How to Solve Maximum Elegance of a K-Length Subsequence

This problem asks you to select a subsequence of length k from an array of items, each with a profit and category. Elegance is defined as the sum of profits and the square of distinct categories. The goal is to maximize the elegance of the subsequence by carefully selecting items with the highest profit while considering the number of distinct categories.

Problem Statement

You are given a 2D integer array items of length n and an integer k. Each item is a pair [profit, category], where profit represents the profit of an item and category represents its associated category.

Your task is to select a subsequence of exactly k items to maximize the elegance, defined as the sum of all profits in the subsequence plus the square of the number of distinct categories in the subsequence.

Examples

Example 1

Input: items = [[3,2],[5,1],[10,1]], k = 2

Output: 17

In this example, we have to select a subsequence of size 2. We can select items[0] = [3,2] and items[2] = [10,1]. The total profit in this subsequence is 3 + 10 = 13, and the subsequence contains 2 distinct categories [2,1]. Hence, the elegance is 13 + 22 = 17, and we can show that it is the maximum achievable elegance.

Example 2

Input: items = [[3,1],[3,1],[2,2],[5,3]], k = 3

Output: 19

In this example, we have to select a subsequence of size 3. We can select items[0] = [3,1], items[2] = [2,2], and items[3] = [5,3]. The total profit in this subsequence is 3 + 2 + 5 = 10, and the subsequence contains 3 distinct categories [1,2,3]. Hence, the elegance is 10 + 32 = 19, and we can show that it is the maximum achievable elegance.

Example 3

Input: items = [[1,1],[2,1],[3,1]], k = 3

Output: 7

In this example, we have to select a subsequence of size 3. We should select all the items. The total profit will be 1 + 2 + 3 = 6, and the subsequence contains 1 distinct category [1]. Hence, the maximum elegance is 6 + 12 = 7.

Constraints

• 1 <= items.length == n <= 105
• items[i].length == 2
• items[i][0] == profiti
• items[i][1] == categoryi
• 1 <= profiti <= 109
• 1 <= categoryi <= n
• 1 <= k <= n

Solution Approach

Array Scanning and Hash Lookup

Scan the items array and use a hash table to track the distinct categories. Select items with higher profits while balancing the category count. This approach ensures that the subsequence has the maximum possible profit and distinct categories.

Greedy Algorithm

A greedy approach helps prioritize items that contribute to both the profit and the distinct category count. Start by selecting the most profitable items, and then evaluate their categories to ensure distinctiveness in the subsequence.

Priority Queue Optimization

Utilize a max-heap or priority queue to manage the selection of items based on their profit. This helps efficiently pick the k items with the highest profits while ensuring that the subsequence has distinct categories.

Complexity Analysis

The time complexity of the solution depends on the specific approach chosen. If a heap is used for optimization, the time complexity is O(n log k). Hashing the categories adds additional O(n) complexity, but the total remains efficient for large inputs.

What Interviewers Usually Probe

• Ability to recognize that the problem combines both profit maximization and distinct category counting.
• Understanding the trade-off between choosing high-profit items and ensuring category diversity.
• Proficiency in greedy algorithms and data structure usage, such as priority queues or hash tables.

Common Pitfalls or Variants

Common pitfalls

• Failing to properly balance profit maximization with distinct category counting.
• Not leveraging efficient data structures like heaps and hash tables, leading to unnecessary recomputation.
• Selecting items without considering the impact of the category count, leading to suboptimal elegance.

Follow-up variants

• Minimize elegance instead of maximizing it.
• Increase the size of the subsequence beyond k to test scalability.
• Change the problem constraints to allow duplicate categories in the subsequence.

How GhostInterview Helps

• GhostInterview provides hints on using efficient algorithms to maximize subsequence elegance by focusing on profit and category management.
• The platform guides you through using greedy techniques and data structures like heaps, showing how to make optimal decisions at each step.
• By analyzing similar problems, GhostInterview helps you improve your ability to handle optimization tasks involving both profit and constraints like category diversity.

Topic Pages

• Array
• Hash Table
• Stack
• Greedy
• Sorting
• Heap (Priority Queue)

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FAQ

What is the pattern for solving the Maximum Elegance of a K-Length Subsequence problem?

The problem relies on array scanning combined with hash lookup to manage category counting while maximizing the profit of selected items.

How does the greedy approach work in this problem?

The greedy approach helps prioritize items that maximize both the profit and the distinct category count, ensuring the elegance is maximized.

What role does a priority queue play in solving this problem?

A priority queue (max-heap) efficiently selects the top k profitable items, considering the distinct category constraint.

What happens if I select items without considering categories?

Failing to consider categories can lead to a subsequence with fewer distinct categories, reducing the overall elegance.

Can this problem be solved without using a hash table?

Using a hash table is crucial for efficiently counting distinct categories, but other methods like sorting or brute-force selection may work with higher time complexity.