arXiv subjects

Soubhik Chakraborty

Publications and source records attributed to Soubhik Chakraborty.

17 recordsLinked to original sources

arXiv2018

Simulating Raga Notes with a Markov Chain of Order 1-2

Semi Natural Algorithmic composition (SNCA) is the technique of using algorithms to create music note sequences in computer with the understanding that how to render them would be decided by the composer. In our approach we are proposing an SNCA2 algorithm (extension of SNCA algorithm) with an illustrative example in Raga Bageshree. For this, Transition probability matrix (tpm) was created for the note sequences of Raga Bageshree, then first order Markov chain (using SNCA) and second order Markov chain (using SNCA2) simulations were performed for generating arbitrary sequences of notes of Raga Bageshree. The choice between first and second order Markov model, is best left to the composer who has to decide how to render these music notes sequences. We have confirmed that Markov chain of order of three and above are not promising, as the tpm of these become sparse matrices.

cs.SD↗

arXiv2013

A Statistical Peek into Average Case Complexity

The present paper gives a statistical adventure towards exploring the average case complexity behavior of computer algorithms. Rather than following the traditional count based analytical (pen and paper) approach, we instead talk in terms of the weight based analysis that permits mixing of distinct operations into a conceptual bound called the statistical bound and its empirical estimate, the so called "empirical O". Based on careful analysis of the results obtained, we have introduced two new conjectures in the domain of algorithmic analysis. The analytical way of average case analysis falls flat when it comes to a data model for which the expectation does not exist (e.g. Cauchy distribution for continuous input data and certain discrete distribution inputs as those studied in the paper). The empirical side of our approach, with a thrust in computer experiments and applied statistics in its paradigm, lends a helping hand by complimenting and supplementing its theoretical counterpart. Computer science is or at least has aspects of an experimental science as well, and hence hopefully, our statistical findings will be equally recognized among theoretical scientists as well.

cs.DS↗

arXiv2013

An Adaptable Fast Matrix Multiplication Algorithm, Going Beyond the Myth of Decimal War

In this paper we present an adaptable fast matrix multiplication (AFMM) algorithm, for two nxn dense matrices which computes the product matrix with average complexity Tavg(n) = d1d2n3 with the acknowledgement that the average count is obtained for addition as the basic operation rather than multiplication which is probably the unquestionable choice for basic operation in existing matrix multiplication algorithms.

cs.DS↗

arXiv2012

The Parameterized Complexity Analysis of Partition Sort for Negative Binomial Distribution Inputs

The present paper makes a study on Partition sort algorithm for negative binomial inputs. Comparing the results with those for binomial inputs in our previous work, we find that this algorithm is sensitive to parameters of both distributions. But the main effects as well as the interaction effects involving these parameters and the input size are more significant for negative binomial case.

cs.DS↗

arXiv2012

Smart Sort: Design and Analysis of a Fast, Efficient and Robust Comparison Based Internal Sort Algorithm

Smart Sort algorithm is a "smart" fusion of heap construction procedures (of Heap sort algorithm) into the conventional "Partition" function (of Quick sort algorithm) resulting in a robust version of Quick sort algorithm. We have also performed empirical analysis of average case behavior of our proposed algorithm along with the necessary theoretical analysis for best and worst cases. Its performance was checked against some standard probability distributions, both uniform and non-uniform, like Binomial, Poisson, Discrete & Continuous Uniform, Exponential, and Standard Normal. The analysis exhibited the desired robustness coupled with excellent performance of our algorithm. Although this paper assumes the static partition ratios, its dynamic version is expected to yield still better results.

cs.DS↗

arXiv2012

Partition Sort Revisited: Reconfirming the Robustness in Average Case and much more!

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.

cs.DS↗

arXiv2012

How does the Shift-insertion sort behave when the sorting elements follow a Normal distribution?

The present paper examines the behavior of Shift-insertion sort (insertion sort with shifting) for normal distribution inputs and is in continuation of our earlier work on this new algorithm for discrete distribution inputs, namely, negative binomial. Shift insertion sort is found more sensitive for main effects but not for all interaction effects compared to conventional insertion sort.

cs.DS↗

arXiv2012

On an Ethical Use of Neural Networks: A Case Study on a North Indian Raga

The paper gives an artificial neural network (ANN) approach to time series modeling, the data being instance versus notes (characterized by pitch) depicting the structure of a North Indian raga, namely, Bageshree. Respecting the sentiments of the artists' community, the paper argues why it is more ethical to model a structure than try and "manufacture" an artist by training the neural network to copy performances of artists. Indian Classical Music centers on the ragas, where emotion and devotion are both important and neither can be substituted by such "calculated artistry" which the ANN generated copies are ultimately up to.

cs.NE↗

arXiv2012

Parameterized Complexity on a New Sorting Algorithm: A Study in Simulation

Sundararajan and Chakraborty (2007) introduced a new sorting algorithm by modifying the fast and popular Quick sort and removing the interchanges. In a subsequent empirical study, Sourabh, Sundararajan and Chakraborty (2007) demonstrated that this algorithm sorts inputs from certain probability distributions faster than others and the authors made a list of some standard probability distributions in decreasing order of speed, namely, Continuous uniform < Discrete uniform < Binomial < Negative Binomial < Poisson < Geometric < Exponential < Standard Normal. It is clear from this interesting second study that the algorithm is sensitive to input probability distribution. Based on these pervious findings, in the present paper we are motivated to do some further study on this sorting algorithm through simulation and determine the appropriate empirical model which explains its average sorting time with special emphasis on parameterized complexity.

stat.CO↗

arXiv2012

How many interchanges does the selection sort make for iid geometric(p) input?

The note derives an expression for the number of interchanges made by selection sort when the sorting elements are iid variates from geometric distribution. Empirical results reveal we can work with a simpler model compared to what is suggestive in theory. The morale is that statistical analysis of an algorithm's complexity has something to offer in its own right and should be therefore ventured not with a predetermined mindset to verify what we already know in theory. Herein also lies the concept of an empirical O, a novel although subjective bound estimate over a finite input range obtained by running computer experiments. For an arbitrary algorithm, where theoretical results could be tedious, this could be of greater use.

stat.CO↗

arXiv2011

K-sort: A new sorting algorithm that beats Heap sort for n <= 70 lakhs!

Sundararajan and Chakraborty (2007) introduced a new version of Quick sort removing the interchanges. Khreisat (2007) found this algorithm to be competing well with some other versions of Quick sort. However, it uses an auxiliary array thereby increasing the space complexity. Here, we provide a second version of our new sort where we have removed the auxiliary array. This second improved version of the algorithm, which we call K-sort, is found to sort elements faster than Heap sort for an appreciably large array size (n <= 70,00,000) for uniform U[0, 1] inputs.

cs.DS↗

arXiv2009

On the Interesting World of Fractals and Their Applications to Music

In this paper we have defined one function that has been used to construct different fractals having fractal dimensions between 1.58 and 2. Also, we tried to calculate the amount of increment of fractal dimension in accordance with the base of the number systems. Further, interestingly enough, these very fractals could be a frame of lyrics for the musicians, as we know that the fractal dimension of music is around 1.65 and varies between a high of 1.68 and a low of 1.60. Further, at the end we conjecture that the switching from one music fractal to another is nothing but enhancing a constant amount fractal dimension which might be equivalent to a kind of different sets of musical notes in various orientations.

cs.OH↗

arXiv2009

On Why and What of Randomness

This paper has several objectives. First, it separates randomness from lawlessness and shows why even genuine randomness does not imply lawlessness. Second, it separates the question -why should I call a phenomenon random? (and answers it in part one) from the patent question -What is a random sequence? -for which the answer lies in Kolmogorov complexity (which is explained in part two). While answering the first question the note argues why there should be four motivating factors for calling a phenomenon random: ontic, epistemic, pseudo and telescopic, the first two depicting genuine randomness and the last two false. Third, ontic and epistemic randomness have been distinguished from ontic and epistemic probability. Fourth, it encourages students to be applied statisticians and advises against becoming armchair theorists but this is interestingly achieved by a straight application of telescopic randomness. Overall, it tells (the teacher) not to jump to probability without explaining randomness properly first and similarly advises the students to read (and understand) randomness minutely before taking on probability.

cs.OH↗

arXiv2008

Which notes are Vadi-Samvadi in Raga Rageshree?

The notes which play the most important and second most important roles in expressing a raga are called Vadi and Samvadi swars respectively in (North) Indian Classical music. Like Bageshree, Bhairavi, Shankara, Hamir and Kalingra, Rageshree is another controversial raga so far as the choice of Vadi-Samvadi selection is concerned where there are two different opinions. In the present work, a two minute vocal recording of raga Rageshree is subjected to a careful statistical analysis. Our analysis is broken into three phases: first half, middle half and last half. Under a multinomial model set up holding appreciably in the first two phases, only one opinion is found acceptable. In the last phase the distribution seems to be quasi multinomial, characterized by an unstable nature of relative occurrence of pitch of all the notes and although the note whose relative occurrence of pitch suddenly shoots is the Vadi swar selected from our analysis of the first two phases, we take it as an outlier demanding a separate treatment like any other in statistics. Selection of Vadi-Samvadi notes in a quasi-multinomial set up is still an open research problem. An interesting musical cocktail is proposed, however, embedding several ideas like melodic property of notes, note combinations and pitch movements between notes, using some weighted combination of psychological and statistical stability of notes along with watching carefully the sudden shoot of one or more notes whenever there is enough evidence that multinomial model has broken down.

cs.SD↗

arXiv2008

How robust is quicksort average complexity?

The paper questions the robustness of average case time complexity of the fast and popular quicksort algorithm. Among the six standard probability distributions examined in the paper, only continuous uniform, exponential and standard normal are supporting it whereas the others are supporting the worst case complexity measure. To the question -why are we getting the worst case complexity measure each time the average case measure is discredited? -- one logical answer is average case complexity under the universal distribution equals worst case complexity. This answer, which is hard to challenge, however gives no idea as to which of the standard probability distributions come under the umbrella of universality. The morale is that average case complexity measures, in cases where they are different from those in worst case, should be deemed as robust provided only they get the support from at least the standard probability distributions, both discrete and continuous. Regretfully, this is not the case with quicksort.

cs.DS↗

arXiv2008

A Statistical Approach to Modeling Indian Classical Music Performance

A raga is a melodic structure with fixed notes and a set of rules characterizing a certain mood endorsed through performance. By a vadi swar is meant that note which plays the most significant role in expressing the raga. A samvadi swar similarly is the second most significant note. However, the determination of their significance has an element of subjectivity and hence we are motivated to find some truths through an objective analysis. The paper proposes a probabilistic method of note detection and demonstrates how the relative frequency (relative number of occurrences of the pitch) of the more important notes stabilize far more quickly than that of others. In addition, a count for distinct transitory and similar looking non-transitory (fundamental) frequency movements (but possibly embedding distinct emotions!) between the notes is also taken depicting the varnalankars or musical ornaments decorating the notes and note sequences as rendered by the artist. They reflect certain structural properties of the ragas. Several case studies are presented.

cs.SD↗

arXiv2008

On an algorithm that generates an interesting maximal set P(n) of the naturals for any n greater than or equal to 2

The paper considers the problem of finding the largest possible set P(n), a subset of the set N of the natural numbers, with the property that a number is in P(n) if and only if it is a sum of n distinct naturals all in P(n) or none in P(n). Here largest is in the set theoretic sense and n is greater than or equal to 2. We call P(n) a maximal set obeying this property. For small n say 2 or 3, it is possible to develop P(n) intuitively but we strongly felt the necessity of an algorithm for any n greater than or equal to 2. Now P(n) shall invariably be a infinite set so we define another set Q(n) such that Q(n)=N-P(n), prove that Q(n) is finite and, since P(n) is automatically known if Q(n) is known, design an algorithm of worst case O(1) complexity which generates Q(n).

cs.DM↗