Sum of decomposition values of all suffixes of an Array

Given an array arr[], the task is to find the sum of the decomposition value of the suffixes subarray.
Decomposition Value: The decomposition value of a subarray is the count of the partition in the subarray possible. The partition in the array at index i    can be done only if the elements of the array before if it less than the current index. That is A[k] < A[i], where k ? i.
Examples: 

Input: arr[] = {2, 8, 4} 
Output: 4 
Explanation: 
All suffixes subarray of arr[] are [2, 8, 4], [8, 4], [4] 
Suffix [4] => only 1 decomposition {4} 
Suffix [8, 4] => only 1 decomposition {8, 4} 
Suffix [2, 8, 4] => 2 decompositions {2, 8, 4}, {2} {8, 4} 
Hence, Sum of Decomposition values = 1 + 1 + 2 = 4
Input: arr[] = {9, 6, 9, 35} 
Output: 8 
Explanation: 
All suffixes of arr are [9, 6, 9, 35], [6, 9, 35], [9, 35], [35] 
Suffix [35] => only 1 decomposition {35} 
Suffix [9, 35] => 2 decompositions {9} {35} 
Suffix [6, 9, 35] => 3 decompositions {6} {9, 35} 
Suffix [9, 6, 9, 35] => 2 decompositions {9, 6, 9} {35} 
Hence, Sum of Decomposition values = 1 + 2 + 3 + 2 = 8 

Approach: The idea is to use Stack to solve this problem. Below is the illustration of the approach
 

• Traverse array from the end to the start.
• Maintain a minimum variable and answer variable.
• If the stack is empty or the current element is less than the top of stack - 
  
  • Push S[i] onto the stack.
  • Increment the answer by the size of the stack.
  • Also, maintain the minimum value till now.
• Otherwise, 
  
  • Keep on popping the blocks as long as top of the stack is less than the current element.
  • Update the minimum value till now with the current element.
  • Push minimum value onto the stack. Because, we want the minimum value of the subarray to represent that subarray
  • Increment the answer by the size of the stack.

Below is the implementation of the above approach:
 

// C++ implementation to find the 
// sum of Decomposition values of 
// all suffixes of an array

#include <bits/stdc++.h>
using namespace std;
#define int long long int

// Function to find the decomposition 
// values of the array 
int decompose(vector<int> S)
{
    // Stack
    stack<int> s;
    int N = S.size();
    int ans = 0;
    
    // Variable to maintain 
    // min value in stack
    int nix = INT_MAX;
    
    // Loop to iterate over the array
    for (int i = N - 1; i >= 0; i--) {
        
        // Condition to check if the
        // stack is empty
        if (s.empty()) {
            s.push(S[i]);
            nix = S[i];
        }
        else {
            
            // Condition to check if the 
            // top of the stack is greater
            // than the current element
            if (S[i] < s.top()) {
                s.push(S[i]);
                nix = min(nix, S[i]);
            }
            else {
                int val = S[i];
                
                // Loop to pop the element out
                while (!s.empty() &&
                       val >= s.top()) {
                    s.pop();
                }
                nix = min(nix, S[i]);
                s.push(nix);
            }
        }
        
        // the size of the stack is the 
        // max no of subarrays for 
        // suffix till index i
        // from the right
        ans += s.size();
    }

    return ans;
}

// Driver Code
signed main()
{
    vector<int> S = { 9, 6, 9, 35 };
    cout << decompose(S) << endl;
    return 0;
}

// Java implementation to find the 
// sum of Decomposition values of 
// all suffixes of an array
import java.util.*;

class GFG{

// Function to find the decomposition 
// values of the array 
static int decompose(Vector<Integer> S)
{
    
    // Stack
    Stack<Integer> s = new Stack<Integer>();
    int N = S.size();
    int ans = 0;
    
    // Variable to maintain 
    // min value in stack
    int nix = Integer.MAX_VALUE;
    
    // Loop to iterate over the array
    for(int i = N - 1; i >= 0; i--)
    {
        
       // Condition to check if the
       // stack is empty
       if (s.isEmpty())
       {
           s.add(S.get(i));
           nix = S.get(i);
       }
       else 
       {
           
           // Condition to check if the 
           // top of the stack is greater
           // than the current element
           if (S.get(i) < s.peek()) 
           {
               s.add(S.get(i));
               nix = Math.min(nix, S.get(i));
           }
           else
           {
               int val = S.get(i);
               
               // Loop to pop the element out
               while (!s.isEmpty() && val >= s.peek())
               {
                   s.pop();
               }
               nix = Math.min(nix, S.get(i));
               s.add(nix);
           }
       }
       
       // The size of the stack is the 
       // max no of subarrays for 
       // suffix till index i
       // from the right
       ans += s.size();
    }
    return ans;
}

// Driver Code
public static void main(String args[])
{
    Vector<Integer> S = new Vector<Integer>();
    S.add(9);
    S.add(6);
    S.add(9);
    S.add(35);
    
    System.out.println(decompose(S));
}
}

// This code is contributed by 29AjayKumar

# Python3 implementation to find the 
# sum of Decomposition values of 
# all suffixes of an array
import sys

# Function to find the decomposition 
# values of the array 
def decompose(S):

    # Stack
    s = []
    N = len(S)
    ans = 0
    
    # Variable to maintain 
    # min value in stack
    nix = sys.maxsize
    
    # Loop to iterate over the array
    for i in range(N - 1, -1, -1):
        
        # Condition to check if the
        # stack is empty
        if (len(s) == 0):
            s.append(S[i])
            nix = S[i]
        
        else:
            
            # Condition to check if the 
            # top of the stack is greater
            # than the current element
            if (S[i] < s[-1]):
                s.append(S[i])
                nix = min(nix, S[i])
            
            else:
                val = S[i]
                
                # Loop to pop the element out
                while (len(s) != 0 and
                          val >= s[-1]):
                    s.pop()
            
                nix = min(nix, S[i]);
                s.append(nix)
        
        # The size of the stack is the 
        # max no of subarrays for 
        # suffix till index i
        # from the right
        ans += len(s)

    return ans

# Driver Code
if __name__ =="__main__":
    
    S = [ 9, 6, 9, 35 ]
    
    print(decompose(S))

# This code is contributed by chitranayal

// C# implementation to find the 
// sum of Decomposition values of 
// all suffixes of an array 
using System;
using System.Collections.Generic;

class GFG{

// Function to find the decomposition 
// values of the array
static int decompose(List<int> S) 
{ 
    
    // Stack 
    Stack<int> s = new Stack<int>(); 
    
    int N = S.Count; 
    int ans = 0; 
    
    // Variable to maintain 
    // min value in stack 
    int nix = Int32.MaxValue; 
    
    // Loop to iterate over the array 
    for(int i = N - 1; i >= 0; i--) 
    { 
        
        // Condition to check if the 
        // stack is empty 
        if (s.Count == 0) 
        { 
            s.Push(S[i]); 
            nix = S[i]; 
        } 
        else
        { 
            
            // Condition to check if the 
            // top of the stack is greater 
            // than the current element 
            if (S[i] < s.Peek()) 
            { 
                s.Push(S[i]); 
                nix = Math.Min(nix, S[i]); 
            } 
            else
            { 
                int val = S[i]; 
                    
                // Loop to pop the element out 
                while (s.Count != 0 && val >= s.Peek()) 
                { 
                    s.Pop(); 
                } 
                nix = Math.Min(nix, S[i]); 
                s.Push(nix); 
            } 
        } 
        
        // The size of the stack is the 
        // max no of subarrays for 
        // suffix till index i 
        // from the right 
        ans += s.Count; 
    } 
    return ans; 
}

// Driver code
static void Main() 
{
    List<int> S = new List<int>();
    S.Add(9); 
    S.Add(6); 
    S.Add(9); 
    S.Add(35); 

    Console.WriteLine(decompose(S));
}
}

// This code is contributed by divyeshrabadiya07

<script>
// Javascript implementation to find the 
// sum of Decomposition values of 
// all suffixes of an array

// Function to find the decomposition 
// values of the array 
function decompose(S)
{

    // Stack
    let s = [];
    let N = S.length;
    let ans = 0;
      
    // Variable to maintain 
    // min value in stack
    let nix = Number.MAX_VALUE;
      
    // Loop to iterate over the array
    for(let i = N - 1; i >= 0; i--)
    {
          
       // Condition to check if the
       // stack is empty
       if (s.length==0)
       {
           s.push(S[i]);
           nix = S[i];
       }
       else 
       {
             
           // Condition to check if the 
           // top of the stack is greater
           // than the current element
           if (S[i] < s[s.length-1]) 
           {
               s.push(S[i]);
               nix = Math.min(nix, S[i]);
           }
           else
           {
               let val = S[i];
                 
               // Loop to pop the element out
               while (s.length!=0 && val >= s[s.length-1])
               {
                   s.pop();
               }
               nix = Math.min(nix, S[i]);
               s.push(nix);
           }
       }
         
       // The size of the stack is the 
       // max no of subarrays for 
       // suffix till index i
       // from the right
       ans += s.length;
    }
    return ans;
}

// Driver Code
let S = [];
S.push(9);
S.push(6);
S.push(9);
S.push(35);

document.write(decompose(S));

// This code is contributed by avanitrachhadiya2155
</script>

8

Time Complexity: O(n)
Auxiliary Space: O(n), where n is the size of the given array.