Minimum operations to make product of adjacent element pair of prefix sum negative

 Given an array arr[ ] of size N, consider an array prefix[ ] where prefix[i] is the sum of the first i elements of arr. The task is to find the minimum number of operations required to modify the given array such that the product of any two adjacent elements in the prefix array is negative. In one operation you can increment or decrement the value of any element by 1.

Input: arr[] = {1, -3, 1, 0}
Output: 4
Explanation: The sequence can be transformed into 1, -2, 2, -2 by 4 operations, the sum of the prefixes are 1, -1, 1, -1 which satisfy all the conditions.

Input: arr[] = {-1 4 3 2 -5 4}
Output: 8

Approach: The idea is to try out two independent possibilities that either even length prefix sums are positive and odd length prefix sums are negative or vice versa. Follow the steps below to solve the problem:

• Initialize a variable res = INT_MAX to store the minimum number of operations.
• Traverse over the range [0, 1] using the variable r. 
  • Initialize variables ans = 0  and sum = 0 to store the count of total operations and the current prefix sum respectively.
  • Traverse over the range [0, N-1] using the variable i,
    • Add arr[i] to the value of sum.
    • If the value of (i+r) is odd and if sum is not positive then add sum+1 to ans and update the value of sum to 1.
    • Otherwise if (i+r) is even and if sum is not negative then add sum+1 to ans update the value of sum to -1.
  • Update the value of res as min(res, ans).
• After completing the above steps print the value of res.

C++

// C++ code for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find minimum operations
// needed to make the product of any
// two adjacent elements in prefix
// sum array negative
void minOperations(vector<int> a)
{
    // Stores the minimum operations
    int res = INT_MAX;
    int N = a.size();

    for (int r = 0; r < 2; r++) {
        // Stores the prefix sum
        // and number of operations
        int sum = 0, ans = 0;

        // Traverse the array
        for (int i = 0; i < N; i++) {
            // Update the value of sum
            sum += a[i];
            // Check if i+r is odd
            if ((i + r) % 2) {

                // Check if prefix sum
                // is not positive
                if (sum <= 0) {

                    // Update the value of
                    // ans and sum
                    ans += -sum + 1;
                    sum = 1;
                }
            }
            else {

                // Check if prefix sum is
                // not negative
                if (sum >= 0) {
                    // Update the value of
                    // ans and sum
                    ans += sum + 1;
                    sum = -1;
                }
            }
        }

        // Update the value of res
        res = min(res, ans);
    }

    // Print the value of res
    cout << res;
}

// Driver Code
int main()
{
    vector<int> a{ 1, -3, 1, 0 };

    minOperations(a);

    return 0;
}

Java

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

class GFG{

// Function to find minimum operations
// needed to make the product of any
// two adjacent elements in prefix
// sum array negative
static void minOperations(ArrayList<Integer> a)
{

    // Stores the minimum operations
    int res = Integer.MAX_VALUE;
    int N = a.size();

    for(int r = 0; r < 2; r++)
    {

        // Stores the prefix sum
        // and number of operations
        int sum = 0, ans = 0;

        // Traverse the array
        for(int i = 0; i < N; i++)
        {

            // Update the value of sum
            sum += a.get(i);

            // Check if i+r is odd
            if ((i + r) % 2 == 1)
            {

                // Check if prefix sum
                // is not positive
                if (sum <= 0)
                {

                    // Update the value of
                    // ans and sum
                    ans += -sum + 1;
                    sum = 1;
                }
            }
            else
            {

                // Check if prefix sum is
                // not negative
                if (sum >= 0)
                {

                    // Update the value of
                    // ans and sum
                    ans += sum + 1;
                    sum = -1;
                }
            }
        }

        // Update the value of res
        res = Math.min(res, ans);
    }

    // Print the value of res
    System.out.print(res);
}

// Driver Code
public static void main(String args[])
{
    ArrayList<Integer> a = new ArrayList<Integer>();
    a.add(1);
    a.add(-3);
    a.add(1);
    a.add(0);

    minOperations(a);
}
}

// This code is contributed by SURENDRA_GANGWAR

Python3

# python code for the above approach
# // Function to find minimum operations
# // needed to make the product of any
# // two adjacent elements in prefix
# // sum array negative
def minOperations(a):

    #Stores the minimum operations
    res = 100000000000
    N = len(a)
    for r in range(0,2):

        # Stores the prefix sum
        # and number of operations
        sum = 0
        ans = 0

        # Traverse the array
        for i in range (0,N):

            # Update the value of sum
            sum += a[i]

            # Check if i+r is odd
            if ((i + r) % 2):

                # Check if prefix sum
                # is not positive
                if (sum <= 0):

                    # Update the value of
                    # ans and sum
                    ans += -sum + 1
                    sum = 1
            else:

                 # Check if prefix sum is
                # not negative
                if (sum >= 0):

                    # Update the value of
                    # ans and sum
                    ans += sum + 1;
                    sum = -1;

        # Update the value of res
        res = min(res, ans)

    # // Print the value of res
    print(res)

    # Driver code
a = [1, -3, 1, 0]
minOperations(a);

# This code is contributed by Stream_Cipher

C#

// C# code for the above approach
using System;
using System.Collections.Generic;

class GFG
{

// Function to find minimum operations
// needed to make the product of any
// two adjacent elements in prefix
// sum array negative
static void minOperations(List<int> a)
{

    // Stores the minimum operations
    int res = Int32.MaxValue;
    int N = a.Count;

    for (int r = 0; r < 2; r++)
    {

        // Stores the prefix sum
        // and number of operations
        int sum = 0, ans = 0;

        // Traverse the array
        for (int i = 0; i < N; i++)
        {

            // Update the value of sum
            sum += a[i];

            // Check if i+r is odd
            if ((i + r) % 2 == 1) {

                // Check if prefix sum
                // is not positive
                if (sum <= 0) {

                    // Update the value of
                    // ans and sum
                    ans += -sum + 1;
                    sum = 1;
                }
            }
            else {

                // Check if prefix sum is
                // not negative
                if (sum >= 0) {
                    // Update the value of
                    // ans and sum
                    ans += sum + 1;
                    sum = -1;
                }
            }
        }

        // Update the value of res
        res = Math.Min(res, ans);
    }

    // Print the value of res
    Console.Write(res);
}

// Driver Code
public static void Main()
{
    List<int> a = new List<int>(){ 1, -3, 1, 0 };

    minOperations(a);
}
}

// This code is contributed by bgangwar59.

Javascript

<script>

// JavaScript program for the above approach

// Function to find minimum operations
// needed to make the product of any
// two adjacent elements in prefix
// sum array negative
function minOperations(a)
{
    // Stores the minimum operations
   let res =Number.MAX_VALUE;
   let N = a.length;

    for (let r = 0; r < 2; r++) {
        // Stores the prefix sum
        // and number of operations
       let sum = 0, ans = 0;

        // Traverse the array
        for (let i = 0; i < N; i++) {
            // Update the value of sum
            sum += a[i];
            // Check if i+r is odd
            if ((i + r) % 2) {

                // Check if prefix sum
                // is not positive
                if (sum <= 0) {

                    // Update the value of
                    // ans and sum
                    ans += -sum + 1;
                    sum = 1;
                }
            }
            else {

                // Check if prefix sum is
                // not negative
                if (sum >= 0) {
                    // Update the value of
                    // ans and sum
                    ans += sum + 1;
                    sum = -1;
                }
            }
        }

        // Update the value of res
        res = Math.min(res, ans);
    }

    // Print the value of res
    document.write(res);
}

// Driver Code

    let a = [1, -3, 1, 0 ];

    minOperations(a);

  // This code is contributed by Potta Lokesh

</script>

Output

4

Time Complexity: O(N)
Auxiliary Space: O(1)