algorithm - Average Runtime of Quickselect

Question

Wikipedia states that the average runtime of quickselect algorithm (Link) is O(n). However, I could not clearly understand how this is so. Could anyone explain to me (via recurrence relation + master method usage) as to how the average runtime is O(n)?

Answer 1

Because

we already know which partition our desired element lies in.

We do not need to sort (by doing partition on) all the elements, but only do operation on the partition we need.

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As in quick sort, we have to do partition in halves *, and then in halves of a half, but this time, we only need to do the next round partition in one single partition (half) of the two where the element is expected to lie in.

It is like (not very accurate)

n + 1/2 n + 1/4 n + 1/8 n + ..... < 2 n

So it is O(n).

_{Half is used for convenience, the actual partition is not exact 50%.}