6.856 — Randomized Algorithms
Spring Term, 2011
Handout #2, Feb. 2, 2011 — Homework 1, Due 2/9
1. Consider the problem of using a source of unbiased random bits to generate a sample
from the set S={1, . . . , n}such that element iis chosen with probability pi.
(a) Suppose n= 2 (so p2= 1 −p1). Give a scheme that uses O(1) bits in expectation
to choose one of the two items. Hint: start with an easy scheme that uses a
possibly infinite number of random bits to choose the item. Then make it lazy,
generating/examining only enough bits to uniquely identify the item that would
be selected by the infinite sequence. What is the probability looking at a new bit
lets you stop? Analyze the expected number of bits you will actually examine.
(b) Generalize this scheme to sample from nelements using O(log n) random bits in
expectation per sample, regardless of the values of the pi.
(c) Prove that for certain piand n(for example, a uniform sample from {1,2,3}), if
you only have unbiased random bits, it is impossible to generate an appropriately
distributed sample using a finite number of bits in the worst case.
2. Consider the following algorithm Find(S, k) to find the kth smallest item in set S. Pick
a random element x∈S, and (as in quicksort) use it to partition the set into S1, the
items smaller than x, and S2, the items larger than x. Let sbe the size of S1. If s≥k,
execute Find(S1, k); else execute Find(S2, k −s−1).
(a) Suppose Shas nelements. Prove that the expected size of the set in the recursive
call is bn for some constant b < 1.
(b) Pick an appropriate c, and argue by induction that the expected runtime of Find
on an n-element set is at most cn.
(c) Explain why your induction in the previous step depended on the fact that you
were proving a linear running time.
(d) Optional: Can you determine the probability that the ith item is compared to
the jth, and use that for an analysis similar to our quicksort analysis? This will be
messier, since it depends on i,j,and k.
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