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The University of Texas at Austin Lecture 6 Department of Computer Sciences Professor Vijaya Ramachandran Randomized Quicksort CS357: ALGORITHMS, Spring 2007
Randomized Partition and Randomized Quicksort
Randomized-Partition is a simple modification of Partition (which was defined earlier) in which a random element in the subarray being considered is moved to the last position in the subarray (hence this element becomes the pivot), and then Partition is invoked.
Rand-Partition(A, p, r)
i := Random(p, r) interchange A[i] and A[r] return Partition(A, p, r)
Correctness of Rand-Partition is immediate once we assume correctness of Partition, since A[i] is guaranteed to be an element in the subarray A[p..r]. The running time remains linear in the size of subarray.
We now use Rand-Partition in the randomized version of Quicksort given below.
Rand-Quicksort(A, p, r)
Input. An array A[1..n] of elements from a totally ordered set; p and r are integers with 1 ≤ p ≤ r ≤ n. Output. The elements in sub-array A[p..r] are rearranged in sorted order; the elements in array A outside of subarray A[p..r] are unchanged. (So initial call is to A[1..n].)
if p < r then q := Rand-Partition(A, p, r) Rand-Quicksort(A, p, q − 1) Rand-Quicksort(A, q + 1, r) fi
- Correctness of Rand-Quicksort, assuming correctness of Rand-Partition.
- Expected running time of Rand-Quicksort
Expected Running Time of Rand-Quicksort
Let X be the number of comparisons between pairs of elements in array A made by Rand- Quicksort on an input array A of size n, where we assume that all elements in the input ar- ray are distinct. We will show that E[X] = Θ(n log n). Since the total number of operations executed by Rand-Quicksort is within a constant factor of the number of comparisons performed by it, this will bound the expected running time as Θ(n log n).
We use a ‘slick’ analysis (from the textook) to evaluate E[X]:
For the purpose of analysis, let the elements in array A in increasing order of value be z 1 < z 2 < · · · < zn.
For 1 ≤ i < j ≤ n, let Zij = {zi, · · · , zj }, i.e., Zij is the set of elements in array A with ranks between i and j, both inclusive.
For each of the above pairs of values i, j, we define an indicator random variable Xij for the event that zi is compared to zj , i.e.,
Xij = 1 if zi is compared to zj ; and 0 otherwise
Observation. Rand-Quicksort compares each pair of elements zi, zj at most once.
Why? — This is because
- every comparison between elements in A occurs within some call to Rand-Partition;
- in every comparison, one of the two elements being compared is the pivot element;
- in a call to Rand-Partition the pivot element is compared to each of the remaining elements in the subarray exactly once; and
- after this call to Rand-Partition terminates, the pivot element is in its final position, and is never examined again.
Given the above observation, it follows that
X =
∑^ n
i=
∑^ n
j=i+
Xij
Hence we have
E[X] = E[
n∑− 1
i=
∑^ n
j=i+
Xij ] =
n∑− 1
i=
∑^ n
j=i+
E[Xij ] (by linearity of expectations)
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