Randomized Algorithms

Content

Over the past 25 years the design and analysis of randomized algorithms, which make random choices during their execution, has become an integral part of algorithm theory. For many problems, surprisingly elegant and fast randomized algorithms can be developed. In this lecture we will (a) study basic tools from probability theory needed in probabilistic analyses and (b) design randomized algorithms for a number of fundamental problems.

• October 19, 2016 (two lectures): Basic results; randomized quicksort; min-cut algorithm; [MR] pages 3-7.
• October 24, 2016: Basic results; min-cut algorithm continued; binary planar partitions; [MR] pages 8-11.
• October 26, 2016 (two lectures): Basic results; binary planar partitions; verifying matrix multiplication; [MR] pages 11-14 and [MU] pages 8-10.
• October 31, 2016: Basic results; a probabilistic recurrence; complexity classes; [MR] pages 15-16 and 21.
• November 2, 2016: Basic results; complexity classes continued; Game-theoretic techniques; game tree evaluation; [MR] pages 22-23 and 28-29.
• November 7, 2016: Game-theoretic techniques; game tree evaluation; von Neumann's minimax theorem; [MR] pages 29-31.
• November 9, 2016: Game-theoretic techniques; von Neumann's minimax theorem continued; Yao's technique; [MR] pages 32-34.
• November 14, 2016: Game-theoretic techniques; Yao's technique continued; a lower bound for game tree evaluation; [MR] pages 34-37.
• November 16, 2016: Game-theoretic techniques; a lower bound for game tree evaluation; randomness and non-uniformity; [MR] pages 37-39.
• November 21, 2016: Game-theoretic techniques; randomness and non-uniformity; Moments and deviations; occupancy problems; [MR] pages 39-40 and 43-44.
• November 28, 2016: Moments and deviations; occupancy problems; bucket sort; [MR] pages 44-45 [MU] pages 92-93.
• November 30, 2016: Moments and deviations; bucket sort; inequalities by Markov and Chebyshev; [MU] pages 94 and 44-49.
• December 5, 2016: Moments and deviations; median algorithm; [MU] pages 53-55 and [MR] pages 47-49.
• December 12, 2016: Moments and deviations; median algorithm continued; two-point sampling; [MU] pages 55-57 and [MR] pages 51-52.
• December 14, 2016: Moments and deviations; two-point sampling continued; coupon collector's problem; [MR] pages 52-53 and [MU] page 33.
• December 19, 2016: Moments and deviations; coupon collector's problem continued; stable marriage problem; [MU] pages 50-55 and [MR] pages 53-54.
• December 21, 2016: Moments and deviations; stable marriage problem continued; Chernoff bounds; moment generating functions; deriving Chernoff bounds; Poisson trials; [MR] page 56-57 and [MU] pages 61-64.
• January 9, 2017: Chernoff bounds; deriving Cheroff bounds for Possoin trials; coin flips; parameter estimation; [MU] pages 64-68.
• January 11, 2017: Chernoff bounds; parameter estimation continued; a better bound for a special case; set balancing; [MU] pages 68-72.
• January 16, 2017: Chernoff bounds; packet routing on the hypercube; [MU] pages 72-74.
• January 18, 2017: Chernoff bounds; packet routing on the hypercube; [MU] pages 75-76.
• January 23, 2017: The probabilistic method; the basic counting argument; [MU] pages 126-128.
• January 25, 2017: The probabilistic method; the expectation argument; [MU] pages 128-132.
• January 30, 2017: The probabilistic method; sample and modify; Data structures; treaps; [MU] page 133 and [MR] 201-208.
• Februar 1, 2017: Data structures; treaps; [MR] 201-208.
• February 6, 2017: Data structures; treaps; [MR] 201-208.

Literature

• [MU] M. Mitzenmacher and E. Upfal. Probability and Computing: Randomized Algorithms and ProbabilisticAnalysis. Cambridge University Press, 2005.
• [MR] R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, 1995.

July 2022: Jens Quedenfeld completed his doctoral degree.

June 2022: Maximilian Janke completed his doctoral degree.

March 2022: Alexander Eckl completed his doctoral degree.

June 2021: Leon Ladwig completed his doctoral degree.