PARTITION SORT REVISITED: RECONFIRMING THE ROBUSTNESS IN AVERAGE CASE AND MUCH MORE!

International Journal of Computer Science, Engineering and Applications (IJCSEA) Vol.2, No.1, February 2012
DOI : 10.5121/ijcsea.2012.2103 23
PARTITION SORT REVISITED:
RECONFIRMING THE ROBUSTNESS IN
AVERAGE CASE AND MUCH MORE!
Niraj Kumar Singh1
, Mita Pal2
and Soubhik Chakraborty3*
1
Department of Computer Science & Engineering, B.I.T. Mesra, Ranchi-835215, India
2,3
Department of Applied Mathematics, B.I.T. Mesra, Ranchi-835215, India
*email address of the corresponding author:
soubhikc@yahoo.co.in (S. Chakraborty)
ABSTRACT
In our previous work there was some indication that Partition Sort could be having a more robust average
case O(nlogn) complexity than the popular Quick sort. In our first study in this paper, we reconfirm this
through computer experiments for inputs from Cauchy distribution for which expectation theoretically does
not exist. Additionally, the algorithm is found to be sensitive to parameters of the input probability
distribution demanding further investigation on parameterized complexity. The results on this algorithm for
Binomial inputs in our second study are very encouraging in that direction.
KEY WORDS
Partition-sort; average case complexity, robustness; parameterized complexity; computer experiments;
factorial experiments
1. Introduction
Average complexity is an important field of study in algorithm analysis as it explains how certain
algorithms with bad worst case complexity perform better on the average like Quick sort. The
danger in making such a claim often lies in not verifying the robustness of the average complexity
in question. Average complexity is theoretically obtained by applying mathematical expectation
to the dominant operation or the dominant region in the code. One problem is: for a complex code
it is not easy to identify the dominant operation. This problem can be resolved by replacing the
count based mathematical bound by a weight based statistical bound that also permits collective
consideration of all operations and then estimate it by directly working on time, regarding the
time consumed by an operation as its weight. A bigger problem is that the probability distribution
over which expectation is taken may not be realistic over the domain of the problem. Algorithm
books derive these expectations for uniform probability inputs. Nothing is stated explicitly that
the results will hold even for non-uniform inputs nor is there any indication as to how realistic the
uniform input is over the domain of the problem. The rejection of Knuth’s proof in [1] and
Hoare’s proof in [2] for non uniform inputs should be a curtain raiser in that direction. Similarly,

24
it appears from [3] that the average complexity in Schoor’s matrix multiplication algorithm is not
the expected number of multiplications O(d1d2 n3
), d1 and d2 being the density (fraction of non
zero elements) of pre and post factor matrices, but the exact number of comparisons which is n2
provided there are sufficient zeroes and surprisingly we don’t need a sparse matrix to get an
empirical O(n2
) complexity! This result is obtained using a statistical bound estimate and shows
that multiplication need not be the dominant operation in every matrix multiplication algorithm
under certain input conditions.
In our previous work [4] we introduced Partition Sort and found it to be having a more robust
average case O(nlogn) complexity than the popular Quick sort. In our first study in this paper, we
reconfirm this through computer experiments for inputs from Cauchy distribution for which
expectation theoretically does not exist! Additionally, the algorithm is found to be sensitive to
parameters of the input probability distribution demanding further investigation on parameterized
complexity on this algorithm. This is confirmed for Binomial inputs in our second study.
The Algorithm Partition Sort
Partition-sort algorithm is based on divide and conquer paradigm. The function “partition” is the
key sub-routine of this algorithm. The nature of partition function is such that when applied on
input A[1…….n] it divides this list into two halves of sizes floor (n/2) and ceiling (n/2)
respectively. The property of the elements in these halves is such that the value of each element in
first half is less than the value of every element in the second half. The Partition-sort routine is
called on each half recursively to finally obtain a sorted sequence of data as required. Partition
Sort was introduced by Singh and Chakraborty [4] who obtained O(nlog2
2
n) worst case count,
(nlog2n) best case count and empirical O(nlog2n) as the statistical bound estimate by working
directly on time, for reasons stated earlier, in the average case.
2. Statistical Analysis
2.1 Reconfirming the robustness of average complexity of Partition Sort
Theorem 1: If U1 and U2 are two independent uniform U [0, 1] variates then Z1 and Z1 defined
below are two independent Standard Normal variates:
Z1= (-2lnU1)1/2
Cos(2ЛU2); Z2= (-2lnU1)1/2
Sin(2ЛU2)
This result is called Box Muller transformation.
Theorem 2: If Z1 and Z2 are two independent standard Normal variates then Z1/Z2 is a standard
Cauchy variate. For more details, we refer to [5].
Cauchy distribution is an unconventional distribution for which expectation does not exist
theoretically. Hence it is not possible to know the average case complexity theoretically for inputs
from this distribution. Working directly on time, using computer experiments, we have obtained
an empirical O(nlogn) complexity in average sorting time for Partition sort for Cauchy
distribution inputs which we simulated using theorems 1 and 2 given above. This result goes a
long way in reconfirming that Partition Sort’s average complexity is more robust compared to
that of Quick Sort. In [4] we have theoretically proved that its worst case complexity is also much

25
better than that of Quick Sort as O(nlog2
2
n) < O(n2
). Although Partition Sort is inferior to Heap
Sort’s O(nlogn) complexity in worst case, it is still easier to program Partition Sort.
Table 1 and figure 1 based on table 1 summarize our results.
Table 1: Average time for Partition Sort for Cauchy distribution inputs
N 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
Mean
Time
(Sec.)
0.05168 0.10816 0.1487 0.17218 0.20494 0.24078 0.2659 0.31322 0.35128 0.39496
Fig. 1: Regression model suggesting empirical O(nlogn) complexity
2.2 Partition Sort subjected to parameterized complexity analysis
Parameterized complexity is a branch of computational complexity theory in computer science
that focuses on classifying computational problems according to their inherent difficulty with
respect to multiple parameters of the input. The complexity of a problem is then measured as a
function in those parameters. This allows to classify NP-hard problems on a finer scale than in the
classical setting, where the complexity of a problem is only measured by the number of bits in the
input. The first systematic work on parameterized complexity was done by Downey & Fellows
[6]. The authors in [7] have strongly argued both theoretically and experimentally why for certain
algorithms like sorting, the parameters of the input distribution should also be taken into account
for explaining the complexity, not just the parameter characterizing the size of the input. The
second study is accordingly devoted to parameterized complexity analysis whereby the sorting
elements of Partition Sort come independently from a Binomial (m, p) distribution. Use is made
of factorial experiments to investigate the individual effect of number of sorting elements (n),
binomial distribution parameters (m and p which give the number of independent trials and the
fixed probability of success in a single trial respectively) and also their interaction effects. A 3-

26
cube factorial experiment is conducted with three levels of each of the three factors n, m and p.
All the three factors are found to be significant both individually and interactively.
In our second study, Table-2 gives the data for factorial experiments to accomplish our study on
parameterized complexity. For clarity, table 2 is presented in three parts- table 2.1, 2.2 and 2.3.
Table 2: Data for 33
factorial experiment for Partition Sort
Partition sort times in second Binomial ( m , p ) distribution input for various n (50000, 100000,
150000) , m ( 100 , 1000, 1500) and p (0.2, 0.5, 0.8).
Each reading is averaged over 50 readings.
Table 2.1 data for n = 50000
m p=0.2 p=0.5 p=0.8
100 0.07248 0.07968 0.07314
1000 0.09662 0.10186 0.09884
1500 0.10032 0.10618 0.10212
Table 2.2 data for n=100000
M p=0.2 p=0.5 p=0.8
100 0.16502 0.1734 0.16638
1000 0.21394 0.22318 0.21468
1500 0.22194 0.23084 0.22356
Table 2.3 data for n = 150000
m p=0.2 p=0.5 p=0.8
100 0.26242 0.27632 0.26322
1000 0.33988 0.35744 0.34436
1500 0.35648 0.37 0.35572
Table-3 gives the results using MINITAB statistical package version 15.

27
Table-3: Results of 33
factorial experiment on partition-sort
General Linear Model: y versus n, m, p
Factor Type Levels Values
n fixed 3 1, 2, 3
m fixed 3 1, 2, 3
p fixed 3 1, 2, 3
Analysis of Variance for y, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
n 2 0.731167 0.731167 0.365584 17077435.80 0.000
m 2 0.056680 0.056680 0.028340 1323846.78 0.000
p 2 0.001440 0.001440 0.000720 33637.34 0.000
n*m 4 0.011331 0.011331 0.002833 132322.02 0.000
n*p 4 0.000283 0.000283 0.000071 3302.87 0.000
m*p 4 0.000034 0.000034 0.000009 397.33 0.000
n*m*p 8 0.000046 0.000046 0.000006 266.70 0.000
Error 54 0.000001 0.000001 0.000000
Total 80 0.800982
S = 0.000146313 R-Sq = 100.00% R-Sq(adj) = 100.00%
3. Discussion and more statistical analysis
Partition- sort is highly affected by the main effects n, m and p. When we consider the interaction
effects, interestingly we find that all interactions are significant in Partition-Sort. Strikingly, even
the three factor interaction n*m*p cannot be neglected. This means Partition Sort is quite
sensitive to parameters of the input distribution and hence qualifies to be a potential candidate for
deep investigation in parameterized complexity both theoretically (through counts) and
experimentally (through weights) for inputs from other distributions. Further, we have obtained
some interesting patterns showing how the Binomial parameters influence the average sorting
time. Our investigations are ongoing for a theoretical justification for the same. The final results
are summarized in tables 4-5 and figures 2A, 2B and 3 based on these tables respectively.
Each entry in the following tables is averaged over 50 readings.
Table 4: Partition Sort, Binomial (m, p) distribution, array size N=50000, p=0.5 fixed
m 100 300 500 700 900 1100 1300 1500
Mean
time
(sec.)
0.07968 0.09066 0.09586 0.09968 0.10154 0.10438 0.10282 0.10618

28
Fig 2A Third degree polynomial fit captures the trend
Fig 2B Fourth degree polynomial appears to be a forced fit (over fit)
(don’t get carried away by the higher value of R2
!)
Although the fourth degree polynomial fit gives a higher value of R2
, it forces the fit to pass
through all the data points. The essence of curve fitting lies in catching the trend (in the
population) exhibited by the observations rather than catching the observations themselves
(which reflect only a sample). Besides, a bound estimate must look like a bound estimate and it is
stronger to write yavg(n, m, p)=Oemp(m3
) than to write yavg(n, m, p)=Oemp(m4
) for fixed n and p.
Avg Time vs m of Binomial (m,p)y = 1E-11x3
- 5E-08x2
+ 7E-05x + 0.0739
R2
= 0.9916
0
0.02
0.04
0.06
0.08
0.1
0.12
0 200 400 600 800 1000 1200 1400 1600
m
AvgTimeofPartitionSortinsec
Avg Time vs m of Binomial (m,p)y = -5E-14x4
+ 2E-10x3
- 2E-07x2
+ 0.0001x + 0.0703
R2
= 0.9987
0
0.02
0.04
0.06
0.08
0.1
0.12
0 200 400 600 800 1000 1200 1400 1600
m
AvgTimeofPartitionSortinsec

29
So we agree to accept the first of the two propositions.
Table 5: Partition Sort, Binomial distribution (m, p), n=50000, m=1000 fixed
p 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Mean
time
(sec.)
0.09084 0.09662 0.09884 0.10198 0.10186 0.10034 0.0989 0.09884 0.09096
Fig. 3 Second degree polynomial fit captures the trend
Fitting higher polynomials lead to over-fitting (details omitted) and from previous arguments we
put yavg(n, m, p)=Oemp(p2
) for fixed n and m.
For definitions of statistical bound and empirical O, we refer to [4]. For a list of properties of a
statistical complexity bound as well as to understand what design and analysis of computer
experiments mean when the response is a complexity such as time, [8] may be consulted.
4. Conclusion and suggestions for future work
We conclude
(i) Partition Sort is more robust than Quick Sort in average case.
Avg time vs p of Binomial (m,p) y = -0.0649x
2
+ 0.0659x + 0.0853
R
2
= 0.9293
0.09
0.092
0.094
0.096
0.098
0.1
0.102
0.104
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
AvgTimeforPartitionSortinsec

30
(ii) Partition Sort is sensitive to parameters of input distribution also, apart from the
parameter that characterizes the input size.
(iii) For n independent Binomial (m, p) inputs, all the three factors are significant both
independently and interactively. All the two factor interactions nxm, nxp and mxp
and even the three factor nxmxp is significant. This last finding is of paramount
importance to excite other researchers on parameterized complexity and is intriguing
if not impossible to be established theoretically. Theoretical analysis might confirm
the influence of the Binomial parameters but how do you confirm the significance of
their interactions? Using computer experiments where cheap and efficient prediction
is the motive [8][9][10], we have settled the imbroglio.
(iv) We have also found yavg(n, m, p)=Oemp(m3
) for fixed n and p while yavg(n, m,
p)=Oemp(p2
) for fixed n and m. It should be kept in mind that these results are
obtained by working on weights and should not be confused with count based
theoretical analysis which need not be identical.
In summary, this paper should convince the reader about the existence of weight based statistical
bounds that can be empirically estimated by merging the quantum of literature in computer
experiments (this literature includes factorial experiments, applied regression analysis and
exploratory data analysis, which we have used here, not to speak of other areas like spatial
statistics, bootstrapping, optimality design and even Bayesian analysis!) with that in algorithm
theory. Computer scientists will hopefully not throw away our statistical findings and will
seriously think about the prospects of building a weight based science theoretically to explain
algorithm analysis given that the current count based science is quite saturated. This was
essentially the central focus in our adventures. So the purpose achieved, we close the paper.
References
[1] S. Chakraborty and S. K. Sourabh, How Robust Are Average Complexity Measures? A Statistical
Case Study, Applied Math. and Compu., vol. 189(2), 2007, 1787-1797
[2] S. K. Sourabh and S. Chakraborty, How robust is quicksort average complexity? arXiv:0811.4376v1
[cs.DS], Advances in Mathematical Sciences Jour. (to appear)
[3] S. Chakraborty, S. K. Sourabh , On Why an Algorithmic Time Complexity Measure can be System
Invariant rather than System Independent, Applied Math. and Compu, Vol. 190(1), 2007, p. 195-204
[4] N. K. Singh and S. Chakraborty, Partition Sort and its Empirical Analysis, CIIT-2011, CCIS 250, pp.
340-346, 2011. © Springer-Verlag Heidelberg 2011.
[5] W. Kennedy and J. Gentle, Statistical Computing, Marcel Dekker Inc., 1980
[6] R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer, (1999)
[7] S. Chakraborty, S. K. Sourabh, M. Bose and K. Sushant, Replacement sort revisited: The “gold
standard” unearthed! , Applied Mathematics and Computation ,vol. 189(2), 2007, p. 384-394
[8] S. Chakraborty and S. K. Sourabh, A Computer Experiment Oriented Approach to Algorithmic
Complexity, Lambert Academic Publishing, (2010)
[9] J. Sacks, W. Weltch, T.Mitchel, H. Wynn, Design and Analysis of Computer Experiments, Statistical
Science Vol.4 (4), (1989)
[10] K. T. Fang, R. Li, A. Sudjianto, Design and Modeling of Computer Experiments, Chapman and Hall
(2006)

PARTITION SORT REVISITED: RECONFIRMING THE ROBUSTNESS IN AVERAGE CASE AND MUCH MORE!

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