Algorithms and Data Structures
Course outline
 This a graduate course on algorithms and data structures. 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. Topics may be added / removed, depending on interest and time available.
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 Tuesday and Thursday 9:30 - 11 am in A-201.

Evaluation will be on the basis of assigments (50%), a mid-term (25%), and a final exam (25%). The weightage of these may be slightly modified later.

Office hours are Wednesdays 9:30 - 11 am.

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.

Lectures
Aug 20
 Logistics. 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 (incomplete).

Other references: Sections 33.4 and 9.3 from CLRS.

Aug 22
 Completing median-finding in O(n) time. Polynomial multiplication via the DFT in O(n log n) time.

Other references: Chapter 30 from CLRS, excluding Section 30.3.

Aug 27
 Dynamic programming: Longest increasing subsequence, and optimal BSTs. Note that the algorithm we actually did in class is different from the algorithm in the notes. In class, we had a one-dimensional table for the DP, where each entry stored the length of the LIS that ended in that entry.

Other references: Chapter 3 from Jeff Erickson's book, available here (particularly Sections 3.6 and 3.9).

Aug 29
 A faster algorithm for Longest Increasing Subsequence. Greedy algorithms - matchings.
Sep 5
 Greedy algorithms: Huffman coding. Matroids and greedy algorithms.

Other references: CLRS Chapter 17.

Sep 17 (lectures 6 and 7)
 Continued with matroids from previous class. Greedy job selection with penalties. Started discussing Dijkstra's algorithm.

Other references: CLRS Chapter 17, and CLRS Section 24.3 for Dijkstra's algorithm.

Sep 19 (lectures 8 and 9)
 Dijkstra's algorithm: with min-heaps, and with Fibonacci heaps.

Other references: CLRS Section 24.3 for Dijkstra's algorithm, Chapter 6 for min heaps, and Chapter 19 for Fibonacci heaps.

Sep 24
 Kruskal's algorithm for MSTs, and the union-find data structure: here.

Other references: Sariel Har-Peled's notes, available here.

Assignment Policies
Please read the assignment policies carefully.

  • 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.
  • Do not try and search for solutions online.
  • Please write in your submission the people with whom you discussed the problems, as well as any references you used.
  • Assignments can be submitted late only in exceptional circumstances. You must inform me, and get my consent to submit an assignment late, at least a day before the assignment submission deadline.
  • The first assignment that is late without prior approval 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.
  • 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