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Minimum Operations to Make Binary Array Elements Equal to One II

Number: 3477

Difficulty: Medium

Paid? No

Companies: Uber

Problem Description

Given a binary array, you are allowed to perform an operation whereby you choose any index i and flip all the bits from that index to the end of the array (i.e. change 0 to 1 and vice versa). The goal is to determine the minimum number of these operations required to make every element in the array equal to 1.

Key Insights

  • Flipping a segment (suffix) of the array changes the state of each bit in that segment.
  • Instead of simulating each flip, maintain a parity indicator to track if the current element is in its original state or flipped.
  • Use a greedy approach: process the array sequentially and flip only when the effective value (taking parity into account) is 0.
  • Toggling the parity when a flip is made allows you to compute each element’s current state without multiple array modifications.

Space and Time Complexity

Time Complexity: O(n) - iterate through the array once.
Space Complexity: O(1) - only constant extra space is used.

Solution

The solution uses a greedy strategy with a parity indicator to determine the effective value of an element considering previous flips. Traverse the array and for each element, compute its effective value (original value if no flips, or inverted if odd number of flips). If the effective value is 0, perform a flip operation by incrementing the operation count and toggling the parity. This ensures that only necessary flips are executed, thereby minimizing the total operations.

Code Solutions

Python:

JavaScript:

C++:

Java:

# Function to compute the minimum operations to convert all elements to 1.
def minOperations(nums):
    operations = 0           # Stores the count of operations performed.
    parity = 0               # Parity indicator: 0 for no inversion, 1 for flipped state.
    
    # Process each element in the array.
    for num in nums:
        # Calculate effective value considering parity.
        current = num if parity == 0 else 1 - num
        
        # If effective value is 0, perform a flip operation.
        if current == 0:
            operations += 1
            parity ^= 1     # Toggle parity for subsequent elements.
    
    return operations

# Example usage:
print(minOperations([0, 1, 1, 0, 1]))  # Expected output: 4
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