CS2800 - Design and Analysis of Algorithms

Meetings  

• Theme 1 : Introduction - 6 meetings

  • Meeting 01 : Thu, Jan 16, 01:00 pm-01:50 pm

  Introduction. Why this course. What is expected from an algorithm. Integer Multiplication Problem.

  References	<a href="./CS2800/introduction.pdf">slides</a> used in the class.
  Exercises
  Reading

  Introduction. Why this course. What is expected from an algorithm. Integer Multiplication Problem.
  

  References

  :

  slides used in the class.

  • Meeting 02 : Mon, Jan 20, 11:00 am-11:50 am

  What does algorithm design involve? Description, termination, correctness, analysis, optimality. Example which demonstrates - JoSAA and SEAT Algorithmic Problem. Stable Matching Problem. Unstable Pairs. Gale-Shapley Algorithm.

  References	<a href="./CS2800/introduction.pdf">slides</a> used in the class.
  Exercises
  Reading

  What does algorithm design involve? Description, termination, correctness, analysis, optimality. Example which demonstrates - JoSAA and SEAT Algorithmic Problem. Stable Matching Problem. Unstable Pairs. Gale-Shapley Algorithm.
  

  References

  :

  slides used in the class.

  • Meeting 03 : Wed, Jan 22, 09:00 am-09:50 am

  Termination, Correctness for Gale-Shapley Algorithm.

  References	<a href="./CS2800/introduction.pdf">slides</a> used in the class.
  Exercises
  Reading

  Termination, Correctness for Gale-Shapley Algorithm.
  

  References

  :

  slides used in the class.

  • Meeting 04 : Mon, Jan 27, 11:00 am-11:50 am

  Optimality of Gale-Shapley Algorithm. Recap of what a good algorithm will consist of. Overview of the course.

  References	<a href="./CS2800/introduction.pdf">slides</a> used in the class.
  Exercises
  Reading

  Optimality of Gale-Shapley Algorithm. Recap of what a good algorithm will consist of. Overview of the course.
  

  References

  :

  slides used in the class.

  • Meeting 05 : Tue, Jan 28, 10:00 pm-10:50 pm

  Review of asymptotic analysis. Common pitfalls and misunderstandings. Fictitious primality checking algorithm.

  References	<a href="./CS2800/introduction.pdf">slides</a> used in the class.
  Exercises
  Reading

  Review of asymptotic analysis. Common pitfalls and misunderstandings. Fictitious primality checking algorithm.
  

  References

  :

  slides used in the class.

  • Meeting 06 : Wed, Jan 29, 09:00 am-09:50 am

  Number theoretic problems. Primality checking problem and connection to LMC course. Algorithm for primality checking. Termination, Correctness, Runtime analysis, Tightness of Analysis, Lowerbounds? Mention AKS algorithm. Algorithms for Addition. High school algorithm for multiplication.

  References
  Exercises
  Reading	<a href="https://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf">AKS Algorithm</a>

  Number theoretic problems. Primality checking problem and connection to LMC course. Algorithm for primality checking. Termination, Correctness, Runtime analysis, Tightness of Analysis, Lowerbounds? Mention AKS algorithm. Algorithms for Addition. High school algorithm for multiplication.
  

  References	:	None
  Reading	:	AKS Algorithm

• Theme 2 : Divide and Conquer Technique - 12 meetings

  • Meeting 07 : Mon, Feb 03, 11:00 am-11:50 am

  Interger Multiplication problem. Upper and Lower Bounds. Recursive approach via the Russian Peasant algorithm. The 4-way recursive approach. Unfolding method on the recursion tree. Quadratic bound.

  References	[E] Chapter 1, section 1.9
  Exercises
  Reading

  Interger Multiplication problem. Upper and Lower Bounds. Recursive approach via the Russian Peasant algorithm. The 4-way recursive approach. Unfolding method on the recursion tree. Quadratic bound.
  

  References

  :

  [E] Chapter 1, section 1.9

  • Meeting 08 : Tue, Feb 04, 10:00 pm-10:50 pm

  Analysis of the recurrence relation. Karastuba Method. Solving the recurrence relation. Masters Theorem.

  References	Section 1.9 of [E]. See worked out examples in <a href="https://jeffe.cs.illinois.edu/teaching/algorithms/notes/99-recurrences.pdf">this document</a>.
  Exercises
  Reading

  Analysis of the recurrence relation. Karastuba Method. Solving the recurrence relation. Masters Theorem.
  

  References

  :

  Section 1.9 of [E]. See worked out examples in this document.

  • Meeting 09 : Wed, Feb 05, 09:00 am-09:50 am

  Seven Examples of applying masters theorem. Proof of Masters theorem.

  References	Proof of Masters theorem [CLRS]. More examples in <a href="https://jeffe.cs.illinois.edu/teaching/algorithms/notes/99-recurrences.pdf">this document</a>
  Exercises	Complete the proof of case 3 of masters theorem from CLRS.
  Reading	(Optional reading) <a href="https://www.imsc.res.in/~vikram/DiscreteMaths/2010/MasterThmGen.pdf">A generalization of masters theorem.</a>

  Seven Examples of applying masters theorem. Proof of Masters theorem.
  

  References	:	Proof of Masters theorem [CLRS]. More examples in this document
  Exercises	:	Complete the proof of case 3 of masters theorem from CLRS.
  Reading	:	(Optional reading) A generalization of masters theorem.

  • Meeting 10 : Thu, Feb 06, 01:00 pm-01:50 pm

  Quick sort Algorithm. Median as the pivot. Finding the median. Recursive algorithm. Need of finding the median. Attempt on the median of medians.

  References	Section 1.5, 12.8 of [E]
  Exercises
  Reading

  Quick sort Algorithm. Median as the pivot. Finding the median. Recursive algorithm. Need of finding the median. Attempt on the median of medians.
  

  References

  :

  Section 1.5, 12.8 of [E]

  • Meeting 11 : Mon, Feb 10, 11:00 am-11:50 am

  Finding the k-th smallest element in linear time. Correctness proofs. Detailed Analysis. Why group by 5 and not by 3?

  References	Section 1.8 of [E]
  Exercises
  Reading

  Finding the k-th smallest element in linear time. Correctness proofs. Detailed Analysis. Why group by 5 and not by 3?
  

  References

  :

  Section 1.8 of [E]

  • Meeting 12 : Tue, Feb 11, 10:00 pm-10:50 pm

  Counting the number of inversions. The divide and conquer by strengthening the induction/recursion.

  References	Section 5.3 in [KT]
  Exercises	<ul> <li>Which sorting algorithm uses exactly Inv(sigma) many comparisons?</li> <li>How do we do the ranking comparisons that we started out with for arbitrary two permutations?</li> </ul>
  Reading	(Optional) <a href="https://www.cs.dartmouth.edu/~deepc/LecNotes/cs31/lec5.pdf">Additional Lecture Notes</a><br> Is there a better algorithm for counting the number of inversions? Here is <a href="https://people.csail.mit.edu/mip/papers/invs/paper.pdf">an O(n sqrt(log n)) algorithm</a>.

  Counting the number of inversions. The divide and conquer by strengthening the induction/recursion.
  

  References	:	Section 5.3 in [KT]
  Exercises	:	Which sorting algorithm uses exactly Inv(sigma) many comparisons? How do we do the ranking comparisons that we started out with for arbitrary two permutations?
  Reading	:	(Optional) Additional Lecture Notes Is there a better algorithm for counting the number of inversions? Here is an O(n sqrt(log n)) algorithm.

  • Meeting 13 : Wed, Feb 12, 09:00 am-09:50 am

  Closest Pair Problem. Trivial O(n^2) algorithm for 2D case. The sorting-based algorithm for 1D case. Counter example in the 2D case. Divide and Conquer strategy. Combining the solution. The trivial O(n^2) algorithm. Identifying the vertical band of candidates and why it is useless. Tightetening the argument using observations on the grid. O(n log n) algorithm

  References	Section 5.4 in [KT]
  Exercises	Write down formal proofs of Observation 2 and 3. We formally proved Observation 1 in class.
  Reading	(Optional)<ul> <li>Can we generalize to d dimnesions? <a href="">For constant d, we can still do it in O(n log n) time</a></li> <li>Can we do better than O(n log n) time? Here is one - <A href="https://www.sciencedirect.com/science/article/pii/0020019079900851?via%3Dihub">A faster algorithm that runs in time O(n log log n)</a> </li>

  Closest Pair Problem. Trivial O(n^2) algorithm for 2D case. The sorting-based algorithm for 1D case. Counter example in the 2D case. Divide and Conquer strategy. Combining the solution. The trivial O(n^2) algorithm. Identifying the vertical band of candidates and why it is useless. Tightetening the argument using observations on the grid. O(n log n) algorithm
  

  References	:	Section 5.4 in [KT]
  Exercises	:	Write down formal proofs of Observation 2 and 3. We formally proved Observation 1 in class.
  Reading	:	(Optional) Can we generalize to d dimnesions? For constant d, we can still do it in O(n log n) time Can we do better than O(n log n) time? Here is one - A faster algorithm that runs in time O(n log log n)

  • Meeting 14 : Tue, Feb 18, 10:00 pm-10:50 pm

  Maximum-sum Subarray Problem.

  References	Section 4.1, Page 68 of [CLRS]
  Exercises	Try out the idea discussed in the class (of strengthening the recursion) to improve the running time of divide and conquer algorithm to O(n).
  Reading	<a href="https://personal.utdallas.edu/~daescu/maxsa.pdf">Linear time algorithm</a>.

  Maximum-sum Subarray Problem.
  

  References	:	Section 4.1, Page 68 of [CLRS]
  Exercises	:	Try out the idea discussed in the class (of strengthening the recursion) to improve the running time of divide and conquer algorithm to O(n).
  Reading	:	Linear time algorithm.

  • Meeting 15 : Wed, Feb 19, 09:00 am-09:50 am

  Matrix Addition and Multiplication. Vector addition and point-wise multiplication. Convolution of vectors. Trivial upper bounds and analysis.

  References
  Exercises
  Reading

  Matrix Addition and Multiplication. Vector addition and point-wise multiplication. Convolution of vectors. Trivial upper bounds and analysis.
  

  References

  :

  None

  • Meeting 16 : Thu, Feb 20, 01:00 pm-01:50 pm

  Convolution as a polynomial multiplication in time. Trivial algorithm. Strategy of evaluation, multiplication and interpolation. Linear time algorithms for evaluating a polynomial in one value. Quadratic evaluations. Choice of evaluation points. Scope for divide and conquer.

  References	[CLRS] Section 30.1
  Exercises
  Reading

  Convolution as a polynomial multiplication in time. Trivial algorithm. Strategy of evaluation, multiplication and interpolation. Linear time algorithms for evaluating a polynomial in one value. Quadratic evaluations. Choice of evaluation points. Scope for divide and conquer.
  

  References

  :

  [CLRS] Section 30.1

  • Meeting 17 : Mon, Feb 24, 11:00 am-11:50 am

  Roots of unity. Properties. The fast Fourier Transform Algorithm.

  References	[CLRS] Section 30.2
  Exercises
  Reading

  Roots of unity. Properties. The fast Fourier Transform Algorithm.
  

  References

  :

  [CLRS] Section 30.2

  • Meeting 18 : Mon, Mar 03, 11:00 am-11:50 am

  Implementing the interpolation when evaluations are roots of unity. Inverse Fourier Transform using FFT Algorithm.

  References	[CLRS] Section 30.2
  Exercises
  Reading

  Implementing the interpolation when evaluations are roots of unity. Inverse Fourier Transform using FFT Algorithm.
  

  References

  :

  [CLRS] Section 30.2

• Theme 3 : Decrease and Conquer Technique - 7 meetings

  • Meeting 19 : Tue, Mar 04, 10:00 pm-10:50 pm

  Decrease and Conquer Strategy. The case of insertion sort. Breadth First Search, as a decrease and conquer strategy. BFS tree.

  References	[CLRS] Textbook Section 20.2
  Exercises
  Reading

  Decrease and Conquer Strategy. The case of insertion sort. Breadth First Search, as a decrease and conquer strategy. BFS tree.
  

  References

  :

  [CLRS] Textbook Section 20.2

  • Meeting 20 : Wed, Mar 05, 09:00 am-09:50 am

  Correctness proof for BFS Algorithm. Invariants maintained by the algorithm. Single source shortest path in undirected graphs.

  References	[CLRS] Textbook Section 20.2
  Exercises
  Reading

  Correctness proof for BFS Algorithm. Invariants maintained by the algorithm. Single source shortest path in undirected graphs.
  

  References

  :

  [CLRS] Textbook Section 20.2

  • Meeting 21 : Thu, Mar 06, 01:00 pm-01:50 pm

  (Compensatory lecture on Jan 21st) DFS. Running time Analysis. Properties of DFS algorithm. Paranethesis theorem. Examples.

  References	[CLRS] Textbook Section 20.3
  Exercises
  Reading

  (Compensatory lecture on Jan 21st) DFS. Running time Analysis. Properties of DFS algorithm. Paranethesis theorem. Examples.
  

  References

  :

  [CLRS] Textbook Section 20.3

  • Meeting 22 : Mon, Mar 10, 11:00 am-11:50 am

  Proof of Paranthesis theorem. White-path theorem and the proof. DFS forest. Edge classification into tree edges, back edges, forward edges, cross edges.

  References	[CLRS] Textbook Section 20.3
  Exercises
  Reading

  Proof of Paranthesis theorem. White-path theorem and the proof. DFS forest. Edge classification into tree edges, back edges, forward edges, cross edges.
  

  References

  :

  [CLRS] Textbook Section 20.3

  • Meeting 23 : Tue, Mar 11, 10:00 pm-10:50 pm

  Application #1 of DFS: Acyclicity testing. Proof of correctness using white-path theorem.

  References	[CLRS] Textbook Section 20.4
  Exercises
  Reading

  Application #1 of DFS: Acyclicity testing. Proof of correctness using white-path theorem.
  

  References

  :

  [CLRS] Textbook Section 20.4

  • Meeting 24 : Thu, Mar 13, 01:00 pm-01:50 pm

  Application #2 of DFS: Topological Sort of Directed Acyclic Graphs. Example. Proof of correctness.

  References	[CLRS] Textbook Section 20.4 and 20.5
  Exercises
  Reading

  Application #2 of DFS: Topological Sort of Directed Acyclic Graphs. Example. Proof of correctness.
  

  References

  :

  [CLRS] Textbook Section 20.4 and 20.5

  • Meeting 25 : Fri, Mar 14, 10:00 pm-10:00 pm

  (compensatory lecture for the lecture on Jan 23, 2025).<br> (planned to be released as a recorded lecture on Mar 17, 2025, 10pm). Strongly connected components in a graph. Application of DFS to find SCCs. Algorithm, Correcteness proof and the Analysis.

  References
  Exercises
  Reading

  (compensatory lecture for the lecture on Jan 23, 2025).
  (planned to be released as a recorded lecture on Mar 17, 2025, 10pm). Strongly connected components in a graph. Application of DFS to find SCCs. Algorithm, Correcteness proof and the Analysis.
  

  References

  :

  None

• Theme 4 : Greedy Technique - 10 meetings

  • Meeting 26 : Mon, Mar 17, 11:00 am-11:50 am

  Optimization problems. Overview of greedy strategy. Interval Scheduling Problem. Greedy Strategies. Failures and counter examples. A working strategy.

  References	Chapter 4 and section 4.1 of [KT]
  Exercises
  Reading

  Optimization problems. Overview of greedy strategy. Interval Scheduling Problem. Greedy Strategies. Failures and counter examples. A working strategy.
  

  References

  :

  Chapter 4 and section 4.1 of [KT]

  • Meeting 27 : Tue, Mar 18, 10:00 pm-10:50 pm

  Proof of correctness of Greedy strategy for interval scheduling problem."Greedy Stays Ahead" technique. Variants of interval scheduling problem. Another example. The single source shortest paths problem. Examples.

  References	Section 4.4 of [KT]
  Exercises
  Reading

  Proof of correctness of Greedy strategy for interval scheduling problem."Greedy Stays Ahead" technique. Variants of interval scheduling problem. Another example. The single source shortest paths problem. Examples.
  

  References

  :

  Section 4.4 of [KT]

  • Meeting 28 : Wed, Mar 19, 09:00 am-09:50 am

  The single source shortest paths problem with positive edge weights. Greedy strategy. Example runs. Dijskstra's Algorithm. Correctness proof using "greedy stays ahead" technique. Implementation using priority queue.

  References	Section 4.4 of [KT]
  Exercises
  Reading

  The single source shortest paths problem with positive edge weights. Greedy strategy. Example runs. Dijskstra's Algorithm. Correctness proof using "greedy stays ahead" technique. Implementation using priority queue.
  

  References

  :

  Section 4.4 of [KT]

  • Meeting 29 : Mon, Mar 24, 11:00 am-11:50 am

  Minimal Spanning Tree Problem. Three greedy strategies. (Kruskal's algorithm, Prim's algorithm, Reverse-delete algorithm). How do we prove correctness? Examples. A generic greedy algorithm using safe edges. Invariant.

  References	Section 4.5 of [KT] and Section 21.1 of CLRS
  Exercises
  Reading

  Minimal Spanning Tree Problem. Three greedy strategies. (Kruskal's algorithm, Prim's algorithm, Reverse-delete algorithm). How do we prove correctness? Examples. A generic greedy algorithm using safe edges. Invariant.
  

  References

  :

  Section 4.5 of [KT] and Section 21.1 of CLRS

  • Meeting 30 : Tue, Mar 25, 10:00 pm-10:50 pm

  How do we identify safe edges? The "cut property". Proof of the cut property. Cycle property.

  References	Section 4.5 of [KT] and Section 21.1 of CLRS
  Exercises
  Reading

  How do we identify safe edges? The "cut property". Proof of the cut property. Cycle property.
  

  References

  :

  Section 4.5 of [KT] and Section 21.1 of CLRS

  • Meeting 31 : Wed, Mar 26, 09:00 am-09:50 am

  Proof of correcteness of Kruskal's Algrithm and Prim's Algorithm using Cut property. Proof of correctness of Reverse delete algorithm using the cycle property. Family of greedy algorithms for MST.

  References	Section 4.5 of [KT] and Section 21.1 of CLRS
  Exercises
  Reading

  Proof of correcteness of Kruskal's Algrithm and Prim's Algorithm using Cut property. Proof of correctness of Reverse delete algorithm using the cycle property. Family of greedy algorithms for MST.
  

  References

  :

  Section 4.5 of [KT] and Section 21.1 of CLRS

  • Meeting 32 : Thu, Mar 27, 01:00 pm-01:50 pm

  (Compensatory Lecture for Feb 25) Implementing Prim's Algorithm. Implemeting Kruskal's Algorithm. Union-Find Data structure. Array implementation. Amortized analysis. MAKESET in O(1), FIND in O(1), UNION in O(log k) amortized over k unions.

  References	Section 4.6 of [KT]
  Exercises
  Reading

  (Compensatory Lecture for Feb 25) Implementing Prim's Algorithm. Implemeting Kruskal's Algorithm. Union-Find Data structure. Array implementation. Amortized analysis. MAKESET in O(1), FIND in O(1), UNION in O(log k) amortized over k unions.
  

  References

  :

  Section 4.6 of [KT]

  • Meeting 33 : Tue, Apr 01, 08:00 am-08:50 am

  Union by rank implementation. Rank based analysis. MAKESET in O(1), UNION in O(1), FIND in O(log n) worst case analysis.

  References	Section 5.1.4 (Page 144) of [DPV]
  Exercises
  Reading

  Union by rank implementation. Rank based analysis. MAKESET in O(1), UNION in O(1), FIND in O(log n) worst case analysis.
  

  References

  :

  Section 5.1.4 (Page 144) of [DPV]

  • Meeting 34 : Wed, Apr 02, 09:00 am-09:50 am

  Union by rank and path compression. Analysis to prove that MAKESET in O(1), UNION in O(1), FIND in O(log* n) amortized cost.

  References	Section 5.1.4 (Page 144) of [DPV]
  Exercises
  Reading

  Union by rank and path compression. Analysis to prove that MAKESET in O(1), UNION in O(1), FIND in O(log* n) amortized cost.
  

  References

  :

  Section 5.1.4 (Page 144) of [DPV]

  • Meeting 35 : Thu, Apr 03, 01:00 pm-01:50 pm

  Implementing Prim's Algorithm. Priority queues. Mention Fibanocci heaps and bounds that can be obtained using an amortized analysis.

  References
  Exercises
  Reading

  Implementing Prim's Algorithm. Priority queues. Mention Fibanocci heaps and bounds that can be obtained using an amortized analysis.
  

  References

  :

  None

• Theme 5 : Dynamic Programming Technique - 6 meetings

  • Meeting 36 : Mon, Apr 07, 11:00 am-11:50 am

  Recap of an example from decrease and conquer. Computing the Fibanacci Number. Recurisve Algorithm. Memoization as a technique to close the recursion faster. Iterative version. Weighted interval scheduling problem. A recurrence relation and proof of correctness.

  References	Section 6.1 (for weighted interval scheduling) in [KT]
  Exercises
  Reading

  Recap of an example from decrease and conquer. Computing the Fibanacci Number. Recurisve Algorithm. Memoization as a technique to close the recursion faster. Iterative version. Weighted interval scheduling problem. A recurrence relation and proof of correctness.
  

  References

  :

  Section 6.1 (for weighted interval scheduling) in [KT]

  • Meeting 37 : Tue, Apr 08, 10:00 pm-10:50 pm

  Algorithm description for Weighted interval scheduling. Iterative version. From memoization to dynamic programming. Structure of Dynamic Programming Solutions. Overlapping Subproblems and the Dependency DAG of subproblems. Requirements for the DP algorithms to be efficient. Maximum sum subarray problem.

  References	Section 6.1 (for weighted interval scheduling) in [KT]
  Exercises
  Reading

  Algorithm description for Weighted interval scheduling. Iterative version. From memoization to dynamic programming. Structure of Dynamic Programming Solutions. Overlapping Subproblems and the Dependency DAG of subproblems. Requirements for the DP algorithms to be efficient. Maximum sum subarray problem.
  

  References

  :

  Section 6.1 (for weighted interval scheduling) in [KT]

  • Meeting 38 : Tue, Apr 15, 10:00 pm-10:50 pm

  Knapsack Problem and Variants. The subset sum problem. Attempts on greedy algorithm. Counter examples. Dynamic Programming algorithm. Proof of correctness. Generalization for the integral knapsack. O(nW) pseudopolynomial algorithm.

  References	Page 266, Section 6.4 of the [KT]
  Exercises
  Reading

  Knapsack Problem and Variants. The subset sum problem. Attempts on greedy algorithm. Counter examples. Dynamic Programming algorithm. Proof of correctness. Generalization for the integral knapsack. O(nW) pseudopolynomial algorithm.
  

  References

  :

  Page 266, Section 6.4 of the [KT]

  • Meeting 39 : Wed, Apr 16, 09:00 am-09:50 am

  Fractional Knapsack Problem. Greedy Approach. Proof of Correctness.

  References	<a href="https://courses.cs.duke.edu/compsci330/spring19/lecture8scribe.pdf">Lecture Notes</a>
  Exercises
  Reading

  Fractional Knapsack Problem. Greedy Approach. Proof of Correctness.
  

  References

  :

  Lecture Notes

  • Meeting 40 : Mon, Apr 21, 11:00 am-11:50 am

  Approximation Algorithms. A modified greedy approach gives a 1/2-approximation algorithm for the integral version.

  References	<a href="https://ac.informatik.uni-freiburg.de/lak_teaching/ws11_12/combopt/notes/knapsack.pdf">Lecture Notes</a>
  Exercises
  Reading

  Approximation Algorithms. A modified greedy approach gives a 1/2-approximation algorithm for the integral version.
  

  References

  :

  Lecture Notes

  • Meeting 41 : Tue, Apr 22, 10:00 pm-10:50 pm

  Shortest paths in negative edge weighted graphs. Bellman Ford Algorithm, Proof of correctness, Running time. Improved analysis using amortization. Space-efficient implementation using rowwise incremental implementation.

  References	Page 290, Section 6.8 in [KT]
  Exercises
  Reading

  Shortest paths in negative edge weighted graphs. Bellman Ford Algorithm, Proof of correctness, Running time. Improved analysis using amortization. Space-efficient implementation using rowwise incremental implementation.
  

  References

  :

  Page 290, Section 6.8 in [KT]

• Theme 6 : Transform and Conquer, Reductions & Intractability - 6 meetings

  • Meeting 42 : Wed, Apr 23, 09:00 am-09:50 am

  Transform and Conquer. Three paradigms (a) reduction to a structured instance of the problem - example - gaussian elimination algorithm (b) change the input representation to make problem easy - example - polynomial multiplication. (c) problem reduction - example - integer multiplication to squaring, matrix multiplication to matrix squaring.

  References	There is no specific textbook reference from which this lecture was based on.
  Exercises
  Reading

  Transform and Conquer. Three paradigms (a) reduction to a structured instance of the problem - example - gaussian elimination algorithm (b) change the input representation to make problem easy - example - polynomial multiplication. (c) problem reduction - example - integer multiplication to squaring, matrix multiplication to matrix squaring.
  

  References

  :

  There is no specific textbook reference from which this lecture was based on.

  • Meeting 43 : Thu, Apr 24, 09:00 am-09:50 am

  (Compensatory Lecture for Missed Lecture on Feb 26, 2025 - the date and time is just a placeholder. This is a recorded lecture to be shared with the class)<br> Transform by preprocessing : input enhancement. Boyer-Moore Algorithm (simplified version).

  References
  Exercises
  Reading

  (Compensatory Lecture for Missed Lecture on Feb 26, 2025 - the date and time is just a placeholder. This is a recorded lecture to be shared with the class)
  Transform by preprocessing : input enhancement. Boyer-Moore Algorithm (simplified version).
  

  References

  :

  None

  • Meeting 44 : Thu, Apr 24, 01:00 pm-01:50 pm

  Another example of a reduction. Counting spanning trees using determinants. Kirchoff's Matrix tree theorem. Yet another example - counting number of paths to matrix powering. <br><br> The problem arena. Optimization vs Decision problems. CLIQUE problem and MaxClique problem. INDSET and the MaxINDSET problem. VC and the MaxVertexCover Problem. Trivial Algorithms. The problem arena. Definition of the class P.

  References	There is no specific textbook reference from which this lecture was based on.
  Exercises
  Reading

  Another example of a reduction. Counting spanning trees using determinants. Kirchoff's Matrix tree theorem. Yet another example - counting number of paths to matrix powering.
  
  The problem arena. Optimization vs Decision problems. CLIQUE problem and MaxClique problem. INDSET and the MaxINDSET problem. VC and the MaxVertexCover Problem. Trivial Algorithms. The problem arena. Definition of the class P.
  

  References

  :

  There is no specific textbook reference from which this lecture was based on.

  • Meeting 45 : Mon, Apr 28, 11:00 am-11:50 am

  More problems seemingly outside class P. Some problems have peculiarities. Short easily verifiable sound certificates for membership. The class NP.

  References	[KT] Chapter on NP-completeness
  Exercises
  Reading

  More problems seemingly outside class P. Some problems have peculiarities. Short easily verifiable sound certificates for membership. The class NP.
  

  References

  :

  [KT] Chapter on NP-completeness

  • Meeting 46 : Tue, Apr 29, 10:00 pm-10:50 pm

  Reductions as a tool to classify the unknown. From Clique to Independent Set. Clique to Vertex Cover.

  References	[KT] Chapter on NP-completeness
  Exercises
  Reading

  Reductions as a tool to classify the unknown. From Clique to Independent Set. Clique to Vertex Cover.
  

  References

  :

  [KT] Chapter on NP-completeness

  • Meeting 47 : Wed, Apr 30, 09:00 am-09:50 am

  Notion of NP-completeness. Satisfiability Problem. Cook-Levin Theorem - Satisfiability problem is NP-complete (without proof). Transitivity of reductions. P vs NP and NP-complete problems. From 3SAT to Vertex Cover Problem.

  References	[KT] Chapter on NP-completeness
  Exercises
  Reading

  Notion of NP-completeness. Satisfiability Problem. Cook-Levin Theorem - Satisfiability problem is NP-complete (without proof). Transitivity of reductions. P vs NP and NP-complete problems. From 3SAT to Vertex Cover Problem.
  

  References

  :

  [KT] Chapter on NP-completeness

• Theme 7 : Tutorials, Eval Meetings - 12 meetings

  • Meeting 48 : Thu, Jan 30, 01:00 pm-01:50 pm

  Tutorial 1 questions released on Moodle. <br> Suggestion : Part A to be attempted in class. Part B is to be attempted at home.

  References
  Exercises
  Reading

  Tutorial 1 questions released on Moodle.
  Suggestion : Part A to be attempted in class. Part B is to be attempted at home.
  

  References

  :

  None

  • Meeting 49 : Thu, Feb 13, 01:00 pm-01:50 pm

  Tutorial 2 (was a takehome tutorial on Feb 6th) <br> Tutorial 3 questions released on Moodle.

  References
  Exercises
  Reading

  Tutorial 2 (was a takehome tutorial on Feb 6th)
  Tutorial 3 questions released on Moodle.
  

  References

  :

  None

  • Meeting 50 : Mon, Feb 17, 11:00 am-11:50 am

  Short Exam 1 (Syllabus : upto and including Feb 13, 2025)<br> Tutorial 4 questions released on Moodle.

  References
  Exercises
  Reading

  Short Exam 1 (Syllabus : upto and including Feb 13, 2025)
  Tutorial 4 questions released on Moodle.
  

  References

  :

  None

  • Meeting 51 : Thu, Feb 27, 01:00 pm-01:50 pm

  Quiz 1

  References
  Exercises
  Reading

  Quiz 1
  

  References

  :

  None

  • Meeting 52 : Thu, Mar 06, 02:00 pm-02:50 pm

  Tutorial 5 released on moodle.

  References
  Exercises
  Reading

  Tutorial 5 released on moodle.
  

  References

  :

  None

  • Meeting 53 : Wed, Mar 12, 09:00 am-09:50 am

  Short Exam 2

  References
  Exercises
  Reading

  Short Exam 2
  

  References

  :

  None

  • Meeting 54 : Thu, Mar 20, 01:00 pm-01:50 pm

  Tutorial 6

  References
  Exercises
  Reading

  Tutorial 6
  

  References

  :

  None

  • Meeting 55 : Thu, Mar 27, 02:00 pm-02:50 pm

  Tutorial 7

  References
  Exercises
  Reading

  Tutorial 7
  

  References

  :

  None

  • Meeting 56 : Wed, Apr 09, 09:00 am-09:50 am

  Tutorial - Shortest paths in DAGs, Longest increasing subsequences

  References
  Exercises
  Reading

  Tutorial - Shortest paths in DAGs, Longest increasing subsequences
  

  References

  :

  None

  • Meeting 57 : Thu, Apr 17, 01:00 pm-01:50 pm

  Short Exam 3

  References
  Exercises
  Reading

  Short Exam 3
  

  References

  :

  None

  • Meeting 58 : Thu, May 01, 01:00 pm-01:50 pm

  Short Exam 4

  References
  Exercises
  Reading

  Short Exam 4
  

  References

  :

  None

  • Meeting 59 : Fri, May 09, 09:00 am-12:00 pm

  Endsemester Examination

  References
  Exercises
  Reading

  Endsemester Examination
  

  References

  :

  None