Problem Description

Given a positive integer n, the task is to transform this number to 1. There are two types of operations that you are allowed to perform:

1. If n is even, you can divide it by 2 (n = n / 2).
2. If n is odd, you can either increase it by 1 (n = n + 1) or decrease it by 1 (n = n - 1).

Each operation counts as a single step, and the goal is to find the minimum number of steps required to reduce n to 1.

Intuition

The intuition behind the solution involves understanding parity (odd or even nature) of n and using bitwise operations for efficiency. For an even n, the decision is straightforward: divide by 2. This halving operation is efficient because it significantly reduces n.

For an odd n, the decision to subtract or add 1 relies on considering the bits of n. Specifically, we observe the following patterns:

• When the least significant two bits of n are 01 (meaning the next division will yield an even number), it is often better to subtract 1 than to add 1, because we want to get to an even number to divide by 2 in the next step.
• However, there is a special case when n is 3; the best operation is to subtract 1 twice to reach 1 rather than add 1 and then divide twice.
• If the least significant two bits are 11, adding 1 converts n to an even number, which can then be halved. This is except for the case when n is 3, as mentioned above.

The approach makes use of bitwise operations:

• We use n & 1 to check if n is even or odd (0 means even and 1 means odd).
• The right shift n >>= 1 effectively divides n by 2.
• We check if the last two bits are 11 by n & 3, and if the result is 3, we know adding 1 is a favorable operation.

The solution employs a loop that iteratively applies the optimal operation until n becomes 1, and a counter ans keeps track of the number of operations performed.

Learn more about Greedy, Memoization and Dynamic Programming patterns.

Solution Approach

The solution provided uses a greedy approach to minimize the number of operations needed to reduce n to 1 with the two allowed operations, substituting and dividing.

Here's a walkthrough of the algorithm using the Reference Solution Approach:

1. Initialize ans to 0. This variable will keep track of the total number of operations performed.

2. Enter a loop that continues until n is equal to 1.

3. Inside the loop, check if n is even by using the bitwise AND operation n & 1. If the result is 0, it means n is even.

4. If n is even, we apply the bitwise right shift operation n >>= 1 to n. This operation is equivalent to dividing n by 2.

5. If n is odd, we have to decide whether to increment or decrement n. To make this decision:

   • Check if n is not equal to 3 and if the last two bits of n are 11 by applying the bitwise AND operation n & 3. If the result is 3, it's better to increment n by 1, because doing so will lead to an even number after this operation.
   • In all other cases (when n is odd, and either n is 3 or the last two bits of n are not 11), it's better to decrement n by 1.

6. After each operation (incrementing, decrementing, or dividing), increment ans by 1 to count the operation.

7. Once n is reduced to 1, exit the loop, and return ans as the total number of operations performed.

The solution applies mathematical operations and understands binary representations to guide the decision-making process. No additional data structures are used, and the algorithm runs in a time complexity that is logarithmic to the value of n, specifically O(log n), because each division by two halves the problem size.

Example Walkthrough

Let's illustrate the solution approach with a small example using n = 15.

1. Initialize ans to 0. At the start, no steps have been taken yet, so ans is 0.

2. Because n is not 1, we enter the loop.

3. As n is odd (15 & 1 is 1), we proceed to check if we should decrement or increment n.

4. Since n equals 15, which is not 3, and the last two bits of n are 11 (15 & 3 gives 3), we increment n by 1, resulting in n = 16. Now ans = 1.

5. Now n is even (16 & 1 is 0), so we right shift n, effectively dividing it by 2. n becomes 8 (n >>= 1). Now ans = 2.

6. n is still even, so we keep right shifting. n becomes 4. Now ans = 3.

7. Continue with the even case; n becomes 2. Now ans = 4.

8. Finally, n is 2, which is even again, and after one more shift, n becomes 1. Now ans = 5.

9. Since n is now equal to 1, we break out of the loop.

10. The value of ans is 5, representing the minimum number of steps needed to transform n from 15 to 1.

The solution successfully applies the steps from the Reference Solution Approach, selecting operations that gradually reduce n to 1 with optimal efficiency, ending with an answer of 5 steps for this example.

Solution Implementation

Time and Space Complexity

The given code implements an algorithm to determine the minimum number of operations to reduce a number n to 1 by either dividing it by 2 if it's even or subtracting 1 or adding 1 if it's odd, which converges to division by 2 when possible.

Time Complexity:

The time complexity of the algorithm is determined by the number of operations needed to reduce n to 1. In the worst case scenario, for each bit in the binary representation of n, the loop might be executed twice (once for subtraction/addition to make it even, and once for the division by 2). However, the addition operation in n += 1 can lead to the removal of multiple trailing '1's in binary, which means it's possible to skip several steps at once. Therefore, the time complexity is O(log n) in the average case, and in the worst case, it's slightly more than O(log n) due to the possible addition steps that can reduce the number of 1's in the binary representation.

Space Complexity:

The space complexity of the code is O(1), as the algorithm uses a fixed amount of space - the variable ans to keep count of the operations, and n is modified in place without using additional memory that scales with the input size.

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