Spring 2008 CMSC 498K: Homework 4 Samir Khuller
Due in class: April 1.
If you cannot come up with algorithms that run in the required time, then provide
(correct) slower algorithms for partial credit. Write your answers using pseudo-code in the
same style as the textbook. These make the algorithm description precise, and easy to read
(as opposed to code in C or some other language).
Please also provide a proof of correctness.
(1) Urn Ahas 5 white and 7 black balls. Urn Bhas 3 white and 12 black balls. We flip
a fair coin. If the outcome is heads, then a ball from urn Ais selected, whereas if
the outcome is tails, then a ball from urn Bis selected. Suppose that a white ball is
selected. What is the probability that the coin landed tails?
(2) Suppose we have a universe Uof elements. The similarity of two sets A,B⊆Uis
defined by s(A, B) = |A∩B|/|A∪B|. Fix krandom orderings σ1, . . . , σkof the
elements of U, and let the sketch of a set S⊆Ube the vector of size kwhose ith
component is the element of Sthat comes first in the ordering σi.
Given the sketches of two sets Aand B, how can we estimate s(A, B)? How large
must we make kto be confident that our estimate is fairly accurate? In other words,
derive a high-probability bound for the error in the estimate in terms of k. (This is
the same kind of bound as the (1 −)F0≤ F0≤(1 + )F0bound for the streaming
heavy-hitters algorithm.)
(3) You are going to explore Mars, and your friend is going to explore Venus. Once you
reach Mars you find life on Mars. You take a DNA sample from this species (think
of this as an nbit binary string). Once you are there you would like to send a short
“summary” of the string to your friend to check if the species he found in Venus has
the same DNA or not. What protocol would you fix so that he can do the check, so
that if the species are identical you can confirm this. You may have to allow for the
possibility that the species are not actually the same, but you conclude otherwise.
(4) Design an efficient algorithm to check if a given graph has a K4(a complete graph on
four vertices).
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