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Maximum Total Beauty of the Gardens

Alice is a caretaker of n gardens and she wants to plant flowers to maximize the total beauty of all her gardens.

You are given a 0-indexed integer array flowers of size n, where flowers[i] is the number of flowers already planted in the ith garden. Flowers that are already planted cannot be removed. You are then given another integer newFlowers, which is the maximum number of flowers that Alice can additionally plant. You are also given the integers target, full, and partial.

A garden is considered complete if it has at least target flowers. The total beauty of the gardens is then determined as the sum of the following:

  • The number of complete gardens multiplied by full.
  • The minimum number of flowers in any of the incomplete gardens multiplied by partial. If there are no incomplete gardens, then this value will be 0.

Return the maximum total beauty that Alice can obtain after planting at most newFlowers flowers.

 

Example 1:

Input: flowers = [1,3,1,1], newFlowers = 7, target = 6, full = 12, partial = 1
Output: 14
Explanation: Alice can plant
- 2 flowers in the 0th garden
- 3 flowers in the 1st garden
- 1 flower in the 2nd garden
- 1 flower in the 3rd garden
The gardens will then be [3,6,2,2]. She planted a total of 2 + 3 + 1 + 1 = 7 flowers.
There is 1 garden that is complete.
The minimum number of flowers in the incomplete gardens is 2.
Thus, the total beauty is 1 * 12 + 2 * 1 = 12 + 2 = 14.
No other way of planting flowers can obtain a total beauty higher than 14.

Example 2:

Input: flowers = [2,4,5,3], newFlowers = 10, target = 5, full = 2, partial = 6
Output: 30
Explanation: Alice can plant
- 3 flowers in the 0th garden
- 0 flowers in the 1st garden
- 0 flowers in the 2nd garden
- 2 flowers in the 3rd garden
The gardens will then be [5,4,5,5]. She planted a total of 3 + 0 + 0 + 2 = 5 flowers.
There are 3 gardens that are complete.
The minimum number of flowers in the incomplete gardens is 4.
Thus, the total beauty is 3 * 2 + 4 * 6 = 6 + 24 = 30.
No other way of planting flowers can obtain a total beauty higher than 30.
Note that Alice could make all the gardens complete but in this case, she would obtain a lower total beauty.

 

Constraints:

  • 1 <= flowers.length <= 105
  • 1 <= flowers[i], target <= 105
  • 1 <= newFlowers <= 1010
  • 1 <= full, partial <= 105

Solution Explanation: Maximum Total Beauty of the Gardens

This problem involves optimizing the planting of flowers in gardens to maximize the total beauty. The total beauty is a function of the number of complete gardens (gardens with at least target flowers) and the minimum number of flowers in incomplete gardens.

The solution employs a clever combination of enumeration and binary search to efficiently explore the solution space.

  1. Sorting: The flowers array is first sorted in ascending order. This allows for easy identification of gardens with the fewest flowers and simplifies the process of determining which gardens to prioritize for additional planting.

  2. Enumeration of Complete Gardens: The algorithm iterates through the possible number of complete gardens (x), ranging from the initial number of complete gardens to the total number of gardens. For each value of x, it determines how many additional flowers are needed to achieve this number of complete gardens.

  3. Binary Search for Optimal Incomplete Gardens: For a given x, the algorithm uses binary search to find the optimal number of incomplete gardens to allocate the remaining newFlowers to. The goal is to maximize the minimum number of flowers in these incomplete gardens, as this directly contributes to the total beauty. The binary search efficiently identifies the best index (l) that balances the number of incomplete gardens and the amount of flowers needed to improve their minimum flower count.

  4. Calculating Total Beauty: Once the number of complete gardens (x) and the minimum flower count among incomplete gardens (y) are determined, the total beauty is calculated using the given formula: x * full + y * partial.

  5. Maximization: The algorithm keeps track of the maximum total beauty encountered so far, updating it whenever a higher beauty is found. The final maximum beauty is returned as the result.

Time Complexity Analysis

  • Sorting the flowers array takes O(n log n) time.
  • The outer loop iterates at most n times.
  • The binary search within the inner loop takes O(log n) time.
  • Therefore, the overall time complexity is O(n log n).

Space Complexity Analysis

  • The algorithm uses extra space for sorting and the s array (prefix sum), which are both proportional to n.
  • Thus, the space complexity is O(n).

Code Examples (Python, Java, C++, Go, TypeScript)

The code examples provided in the original response demonstrate the algorithm effectively. Each language version follows the same core logic: sorting, enumeration, binary search, beauty calculation, and maximization. The differences are mainly in syntax and data structure handling specific to each language. They all achieve the O(n log n) time and O(n) space complexity.