QuestionQuestion

In RandSelect, we use a pivot to partition (aka split) the data, in dn time, where d is a constant. Suppose that the partition is called “balanced” if each side of the pivot ends up with no more than 51% of the data. Otherwise the partition is unbalanced.Assume that we have truly random data.

(a) Provide a quick analysis to show that RandSelect will take O(n) expected time, by dealing with only two types of partitions (balanced vs unbalanced).

(b) What is the leading constant that you get, that is normally hidden in the big-O?

(c) What leading constant would you get if we had said that a balanced partition yields at most 99% of the data on one side?

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Problem 1
a).
We use the same approach as in the class notes.
The worst case if the split is balanced leads to T(n) ≤ T(51n/100) + dn
The worst case if the split is unbalanced leads to T(n) ≤T(n-1)+dn < T(n)+dn
Because each split has 50% chances to be balanced =>
T(n) ≤ 0.5*(T(51n/100) +dn) + 0.5*(T(n)+dn) =>
T(n) ≤ 0.5*T(n) + 0.5*T(51n/100) + dn =>
0.5*T(n) ≤ 0.5*T(51n/100) + dn =>
T(n) ≤ T(51n/100) + 2dn...

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