Computer Science 308-690A
Probabilistic Analysis of Algorithms and Data Structures
September 2, 2014

Fall 2014 --- Course Syllabus





Instructor  


Luc Devroye | Email | Tel: (514) 398-3738 (office) | McConnell Engineering Building, Room 300N | Office hours: Monday and Tuesday, 10-11:30am


Teaching Assistant  


Neil Girdhar (Google search for Neil) | Email to neil.girdhar@mail.mcgill.ca | Tel: TBA | McConnell Engineering Building, Room 109 | Office hours: Tuesday, Friday, 11-12am, or by email appointment.


Location  


Monday, Wednesday 8:30--10am. McConnell Engineering, room 320. Lectures start September 3, 2014 and run until December 4, 2014.

Special dates:

  • Lectures that will be moved to another time: Sep 22, 24; Oct 6, 8; Nov 3, 5.
  • Replacement lectures, on Fridays, 8:30-10am: Sep 12, 19; Oct 17, 24, 31; Nov 14.
  • No lecture on Thanksgiving, Oct 13.
  • Special McGill-imposed lecture on Thursday, Dec 4 from 8:30-10. This is also the last day of classes.


Description  


This course looks at basic methods for analyzing the average behavior of algorithms and data structures. It is shown how conventional and modern probability theoretical techniques can be used in this respect. The list of topics does not pretend to be exhaustive; rather, it is selected to give a broad horizontal view of possible applications. The students should be familiar with elementary concepts in probability theory and data structures.


Contents  


Binary search trees Connection with the theory of records.
Analysis of depth and height.
Quadtrees, k-d trees, union-find trees.
Introduction to branching processes, branching random walks.
Divide-and-conquer Expected time analysis of divide-and-conquer methods.
Algorithms for outer layers and convex hulls.
Analysis of the cardinality of the random convex hull.
Randomized algorithms Introduction to the methodology.
Finding the k-th largest quickly on the average.
Exponential large deviation inequalities.
Closest point problems.
Random incremental algorithms.
Randomzed approximation algorithms.
Conditional branching processes Analysis of simple families of random trees.
Random graphs Random graphs, independent sets, coloring.
The second moment method.
Analysis of simple heuristics for graph problems.
Linear expected time connectivity algorithm.
Properties of sparse random graphs.
The Erdös-Rényi theorem on connected graphs.
Random geometric graphs, the Gilbert disc model, percolation.
Combinatorial search problems Euclidean traveling salesman problem.
Assignment problems.
Martingales and the bounded differences method.
Concentration inequalities.
Markov chains Basic properties.
Markov chain Monte Carlo.
Generating random combinatorial objects.
Rapid mixing. Mixing time.
Entropy Entropy, coding and compression.
Entropy and random tries.
Digital search trees.
Random walks Random walks for analyzing trees.
Bin packing heuristics.
Hashing, bucketing Analysis of various hashing algorithms.
Influence of non-uniform distributions.
Maximal occupancy.
Paradigm of two choices.


Evaluation  


About 9 sets of theoretical problems will be assigned.


Textbook  


There is no specific textbook. Course notes will be handed out.


Selected references  


Some of the course material is based upon parts of the following references.

N. Alon, J. Spencer, and P. Erdös, The Probabilistic Method, John Wiley, New York, 1992.

B. Bollobas, Random Graphs, Academic Press, New York, 1985.

L. Devroye, "Branching processes in the analysis of the heights of trees," Acta Informatica, vol. 24, pp. 277--298, 1987.

L. Devroye, "Applications of the theory of records in the study of random trees," Acta Informatica, vol. 26, pp. 123--130, 1988.

S. Janson, T. Luczak, A. Rucinski, Random Graphs, Wiley-Interscience, New York, 2000.

R. M. Karp, "The probabilistic analysis of some combinatorial search algorithms," in: Algorithms and Complexity, ed. J. F. Traub, pp. 1--19, Academic Press, New York, 1976.

R. M. Karp and J. M. Steele, "Probabilistic analysis of heuristics," in: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, ed. E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan and D. B. Shmoys, pp. 181--205, John Wiley, New York, 1985.

D. E. Knuth, The Art of Computer Programming, Vol. 3 : Sorting and Searching, Addison-Wesley, Reading, Mass, 1973.

H. M. Mahmoud, Evolution of Random Search Trees, John Wiley, New York, 1992.

R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, 1995.

E. M. Palmer, Graphical Evolution, John Wiley, New York, 1985.

J. Pearl, Heuristics. Intelligent Search Strategies for Computer Problem Solving (Chapter 5), Addison-Wesley, Reading, Mass, 1984.

W. Szpankowski, Average Case Analysis of Algorithms on Sequences, Springer-Verlag, New York, 2001.

J. S. Vitter and P. Flajolet, "Average-case analysis of algorithms and data structures," in: Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity, ed. J. van Leeuwen, pp. 431--524, MIT Press, Amsterdam, 1990.