Randomized AlgorithmsCS648

Hints for Theoretical Assignment 2

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Problem 1

• Toss the coin times and use the outcome to compute the estimate .

• Use Chernoff bound suitably and in both directions to find an expression containing .

• You will need to use the fact that to get an expression for .

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Problem 2

1. Observe using suitable arguments that there are at most cuts only.2. Show that there is a very simple (in fact trivial) time deterministic algorithm to

compute an exact ham-sandwich cut.3. The randomized algorithm will look like the following: Select a random sample of

blue points and of red points and compute exact ham- sandwich cut for . (Think how will you form the samples and and how large should they be). You might use points 1 and 2 above.

4. The output of the algorithm could be wrong. To formalize it, introduce the notion/definition of a bad cut suitably.

5. Show that the probability of a particular bad cut to happen is inverse polynomial (in fact it is inverse exponential but we don’t require it here).

6. How many bad cuts can there be ? Proceed …7. Convert Monte Carlo to Las Vegas along similar lines we used on various

occasions in the Lecture class.Tools you will need for this problem: Union theorem, Chernoff bound

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Problem 3

1. First precisely characterize those pairs of points whose distance is implicitly stored in a BFS tree rooted at a vertex. How efficiently can you retrieve it ?

2. Randomized algorithm will have the sketch: Select a random sample of vertices, build BFS tree on each of them, and use each BFS tree to compute distance (may be wrong distance) between each pair of vertices. (Think how will you form the sample and how large should be).

3. Consider any pair of vertices whose distance is more than . Let be the shortest path between them. Formulate a condition for the event that the randomized algorithm fails to compute exact distance between and . This condition will be in terms of the sample and the path .

4. Show that the probability that the algorithm fails to compute exact distance between and is inverse polynomial.

5. Proceed …Tools needed to solve this problem: Union theorem, Chernoff bound (although there is a direct and simple way avoiding Chernoff bound for this problem)

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