#2813

Hard

auto_awesomeArray scanning plus hash lookup

LeetCode Problem Workspace

Maximum Elegance of a K-Length Subsequence

Maximize elegance of a k-length subsequence from a list of items with profits and 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

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

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.

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.

terminal

Solution

Solution 1: Greedy

We can sort all items by profit from large to small. First choose the first $k$ items and calculate the total profit $tot$. Use a hash table $vis$ to record the categories of these $k$ items, use a stack $dup$ to record the profits of the repeated categories in order, and use a variable $ans$ to record the current maximum elegance.

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class Solution:
    def findMaximumElegance(self, items: List[List[int]], k: int) -> int:
        items.sort(key=lambda x: -x[0])
        tot = 0
        vis = set()
        dup = []
        for p, c in items[:k]:
            tot += p
            if c not in vis:
                vis.add(c)
            else:
                dup.append(p)
        ans = tot + len(vis) ** 2
        for p, c in items[k:]:
            if c in vis or not dup:
                continue
            vis.add(c)
            tot += p - dup.pop()
            ans = max(ans, tot + len(vis) ** 2)
        return ans

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