Threshold Majority Queries

The problem requires identifying the most frequent element in subarrays based on a threshold value. You must choose the element with the highest frequency or -1 if no element meets the threshold. The efficient solution involves using sqrt decomposition to optimize query processing.

Problem Statement

You are given an integer array nums of length n and an array queries, where each query queries[i] = [li, ri, thresholdi]. For each query, find the element in the subarray nums[li...ri] that appears at least thresholdi times. If such an element exists, return the one with the highest frequency; if multiple elements have the same frequency, return the smallest one. If no element satisfies the condition, return -1.

The challenge lies in efficiently processing multiple queries, where brute force methods could be too slow given the constraints. The optimal solution requires advanced techniques, such as sqrt decomposition, to balance time complexity with query requirements.

Examples

Example 1

Input: nums = [1,1,2,2,1,1], queries = [[0,5,4],[0,3,3],[2,3,2]]

Output: [1,-1,2]

Example 2

Input: nums = [3,2,3,2,3,2,3], queries = [[0,6,4],[1,5,2],[2,4,1],[3,3,1]]

Output: [3,2,3,2]

Constraints

• 1 <= nums.length == n <= 104
• 1 <= nums[i] <= 109
• 1 <= queries.length <= 5 * 104
• queries[i] = [li, ri, thresholdi]
• 0 <= li <= ri < n
• 1 <= thresholdi <= ri - li + 1

Solution Approach

Sqrt Decomposition

Break the array into blocks of size approximately sqrt(n), so that the queries can be processed efficiently. Sorting the queries by block number helps optimize the time spent on query processing, reducing redundant calculations for overlapping subarrays.

Frequency Count with Sliding Window

For each query, maintain a sliding window over the subarray and track the frequency of each element. This technique minimizes recalculations by reusing frequency data from previous queries, especially when combined with sqrt decomposition.

Efficient Query Sorting

Sort queries based on the left index divided by the block size and the right index. This sorting ensures that queries are processed in an order that minimizes changes to the sliding window, leading to better performance.

Complexity Analysis

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

The time complexity depends on the sqrt decomposition approach and query sorting. With sqrt decomposition, each query can be processed in approximately O(sqrt(n)) time, leading to a total time complexity of O((n + q) * sqrt(n)), where q is the number of queries. Space complexity is O(n) due to the storage of frequency counts and the array partitions.

What Interviewers Usually Probe

• Focus on how the candidate handles the query sorting and efficient sliding window techniques.
• Look for an understanding of advanced query optimization methods like sqrt decomposition.
• Check if the candidate can identify and address the potential pitfalls in query overlapping and frequency counting.

Common Pitfalls or Variants

Common pitfalls

• Failure to handle overlapping subarrays efficiently, leading to high time complexity.
• Incorrectly selecting the smallest element in case of a frequency tie.
• Not optimizing the solution with techniques like sqrt decomposition, resulting in poor performance on large inputs.

Follow-up variants

• Consider handling dynamic updates to the array, where elements can be changed between queries.
• Modify the problem to return the element with the highest frequency for all subarrays, not just those satisfying the threshold.
• Extend the problem to multiple thresholds for each query, requiring additional logic for comparison.

FAQ

What is the optimal solution approach for Threshold Majority Queries?

The optimal approach involves using sqrt decomposition and efficient query sorting to minimize time complexity when processing multiple queries.

How do I handle overlapping subarrays in Threshold Majority Queries?

Use a sliding window technique along with frequency counting to efficiently manage overlapping subarrays, especially when combined with sqrt decomposition.

What is sqrt decomposition in the context of this problem?

Sqrt decomposition breaks the array into blocks of size sqrt(n) to reduce redundant calculations when processing multiple queries, optimizing the overall performance.

What happens if no element meets the threshold in a query?

If no element appears at least thresholdi times, return -1 for that query.

Can the problem be solved without sorting queries?

While it's possible to solve without sorting, not sorting the queries can lead to inefficient query handling, especially with large inputs.