#2749

Medium

auto_awesomeBit Manipulation plus Brainteaser

LeetCode Problem Workspace

Minimum Operations to Make the Integer Zero

This problem challenges you to compute the minimum operations to reduce an integer to zero using bit manipulation and strategic subtraction.

bit manipulation brainteaser enumeration

Problem Statement

Given two integers num1 and num2, you are allowed to perform an operation where you select an integer i between 0 and 60 and subtract 2^i plus num2 from num1. Your task is to determine the fewest number of operations needed to reduce num1 to zero.

If it is impossible to make num1 equal to zero using any sequence of these operations, return -1. Focus on bit-level analysis and iterative exploration to track all possible intermediate states and identify minimal sequences.

Examples

Example 1

Input: num1 = 3, num2 = -2

Output: 3

We can make 3 equal to 0 with the following operations:

We choose i = 2 and subtract 22 + (-2) from 3, 3 - (4 + (-2)) = 1.
We choose i = 2 and subtract 22 + (-2) from 1, 1 - (4 + (-2)) = -1.
We choose i = 0 and subtract 20 + (-2) from -1, (-1) - (1 + (-2)) = 0. It can be proven, that 3 is the minimum number of operations that we need to perform.

Example 2

Input: num1 = 5, num2 = 7

Output: -1

It can be proven, that it is impossible to make 5 equal to 0 with the given operation.

Constraints

1 <= num1 <= 109
-109 <= num2 <= 109

Solution Approach

Bitwise Enumeration

Enumerate all relevant powers of 2 up to 2^60 and combine them with num2 to generate potential subtractions. Track all reachable states of num1 and count operations to find the minimal sequence.

Breadth-First Search

Use BFS starting from num1 to explore all states reachable by subtracting 2^i + num2. Each BFS level represents one operation, allowing early termination when zero is reached.

Pruning Impossible Paths

Detect impossible scenarios early by checking if num1 cannot reach zero modulo small powers of 2 or if num2 prevents convergence. This avoids exhaustive exploration of futile sequences.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on the number of states explored in BFS or enumeration, which can grow with the range of num1 and 2^i values. Space complexity is tied to storing visited states and the BFS queue.

What Interviewers Usually Probe

Look for bit-level patterns in num1 that align with powers of 2 subtractions.
Consider sequences where operations alternate between positive and negative deltas due to num2.
Check for early impossibility to avoid unnecessary computation.

Common Pitfalls or Variants

Common pitfalls

Failing to handle negative intermediate values when subtracting 2^i + num2.
Ignoring the full range of i up to 60, which can miss optimal paths.
Overlooking scenarios where no sequence of operations can reach zero.

Follow-up variants

Restrict i to a smaller range, forcing tighter enumeration constraints.
Change the fixed addition term num2 dynamically per operation.
Require tracking the exact sequence of operations rather than just the minimum count.

FAQ

What is the main strategy to solve Minimum Operations to Make the Integer Zero?

Use bit manipulation combined with BFS or enumeration to subtract powers of 2 plus num2 until num1 reaches zero or detect impossibility.

Why is considering i up to 60 necessary?

Because 2^60 covers the largest possible num1 within constraints, ensuring no optimal operation is missed.

Can negative num1 values occur during operations?

Yes, intermediate negative values are possible and must be tracked to avoid invalid sequences or missed solutions.

How does num2 affect the minimum operation count?

num2 shifts all subtractions by a fixed amount, altering reachable states and potentially making some sequences impossible.

Is this problem purely a bit manipulation task?

No, it combines bit manipulation with a brainteaser pattern and careful enumeration to find the minimum operations.

terminal

Solution

Solution 1: Enumeration

If we operate $k$ times, then the problem essentially becomes: determining whether $\textit{num1} - k \times \textit{num2}$ can be split into the sum of $k$ $2^i$s.

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class Solution:
    def makeTheIntegerZero(self, num1: int, num2: int) -> int:
        for k in count(1):
            x = num1 - k * num2
            if x < 0:
                break
            if x.bit_count() <= k <= x:
                return k
        return -1

Continue Practicing

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