Problem Description

In this problem, we are working with a garden represented by a linear arrangement of flowers, each flower having a beauty value, which is a numerical value associated with it.

A garden is considered valid if:

• It contains at least two flowers.
• The beauty value of the first flower is equal to the beauty value of the last flower in the garden.

As the gardener, your job is to possibly remove any number of flowers from the garden to ensure that it becomes valid while maximizing the total beauty of the garden. The total beauty of the garden is defined as the sum of the beauty values of all the flowers that are left.

To summarize, the objective is to maximize the sum of the beauty values of a valid garden by selectively removing flowers. The solution to this challenge will require an efficient algorithm to calculate the maximum possible beauty without having to try all possible combinations of flower removal, which would be inefficient.

Intuition

To find the maximum beauty of a valid garden, we analyze the following key points:

1. As the valid garden must start and end with flowers having the same beauty value, and we aim to maximize beauty, we must identify pairs of flowers with the same value that are as far from each other as possible, since all flowers between them can contribute to the total beauty.

2. Given point 1, removing any flowers with a negative beauty value from the middle of our garden will only increase the total beauty since they detract from it.

With these points in mind, we can sketch an algorithm that:

A. Keeps a running sum (s) of the beauty values (ignoring negative values as they are never beneficial to our sum), to quickly calculate the total beauty including any range of flowers.

B. Utilizes a dictionary (d) to keep track of the last occurrence index for each beauty value we come across as we iterate through the flower array.

As we iterate over the flowers array:

• If we encounter a flower with a beauty value we've seen before, we have a potential garden that starts at the previous occurrence index and ends at the current index. We calculate the potential beauty for this garden by adding the total beauty value twice (once for each flower at the ends) and subtracting the sum of values between them (as stored in s).

• We update our maximum beauty (ans) if the current potential beauty is greater than what we have found so far.

• If we encounter a flower with a beauty value for the first time, we simply record its index in the dictionary.

• The running sum (s) is updated at every step, adding the current flower's beauty value, but skipping negative values.

By iterating through the entire array just once, we are able to calculate the maximum possible garden beauty by considering all pairs of same-beauty flowers and calculating the maximum sum efficiently.

The given solution code implements this approach and correctly calculates the answer.

Learn more about Greedy and Prefix Sum patterns.

Solution Approach

The solution to this LeetCode problem involves finding subarrays with the same starting and ending beauty values and maximizing the sum of the beauty values of the elements in between. The algorithm can be broken down into the following steps, incorporating efficient data structures and patterns:

1. Prefix Sum Array: As an optimization for quick sum queries, a prefix sum array s is created. s[i] represents the sum of all beauty values from flowers[0] up to flowers[i-1], excluding negative values, as they reduce the total beauty. The prefix sum array allows for constant-time range sum queries, which is crucial for efficient computation.

2. Dictionary Tracking: A dictionary d is used to keep track of the last index where each beauty value appears. This is used to find the most distant matching pair of the same beauty value, which potentially forms the start and end of a valid garden.

3. Finding Maximum Beauty Garden: As we iterate over the flowers array:

   • If we encounter a beauty value that has appeared before, it means there's a potential valid garden ending with the current flower. The beauty of this potential garden is calculated as the sum of all the values between the matching pair, including the value at the end points themselves (which are same due to the valid garden condition). We calculate this by subtracting the prefix sum at the start index from the prefix sum at the end index and adding the value of the end points twice.
   • If it’s the first time we encounter a certain beauty value, we record its index in the dictionary d, marking it as the start point for potential gardens with that beauty value.

4. Maintaining Maximum Beauty Value: Throughout the iteration, we maintain a variable ans, which stores the maximum beauty value found so far. Whenever we find a potential garden with a higher beauty value than our current maximum, we update ans.

5. Update Rules:

   • For each flower flowers[i], the current flower's beauty is added to the prefix sum only if it is non-negative.
   • If flowers[i] has occurred before, compute the potential beauty by subtracting s[d[flowers[i]] + 1] from s[i] and adding flowers[i] * 2 (for the start and end flowers).
   • Update the dictionary d for the current beauty value to the current index so that we can use the most recent index for calculating future gardens.

By following these steps, the provided Python code efficiently loops through the flower array once and finds the maximum possible beauty of a valid garden. This approach leverages the usage of a prefix sum array for fast range sum calculations and a dictionary for O(1) look-up of indices, making the overall time complexity O(n), where n is the number of flowers.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have the following array of flowers with their respective beauty values:

flowers = [1, -2, -1, 1, 3, 1]

We wish to identify a valid garden with the maximum total beauty. We will use the given solution approach to solve it.

1. Prefix Sum Array (s): We will construct this array on the fly as we iterate through the flowers array. Let's initialize the prefix sum array with 0.

2. Dictionary Tracking (d): We'll start with an empty dictionary.

3. Finding Maximum Beauty Garden:

• Iterating over the given flowers array:
  • For flowers[0] which has a beauty value of 1, d[1] does not exist yet, so we add it to d with a value of 0 and set the prefix sum s[0] to 1.
  • flowers[1] with a beauty value of -2 is negative, so we do not add it to our prefix sum. Prefix sum remains s[0] = 1.
  • flowers[2] has a beauty of -1, again negative, so we continue. Prefix sum is still s[0] = 1.
  • When we reach flowers[3] with the beauty value of 1, d[1] exists and suggests a potential valid garden from index 0 to 3. We compute the beauty by checking the prefix sum at the start index 0 and end index 3. The total beauty of this potential garden is s[3] - s[d[1] + 1] + flowers[3] * 2 = 1 - 1 + 2 = 2. This is our first potential garden, so we set ans to 2.
  • Next, flowers[4] has a beauty value of 3. We add its index to d as d[3] = 4 and update our prefix sum to include this flower's positive beauty value, making s[4] = 4.
  • Lastly, flowers[5] has a beauty value of 1. Again, d[1] exists, so we check this potential garden which spans from index 0 to 5. The total beauty is s[5] - s[d[1] + 1] + flowers[5] * 2 = 4 - 1 + 2 = 5. This potential garden has a higher beauty than our current ans, so we update ans to 5.

4. Maintaining Maximum Beauty Value (ans): Throughout the iteration, we have updated ans whenever we found a potential garden with a higher beauty value. First, ans was 2, but after considering the full array, we updated ans to 5.

5. Update Rules:

• For each flower flowers[i], if the flower's beauty value is positive, it is added to the prefix sum. If not, it is ignored.
• When a flower's beauty value has been encountered before, calculate the potential beauty as described above.
• Update the dictionary d with the most recent index for the current beauty value.

By the end of this process, we determine that taking the entire array from index 0 to 5 gives us the maximum beauty of 5, resulting in a valid garden with [1, -2, -1, 1, 3, 1].

Solution Implementation

Time and Space Complexity

The given Python code snippet defines a Solution class with a method maximumBeauty that calculates a specific value related to the beauty of a sequence of flowers represented by an integer array. The method employs a dynamic programming approach and uses additional storage to keep track of sum results and indices.

Time Complexity:

The time complexity of the method maximumBeauty can be analyzed based on the single loop through the flowers array and the operations performed within this loop.

• There is a loop that iterates through each element of the input list flowers, which gives us an O(n) where n is the length of flowers.
• Within this loop, operations such as checking if a value is in a dictionary, updating the dictionary, computing a maximum, and updating the prefix sum array are all O(1) operations.
• Therefore, the dominating factor of time complexity is the loop itself, which gives us a total time complexity of O(n).

Space Complexity:

The space complexity of the method maximumBeauty is determined by the additional space used by the data structures s and d.

• The prefix sum array s is of length len(flowers) + 1, which introduces an O(n) space complexity where n is the length of flowers.
• The dictionary d stores indices of distinct elements encountered in flowers. In the worst case, all elements of flowers are distinct, resulting in an O(n) space complexity, where n is the length of flowers.

The final space complexity is the sum of the complexities of the prefix sum array and the dictionary, both of which are O(n). Thus, the overall space complexity of the method maximumBeauty is O(n) as well.

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