Deterministic Selection Algorithm Python Code

Through this post, I’m sharing Python code implementing the median of medians algorithm, an algorithm that resembles quickselect, differing only in the way in which the pivot is chosen, i.e, deterministically, instead of at random.

Its best case complexity is O(n) and worst case complexity O(nlog2n)

I don’t have a formal education in CS, and came across this algorithm while going through Tim Roughgarden’s Coursera MOOC on the design and analysis of algorithms. Check out my implementation in Python.

def merge_tuple(a,b):
""" Function to merge two arrays of tuples """
c = []
while len(a) != 0 and len(b) != 0:
if a[0][0] < b[0][0]:
if len(a) == 0:
c += b
c += a
return c
def mergesort_tuple(x):
""" Function to sort an array using merge sort algorithm """
if len(x) == 0 or len(x) == 1:
return x
middle = len(x)/2
a = mergesort_tuple(x[:middle])
b = mergesort_tuple(x[middle:])
return merge_tuple(a,b)
def lol(x,k):
""" Function to divide a list into a list of lists of size k each. """
return [x[i:i+k] for i in range(0,len(x),k)]
def preprocess(x):
""" Function to assign an index to each element of a list of integers, outputting a list of tuples"""
return zip(x,range(len(x)))
def partition(x, pivot_index = 0):
""" Function to partition an unsorted array around a pivot"""
i = 0
if pivot_index !=0: x[0],x[pivot_index] = x[pivot_index],x[0]
for j in range(len(x)-1):
if x[j+1] < x[0]:
x[j+1],x[i+1] = x[i+1],x[j+1]
i += 1
x[0],x[i] = x[i],x[0]
return x,i
def ChoosePivot(x):
""" Function to choose pivot element of an unsorted array using 'Median of Medians' method. """
if len(x) <= 5:
return mergesort_tuple(x)[middle_index(x)]
lst = lol(x,5)
lst = [mergesort_tuple(el) for el in lst]
C = [el[middle_index(el)] for el in lst]
return ChoosePivot(C)
def DSelect(x,k):
""" Function to """
if len(x) == 1:
return x[0]
xpart = partition(x,ChoosePivot(preprocess(x))[1])
x = xpart[0] # partitioned array
j = xpart[1] # pivot index
if j == k:
return x[j]
elif j > k:
return DSelect(x[:j],k)
k = k - j - 1
return DSelect(x[(j+1):], k)
arr = range(100,0,-1)
print DSelect(arr,50)
%timeit DSelect(arr,50)
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I get the following output:

100 loops, best of 3: 2.38 ms per loop

Note that on the same input, quickselect is faster, giving us:

1000 loops, best of 3: 254 µs per loop

Karatsuba Multiplication Algorithm – Python Code

Motivation for this blog post

I’ve enrolled in Stanford Professor Tim Roughgarden’s Coursera MOOC on the design and analysis of algorithms, and while he covers the theory and intuition behind the algorithms in a surprising amount of detail, we’re left to implement them in a programming language of our choice.

And I’m ging to post Python code for all the algorithms covered during the course!

The Karatsuba Multiplication Algorithm

Karatsuba’s algorithm reduces the multiplication of two n-digit numbers to at most  n^{\log_23}\approx n^{1.585} single-digit multiplications in general (and exactly n^{\log_23} when n is a power of 2). Although the familiar grade school algorithm for multiplying numbers is how we work through multiplication in our day-to-day lives, it’s slower (\Theta(n^2)\,\!) in comparison, but only on a computer, of course!

Here’s how the grade school algorithm looks:
(The following slides have been taken from Tim Roughgarden’s notes. They serve as a good illustration. I hope he doesn’t mind my sharing them.)


…and this is how Karatsuba Multiplication works on the same problem:



A More General Treatment

Let x and y be represented as n-digit strings in some base B. For any positive integer m less than n, one can write the two given numbers as

x = x_1B^m + x_0
y = y_1B^m + y_0,

where x_0 and y_0 are less than B^m. The product is then

xy = (x_1B^m + x_0)(y_1B^m + y_0)
xy = z_2B^{2m} + z_1B^m + z_0


z_2 = x_1y_1
z_1 = x_1y_0 + x_0y_1
z_0 = x_0y_0

These formulae require four multiplications, and were known to Charles Babbage. Karatsuba observed that xy can be computed in only three multiplications, at the cost of a few extra additions. With z_0 and z_2 as before we can calculate

z_1 = (x_1 + x_0)(y_1 + y_0) - z_2 - z_0

which holds since

z_1 = x_1y_0 + x_0y_1
z_1 = (x_1 + x_0)(y_1 + y_0) - x_1y_1 - x_0y_0

A more efficient implementation of Karatsuba multiplication can be set as xy = (b^2 + b)x_1y_1 - b(x_1 - x_0)(y_1 - y_0) + (b + 1)x_0y_0, where b = B^m.


To compute the product of 12345 and 6789, choose B = 10 and m = 3. Then we decompose the input operands using the resulting base (Bm = 1000), as:

12345 = 12 · 1000 + 345
6789 = 6 · 1000 + 789

Only three multiplications, which operate on smaller integers, are used to compute three partial results:

z2 = 12 × 6 = 72
z0 = 345 × 789 = 272205
z1 = (12 + 345) × (6 + 789) − z2z0 = 357 × 795 − 72 − 272205 = 283815 − 72 − 272205 = 11538

We get the result by just adding these three partial results, shifted accordingly (and then taking carries into account by decomposing these three inputs in base 1000 like for the input operands):

result = z2 · B2m + z1 · Bm + z0, i.e.
result = 72 · 10002 + 11538 · 1000 + 272205 = 83810205.

Pseudocode and Python code

procedure karatsuba(num1, num2)
if (num1 < 10) or (num2 < 10)
return num1*num2
/* calculates the size of the numbers */
m = max(size_base10(num1), size_base10(num2))
m2 = m/2
/* split the digit sequences about the middle */
high1, low1 = split_at(num1, m2)
high2, low2 = split_at(num2, m2)
/* 3 calls made to numbers approximately half the size */
z0 = karatsuba(low1,low2)
z1 = karatsuba((low1+high1),(low2+high2))
z2 = karatsuba(high1,high2)
return (z2*10^(2*m2))+((z1-z2-z0)*10^(m2))+(z0)

def karatsuba(x,y):
"""Function to multiply 2 numbers in a more efficient manner than the grade school algorithm"""
if len(str(x)) == 1 or len(str(y)) == 1:
return x*y
n = max(len(str(x)),len(str(y)))
nby2 = n / 2
a = x / 10**(nby2)
b = x % 10**(nby2)
c = y / 10**(nby2)
d = y % 10**(nby2)
ac = karatsuba(a,c)
bd = karatsuba(b,d)
ad_plus_bc = karatsuba(a+b,c+d) - ac - bd
# this little trick, writing n as 2*nby2 takes care of both even and odd n
prod = ac * 10**(2*nby2) + (ad_plus_bc * 10**nby2) + bd
return prod
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Machine Learning — New Coursera Specialization from the University of Washington

I have finally embarked on my first machine learning MOOC / Specialization. I love Python, and this course uses Python as the language of choice. Also, the instructors assert that Python is widely used in industry, and is becoming the de facto language for data science in industry. They use IPython Notebook in their assignments and videos.

The specialization offered by the University of Washington consists of 5 courses and a capstone project spread across about 8 months (September through April). The specialization’s first iteration kicked off yesterday.

washingtonMachineLearningThe first course, Machine Learning Foundations: A Case Study Approach is 6 weeks long, running from September 22 through November 9.

The Instructors:

Emily Fox and Carlos Guestrin

Key Learning Outcomes
– Identify potential applications of machine learning in practice.
– Describe the core differences in analyses enabled by regression, classification, and clustering.
– Select the appropriate machine learning task for a potential application.
– Apply regression, classification, clustering, retrieval, recommender systems, and deep learning.
– Represent your data as features to serve as input to machine learning models.
– Assess the model quality in terms of relevant error metrics for each task.
– Utilize a dataset to fit a model to analyze new data.
– Build an end-to-end application that uses machine learning at its core.
– Implement these techniques in Python.

Week 1: Introductory welcome videos and the instructors’ views on the future of intelligent applications
Week 2: Predicting House Prices (Regression)
Week 3: Classification (Sentiment Analysis)
Week 4: Clustering and Similarity: Retrieving Documents
Week 5: Recommending Products
Week 6: Deep Learning: Searching for Images


It’s been 3 days since the course began, and here’s how the classmate demographic looks like:


Object Oriented Programing with Python – Particle Diffusion Simulation

I’m a newbie to the programming world. I first started programming in Python in May this year, a month after I started this blog, so I still haven’t learnt enough to contribute to economics as is the stated goal of this blog. But I know I’ll get there in a year or less.

This blog was also meant to document my learning. In May, I would have called myself Newb v0.0. Today, 3 months later, I’d like to call myself Newb v0.3 and the goal is to be at least Expert v1.0 by January 2016.

With the help of Rice University’s awesome classes on Python programming I created a cool simulation of particles diffusing into space, using the concept of Classes, which I learnt just yesterday!

Click to check out the code !

Screenshot from 2015-07-23 11:49:00

Screenshot from 2015-07-23 11:49:10

Screenshot from 2015-07-23 11:49:39

Number of Inversions in an Unsorted Array: Python Code

This is my solution to the first programming assignment of Tim Roughgarden’s course on Algorithms  that was due 12:30 PM IST today. Here’s the question quoted as it is:

Programming Question-1
Download the text file here. (Right click and save link as) This file contains all of the 100,000 integers between 1 and 100,000 (inclusive) in some order, with no integer repeated.

Your task is to compute the number of inversions in the file given, where the ith row of the file indicates the ith entry of an array.
Because of the large size of this array, you should implement the fast divide-and-conquer algorithm covered in the video lectures. The numeric answer for the given input file should be typed in the space below.
So if your answer is 1198233847, then just type 1198233847 in the space provided without any space / commas / any other punctuation marks. You can make up to 5 attempts, and we’ll use the best one for grading.
(We do not require you to submit your code, so feel free to use any programming language you want — just type the final numeric answer in the following space.)

My Solution

I modified an earlier code I wrote for merge sort to arrive at the solution. It needed just a couple of modifications, and if you look carefully, it turns out that the number of inversions are unearthed each and every time we merge two sorted sub-arrays. So, intuitively, if the merge sort algorithm was O(nlog2 n), it would take almost as many operations for counting inversions. In python, the code to sort and count inversions in an array of 10,000 integers took less than 3 seconds.

# load contents of text file into a list
# numList
NUMLIST_FILENAME = "IntegerArray.txt"
inFile = open(NUMLIST_FILENAME, 'r')
with inFile as f:
numList = [int(integers.strip()) for integers in f.readlines()]
count = 0
def inversionsCount(x):
global count
midsection = len(x) / 2
leftArray = x[:midsection]
rightArray = x[midsection:]
if len(x) > 1:
# Divid and conquer with recursive calls
# to left and right arrays similar to
# merge sort algorithm
# Merge sorted sub-arrays and keep
# count of split inversions
i, j = 0, 0
a = leftArray; b = rightArray
for k in range(len(a) + len(b) + 1):
if a[i] <= b[j]:
x[k] = a[i]
i += 1
if i == len(a) and j != len(b):
while j != len(b):
k +=1
x[k] = b[j]
j += 1
elif a[i] > b[j]:
x[k] = b[j]
count += (len(a) - i)
j += 1
if j == len(b) and i != len(a):
while i != len(a):
k+= 1
x[k] = a[i]
i += 1
return x
# call function and output number of inversions
print count

Test my solution here with your own test cases.

Review: An Introduction to Interactive Programming in Python (Part 1)

This class (Part 1 of a 2-part course on interactive programming using Python – and the first course of the Fundamentals of Computing Specialization offered by RICE Unviersity) was an excellent introduction to programming because of its focus on building interactive (and fun) applications with the lessons learned each week. Most introductory coding classes start with text based (boring?) programs, while all through this course you’re required to build a series of projects that get progressively complicated with every passing week. I’m not to be mistaken to be trashing conventional pedagogy, but then again, how many gifted coders do you know who learned to code after completing all the exercises, cover-to-cover of some programming textbook? The best way to learn to enjoy coding would be to build interactive stuff, and this course scores full points on that.

A short introduction to the class in a charmingly nerdy way

The mini-projects / assignments during the course are implemented on a cloud-based environment called CodeSkulptor (built by Scott Rixner, one of the instructors for this course). I found CodeSkulptor unique, in that it allows you to share your code (because it’s browser based) with just about anyone with an Internet connection and makes you work with a graphic user interface (GUI) module similar to Pygame, called Simplegui. It also had a debugging tool, called Viz Mode that helped visualize the process. It eases the task of debugging your code and you’ll realize how cool it is as you start using it more.

Since the course mini-projects were peer-reviewed, evaluating other people’s code also became a more straight-jacket affair, as everyone has their code on the same version of Python. This ensures that the focus is on learning to code, without wasting time on the logistics of programming environment (tuning differences in versions or IDEs). I especially enjoyed peer grading – for each mini project we completed, we had to evaluate and grade the work of 5 others. This was very rewarding – because I got the opportunity to fix bugs in others’ code (which makes you a better coder, I guess) and also got to see better implementations than the ones I had coded, further enriching the learning experience. Indeed, the benefits of peer grading and assessment have been well studied and documented.

Of all the assignments, the one I loved the most was implementing the classic arcade game Pong. You could try playing a version of the game I implemented here. It is a 2-player implementation, but you can play it as a single-player game, only if you imagine yourself to be answering this somewhat cheeky question! Which Pong character are you? Left or Right?

which pong


Pong v1.1

The principal reason behind my joining this course was the way it is structured and taught. We had to watch two sets of videos (part a and part b) and then complete one quiz for each set. The main task for each week was to complete a mini-project that was due along with the quizzes early Sunday morning, followed by assessment of peers’ mini-projects on the following Sunday-Wednesday. The instructors clearly put in A LOT OF WORK to make the lecture videos interesting, laced with humor, with just enough to get you going on your own with the week’s mini-project. That way you’d spend less time viewing the lecture videos, spending more time on actually getting the code for your mini-project to work. So in a way, one might say this course doesn’t follow standard pedagogy for an introductory programming course, but then, as Scott Rixner assures, “You’d know enough to be dangerous!

The projects that were completed in Part 1 of this course were indeed exciting:

Rock Paper Scissors Lizard Spock: A simple implementation played with the computer. This project covers basics on statements, expressions and variables, functions, logic and conditionals [I’m a huge fan of The Big Bang Theory, so I was obviously eager to complete this game. Instead of a series of if-elif-else clauses, this implementation used modular logic, all of which is taught in a really fun way. A great way to start off the course].
Guess the Number: Computer chooses a random number between 1 and 100 and you guess that number. It covered event-driven programming, local and global variables, buttons and input fields [This game although fun, might have been more interesting to code if the computer had to guess the number that the player chose, using bisection search].
Stopwatch: This was the first project that used a graphic user interface, using some modular arithmetic to get the digits of the ticking seconds in place. A game was also built on it where the player had to stop the watch right at the start of a second to score points. This game tested your reaction-time. It covered static drawing, timers and interactive drawing.
Pong: The last project of Part 1 and the most fun. Creating the game required only a minor step-up from learnings from previous weeks. It covered knowledge of lists, keyboard input, motion, positional/velocity control. Coding the ball physics where you put to use high-school physics knowledge of elasticity and collisions was very enjoyable. In my game, I set elasticity = 1 (for perfectly elastic collisions)


In an interview with the founders of this MOOC, who spent they say that they spent over 1000 hours building it (Part 1 and Part 2 combined, I guess). That’s an awful lot of effort and it all shows in how brilliantly the class is executed. The support system in the class is excellent. You’ll always find help available within minutes of posting your doubts and queries on the forums. I’ve seen Joe Warren (one of the main instructors of the course) replying to forum posts quite regularly. In addition, there was enough supplementary material in the form of pages on concepts and examples, practice exercises, and video content created by students from previous iterations of the class to better explain concepts and aspects of game-building, improving upon the lecture material.

Concepts and Examples


Practice Exercises


Student-created Videos Explaining Concepts


Overall, I had a great learning experience. I completed Part 1 with a 100 per cent score even though I had a minor hiccup while building the game Pong, which was the most satisfying of all the projects in Part 1. I would review Part 2 when I’m done with that in August this year. I’d easily recommend this course to anyone wishing to start off with Python. It is a great place to be introduced to Python, but it shouldn’t be your ONLY resource. I have been taking MIT’s 6.01x introductory Python course side-by-side. I shall review that course as soon as I’m through with it. That course is pedagogically more text-bookish, and indeed they do profess the use of their textbook to complement the course. I’m 4 weeks into that course and finding that enjoyable too – albeit in a different way. I still haven’t lost a point on any of the assignments or finger exercises there, and hope the trend continues:


PS: In one of the forum threads, Joe posted a list of resources that could be referred to in addition to the class.


Python Books:

Another List of Books:

  •  – about 50 books –  Another good list of free python books that is kept up to date, and I believe are all free or open-source: (I won’t repeat all the books on the list here, just go check it out! Some are also on the list above, but not all)

Further Online Learning:

Getting Started

I have been searching for good MOOCs to get me started with R and Python programming languages. I’ve already begun the Johns Hopkins University Data Science Specialization on Coursera. It consists of 9 courses (including Data Scientist’s Toolbox, R programming, Getting and Cleaning Data, Exploratory Data Analysis, Reproducible Research, Statistical Inference, Regression Models, Practical Machine Learning and Developing Data Products), ending with a 7-week Capstone Project that I’m MOST excited about. I want to get there fast.

The Capstone would consist of :
  • Building a predictive data model for analyzing large textual data sets
  • Cleaning real-world data and perform complex regressions
  • Creating visualizations to communicate data analyses
  • Building a final data product in collaboration with SwiftKey, award-winning developer of leading keyboard apps for smartphones

I started with the R programming course where I found the programming assignments to be moderately difficult. They were good practice and also time-consuming for me since I haven’t yet gotten used to the R syntax, which is supposedly unintuitive. Anyway, I completed the course with distinction (90+ marks) scoring 95 on 100, losing 5 because I hadn’t familiarized myself with Git / GitHub. I did this course for a verified certificate, which cost me $29, and looks like this:

Coursera rprog 2015

I won’t be paying for any of the remaining courses though, but still will get a certificate of accomplishment for each course I pass. I have alredy begun with Getting and Cleaning Data and Data Scientist’s Toolbox.

I checked today, and it seems Andrew Ng’s Machine Learning course has gone open to all and is self-paced. A lot of people have gone on to participate in Kaggle competitions with what they learnt in his course, so I’d like to experience it — even though it’s taught with Octave / MATLAB. My very short term goal is to start participating in these competitions ASAP.

Kaggle Competitions

I will be learning the basics of Git this week and along with that, about reading from MySQL, HDF5, the web and APIs. I intend to start reading Trevor Hastie’s highly recommended book, Introduction to Statistical Learning.

ISL Cover 2


Meanwhile, I need to get started with Git and GitHub too, and I found a very useful blog by Kevin Markham and his short concise videos are great introductory material.

Incidentally, I was in a dilemma whether to start with Hastie’s material or Andrew Ng’s course first. This is what Kevin had to say

Hastie or Ng

The only reason I have reservations against Andrew Ng’s course is that its instruction isn’t in R or Python. Also, CTO and co-founder of Kaggle, Ben Hamner mentions here how useful R and Python are vis-à-vis Matlab / Octave.

Ben Hamner on Python R Matlab v2