Minimize Length of Array Using Operations — LeetCode #3012 (Medium)
Tags: Array, Math, Greedy, Number Theory
Related patterns: Greedy algorithms, Mathematical properties of numbers (modulus)
Brute Force approach
Time complexity: O(n²). Space complexity: O(1).
The brute-force approach involves trying all possible pairs of indices in the array and performing the modulus operation. This will help us understand how the array can be reduced step by step.
The time complexity is O(n²) because we check all pairs of indices in a nested loop. The space complexity is O(1) since we are modifying the array in place without using additional data structures.
- Step 1: Iterate through all pairs of indices (i, j) where i != j.
- Step 2: For each pair, calculate nums[i] % nums[j] and append it to the end of the array.
- Step 3: Remove the elements at indices i and j from the array.
- Step 4: Repeat until no valid pairs can be selected.
Initial nums: [1, 4, 3, 1]
Step 1: Select (1, 4) -> Append 1 % 4 = 1 -> New nums: [1, 4, 3, 1, 1]
Step 2: Select (4, 3) -> Append 4 % 3 = 1 -> New nums: [1, 3, 1, 1]
Step 3: Select (3, 1) -> Append 3 % 1 = 0 -> New nums: [1, 1, 0]
Step 4: Select (1, 1) -> Append 1 % 1 = 0 -> New nums: [0]
Final length: 1
Optimal Solution approach
Time complexity: O(n). Space complexity: O(1).
The optimal solution leverages the fact that the minimum length of the array can be determined by the smallest number in the array. If we can reduce all other numbers to this minimum, we can achieve the smallest possible array length.
The time complexity is O(n) because we traverse the array twice: once to find the minimum and once to count its occurrences. The space complexity is O(1) as we use only a few variables.
- Step 1: Find the minimum value in the array, denoted as x.
- Step 2: Count how many times x appears in the array.
- Step 3: The minimum length of the array will be the count of x, since all other numbers can be reduced to x.
Initial nums: [1, 4, 3, 1]
Step 1: Find min_value = 1
Step 2: Count occurrences of 1 = 2
Final length: 2
Key Insights
- The minimum length of the array is determined by the smallest number present.
- All other numbers can be reduced to the minimum value through the modulus operation.
Common Mistakes
- Not considering the case where all numbers are the same, leading to unnecessary operations.
- Failing to optimize the approach after understanding the brute force method.
Interview Tips
- Start with a brute-force solution to demonstrate understanding, but quickly pivot to optimization.
- Explain your thought process clearly, especially when identifying patterns in the problem.
- Practice similar problems to recognize when to apply the optimal strategy.
