Algorithms and Data Structures
Course outline
 This a graduate course on algorithms and data structures. It will be co-taught by Kavitha Telikepalli and me; I will teach the first half, and Kavitha the second. Topics that we plan to cover include: basic programming paradigms (recursion, divide and conquer, greedy, dynamic programming), data structures (union-find, heaps), graph algorithms (shortest paths, spanning trees, network flows, matching, min-cut), randomized algorithms, linear programming and duality, semi-definite programming, approximation algorithms, online algorithms.
Prerequisites
 We will assume knowledge of the basics of algorithms and analysis and data structures, including basic sorting and searching algorithms, graph traversal, solving recurrences, big-oh notation, and NP-completeness. These prerequisites may be obtained from the CLRS reference book.
Details
 Classes will be held Mon and Wed 2-3:30 pm.

Evaluation will be on the basis of assigments (50%), a mid-term (20%), and a final exam (30%). The weightage of these may be slightly modified later.

Reference material
 The topics we cover will mostly be from the book Introduction to Algorithms by Cormen, Leiserson, Rivest, and Stein, for the first part of the course. However, our treatment and notation will differ. Other good reference materials are

  • Algorithm Design by Kleinberg and Tardos
  • Algorithms by Dasgupta, Papadimitriou and Vazirani
  • Introduction to Linear Optimization by Dimitris Bertsimas
  • The Design of Approximation Algorithms by Williamson and Shmoys
  • Approximation Algorithms by Vazirani
  • Randomized Algorithms by Motwani and Raghavan

Our treatment will often differ from that in books, and I may cover topics out of order. I will post an outline of topics covered in each lecture below, including source material.

Aparna Shankar made some nice notes on primality testing, available here.

Lectures
Aug 19
 An O(n log n) algorithm for finding the closest pair of points in the plane. An O(n) algorithm for finding the median of a set of numbers.

References: Sections 33.4 and 9.3 from CLRS. My notes.

Aug 21
 An O(n log n) for polynomial multiplication using the Fast Fourier Transfom. Integer multiplication by Karatsuba's algorithm.

References: Chapter 30 from CLRS. My notes. Further references: for the Karatsuba algorithm, see the course handout by Babai.

Aug 26
 Dynamic programming - computing Fibonacci numbers, longest increasing subsequence, and optimal binary search trees.

References: My notes. Chapter 3 from Jeff Erickson's book, available here (leaving out Sections 3.7, 3.8, 3.10).

Aug 28
 Huffman coding. Matroids and greedy algorithms.

References: My notes. CLRS Chapter 17 for greedy algorithms and matroids.

Sep 4
 Matroids continued - The unit job scheduling problem. Submodular functions and the greedy algorithm.

References: My notes. CLRS Chapter 17 for greedy algorithms and matroids. Lecture notes by Jan Vondrak.

Sep 9
 Faster algorithms using better data structures for Dijkstra's algorithm: min heaps and amortized analysis.

References: CLRS Section 24.3 for Dijkstra's algorithm, Chapter 6 for min heaps. My notes.

Sep 11
 Fibonacci heaps.

References: CLRS Chapter 19 for Fibonacci heaps. My notes.

Sep 16
 Kruskal's algorithm for minimum spanning trees, and the union-find data structure.

References: Sariel Har-Peled's notes, available here. My notes.

Sep 18
 The union-find data structure. Red-black trees.

My notes: 1 and 2. There are a number of youtube videos which may also be helpful.

Sep 26
 Maximum flows and minimum cuts. Ford-Fulkerson, Edmonds-Karp.

References: Class notes, as well as Sections 26.1, 26.2 from CLRS.

Oct 3
 Edmonds-Karp continued. The preflow-push algorithm.

References: Class notes, example, as well as Sections 26.4 from CLRS.

Oct 7
 The preflow-push algorithm continued. All-pairs shortest paths algorithm (Johnson's algorithm).

References: Class notes, an example, as well as Sections 25.3, 26.4 from CLRS.

Assignment Policies
  • For the portion of the course taught by me, each student gets 6 late days for assignments, which can be used in any combination, and for any reason (e.g., you could turn in assignment 2 one day late and assignment 3 three days late).
  • The first assignment that is late after exhausting these late days will be graded out of 50% of the total marks. Any further late assignments will not be graded at all. Any assignment turned in more than 6 days late will not be graded at all.
  • You are encouraged to discuss the problems with others in the class, but you must write up the solution by yourself, in your own words.
  • Please write in your submission the people with whom you discussed the problems, as well as any references you used.
  • Please write clearly and legibly, and include how you arrived at the solution! All algorithms must be accompanied by proofs of correctness and runtime analysis, unless otherwise stated.
Assignments