LECTURES: Tuesday, Thursday 9:30-11:00 in 310 Soda

OFFICE HOURS: Monday 1:00-2:00, Tuesday 11:00-12:00 in 677 Soda

TA: Greg Valiant (gvaliant@eecs, 615 Soda)

OFFICE HOURS: Monday 2:00-3:00 in 611 Soda, Friday 2:00-3:00 in 751 Soda

1. First, you can make your life alot easier by assuming that the points are distributed in the unit square according to a Poisson Point Process (PPP) of intensity n. This means that the number of points in any subregion A has a Poisson distribution with parameter n x area(A), and the numbers of points in disjoint subregions are independent. (This independence makes things much simpler.) Since the property of being connected is obviously monotonically increasing with the number of points, it follows by exactly the same argument as in the proof of Theorem 14.7 in Lecture 14 that Pr[G is connected] <= 4 x Pr'[G is connected] where Pr denotes the probability in the original n-point model, and Pr' denotes the probability in the PPP model. Thus we can work in the PPP model and show that Pr'[G is connected] -> 0.

2. Since we're working in the PPP model, you will probably need a Chernoff-type bound for a Poisson r.v. You may assume that a Poisson r.v. X satisfies exactly the same form of tail bounds as the Angluin bounds for a binomial r.v., as given in Corollary 13.3. (This bound for the upper tail follows immediately by substituting \lambda=\beta\mu in the bound you derived in Q1(c) of the present HW. The bound for the lower tail follows by a completely analogous argument.)

3. The strategy outlined in the original hint is still valid, except that condition (iii) in the definition of a "bad" set of discs should be modified slightly as follows. Condition (iii) should read: "The intersection of D_5 -D_3 with each disc of radius 1.5r centered at a set of points spaced equally at distance 0.01r around the boundary of D_3 contains at least (k+1) points." This is the same as the previous condition, except that the radius of the discs is a bit smaller and (most important) the number of discs involved is small (actually constant). In addition to verifying the claimed lower bound on the probability that a given set of three discs is bad, you should explain clearly why the presence of a bad set of discs ensures that G is not connected.

- Lecture 1 (8/25)
- Lecture 2 (8/30)
- Lecture 3 (9/1)
- Lecture 4 (9/6)
- Lecture 5 (9/8)
- Lecture 6 (9/13)
- Lecture 7 (9/15)
- Lecture 8 (9/20)
- Lecture 9 (9/27)
- Lecture 10 (9/29)
- Lecture 11 (10/4)
- Lecture 13 (10/6) [Note: Lecture 12 is omitted]
- Lecture 14 (10/11)
- Lecture 15 (10/13)
- Lecture 16 (10/18)
- Lecture 17 (10/20)
- Lecture 18 (10/25)
- Lecture 19 (10/27)
- Lecture 20 (11/1)
- Lecture 21 (11/8)
- Lecture 22 (11/10)
- Lecture 23 (11/17)
- Lecture 24 (11/22)
- Lecture 25 (11/29)

- Problem Set 1 (due 10/14)
- Problem Set 1 Solutions
- Problem Set 2 (due 11/14)
- Problem Set 2 Solutions
- Problem Set 3 (due 12/14)
- Problem Set 3 Solutions

**Elementary examples:**e.g., checking identities, fingerprinting and pattern matching, primality testing.**Moments and deviations:**e.g., linearity of expectation, universal hash functions, second moment method, unbiased estimators, approximate counting.**The probabilistic method:**e.g., threshold phenomena in random graphs and random k-SAT formulas; Lovász Local Lemma.**Chernoff/Hoeffding tail bounds:**e.g., Hamilton cycles in a random graph, randomized routing, occupancy problems and load balancing, the Poisson approximation.**Martingales and bounded differences:**e.g., Azuma's inequality, chromatic number of a random graph, sharp concentration of Quicksort, optional stopping theorem and hitting times.**Random spatial data:**e.g, subadditivity, Talagrand's inequality, the TSP and longest increasing subsequences.**Random walks and Markov chains:**e.g., hitting and cover times, probability amplification by random walks on expanders, Markov chain Monte Carlo algorithms.**Miscellaneous additional topics as time permits:**e.g., statistical physics, reconstruction problems, rigorous analysis of black-box optimization heuristics,...

- Noga Alon and Joel Spencer,
*The Probabilistic Method*(3rd ed.), Wiley, 2008. - Svante Janson, Tomasz Łuczak and Andrzej Ruciński,
*Random Graphs*, Wiley, 2000. - Geoffrey Grimmett and David Stirzaker,
*Probability and Random Processes*(3rd ed.), Oxford Univ Press, 2001. - Michael Mitzenmacher and Eli Upfal,
*Probability and Computing: Randomized Algorithms and Probabilistic Analysis*, Cambridge Univ Press, 2005. - Rajeev Motwani and Prabhakar Raghavan,
*Randomized Algorithms*, Cambridge Univ Press, 1995.