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Number Of Longest Increasing Subsequence

Problem Statement

The problem “Number Of Longest Increasing Subsequence” states that you are given an array a[ ] of size n. Print the number of longest increasing subsequences in it.

Example

`a[ ] = {1, 2, 5, 4, 7}`
`2`

Explanation: The longest increasing subsequences can be seen in the above image.

Input :Â a[ ] = {1, 2, 3, 4, 5}

Output : 1Â  Â  (1,2,3,4,5 is the longest sub-sequence here)

Algorithm for Number Of Longest Increasing Subsequence

1. Initialize an array a[ ] of integer type of size n.
2. Create a function to find number of the longest increasing sub-sequences which accept an array of integer type and it’s size as it’s parameters.
3. Create two arrays of integer type len and cnt of size n and initialize every element of both the arrays as 1. Also, initialize an integer variable lis = 1.
4. Traverse from 1 till n-1 using i and create an inner loop from 0 to i-1.
5. Check if the element in array a[ ] at current index of outer loop is greater than the element in array a[ ] at the current index of the inner loop, check if the value + 1 in array len[ ] at current index of inner loop is greater than the element in array len[ ] at current index of outer loop, update the value in array len[ ] at current index of outer loop as the value + 1 in array len[ ] at current index of inner loop and the value in array cnt[ ] at current index of outer loop as the value in array cnt[ ] at current index of inner loop.
6. Else if the value + 1 in array if len[ ] at current index of inner loop is equal to the element in array len[ ] at current index of outer loop, update the value in array cnt[ ] at current index of outer loop as the sum of the value in array cnt[ ] at current index of inner loop and the outer loop itself.
7. Store the maximum of lis and len[i] in lis.
8. Initialize a variable ans as 0.
9. Traverse again from 0 to n-1 and check if len[i] is equal to lis add the value at the current index of cnt in ans.
10. Return ans.

Code

C++ Program to find number of longest increasing subsequence

```#include <bits/stdc++.h>
using namespace std;

class Longestsubseq{
public:
int numOfIncSubseq(int a[], int n){

int len[n], cnt[n];

for(int i=0; i<n; i++){
len[i]=1;
cnt[i]=1;
}

int lis = 1;
for(int i=1; i<n; i++){
for(int j=0; j<i; j++){
if(a[i] > a[j]){

if(len[j]+1 > len[i]){
len[i] = len[j]+1;
cnt[i] = cnt[j];
}

else if(len[j]+1 == len[i]){
cnt[i] += cnt[j];
}
}

lis = max(lis, len[i]);
}
}

int ans = 0;

for(int i=0; i<n; i++){
if(len[i] == lis)ans += cnt[i];
}

return ans;
}
};

int main(){
Longestsubseq ls;

int a[] = {1,2,5,4,7};
int n = sizeof(a)/sizeof(a[0]);

cout<<(ls.numOfIncSubseq(a, n));

return 0;
}```
`2`

Java Program to find number of longest increasing subsequence

```import java.util.*;

class Longestsubseq{

static int numOfIncSubseq(int[] a, int n){

int[] len = new int[n];
int[] cnt = new int[n];

for(int i=0; i<n; i++){
len[i]=1;
cnt[i]=1;
}

int lis = 1;
for(int i=1; i<n; i++){
for(int j=0; j<i; j++){
if(a[i] > a[j]){

if(len[j]+1 > len[i]){
len[i] = len[j]+1;
cnt[i] = cnt[j];
}

else if(len[j]+1 == len[i]){
cnt[i] += cnt[j];
}
}

lis = Math.max(lis, len[i]);
}
}

int ans = 0;

for(int i=0; i<n; i++){
if(len[i] == lis)ans += cnt[i];
}

return ans;
}

public static void main (String[] args){
int[] a = {1,2,5,4,7};
int n = a.length;

System.out.println(numOfIncSubseq(a, n));
}
}
```
`2`

Complexity Analysis

Time Complexity

O(n*n) where n is the number of elements in the given array a[ ]. The time complexity is the same as what is required to find the longest increasing subsequence.

Space Complexity

O(n) because we used extra n space. We have used a len[] array which stores the longest increasing subsequence ending at ith element.

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