Sum of decomposition values of all suffixes of an Array

Given an array arr[], the task is to find the sum of decomposition value of the suffixes subarray.

Decomposition Value: 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++

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// 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;
}

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Java

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// 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

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Python3

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# 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

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C#

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// 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

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Output:

8

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