#3012

Medium

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Minimize Length of Array Using Operations

Minimize the length of an integer array through a series of operations, using a greedy approach with modular arithmetic.

array math greedy number theory

Problem Statement

You are given a 0-indexed integer array nums containing positive integers. Your task is to minimize the length of nums by performing the following operations any number of times (including zero):

Select two indices i, j (i != j) in the array, and insert nums[i] % nums[j] at the end of the array. Then, delete the elements at indices i and j. Return the minimum length of nums after performing the operation any number of times.

Examples

Example 1

Input: nums = [1,4,3,1]

Output: 1

One way to minimize the length of the array is as follows: Operation 1: Select indices 2 and 1, insert nums[2] % nums[1] at the end and it becomes [1,4,3,1,3], then delete elements at indices 2 and 1. nums becomes [1,1,3]. Operation 2: Select indices 1 and 2, insert nums[1] % nums[2] at the end and it becomes [1,1,3,1], then delete elements at indices 1 and 2. nums becomes [1,1]. Operation 3: Select indices 1 and 0, insert nums[1] % nums[0] at the end and it becomes [1,1,0], then delete elements at indices 1 and 0. nums becomes [0]. The length of nums cannot be reduced further. Hence, the answer is 1. It can be shown that 1 is the minimum achievable length.

Example 2

Input: nums = [5,5,5,10,5]

Output: 2

One way to minimize the length of the array is as follows: Operation 1: Select indices 0 and 3, insert nums[0] % nums[3] at the end and it becomes [5,5,5,10,5,5], then delete elements at indices 0 and 3. nums becomes [5,5,5,5]. Operation 2: Select indices 2 and 3, insert nums[2] % nums[3] at the end and it becomes [5,5,5,5,0], then delete elements at indices 2 and 3. nums becomes [5,5,0]. Operation 3: Select indices 0 and 1, insert nums[0] % nums[1] at the end and it becomes [5,5,0,0], then delete elements at indices 0 and 1. nums becomes [0,0]. The length of nums cannot be reduced further. Hence, the answer is 2. It can be shown that 2 is the minimum achievable length.

Example 3

Input: nums = [2,3,4]

Output: 1

One way to minimize the length of the array is as follows: Operation 1: Select indices 1 and 2, insert nums[1] % nums[2] at the end and it becomes [2,3,4,3], then delete elements at indices 1 and 2. nums becomes [2,3]. Operation 2: Select indices 1 and 0, insert nums[1] % nums[0] at the end and it becomes [2,3,1], then delete elements at indices 1 and 0. nums becomes [1]. The length of nums cannot be reduced further. Hence, the answer is 1. It can be shown that 1 is the minimum achievable length.

Constraints

1 <= nums.length <= 105
1 <= nums[i] <= 109

Solution Approach

Greedy Approach with Modulus

To minimize the length of the array, apply a greedy approach where each operation selects the best possible pair for modulus to maximize reduction. The focus is to continually minimize the array size by choosing the most effective pairs to reduce the length efficiently.

Invariant Validation

The key insight for solving this problem is recognizing that after each operation, the array length is reduced only if valid operations are chosen. This validation requires checking that the modulus operation effectively reduces the length and doesn't result in redundant operations that do not reduce the size further.

Edge Case Handling

Consider edge cases where the array is already at its minimal possible length, such as when all elements are identical or when the modulus operation leads to minimal reductions quickly. Efficiently handling these cases ensures that unnecessary operations are avoided.

Complexity Analysis

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

The time and space complexity depend on the specific approach chosen for reducing the array. A greedy strategy generally leads to an efficient solution but requires careful handling of modulus operations and array management to ensure minimal computational overhead.

What Interviewers Usually Probe

Look for a solid understanding of greedy strategies and their implementation in reducing array length.
Evaluate how well the candidate handles edge cases, such as arrays with repeated elements or when the modulus operation does not significantly reduce the array.
Assess the candidate's ability to justify the sequence of operations and whether they can explain the rationale behind choosing specific pairs for modulus.

Common Pitfalls or Variants

Common pitfalls

Failing to account for the most optimal pairings that reduce the array length effectively.
Overlooking edge cases where the array cannot be minimized further due to redundant modulus operations.
Misunderstanding the operation's effect on the array, especially when it seems like the array length will not decrease significantly.

Follow-up variants

Allow only a limited number of operations and ask for the minimal achievable length after those operations.
Change the operation to insert nums[i] * nums[j] instead of modulus and test how that affects the minimal length.
Limit the number of times any element can be involved in an operation and ask for the minimum array length under that constraint.

FAQ

What is the main approach to solving the 'Minimize Length of Array Using Operations' problem?

The problem is solved using a greedy approach, choosing optimal pairs of elements for the modulus operation to reduce the array size most effectively.

How do you handle edge cases in the Minimize Length of Array problem?

Edge cases include arrays where no further reduction is possible or where all elements are the same. These cases require careful handling to avoid unnecessary operations.

What are the time and space complexities of this problem?

The time and space complexities depend on the approach chosen, with a greedy method typically leading to efficient solutions, though exact complexities depend on implementation details.

Can we solve the 'Minimize Length of Array Using Operations' problem without using the modulus operation?

The modulus operation is key to the problem, but alternate operations can be explored as variants of the problem, such as multiplication or addition, to test different behaviors.

How does GhostInterview help with problems like Minimize Length of Array Using Operations?

GhostInterview provides detailed feedback and practice problems, ensuring you can master the greedy approach and handle all edge cases efficiently in problems like this.

terminal

Solution

Solution 1: Case Discussion

Let's denote the smallest element in the array $nums$ as $mi$.

class Solution:
    def minimumArrayLength(self, nums: List[int]) -> int:
        mi = min(nums)
        if any(x % mi for x in nums):
            return 1
        return (nums.count(mi) + 1) // 2

Continue Practicing

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