Subsequences with a Unique Middle Mode I

The problem requires finding the number of subsequences of size 5 with a unique middle mode from an integer array. The mode is the element appearing the most in the subsequence. A hash table can be utilized to efficiently track and count subsequences that meet the criteria, and the result is returned modulo 10^9 + 7.

Problem Statement

Given an integer array nums, you need to find how many subsequences of size 5 have a unique middle mode. The middle mode of a sequence is the element that appears the most times in that sequence. If the frequency of the middle mode is tied with other numbers in the subsequence, it does not qualify as a unique middle mode.

To solve this, the subsequences are of size 5, and the mode must appear more frequently than any other element in the subsequence. The output should be the count of such subsequences modulo 10^9 + 7.

Examples

Example 1

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

Output: 6

[1, 1, 1, 1, 1] is the only subsequence of size 5 that can be formed, and it has a unique middle mode of 1. This subsequence can be formed in 6 different ways, so the output is 6.

Example 2

Input: nums = [1,2,2,3,3,4]

Output: 4

[1, 2, 2, 3, 4] and [1, 2, 3, 3, 4] each have a unique middle mode because the number at index 2 has the greatest frequency in the subsequence. [1, 2, 2, 3, 3] does not have a unique middle mode because 2 and 3 appear twice.

Example 3

Input: nums = [0,1,2,3,4,5,6,7,8]

Output: 0

There is no subsequence of length 5 with a unique middle mode.

Constraints

• 5 <= nums.length <= 1000
• -109 <= nums[i] <= 109

Solution Approach

Array Scanning with Hash Table

By scanning through the array, we can use a hash table to store frequencies of each element. For each index in the array, we identify subsequences that contain the current number as the unique middle mode.

Combinatorial Counting

Once the frequency of each element is known, we calculate the number of valid subsequences of size 5 in which that element appears as the mode. The problem then reduces to counting valid combinations where the element with the highest frequency occupies the middle position.

Modulo Arithmetic

Since the answer can be large, each count of valid subsequences is taken modulo 10^9 + 7 to ensure that the result remains within the problem's constraints.

Complexity Analysis

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

The time and space complexity depend on the approach used to track frequencies and calculate valid subsequences. The complexity can be reduced by efficiently scanning the array and performing combinatorial counting based on precomputed frequencies.

What Interviewers Usually Probe

• Strong understanding of hash tables for frequency counting and array scanning.
• Ability to reduce a complex combinatorics problem into manageable subproblems.
• Comfort with handling large numbers and modulo operations for result constraints.

Common Pitfalls or Variants

Common pitfalls

• Not properly accounting for tied modes where no element appears more times than others.
• Misunderstanding the problem constraints and failing to handle large subsequences.
• Incorrectly calculating combinations when considering the middle mode element.

Follow-up variants

• Generalizing the problem to subsequences of other sizes.
• Handling cases where the array length is much larger than 5.
• Optimizing the algorithm to avoid unnecessary recalculations of frequencies.

FAQ

What is the primary pattern for solving this problem?

The primary pattern involves array scanning combined with hash table frequency lookup to efficiently count subsequences.

How do we handle large numbers in this problem?

The result should be calculated modulo 10^9 + 7 to prevent overflow and ensure the output fits within the problem constraints.

What makes the middle mode unique in the subsequence?

A unique middle mode means that the most frequent element in the subsequence occurs more times than any other element.

Can we use dynamic programming for this problem?

Dynamic programming is not necessary for this problem. The main approach relies on hash tables and combinatorial counting.

What is the difficulty level of the 'Subsequences with a Unique Middle Mode I' problem?

This problem is considered 'Hard' due to its combinatorial nature and the requirement to handle large arrays and numbers efficiently.