Problem Description

Given an integer num, the task is to find out how many steps it would take to reduce this number to zero. The rule for each step of the reduction process is simple:

• If num is even, divide it by 2.
• If num is odd, subtract 1 from it.

The process is repeated until num becomes zero and we need to count and return the total number of steps taken.

Intuition

The intuition behind the solution is to use a loop that keeps running until the number becomes zero. Inside the loop, we check if the current number is odd or even. The bitwise AND operation num & 1 is a quick way to check if a number is odd (1) or even (0). If the number is odd (last bit is 1), we subtract 1 from it. If it is even (last bit is 0), we shift the bits to the right by one position using num >>= 1, which effectively divides the number by 2. After each operation, we increment a counter ans by 1 to keep track of the number of steps taken. Once the loop ends (when num is zero), we return the counter as our result.

So, the intuition leverages simple bitwise operations for efficiency, and a while loop to iteratively reduce the number to zero, counting the steps along the way.

Learn more about Math patterns.

Solution Approach

The solution is implemented using a straightforward approach that doesn't rely on any complex algorithms or advanced data structures. The primary tools used here are bitwise operations and a loop. Here's how the implementation works:

1. We define a function numberOfSteps that takes an integer num as an input and initializes a counter ans to 0. This counter will hold the total number of steps required to reduce num to zero.

2. The solution employs a while loop that runs as long as num is not zero. Within the loop, we perform one of the two operations in each iteration:

   • If num is odd (which we check using the bitwise AND operation num & 1), we subtract 1 from it. This step ensures that an odd number becomes even, and thus can be halved in the next step.
   • If num is even, we use the right shift bitwise operation num >>= 1 to divide num by 2. Bit shifting to the right by one position is synonymous with integer division by two for non-negative integers.

3. After each operation inside the loop, we increment the counter ans by 1. This accounts for the step we've just performed.

4. Once num is zero, the loop exits, and the function returns the ans counter, which by then has accumulated the total number of steps taken to reduce num to zero.

The solution uses no auxiliary data structures; it only manipulates the given num and maintains a simple integer counter. The approach is efficient both in terms of time and space complexity, as it operates in linear time with respect to the number of bits in num and only uses a constant amount of additional space for the counter.

Example Walkthrough

Let's walk through a small example to clearly illustrate the solution approach using the integer 14 as our num.

1. We call the function numberOfSteps with num = 14.

2. Since the while loop condition (num != 0) holds true, we enter the while loop.

3. We check if 14 is odd or even using num & 1. Since 14 & 1 equals 0, 14 is even.

4. We divide 14 by 2 using num >>= 1, which results in 7. Now, num = 7. We increment ans to 1.

5. Since 7 is odd (7 & 1 equals 1), we subtract 1 from it resulting in 6. Now, num = 6. We increment ans to 2.

6. 6 is even (6 & 1 equals 0), so we divide it by 2 using num >>= 1, resulting in 3. Now, num = 3. We increment ans to 3.

7. 3 is odd (3 & 1 equals 1), so we subtract 1 from it, resulting in 2. Now, num = 2. We increment ans to 4.

8. 2 is even (2 & 1 equals 0), so we divide it by 2 using num >>= 1, yielding 1. Now, num = 1. We increment ans to 5.

9. 1 is odd (1 & 1 equals 1), so we subtract 1 from it, resulting in 0. Now, num = 0. We increment ans to 6.

10. The while loop condition num != 0 no longer holds true, so we exit the loop.

11. Finally, numberOfSteps returns ans which is 6. Therefore, it takes 6 steps to reduce the number 14 to zero.

Here we took six steps to reduce the number from 14 to 0. The approach involved alternating bitwise operations and arithmetic subtractions until reaching zero, tracking each operation as a step.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(log n). This is because each operation either decreases the number by one if it is odd or divides the number by two if it is even. In the worst-case scenario, for a number with all bits set to 1, it will take at most 2 * log2(n) steps to reduce the number to 0, since each bit position would need a subtraction and a right shift to clear it.

The space complexity of the code is O(1) since the algorithm uses a fixed amount of space. The extra space used by the algorithm does not grow with the input size n. Only a constant number of variables (num and ans) are used regardless of the size of the input.

Learn more about how to find time and space complexity quickly using problem constraints.