Computer Science 690
Probabilistic Analysis of Algorithms and Data Structures
August 23, 2019

Fall 2019 --- Course Syllabus


Luc Devroye | Email | Tel: (514) 398-3738 (office) | McConnell Engineering Building, Room 300N | Office hours: Tuesday, 10-11:30am, and Wednesday, 8:30-10am.


Tuesday, Thursday 8:30--10am. McConnell Engineering, room 321 (the lounge). Lectures start on Tuesday, September 3, 2019 and run until Thursday, November 28, 2019.

Teaching Assistant

Gavin McCracken, McConnell Engineering, room 310. Email. Office hours Thursday and Friday, 10-11:30am.


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.


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.


About 9 sets of theoretical problems will be assigned.


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.