# Randomized Methods in Complexity

By Prof. Nitin Saxena   |   IIT Kanpur
Learners enrolled: 69
In this course we will study how randomness helps in designing algorithms and how randomness can be removed from algorithms.We will start by formalizing computation in terms of algorithms and circuits. We will see an example of randomized algorithms-- identity testing --and prove that eliminating randomness would require proving hardness results. We prove hardness results for the problems of parity and clique using randomized methods. We construct `highly’-connected graphs called expanders that are useful in reducing randomness in algorithms. These lead to a surprising logarithmic-space algorithm for checking connectivity in graphs. We show that if there is hardness in nature then randomness cannot exist! This we prove by developing pseudo-random generators and error-correcting codes.

INTENDED AUDIENCE  : Computer Science & Engineering, Mathematics, Electronics, Physics, & similar disciplines.
PRE REQUISITE : Preferable (but not necessary) - Theory of Computation, or Algorithms, or Discrete Mathematics
INDUSTRY SUPPORT  : Discrete Optimization, Cryptography, Coding theory, Computer Algebra, Symbolic Computing Software, Cyber Security,Learning Software
Summary
 Course Status : Upcoming Course Type : Elective Duration : 12 weeks Start Date : 23 Jan 2023 End Date : 14 Apr 2023 Exam Date : 29 Apr 2023 IST Enrollment Ends : 30 Jan 2023 Category : Computer Science and Engineering Mathematics Credit Points : 3 Level : Undergraduate/Postgraduate

### Course layout

Week 1: Outline. Introduction to Complexity.
Week 2: Circuits. Polynomial Identity Testing (PIT).
Week 3: Derandomize &amp; get a lower bound.
Week 4: Constant-depth circuits are weak.
Week 5: Monotone circuits are weak.
Week 6: Random Walk converges fast.
Week 7: Expansion properties.
Week 8: Construct Explicit Expanders.
Week 9: Pseudorandom generator (prg) &amp; hardness.
Week 10: Error-correcting codes.
Week 11: List Decoding. Local List Decoding.
Week 12: Error-correcting codes amplify hardness.

### Prof. Nitin Saxena

IIT Kanpur
I completed my Bachelors in Computer Science from the Indian Institute of Technology, Kanpur in 2002 and completed my PhD under Manindra Agrawal in 2006. I am broadly interested in Computational Complexity Theory, Algebra, Geometry and Number Theory. I have been a visiting graduate student in Princeton University (2003-2004) and National University of Singapore (2004-2005); a postdoc at CWI, Amsterdam (2006-2008) and a Bonn Junior Fellow (W2 Professor) at Hausdorff Center for Mathematics, Bonn (2008-2013). Since April 2013, I have a faculty position in the department of CSE, IIT Kanpur.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams:
29 April 2023 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kanpur .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team