Modify array to another given array by replacing array elements with the sum of the array | Set-2

Given an Array input[] consisting only of 1s initially and an array target[] of size N, the task is to check if the array input[] can be converted to target[] by replacing input[i] with the sum of array elements in each step.

Examples:

Input: input[] = { 1, 1, 1 }, target[] = { 9, 3, 5 } 
Output: YES 
Explanation:
Replacing input[1] with (input[0] + input[1] + input[2]) modifies input[] to { 1, 3, 1 } 
Replacing input[2] with (input[0] + input[1] + input[2]) modifies input[] to { 1, 3, 5 } 
Replacing input[0] with (input[0] + input[1] + input[2]) modifies input[] to { 9, 3, 5 } 
Since the array input[] equal to the target[] array, the required output is “YES”.

Input: input[] = { 1, 1, 1, 1 }, target[] = { 1, 1, 1, 2 } 
Output: NO

Naive Approach: The naive approach and the Greedy approach are already mentioned in Set-1 of this article.

Efficient Approach: The idea to solve the problem efficiently is based on the below intuition:

• Instead of trying to check if target[] array can be reached, work backward and try to generate the array of 1s from the target[].
• While working backward the maximum element of the array will be the sum of elements after the last turn. To keep track of the maximum element, use a max heap.
• After every turn, remove the maximum element from the heap and determine the previous maximum element value. To do this find the sum of all elements of the array.

Follow the steps to solve the problem:

• Create variables sum and lastSum to store the sum of all elements the sum of the array on previous step. 
• To determine the previous element, find the difference of “sum” and “lastSum” from “lastSum”, i.e (lastSum – (sum – lastSum)).
• Then put this value back to the heap and update the sum.
• Continue this until either the sum is equal to one or the lastSum is equal to one.
• In case, lastSum becomes less than sum or sum becomes equal to zero or the difference of lastSum and sum becomes zero, return false.

Below is the implementation of the above approach :

C++

// C++ code for the above approach:

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

// Function to find if target[] can be reached
bool createTarget(vector<int>& target)
{

    // Initialise size of target array
    int n = target.size();

    // Initialise variable to store
    // sum of values
    int sum = 0;

    // Initialise variable to store
    // last sum
    int lastSum;

    // Initialise a max-heap to keep track
    // of the maximum element
    priority_queue<int> maxHeap(target.begin(),
                                target.end());

    // Start traversing to find the sum
    for (int i = 0; i < n; i++) {
        sum = sum + target[i];
    }

    // While heap has element traverse
    while (true) {

        // Update last sum with
        // maximum value of heap
        lastSum = maxHeap.top();

        // Pop the maximum element
        // of the heap
        maxHeap.pop();

        // Update sum of values
        sum = sum - lastSum;

        // If either sum or last sum is
        // equal to 1, then
        // target array possible
        if (lastSum == 1 || sum == 1) {

            // Return true
            return true;
        }

        // If last sum becomes less than
        // sum or if sum becomes equal
        // to 0 or if difference of last
        // sum and sum becomes 0
        if (lastSum < sum || sum == 0
            || lastSum - sum == 0) {

            // Return false
            return false;
        }

        // Update last sum
        lastSum = lastSum - sum;

        // Update sum
        sum = sum + lastSum;

        // Push last sum into the queue
        maxHeap.push(lastSum);
    }
}

// Driver code
int main()
{
    int N = 2;
    vector<int> target = { 2, 3 };
    bool ans = createTarget(target);
    if (ans)
        cout << "YES";
    else
        cout << "NO";
    return 0;
}

Java

// Java code for the above approach:
import java.util.*;

// Function to find if target[] can be reached
class GFG {
  static boolean createTarget(int[] target)
  {

    // Initialise size of target array
    int n = target.length;

    // Initialise variable to store
    // sum of values
    int sum = 0;

    // Initialise variable to store
    // last sum
    int lastSum = 0;

    // Initialise a max-heap to keep track
    // of the maximum element
    PriorityQueue<Integer> maxHeap = new PriorityQueue<Integer>();
    for (int i = 0; i < target.length; i++)

      // we are using negative values as we want to remove the maximum element
      // from the priority queue, however, by default the minimum element i removed using poll()
      maxHeap.add(-1 * target[i]);

    // Start traversing to find the sum
    for (int i = 0; i < n; i++)
      sum += target[i];

    // While heap has element traverse
    while (true)
    {

      // Update last sum with
      // maximum value of heap and also
      //remove the maximum value from heap
      lastSum = -1 * maxHeap.poll();

      // Update sum of values
      sum -= lastSum;

      // If either sum or last sum is
      // equal to 1, then
      // target array possible
      if (lastSum == 1 || sum == 1)
      {
        return true;
      }

      // If last sum becomes less than
      // sum or if sum becomes equal
      // to 0 or if difference of last
      // sum and sum becomes 0
      if (lastSum <= sum || sum == 0 )
      {
        return false;
      }

      // update lastsum and sum
      lastSum = lastSum - sum;
      sum += lastSum;

      // Push last sum into the queue
      maxHeap.add(-1 * lastSum);
    }
  }

  // Driver code
  public static void main(String[] args) {
    int N = 2;
    int[] target = {2, 3};
    boolean ans = createTarget(target);
    if (ans)
      System.out.println("YES");
    else
      System.out.println("NO");
  }
}

// This code is contributed by phasing17

Python3

# python3 code for the above approach:
from queue import PriorityQueue

# Function to find if target[] can be reached
def createTarget(target):

    # Initialise size of target array
    n = len(target)

    # Initialise variable to store
    # sum of values
    sum = 0

    # Initialise variable to store
    # last sum
    lastSum = 0

    # Initialise a max-heap to keep track
    # of the maximum element
    maxHeap = PriorityQueue()

    for itm in target:
        maxHeap.put(-itm)

        # Start traversing to find the sum
    for i in range(0, n):
        sum = sum + target[i]

        # While heap has element traverse
    while True:

                # Update last sum with
                # maximum value of heap
        lastSum = -maxHeap.get()

        # Pop the maximum element
        # of the heap

        # Update sum of values
        sum = sum - lastSum

        # If either sum or last sum is
        # equal to 1, then
        # target array possible
        if (lastSum == 1 or sum == 1):

                        # Return true
            return True

            # If last sum becomes less than
            # sum or if sum becomes equal
            # to 0 or if difference of last
            # sum and sum becomes 0
        if (lastSum < sum or sum == 0
                or lastSum - sum == 0):

                        # Return false
            return False

            # Update last sum
        lastSum = lastSum - sum

        # Update sum
        sum = sum + lastSum

        # Push last sum into the queue
        maxHeap.put(-lastSum)

# Driver code
if __name__ == "__main__":

    N = 2
    target = [2, 3]
    ans = createTarget(target)
    if (ans):
        print("YES")
    else:
        print("NO")

    # This code is contributed by rakeshsahni

C#

// C# program to implement
// the above approach
using System;
using System.Collections.Generic;

class GFG
{

  static bool createTarget(int[] target)
  {

    // Initialise size of target array
    int n = target.Length;

    // Initialise variable to store
    // sum of values
    int sum = 0;

    // Initialise variable to store
    // last sum
    int lastSum = 0;

    // Initialise a max-heap to keep track
    // of the maximum element
    List<int> maxHeap = new List<int>();
    for (int i = 0; i < target.Length; i++)

      // we are using negative values as we want to remove the maximum element
      // from the priority queue, however, by default the minimum element i removed using poll()
      maxHeap.Add(-1 * target[i]);

    // Start traversing to find the sum
    for (int i = 0; i < n; i++)
      sum += target[i];

    // While heap has element traverse
    while (true)
    {

      // Update last sum with
      // maximum value of heap and also
      //remove the maximum value from heap

      // Update last sum with
      // maximum value of heap
      lastSum = maxHeap[0];

      // Pop the maximum element
      // of the heap
      maxHeap.RemoveAt(0);


      // Update sum of values
      sum -= lastSum;

      // If either sum or last sum is
      // equal to 1, then
      // target array possible
      if (lastSum == 1 || sum == 1)
      {
        return true;
      }

      // If last sum becomes less than
      // sum or if sum becomes equal
      // to 0 or if difference of last
      // sum and sum becomes 0
      if (lastSum <= sum || sum == 0 )
      {
        return false;
      }

      // update lastsum and sum
      lastSum = lastSum - sum;
      sum += lastSum;

      // Push last sum into the queue
      maxHeap.Add(-1 * lastSum);
    }
  }

  // Driver Code
  public static void Main()
  {
    int N = 2;
    int[] target = {2, 3};
    bool ans = createTarget(target);
    if (ans == false)
      Console.Write("YES");
    else
      Console.Write("NO");
  }
}

// This code is contributed by sanjoy_62.

Javascript

// JavaScript program to implement
// the above approach
function createTarget(target)
{

    // Initialise size of target array
    var n = target.length;

    // Initialise variable to store
    // sum of values
    var sum = 0;

    // Initialise variable to store
    // last sum
    var lastSum = 0;

    // Initialise a max-heap to keep track
    // of the maximum element
    maxHeap = [];
    for (var i = 0; i < target.length; i++)

        // we are using negative values as we want to remove
        // the maximum element from the priority queue,
        // however, by default the minimum element i removed
        // using poll()
        maxHeap.push(-1 * target[i]);

    // Start traversing to find the sum
    for (var i = 0; i < n; i++)
        sum += target[i];

    // While heap has element traverse
    while (true) {

        // Update last sum with
        // maximum value of heap and also
        // remove the maximum value from heap

        // Update last sum with
        // maximum value of heap
        lastSum = maxHeap[0];

        // Pop the maximum element
        // of the heap
        maxHeap.splice(0, 1);

        // Update sum of values
        sum -= lastSum;

        // If either sum or last sum is
        // equal to 1, then
        // target array possible
        if (lastSum == 1 || sum == 1) {
            return true;
        }

        // If last sum becomes less than
        // sum or if sum becomes equal
        // to 0 or if difference of last
        // sum and sum becomes 0
        if (lastSum <= sum || sum == 0) {
            return false;
        }

        // update lastsum and sum
        lastSum = lastSum - sum;
        sum += lastSum;

        // Push last sum into the queue
        maxHeap.pushh(-1 * lastSum);
    }
}

// Driver Code
var N = 2;
var target = [ 2, 3 ];
var ans = createTarget(target);
if (ans == false)
    console.log("YES");
else
    console.log("NO");

// This code is contributed by  phasing17

Output

true

Time Complexity: O(N * log(N) + (K / N * log(N))), where K is the maximum element of the array.
Auxiliary Space: O(N)