6.856 — Randomized Algorithms
David Karger
Handout #4, Feb. 16, 2011 — Homework 3, Due 2/23
1. Consider a collection of nrandom variables Xidrawn independently from the geometric
distribution with mean 2 – that is, Xiis the number of flips of an unbiased coin up to
and including the first occurrence of heads. Let X=PXi. Use two different methods
to derive bounds on the probability that X > (1 + )(2n) for any fixed :
(a) Figure out how to reduce this to a question involving just the sum of independent
Bernoulli (i.e. indicator) variables, allowing you to apply the Chernoff bound we
already know.
(b) Use the method of the in-class Chernoff bound analysis to derive ab initio an upper
bound for deviation of sums of geometric random variables.
2. This problem should be done without collaboration. MR 4.14. Show that the
Quicksort algorithm of Chapter 1 runs in O(nlog n) time with high probability. Do
so by bounding the number of pivots to which each element is compared. Hint: for a
given item x, call a pivoting round good if xends up in the smaller subproblem. How
many good rounds can xbe in? How long will that take to happen? You may use the
previous problem.
3. Some problems with bit-fixing:
(a) Based on MR Exercise 4.2. Consider the transpose permutation: writing ias the
concatenation of two n/2-bit strings aiand bi, we want to route aibito biai. Show
the bit fixing strategy takes Ω(√N) steps on this permutation.
(b) MR 4.9. Consider the following randomized variant of the bit fixing algorithm.
Each packet randomly orders the bit positions in the label of its source and then
corrects the mismatched bits in that order. Show that there is a permutation for
which with high probability that algorithm uses 2Ω(n)steps to route. An inequality
that might be helpful:
n
kk
≤n
k≤en
kk
.
4. In class we showed that nballs in nrandom bins see a maximum load of O(log n/ log log n).
Show this bound is tight:
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