533 .3092 .1908
800000 .5188 .3672 .219
900000 .902 .425 .2534
10000 1.0154 .4750 .2842
1000000
800000
600000
400000
200000
0
1.0
0.8
0.6
0.4
0.2
0.0
n
Y-Data
H S
KS
Q S
Variab le
Relative performance plot for Uniform Continuous distribution
Figure 4. Relative plots of quick heap and K-sort algorithms [0-1]
Here the scenario has just changed. Quick sort outperforms the two algorithms in its performance.
The K-sort for all values of N in selected range out performance the heap algorithm . However
all the three algorithms suggest Oemp(Nlog2N) complexity.
3. STATISTICAL ANALYSIS OF DIFFERENT ALGORITHMS FOR
DISCRETE DISTRIBUTIONS
In this section we test the hypothesis whether the average performance of the algorithms for
different input distributions is same or not. For this we apply two way analysis of variance. A
value of p = 0.358 greater than 0.05 as shown in table 5, is indicative of the fact that as for as
their average performance is concerned there is no reason to differentiate between the three.
Table 5. Results of Two Way ANOVA
Source DF SS MS F P
PRODIS 3 163.354 54.4514 2.52 0.155
SORTALG 2 52.955 26.4776 1.23 0.358
Error 6 129.602 21.6003
Total 11345.911

27
3.1 Parametric complexity
Parametric complexity is one of the important criterions for selection of an algorithm among the
several algorithms, since besides the size of input, parameters of input distribution has direct
effect on execution time of an algorithm. We have examined the parameterized complexity of the
three algorithms for binomial inputs, the reason being that distribution has two parameters . A 32
factorial experiment with three repeated set of data elements at same combination of level of
factors m and p has been employed. The results given in table 6 below reveal some interesting
findings:
Table 6. Parameterized Complexity of heap sort, k-sort and quick sort
sources Heap
sort
(HS)
K-sort
(KS)
Quick
sort
(QS)
df F P F P F P
M 2 0.13 .876 3.85 .04 1.11 .351
P 2 0.15 .866 37.37 0.00 26.1
8
.000
MP 4 0.18 .948 1.62 .213 0.31 .867
The F and P values revealed that in case of Heap Sort the two parameters neither singularly nor
jointly has any effect on sorting time while in case of K-sort though both the factors have
independent effects on the complexity , probability of success being highly significant. While
applying the quick sort algorithm, the number of trials (m) shows highly non significant effect
while the probability of success p delivers significantly high effect on complexity .Thus the
proper selection of input parameters can have rewarding effect on reducing the complexity of an
algorithm.. For different values of p, the average execution time is given in the table 7 below
Table 7. optimal value of p m=5000, n=50000
P K-sort Quick sort
0.1 .188 .056
0.2 .144 .047
0.3 .126 .040
0.4 .127 .044
0.5 .115 .0438
06 .117 .0566
0.7 .125 .063
o.8 .1406 ..072
0.9 .183 ..072
Here we find that in both the cases initially the execution time decreases as the value of p is
increased, at p=.5 , execution time is minimum and then again it goes on increasing. Thus the
optimal value of p is .5. But as for the optimal value of m is concerned in case of k sort, as shown
in the following table 8, the execution time decreases with increase in the value of m Thus in
this case a high value of m is preferable.

28
Table 8. optimal value of m for K sort (p=.5, n=50000)
M Ksort
2000 .174
3000 ,1456
4000 .1275
5000 .0782
6000 .0704
7000 .0666
8000 .0644
9000 .061
10000 .056
4. CONCLUSIONS
The three sorting algorithms, heap sort, K-sort and quick sort though theoretically deliberating to
same complexity O(Nlog2N) has been supported by the statistical analysis in section 3. We have
no evidence against rejection of the hypothesis of homogeneity of algorithms as far as their
average performance is considered. But unfortunately in worst case quick sort and K-sorts have
complexity O(N2
) than heap sort which exhibits a complexity of O(Nlog2N) since we have the
relation that O(Nlog2N)<O(N2
) .
However as far as Empirical-O estimates are considered , quick sort for N less than 200000 gives
better performance for some discrete distributions such as Binomial and Uniform distribution
while for N>300000 heap sort is best, K-sort for these distributions does not work good. But
sorting an array generated randomly (not generated from any standard probability distribution, K-
sort works good. It can sort an array of size never greater than 7000000 in less time than heap sort
[2].
For continuous uniform distribution, quick sort gives the good performance as compared to other
two. This result is quite expected as in case of a continuous distribution the probability of getting
similar valued elements (ties) is theoretically zero. It is well known through various results [8]
that quick sort behaves exceptionally good in such cases. Whereas in the very same scenario heap
sort is expected to perform relatively poor (however the time complexity of heap sort remains of
the order of Nlog2N) [9].
The different behaviour of the algorithms to input data can be supplemented by the parameterised
complexity analysis since the true potential of an algorithm is related to parameters of the input
distribution. As far as the parameterised complexity of heap sort is considered, it is in favour of
its worst case complexity which is less than the other two algorithms. The reason being obvious
as execution time does not depend on the binomial parameters. But in case of quick sort the
parameter p has highly significant effect on execution time. Though the two parameters m and p
have independent effects on complexity in case of K sort, but we find that parameter m has a
very little effect where as p has highly significant . In both the cases of K sort and Quick sort, the
complexity is minimum at p=.5 but a high value of p is preferable to reduce the complexity while
using the K-sort algorithm. Thus proper selection of input parameters can have rewarding effect
on reducing complexity of an algorithm.

29
REFERENCES
[1] Sundararajan, KK & Chakraborty S, (2007) "A New sorting algorithm" Journal of Applied
Ma1thematics and Computation 188, pp 1037-1041.
[2] Sundararajan, KK, Pal, M, Chakarborty ,S., & Mahanti, NC, (2013) "K-Sort: A New Sorting
Algorithm that beats Heap Sort for n ≤ 70 lakhs".
[3] Singh, N K, Chakraborty, S, and Mallick, DK, (2014) "A Statistical Peek into Average Case
Complexity", IJEDPO, Vol. 4, No 2, Int. J. on Recent Trends in Engineering and Technology 8(1),
64-67.
[4] Fang, KT, Li, R, and Sudjianto, (2000) Design and Modelling of computer Experiments, Taylor &
Francis Group, Boca Raton.
[5] Sacks, J, Welch,W, Mitchel, T, and Wynn, (1989) "Design and Analysis of Computer Experiments",
Statistical Science 4.
[6] Chakraborty , S, & Sourabh, SK, (2010) A Computer Oriented Approach to Algorithmic Complexity,
Lambert Academic Publishing.
[7] Chakraborty, S, Modi, N and Panigrahi, S, (2009) "Will the weight based Statistical Bounds
Revolutionize the It?", International Journal of Computational Cognition 7(3).
[8] Singh, NK & Chakraborty, S, (2012) "Smart Sort: A Fast, Efficient and Robust Sorting Algorithm",
IJACMS, ISSN 2230-9624, Vol 3, Issue 4, pp 487-493.
[9] Sourabh, SK ,and Chakraborty, S, (2009) "Empirical Study on the Robustness of Average Complexity
and Parameterized Complexity Measure for Heap sort Algorithm", International Journal of
Computational Cognition.Vol.7, No.4.
Authors
Anchala Kumari is a Professor in the department of statistics, Patna University, India. Her
research area includes: Operations research, Design of Experiments and Computer
Programming.
Niraj Kumar Singh is a Teaching Cum Research Fellow at department of CSE, BIT
Mesra, India. His research interest includes: Design and Analysis of Algorithms and
Algorithmic Analysis through Statistical bounds.
Soubhik Chakraborty is a Professor in the department of Mathematics, BIT Mesra, India.
His research interest includes: Music Analysis, Algorithmic Complexity and Statistical
Computing.

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