Maximum Total Beauty of the Gardens

Maximum Total Beauty of the Gardens

1. What is this problem about?

The Maximum Total Beauty of the Gardens problem is a complex optimization challenge. You are given an array of gardens, each with a certain number of flowers, and a target number of flowers for a garden to be considered "complete." You also have a fixed number of "new flowers" that you can plant. Your total beauty score is the sum of: (Number of complete gardens * full_beauty) + (Minimum number of flowers in any incomplete garden * partial_beauty). The goal is to distribute your new flowers to maximize this total beauty score.

2. Why is this asked in interviews?

This "Maximum Total Beauty of the Gardens interview question" is a "Hard" problem often seen in Intuit interviews. It tests a candidate's ability to handle multi-faceted optimization. It requires balancing two different rewards (full vs. partial) and making decisions on how many gardens to complete versus how much to raise the floor of the remaining incomplete gardens. It evaluates proficiency in sorting, prefix sums, and two-pointer/binary search strategies.

3. Algorithmic pattern used

The "Array, Sorting, Two Pointers, Binary Search, Enumeration, Greedy, Prefix Sum interview pattern" is the way to go.

1. Sort the gardens to handle them systematically.
2. Use prefix sums to quickly calculate how many flowers are needed to bring a range of incomplete gardens up to a certain minimum level.
3. Enumerate the number of gardens you will make "complete" (starting from the most full gardens).
4. For the remaining new flowers, use a two-pointer approach or binary search to find the maximum possible "minimum level" you can achieve for the incomplete gardens.

4. Example explanation

Target = 10, New Flowers = 5, Full Beauty = 100, Partial Beauty = 10. Gardens: [8, 5]

• Scenario 1: Complete the first garden (needs 2 flowers). 3 flowers left. Use them on the 5-flower garden -> it becomes 8. Beauty = (1 * 100) + (8 * 10) = 180.
• Scenario 2: Don't complete any. Use all 5 flowers to raise the minimum. 5+8=13. We can make both gardens have at least 6.5? No, integers. We can make both 6. Min is 6. Beauty = (0 * 100) + (6 * 10) = 60. The maximum beauty is 180.

5. Common mistakes candidates make

In the "Maximum Total Beauty of the Gardens coding problem," the most common mistake is not considering all possible numbers of "full" gardens. Candidates often pick a greedy approach that either fills as many as possible or fills none, missing the middle ground. Another error is inefficiently calculating the flowers needed to raise the minimum level, leading to O(N^2) complexity. You must also be careful with the "target - 1" limit for incomplete gardens—if you accidentally fill an incomplete garden beyond the target, it must be counted as a complete garden.

6. Interview preparation tip

Practice the "Two-Pointer + Prefix Sum" optimization. This is a common way to solve problems where you need to calculate costs for a sliding range. Also, get comfortable with "Enumeration" as a technique—sometimes the best way to find a global maximum is to iterate through one specific variable (like the number of full gardens) and optimize the rest of the state relative to that variable.