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Maximum Elegance of a K-Length Subsequence

Number: 2894

Difficulty: Hard

Paid? No

Companies: N/A

Problem Description

Given a list of items where each item is represented as [profit, category] and an integer k, the goal is to select a subsequence of exactly k items (preserving their original order) such that the "elegance" is maximized. The elegance of a subsequence is defined as the sum of profits of the selected items plus the square of the number of distinct categories present in the subsequence.

For example, if the subsequence has a total profit of P and contains D distinct categories, its elegance is calculated as P + D². You need to determine the maximum elegance that can be achieved.

Key Insights

  • A greedy approach is used by first sorting the items in descending order of profit.
  • Initially pick the top k items to maximize profit, but this may include multiple items from the same category.
  • To increase the number of distinct categories (which adds D² to the elegance), try to replace duplicate-category items (using a min-heap to quickly get the smallest profit among duplicates) with items from categories not yet selected.
  • The trade-off is between decreasing the profit slightly (by replacing a low-profit duplicate) and increasing the bonus from having an additional unique category.
  • Sorting and heap operations ensure efficient swaps and profit updates.

Space and Time Complexity

Time Complexity: O(n log n) due to sorting and heap operations.
Space Complexity: O(n) for storing counts, the duplicate min-heap, and additional data structures.

Solution

The solution starts by sorting items in descending order of profit. Then, pick the first k items to maximize profit. Track the count of each category among the selected items. For items from categories that appear more than once, add their profit to a min-heap (this heap will help you quickly determine the candidate for replacement having the smallest profit among duplicates).

After this initial selection, traverse through the rest of the sorted items and whenever you find an item whose category hasn't been used yet, consider it as a candidate to increase the distinct category count. Replace a duplicate (extracted from the min-heap) with this candidate, updating the total profit and count of distinct categories accordingly. Each swap may improve the elegance by sacrificing a small profit loss for a potentially large increase thanks to the square term.

Data structures used include a min-heap (to track duplicate items for replacement) and a hash map/dictionary (to count frequency of categories). This combination ensures that the extra bonus from additional unique categories is maximized while affecting the profit as little as possible.

Code Solutions

# Python solution for Maximum Elegance of a K-Length Subsequence

import heapq

def maximum_elegance(items, k):
    # Sort items by profit in descending order
    items.sort(key=lambda x: x[0], reverse=True)
    
    total_profit = 0        # Sum of selected profits
    distinct_count = 0      # Count of distinct categories selected
    category_count = {}     # Frequency map for categories in the selection
    duplicate_heap = []     # Min-heap for duplicate items (profit values)
    
    # First pass: select first k items and track duplicates
    for profit, category in items[:k]:
        total_profit += profit
        if category in category_count:
            category_count[category] += 1
            # Push duplicate candidate into heap (profit value)
            heapq.heappush(duplicate_heap, profit)
        else:
            category_count[category] = 1
            distinct_count += 1
    
    # Calculate initial elegance
    max_elegance = total_profit + distinct_count * distinct_count
    
    # Try to upgrade the subsequence by replacing duplicates with new categories
    for profit, category in items[k:]:
        # If category is new, then we can try to replace one duplicate
        if category in category_count:
            continue  # already present, skip this candidate

        # Only perform replacement if we have any duplicate to swap out
        if duplicate_heap:
            # Remove the duplicate with the smallest profit
            removed_profit = heapq.heappop(duplicate_heap)
            total_profit = total_profit - removed_profit + profit
            # Update distinct category info
            category_count[category] = 1
            distinct_count += 1
            # Update maximum elegance after swap
            max_elegance = max(max_elegance, total_profit + distinct_count * distinct_count)
        else:
            # No duplicate available, cannot upgrade further
            break
    return max_elegance

# Example usage:
# items = [[3,2],[5,1],[10,1]]
# k = 2
# print(maximum_elegance(items, k))  # Output: 17
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