Leetcode 3228. Maximum Number of Operations to Move Ones to the End

Let's understand what we're being asked to do. We have a binary string like '10100' and can perform a specific operation: find any '10' substring and slide the '1' rightward over all following zeros until it hits another '1' or the end.

Tracing through '10100': We could pick the first '10' (positions 0-1) and slide the '1' right, transforming it to '01100'. Or we could pick the '10' at positions 2-3 and slide that '1' right to get '10010'. Each valid slide counts as one operation.

Important constraint: We can only slide a '1' that is immediately followed by a '0' (forming '10'). The '1' slides over consecutive zeros until blocked.

Edge cases to consider: What if the string has no '10' patterns? (no moves possible, return 0). What if all '1's are already at the right? What if we have '111000' (multiple ones can each slide)?

The straightforward approach is to simulate the process: repeatedly scan the string for any '10' pattern, perform the slide operation (move the '1' rightward over zeros), increment the operation counter, and repeat until no more '10' patterns exist. This approach directly models the problem description but requires multiple passes through the string and string manipulation for each operation, making it inefficient for large inputs.

def max_slide_operations(s):
    """
    Brute force: repeatedly find '10' patterns and slide the '1' rightward
    over all zeros until it hits another '1' or the end.
    Count total operations until no more '10' patterns exist.
    """
    # Convert string to list for mutability
    chars = list(s)
    operations = 0
    
    # Keep scanning until no more '10' patterns exist
    while True:
        found = False
        
        # Scan for any '10' pattern
        for i in range(len(chars) - 1):
            if chars[i] == '1' and chars[i + 1] == '0':
                found = True
                
                # Slide the '1' rightward over all consecutive zeros
                j = i + 1
                while j < len(chars) and chars[j] == '0':
                    j += 1
                
                # Move the '1' to position j-1 (just before next '1' or end)
                chars[i] = '0'
                chars[j - 1] = '1'
                
                # Increment operation count
                operations += 1
                break
        
        # If no '10' pattern found, we're done
        if not found:
            break
    
    return operations

Each '1' in the string can potentially slide rightward over zeros. The key insight is that a '1' at position i can slide over all zeros to its right until it hits another '1' or the end.

The order in which we perform slides matters! If we have '1010', sliding the first '1' blocks the second '1' from sliding as far.

By thinking about each '1' independently and counting how many positions it can slide, we can compute the total number of operations.

The crucial realization: processing '1's from right to left (or tracking how many zeros are available) lets us count moves without simulating each step. Each '1' can slide over all zeros to its right that aren't already "claimed" by '1's further right.

Imagine cars (ones) on a highway with empty spaces (zeros) ahead. Each car wants to move forward into empty spaces.

If we process cars from back to front, each car can count exactly how many empty spaces it can claim. The first car (rightmost '1') claims all zeros to its right. The second car claims zeros between it and the first car, and so on.

This is like counting inversions - how many zeros does each '1' need to "jump over" to reach its final position?

The optimal approach scans the string from right to left, tracking how many zeros we've seen so far. Each time we encounter a '1', it can slide over all the zeros to its right (the count we've been tracking), and we add this count to our total operations. After counting a '1''s contribution, we increment the zero count (because this '1' will act as a zero-space for '1's further left after it slides). This greedy single-pass approach counts all possible operations without simulation.

def max_slide_operations(s):
    """
    Count maximum number of '10' -> '01' slide operations.
    Scan right-to-left, tracking zeros; each '1' can slide over all zeros to its right.
    """
    total_operations = 0
    zero_count = 0
    
    # Scan from right to left
    for i in range(len(s) - 1, -1, -1):
        if s[i] == '0':
            # Count zeros for '1's to the left
            zero_count += 1
        else:
            # This '1' can slide over all zeros to its right
            total_operations += zero_count
            # After sliding, this '1' acts as a zero-space for '1's further left
            zero_count += 1
    
    return total_operations

• Greedy right-to-left scanning: When elements need to move to the end based on relative positions, scan from right to left to count how many positions each element can move - this naturally accumulates the total operations because later '1's block earlier ones, making a single pass sufficient.
• Inversion counting pattern: This problem is fundamentally about counting inversions (how many '0's each '1' must pass over) - recognize that 'moving elements to the end' problems often reduce to counting how many obstacles each element encounters, summed across all movable elements.
• Cumulative zero tracking: Maintain a running count of zeros encountered so far (scanning right-to-left) - when you hit a '1', add the zero count to your result because that '1' can slide over all those zeros, then increment the zero count to account for this '1' becoming an obstacle for future '1's.
• Order independence insight: The key realization is that the total number of operations is independent of the order you perform them - each '1' will eventually move past all '0's to its right, so you can simply count all possible moves rather than simulating the actual movement sequence.
• Single-pass O(n) optimization: Avoid simulating the actual string modifications (which would be O(n²) or worse) - instead, recognize that you can compute the answer analytically by counting relationships between '1's and '0's in one linear scan, trading simulation for mathematical reasoning.
• Relative positioning problems: Apply this counting approach to any problem where elements move based on relative order constraints (bubble sort operations, minimum swaps, array rearrangement) - often the total operations needed equals the sum of 'distances' each element must travel past blocking elements.