You are given two integers num1 and num2.
In one operation, you can choose integer i in the range [0, 60] and subtract 2i + num2 from num1.
Return the integer denoting the minimum number of operations needed to make num1 equal to 0.
If it is impossible to make num1 equal to 0, return -1.
Example 1:
Input: num1 = 3, num2 = -2 Output: 3 Explanation: We can make 3 equal to 0 with the following operations: - We choose i = 2 and substract 22 + (-2) from 3, 3 - (4 + (-2)) = 1. - We choose i = 2 and substract 22 + (-2) from 1, 1 - (4 + (-2)) = -1. - We choose i = 0 and substract 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 Explanation: It can be proven, that it is impossible to make 5 equal to 0 with the given operation.
Constraints:
1 <= num1 <= 109-109 <= num2 <= 109This approach involves using a greedy method to subtract the largest possible power of 2 plus num2 from num1. At each step, we try to maximize the decrease in num1, aiming to reach exactly zero. If it becomes impossible to subtract without going below zero or exceeding the potential operations limit, we return -1.
The function iteratively subtracts the largest valid (2^i + num2) from num1. It counts the operations until num1 becomes zero. If num1 becomes negative and no valid i can be found, it returns -1.
Java
Time Complexity: O(1) due to loop limits.
Space Complexity: O(1), only a few variables are used.
This approach leverages dynamic programming to store intermediate results of subproblems, reducing redundant calculations. We systematically explore the impact of subtracting each (2^i + num2) on num1, storing the minimum steps required to reach zero. This method is effective but requires more memory management.
This DP solution initializes a DP array where each index represents the minimum operations needed for that particular num1 value. Each valid operation updates the DP based on historical values.
JavaScript
Time Complexity: O(num1 * 61), for each num1 value, checks up to 61 possibilities for i.
Space Complexity: O(num1), required to store DP results for each subproblem.
| Approach | Complexity |
|---|---|
| Approach 1: Greedy Subtraction with Powers of 2 | Time Complexity: O(1) due to loop limits. |
| Approach 2: Dynamic Programming (DP) for Minimization | Time Complexity: O(num1 * 61), for each num1 value, checks up to 61 possibilities for i. |
Minimum Operations to Reduce X to Zero - Leetcode 1658 - Python • NeetCodeIO • 16,975 views views
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