1. Median Finding, Order Statistics & Quick Sort
Vanithadevi Devarajan

2. Order Statistics 7th order statistic 3 4 13 14 23 27 41 54 65 75
The k-th order statistic is the k-th smallest element of an array.
7th order statistic 3 4 13 14 23 27 41 54 65 75
upper median 3 4 13 14 23 27 41
lower median 54 65 75
The lower median is the th order statistic
The upper median is the th order statistic
If n is odd, lower and upper median are the same

3. Given an unsorted array, how quickly one can find the i-th order statistics = A[i] ?
By Sorting the array and retrieving the i-th value takes (n lg n),
even though max (i=n) and min(i=1) can be done in (n).
Can we do better in Linear Time?
- Solved affirmatively in 1972 by (Manuel)Blum, Floyd, Pratt, Rivest, and Tarjan.
- Described two Linear-Time algorithms
- One Randomized and One Deterministic

4. Selection ( Deterministic and Randomized ) –
Finding the Median in Linear Time
Randomized Algorithm
After the partitioning step, one can tell which subarray has the
item, one is looking for, by just looking at their sizes.
So, only need to recursively examine one subarray, not two.
For instance,
1. If one is looking for the 87th-smallest element in
an array of size 200, and after partitioning the “LESS”
subarray (of elements less than the pivot), then just
need to find the 87th smallest element in LESS.
2. On the other hand, if the “LESS” subarray has size 40, then
just need to find the 87−40−1 = 46th smallest element in
GREATER.
3. And if the “LESS” subarray has size exactly 86 then, just
return the pivot. (

5. Randomized/QuickSelect:
Given an array A of size n and integer k ≤ n,
Pick a pivot element p at random from A.
Split A into subarrays - LESS and GREATER, by comparing
each element to p as in Quicksort. While we are at it, count the number of elements ( L ) going into LESS.
3. (a) If L = k − 1, then output p.
(b) If L > k − 1, output QuickSelect(LESS, k).
(c) If L < k − 1, output QuickSelect(GREATER, k − L − 1)
The expected number of comparisons for QuickSelect is O(n).

6. Deterministic Linear Time Algorithm
Deterministic Select
Given an array A of size n and integer k ≤ n,
Group the array into n/5 groups of size 5 and find the median
of each group.
2. Each group is then sorted and its median is selected.
3. Recursively, find the median of the medians. Call this as ‘p’.
Use p as a pivot to split the array into subarrays - LESS and
GREATER.
Recurse on the appropriate array to find the K-th smallest
element.
DeterministicSelect makes O(n) comparisons to find the kth smallest in an array of size n.

7. Quick Sort Given array of some length n,
Pick an element p of the array as the pivot (or halt if the
array has size 0 or 1).
2. Split the array into sub-arrays LESS, EQUAL, and GREATER
by comparing each element to the pivot. (LESS has all
elements less than p, EQUAL has all elements equal
to p, and GREATER has all elements greater than p).
Recursively sort LESS and GREATER.
Worst-case running time is O(n^2 )
Randomized-Quicksort
Run the Quicksort algorithm as given above, each time
picking a random element in the array as the pivot.
Worst-case Expected-Time bound is O(n log n)

8. Project Implementations
Implementation of QuickSort , where the pivot is chosen from the previous Order Statistics Algorithm.
Implementation of Randomized QuickSort, that chooses a random element as the Pivot.
Implementation of Real World Quick Sort with the following heuristics.
If L and R partitions are of unequal sizes, sort the smallest
partition first.
2. If array size <= 7, use insertion sort
3. Use the following idea for getting the pivot.
middle = (start+end)/2
if array size > 40
length = array size / 8
pivot1 = median(start, start - length, start - (2*length))
pivot2 = median(end, end+length, end + 2*length)
pivot3 = median(middle,middle-length, middle+length)
pivot = median(pivot1,pivot2,pivot3)
else
pivot = median(start,middle,end)

9. Performance Analysis Array Size Quick Sort (Linear Median)
(milliseconds)
(Random Median)
Randomized Quick Sort
RealWorld
QuickSort
100
1000
5000 15
10000 16 32
50000 63 31 78
75000
110 47 62
100000
156 79

10. Performance Analysis

11. DEMO

12. THANK YOU !