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Algorithms and Data Structures (WS15/16)Example Solutions for Unit 12-13
Question 2
In this problem, we will not distinguish the key xi and the node which store xi.
Let Xij be an indicator random variable where Xij = 1 iff xi is an ancestor of xj . Let Sk bethe size of subtree rooted at xk. We want to find E[Sk].
By definition, Sk =∑n
i=1Xki . From class, we know E[Xki] = Pr[Xki = 1] = Pr[xk hasminimum priority among elements between xi and xk] =
1|k−i|+1 .
Now we can compute E[Sk] directly.
E[Sk] = E[
n∑i=1
Xki]
=
n∑i=1
E[Xki]
=k∑
i=1
1
k − i+ 1+
n∑i=k+1
1
i− k + 1from class
= Hk +Hn−k − 1 = O(log n)
1
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