find ith smallest element in an array

Order Statistics

given n elements in array
find kthsmallest element(element of rank k)

Naive algorithm:
1. sort A
2. return A[k-1]
T(n)=Θ(nlog2n)

Here find median of an array.

Randomized divide and conquer

Intuition for analysis

(Assume elements are distinct)
Lucky case: 1/10 9/10
T(n)≤T(9/10)+Θ(n)
We’re doing the worst-case analysis within the lucky cases.
T(n)=Θ(n)

Unlucky case: 0 :n-1
T(n)=Θ(n)+T(n−1)=Θ(n2)

Analysis of expected time
- Let T(n) be the random variable for running time of rand-select on input of size n.Assume random numbers are independent.
- Define indicator random variable Xk k from 0 to n-1
- Xk=

⎧⎩⎨1,0,if partition generates an k to n-k-1 splitotherwise

T(n)=

⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪T(max(0:n−1))+Θ(n),T(max(1:n−2))+Θ(n),...T(max(n−1:0))+Θ(n)if 0:n-1 splitif 1:n-2 splitif n-1:0 split

⇒T(n)=∑n−10Xk∗(T(max(k,n−k−1))+Θ(n))
E[T(n)]=E[∑n−10Xk∗(T(max(k,n−k−1))+Θ(n))]=∑n−10E[Xk∗(T(max(k,n−k−1))+Θ(n))]=∑n−10E[Xk]∗E[T(max(k,n−k−1))+Θ(n))]=∑n−101/n∗E[T(max(k,n−k−1))+Θ(n))]=∑n−101/n∗E[T(max(k,n−k−1))]+∑n−101/n∗Θ(n)≤2/n∗∑n−1[n/2]E[T(k)]+Θ(n)

Claim:E[T(n)]≤c∗n for constant c>0

Proof

Substitution Method
Assume true for <n
E[T(n)]≤2/n∗∑n−1[n/2]E[T(k)]+Θ(n)≤2/n∗∑n−1[n/2]c∗k+Θ(n)≤2c/n∗3/8n2+Θ(n)=cn−(1/4cn−Θ(n))≤cn if 1/4cn≥Θ(n)

Rand-Select has Θ(n) expect running time.
Θ(n2)worst running time.

Worst-case linear-time order statistics

[Blum,Floyd,Pratt,Rivest and Tarjan]
- guarantee good pivot recursively
Select(i,n):

1. Divide the n elements
   [n/5] groups of 5 elements.
   Find the median of each group. Θ(n)
   
2. Recursively select the median x of the [n/5] group medians. T(n/5)
3. Partition with x as pivot.Let k = rank(x)

4. if i = k then return x
   if i<k then recursively select ith smallest element in the left .
   if i <k then recursively select ith smallest element in the right. T(3n/4)

   
   
   So running time in the step 4 is T(3n/4)
   T(n)≤T(3n/4)+T(n/5)+Θ(n)
   Claim: T(n)≤cn

Proof:substitution

Assume true for <n
T(k)≤ck
So
T(n)≤3cn/4+cn/5+Θ(n)≤19/20cn+Θ(n)≤cn−(1/20cn−Θ(n))≤cn
if 1/20cn>Θ(n)
So the running time is Θ(n)

Code for IthElement in github:
code in github