Smallest K Integers V in O(n log maxVal) Time Complexity using Java

Input: nums = [5, 9, 3, 6, 2, 1, 3, 2, 7, 5], k = 4
Output: [1, 2, 2, 3]
Explanation: Smallest number is 1, 2nd smallest is 2, 
            3rd smallest is 2, 4th smallest is 3

The result can be in any order, [2, 1, 3, 2] is also a correct answer.

Understanding the Problem

The core challenge of this problem is to find the smallest k integers from an array of positive integers. This problem is significant in scenarios where we need to filter out the smallest elements from a large dataset, such as in data analysis or competitive programming.

Potential pitfalls include misunderstanding the requirement to return the smallest k values in any order and not optimizing the solution to meet the time complexity constraint.

Approach

To solve this problem, we can consider several approaches:

Naive Solution

A naive solution would be to sort the array and then return the first k elements. However, this approach has a time complexity of O(n log n), which is not optimal for this problem.

Optimized Solution

We can use a min-heap (priority queue) to efficiently find the smallest k elements. The min-heap allows us to keep track of the smallest elements as we iterate through the array. This approach has a time complexity of O(n log k), which is better but still not optimal for the given constraints.

Optimal Solution

To achieve the required O(n log maxVal) time complexity, we can use a counting sort-like approach. This approach leverages the fact that the array contains positive integers, and we can use an array to count the occurrences of each integer. This method ensures we meet the time complexity constraint and uses O(1) extra space.

Algorithm

Here is a step-by-step breakdown of the optimal algorithm:

1. Find the maximum value in the array to determine the size of the counting array.
2. Create a counting array to store the frequency of each integer in the input array.
3. Iterate through the input array and populate the counting array.
4. Iterate through the counting array to collect the smallest k integers.

Code Implementation

import java.util.*;

public class SmallestKIntegers {
    public static List<Integer> findSmallestK(int[] nums, int k) {
        // Step 1: Find the maximum value in the array
        int maxVal = Arrays.stream(nums).max().getAsInt();
        
        // Step 2: Create a counting array
        int[] count = new int[maxVal + 1];
        
        // Step 3: Populate the counting array
        for (int num : nums) {
            count[num]++;
        }
        
        // Step 4: Collect the smallest k integers
        List<Integer> result = new ArrayList<>();
        for (int i = 0; i < count.length && result.size() < k; i++) {
            while (count[i] > 0 && result.size() < k) {
                result.add(i);
                count[i]--;
            }
        }
        
        return result;
    }

    public static void main(String[] args) {
        int[] nums = {5, 9, 3, 6, 2, 1, 3, 2, 7, 5};
        int k = 4;
        List<Integer> smallestK = findSmallestK(nums, k);
        System.out.println(smallestK);
    }
}

Complexity Analysis

The time complexity of the optimal solution is O(n log maxVal) because we iterate through the input array to populate the counting array and then iterate through the counting array to collect the smallest k integers. The space complexity is O(1) extra space since the counting array size depends on the maximum value in the input array, not the input size.

Edge Cases

Potential edge cases include:

• k is greater than the length of the array: The algorithm should handle this by returning all elements in the array.
• All elements in the array are the same: The algorithm should correctly return k instances of that element.

Testing

To test the solution comprehensively, consider the following test cases:

• Simple cases with small arrays.
• Arrays with duplicate elements.
• Arrays where k is larger than the array length.
• Arrays with all elements being the same.

Thinking and Problem-Solving Tips

When approaching such problems, consider the following tips:

• Understand the problem constraints and requirements thoroughly.
• Think about different approaches and their time and space complexities.
• Optimize your solution step-by-step, starting from a naive approach.
• Practice solving similar problems to improve your problem-solving skills.

Conclusion

In this blog post, we discussed how to find the smallest k integers from an array of positive integers. We explored different approaches, including a naive solution and an optimal solution using a counting sort-like approach. We also provided a detailed algorithm, code implementation, complexity analysis, and testing strategies. Understanding and solving such problems is crucial for improving your algorithmic thinking and coding skills.

Additional Resources

For further reading and practice, consider the following resources:

• LeetCode - Practice coding problems and challenges.
• GeeksforGeeks - Tutorials and articles on algorithms and data structures.
• Coursera - Online courses on algorithms and computer science.