6.856 — Randomized Algorithms
David Karger
Handout #3, February 9, 2009 — Homework 2, Due 2/16
1. MR 2.3. Consider a uniform rooted tree of height h(every leaf is at
distance hfrom the root). The root, as well as any internal node, has 3
children. Each leaf has a boolean value associated with it. Each internal
node returns the value returned by the majority of its children. The
evaluation problem consists of determining the value of the root; at each
step, an algorithm can choose one leaf whose value it wishes to read.
(a) Show that for any deterministic algorithm, there is an instance (a set
of boolean values for the leaves) that forces it to read all n= 3hleaves.
(b) Show that there is a nondeterministic algorithm can determine the
value of the tree by reading at most nlog32leaves. In other words,
prove that one can present a set of this many leaves from which the
tree value can be determined.
(c) Consider the recursive randomized algorithm that evaluates two sub-
trees of the root chosen at random. If the values returned disagree, it
proceeds to evaluate the third sub-tree. Show the expected number
of leaves read by the algorithm on any instance is at most n0.9.
2. MR 2.6. Use Yao’s minimax principle to prove a lower bound on the ex-
pected running time of any Las Vegas algorithm for sorting nnumbers that
uses only comparisons. You might want to review deterministic sorting
lower bounds from, e.g., [CLR] chapter 9.
3. Consider a sequence of nunbiased coin flips. Consider the length of the
longest contiguous sequence of heads.
(a) Show that you are unlikely to see a sequence of length c+ log2nfor
c > 1 (give a decreasing bound as a function of c).
(b) Show that with high probability you will see a sequence of length
log2n−O(log2log2n). Note: this observation can be used to detect
cheating. When told to fake a random sequence of coin tosses, most
humans will avoid creating runs of this length under the mistaken
assumption that they don’t look random.
Use specifics of the problem, not the general deviation bounds.
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