Problem Description

In this problem, there is a set of n items, each with an associated value and label. Two arrays values and labels represent these, where values[i] is the value and labels[i] is the label for the i-th item. We are given two additional integers, numWanted which represents the maximum number of items to choose, and useLimit which represents the maximum number of items with the same label we are allowed to choose.

The goal is to find a subset s of these items that maximizes the total value (or "score") while adhering to the constraints:

1. The size of s is less than or equal to numWanted.
2. No more than useLimit items in s can have the same label.

The desired output is the maximum score that can be achieved following these rules.

Intuition

The intuitive approach to solving this problem is to maximize the value while respecting the constraints given. To start, since we want the maximum value, we should consider the items with the highest values first. Hence, we sort the items by value in descending order. Once sorted, a simple strategy can be employed:

• Iterate over the list of items in order of descending value.
• Keep a count of how many items with each label have been added to our subset s.
• Add the value of the current item to our score if the count of items with the same label is less than the useLimit.
• Increment the count of items chosen.
• If the number of items chosen reaches numWanted, we've found the best possible subset, and we can stop and return the score.

This approach ensures that we are always adding the item with the highest available value while respecting the useLimit condition for labels and not exceeding the desired subset size numWanted.

Learn more about Greedy and Sorting patterns.

Solution Approach

The solution approach relies on a greedy algorithm that selects items with higher values first, while ensuring that it does not select more than useLimit items with the same label. To accomplish this effectively, the following steps are taken:

• Sorting: We need to sort the items by their values in descending order. To do so, we use Python's built-in sorted function, pairing each value with its corresponding label using zip(values, labels). The reverse=True parameter is used to sort the pairs in descending order of value.

• Counting: To keep track of the number of items with the same label selected, we use a Counter from Python's collections module. This data structure allows easy increments and tracking of counts corresponding to each label.

• Selection Loop: Iterate through the sorted pairs of (value, label). For each pair, check if the current label's count is less than the useLimit using the previously initialized Counter.

• If the label's count is within the limit, we proceed to include the item's value in our current subset score ans and increment the count for that label in the Counter.

• We also keep a tally of the total number of items selected in num. When num reaches numWanted, we have selected the maximum allowed number of items, and we break out of the loop since we cannot add any more items to the subset.

• Finally, after the loop (either once we've reached numWanted or gone through all items), we return ans as the maximum score obtained from the selected subset of items.

The combination of sorting, counting, and a selection loop makes this greedy algorithm efficient and ensures that at each step, the best choice is made towards achieving the optimal solution (maximizing the subset score under the given constraints).

Mathematically, if we denote v as the value and l as the label of the current item being considered, and cnt as the Counter for labels, the pseudo-code can be described as follows:

sort items by value in descending order
initialize ans to 0, num to 0, and cnt to Counter()
for each (v, l) in sorted items:
    if cnt[l] < useLimit:
        cnt[l] += 1
        num += 1
        ans += v
        if num == numWanted:
            break
return ans

This implementation ensures that our greedy algorithm selects items in a way that maximizes the score while conforming to the constraints of numWanted and useLimit.

Example Walkthrough

Consider the following small example to illustrate the solution approach:

Suppose we have the following input:

• values = [5, 4, 3, 2, 1]
• labels = [1, 1, 2, 2, 3]
• numWanted = 3
• useLimit = 1

Following the solution approach, we would execute these steps:

1. Sort the items by value in descending order:

   We pair each value with its corresponding label and sort:

   Before sorting:

   • values: [5, 4, 3, 2, 1]
   • labels: [1, 1, 2, 2, 3]

   After sorting:

   • sorted_pairs: [(5, 1), (4, 1), (3, 2), (2, 2), (1, 3)]

2. Initialize the counter and other variables:

   We start with an empty Counter to track the labels, ans as 0 (our cumulative value), and num as 0 (the number of items in our subset):

   • ans (max score): 0
   • num (number of items selected): 0
   • cnt (the Counter for labels): {}

3. Selection Loop:

   We iterate over the sorted pairs and apply the constraints:

   a) (5, 1): Count for label 1 is 0, which is less than useLimit. We select this item.

   • ans becomes 5 (0 + 5)
   • num becomes 1 (0 + 1)
   • cnt becomes {1: 1}

   b) (4, 1): Count for label 1 is currently 1, equal to useLimit. We cannot select this item.

   c) (3, 2): Count for label 2 is 0, which is less than useLimit. We select this item.

   • ans becomes 8 (5 + 3)
   • num becomes 2 (1 + 1)
   • cnt becomes {1: 1, 2: 1}

   d) (2, 2): Count for label 2 is currently 1, equal to useLimit. We cannot select this item.

   e) (1, 3): Count for label 3 is 0, which is less than useLimit. We select this item.

   • ans becomes 9 (8 + 1)
   • num becomes 3 (2 + 1) --> This reaches numWanted, we stop selecting items.
   • cnt becomes {1: 1, 2: 1, 3: 1}

4. Return the maximum score:

   Since we have selected the maximum number of items (numWanted), we return ans which is 9. This is the maximum score we can achieve by selecting a subset of 3 items following the given constraints.

   The selected subset is represented by the pairs (5, 1), (3, 2), and (1, 3), leading to the maximum value of 9.

Applying this approach has allowed us to maximize our score under the constraints of numWanted and useLimit.

Solution Implementation

Time and Space Complexity

The given Python code snippet is a function that selects the largest values from a list values, each associated with a label from the list labels, with two constraints: a maximum number of items selected (numWanted) and a maximum number of times each label can be used (useLimit). The function returns the sum of the largest values selected according to these constraints.

Time Complexity

The time complexity of the code can be broken down as follows:

1. Sorting the combined list of values and labels: sorted(zip(values, labels), reverse=True) takes O(N log N) time, where N is the length of the values (and labels) list.

2. The for loop iterates over the sorted list, which has N elements. In the worst case, it will iterate over all N elements once.

3. Inside the loop, updating the Counter and performing basic arithmetic operations are O(1) operations.

Combining these operations, the overall time complexity of the code is dominated by the sorting step, resulting in O(N log N).

Space Complexity

The space complexity can be analyzed as follows:

1. The additional space used by the sorted list of zipped values and labels is O(N), where N is the length of the original lists.

2. The Counter object cnt which keeps track of the number of times each label has been used will, in the worst case, have as many elements as there are distinct labels. In the worst case, this can also be O(N) if every value has a unique label.

Therefore, the overall space complexity is O(N) due to the space required to store the sorted list and the counter.

Learn more about how to find time and space complexity quickly using problem constraints.