Lecture Notes for COMP572 -- Fall 2007

All  revisions to the lecture notes will be recorded here.

• Introduction   PS  PDF
  The fact that there are n^{n-2} spanning trees on K_n is Cayley's theorem. One standard proof of Cayley's theorem uses Prufer encodings. See these two links for more information:  Link1  Link2
  
   
• Maximum Flows PS PDF   
  The description in the slides follows  Sections 26.1-26.3 of the CLRS book.
  More advanced material can be found in Data structures and network algorithms by Robert Endre Tarjan
  
   
• Binomial & Fibonnacci Heaps and an Introduction to Amortized Analysis
  Our description will follow  Chapter 19, Sections 17.1-17.3 and Chapter 20 of the CLRS book.
  Another (possibly simpler) reference is Chapter 11 of Data Structures and Algorithm Analysis in C, Mark Allen Weiss, 1993, Benjamin Cummings. Also check out this animation of Fibonacci Heaps
  Binomial Heaps  PS PDF
  Amortized Analysis  PS PDF
  Fibonacci Heaps PS PDF 
  
   
• The Hungarian Algorithm for the Assignment Problem PS PDF
  This extra handout (PS PDF)  proves  lemma on page 13
  This extra handout illustrates (PDF) how updating can lose an edge in equality graph
  For a nice historical introduction to the development of the algorithm see pages 4-10 of
  On the history of combinatorial optimization (till 1960) by Alexander Schrijver
  Also, see Being in the Right Place at the Right Time by Harold Kuhn (the creator of the Hungarian Algorithm)
      (the above link might only work within the IP domains of institutions that subscribe to JSTOR, such as HKUST)
  
   
• Linear Programming
  This (long) section is based on Chapter 2+ of Papadimitriou and Steiglitz
  After learning about LP check out the short guide What Can’t You Do With LP? by Devdatt P. Dubhashi for an interesting perspective.
  Before Starting: Review of Basic Matrix Mainpulations and Linear Algebra PS PDF
  Simplex Part 1: Introduction and Basic Feasible Solutions PS PDF
      Simplex Part 1 (extra) Correspondence between polytopes and feasible sets  PS PDF
                  Last Revised 16/10/07
  Simplex Part 2: Changing BFSs and Pivots PS PDF
                   Last Revised 16/10/07
  Simplex Part 3: The Simplex Algorithm  PS PDF
                          Last Revised 16/10/07
      A worked simplex example with polytope correspondence PDF
  Odds & Ends:  Variations on Simplex & LP PS PDF
  
   
• Linear Programming Duality  PS PDF
                          Last Revised 16/10/07
  This section is based on Chapter 3 of Papadimitriou and Steiglitz
  Some more applications of Duality PS PDF

• The Primal-Dual Algorithm
  This section is based on Chapter 5 of Papadimitriou and Steiglitz and Chapter 18 of Vazirani
  Primal-Dual Part 1: Introduction and Basic Concepts PS PDF
  Primal-Dual Part 2: Shortest Path & Max-Flow PS PDF
  Primal-Dual Part 3: (Extra Material) Approximation Algorithms  I PS PDF
              Set-Cover Example PDF  Last Revised 1/11/07
  Primal-Dual Part 4: (Extra Material) Approximation Algorithms  II PS PDF

• Two-Person Zero-Sum Games & the MinMax Theorem
  This will be taught on the whiteboard.  Class presentation is based on Chapter 15 of  Linear programming  by Vašek Chvátal.  A copy of the relevant pages is here (a larger version with white background is also available here; note that all these pages can only be accessed within the UST domain).  For more on applications of Game Theory to Computer Science please see the COMP670O (Spring 2006) home page.

• Randomized (Approximation) Algorithms
  This section is based on  Vazirani and  Lecture Notes on  Approximation Algorithms (IBM Research Report RC21409) by David Williamson
  Introduction To Randomized Algorithms I   PS PDF
  Introduction To Randomized Algorithms II  PS PDF
  The Best-of-Two Algorithm PS PDF
  Randomized Rounding for Set-Cover PS PDF

• Improvements in Dynamic programming
  • Saving Space in Dynamic Programming
    This will be taught on the whiteboard. Class presentation is based on the paper
        A Linear Space Algorithm for Computing Maximal Common Subsequences
        by D.S. Hirschberg in Communications of the ACM, Volume 18, Issue 6, June 1975, Pages 341-343.
        available through the ACM Digital Library via the HKUST library
    and also
        Reducing Space Requirements for Shortest Path Problems
        by J. Ian Munro and  Raul J. Ramirez
        Operations Research. (30)5, 1982. pp 1009-1013.
     
  • Speeding up Dynamic Programming
    This section is based on papers referenced in the notes. In particular, check out,
        Efficient dynamic programming using quadrangle inequalities,
        by F.F. Yao in the Proceedings of the 12th Annual ACM Symposium on Theory of Computing (1980)
        available through the ACM Digital Library via the HKUST library.
    DP Speedups 1: Introduction and the Quadrangle Inequality PS PDF
          A worked example PS PDF
    DP Speedups II: Totally Monotone Matrices PS PDF
    Note: The PS version of DPII doesn't display properly on some GS viewers. The PDF version is recommended

• Introduction to Polynomial Time Approximation Schemes (PTASs)
  • A Fully Polynomial Time Approximation Scheme for Subset Sum  PS PDF
        Proof of Statement in the Notes PS PDF
  • A Polynomial Time Approximation Scheme for Minimum Makespan PS PDF
        A Worked example of Restricted Bin Packing PS PDF
     

   

• A 4-Approximation Algorithm for The  Shortest Superstring Problem   PS PDF
  • A proof of the Overlap Lemma  PS PDF

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Page last updated on 30 November, 2007 03:04:06 PM