LeetCode 1793. Maximum Score of a Good Subarray (Solution)

Introduction

The world of algorithms is filled with captivating challenges, and today, we’re diving deep into one such intriguing puzzle: the “Maximum Score of a Good Subarray.” At first glance, the problem might seem like a simple array traversal task. But beneath its surface lies a rich tapestry of patterns and logic, waiting to be unraveled.

The essence of the problem revolves around subarrays and their scores. We’re on a quest to find that one ‘good’ subarray that stands out from the rest, boasting the highest score. But what makes this journey truly fascinating is the presence of our trusted guide: the two-pointer algorithm.

For those unfamiliar, the two-pointer technique is akin to a choreographed dance on an array stage. Two pointers, often starting from different positions, move in harmony, adjusting their steps based on the rhythm of the data. In the context of our problem, these pointers will help us efficiently explore potential subarrays, ensuring we never miss a beat (or in this case, the maximum score).

As we embark on this algorithmic adventure, we’ll witness the elegance of the two-pointer method in action. It’s a dance of precision, efficiency, and beauty. So, let’s lace up our dancing shoes and dive into the mesmerizing world of “Maximum Score of a Good Subarray.”

Intuition

Understanding Two-Pointer Algorithms

In the world of algorithmic problem-solving, the two-pointer technique stands out as both elegant and efficient. But what exactly is it?
At its core, the two-pointer algorithm involves using two pointers (or indices) to traverse an array or list. These pointers often move towards each other, away from each other, or at different speeds, depending on the problem at hand. The primary benefit of this approach is that it can reduce the time complexity of certain problems from quadratic (O(n^2)) to linear (O(n)).

Types of Two-Pointer Approaches:

1. Classic Two-Pointer: This is the most straightforward application, where two pointers start at opposite ends of an array and move towards each other. It’s commonly used in problems where you need to find a pair of elements meeting a specific condition, such as the “Two Sum” problem.
   
2. Slow and Fast Pointer: This variant involves two pointers moving at different speeds, usually in linked list problems. A classic example is detecting a cycle in a linked list, where one pointer moves one step at a time, and the other moves two.
   
3. Sliding Window: This approach involves maintaining a ‘window’ of elements in an array or list. As you progress, this window might grow, shrink, or slide, depending on the problem’s requirements. It’s particularly useful for problems involving contiguous subarrays or substrings, such as “Maximum Sum Subarray of Size K.”

Why Use Two-Pointer Algorithms?

1. Efficiency: They can drastically reduce time complexity, especially in problems that might seem to require nested loops.
   
2. Simplicity: Once you grasp the concept, the logic is often more straightforward than alternative approaches.
   
3. Versatility: They’re applicable to a wide range of problems, from array manipulations to linked list challenges.

In conclusion, the two-pointer technique is a powerful tool in a coder’s toolkit. Whether you’re a beginner just delving into algorithms or a seasoned programmer, mastering this strategy will undoubtedly prove beneficial.

Code

1. Start with left = right = k.
2. Initialize a variable ans to store the maximum score and set it to nums[k] initially.
3. Keep track of the current minimum value in the subarray, initialized to nums[k].
4. In each step, compare nums[left-1] and nums[right+1].
   Expand the subarray towards the direction of the larger value (if left can be decremented or right can be incremented). Update the current minimum value based on the newly added element.
5. Calculate the score for the current subarray and update ans if the current score is higher.
6. Repeat the above two steps until left reaches 0 and right reaches the end of the array.
7. Return ans.

class Solution:
    def maximumScore(self, nums, k):
        n = len(nums)
        left, right = k, k
        ans = c_min = nums[k]

        while 0 != left or right != n-1:
            if not left or (right != n-1 and nums[left-1] < nums[right+1]):
                right += 1
            else:
                left -= 1

            c_min = min(c_min, nums[right], nums[left])
            ans = max(ans, c_min * (right-left+1))

        return ans

Conclusion

Time Complexity: O(n)
In the worst case, one of the pointers might move to the start of the array, and the other might move to the end of the array.
Each pointer only traverses the array once. So, in the worst case, the total number of operations will be proportional to the length of the array, n.
Thus, the time complexity is O(n).

Space Complexity: O(1)
Throughout the algorithm, we only use a constant amount of extra space to store variables like left, right, and, and c_min.
We don’t use any additional data structures that grow with the input size.
Hence, the space complexity is O(1), which means it’s constant space.

Understanding the Problem’s Constraints:

1. A “good subarray” must contain the element at index k.
2. The score of any subarray is defined by its minimum element multiplied by its length.

Given these constraints, the optimal strategy is to expand outwards from k and always incorporate the highest possible values into our subarray to maximize the score.

Perhaps some of you might have received a TLE and are wondering why this linear approach guarantees the optimal solution. That’s an excellent observation; see the following explanation:

Why the Two-Pointer Approach Guarantees an Answer:

1. Starting Point: We start with both pointers at k, ensuring our subarray always contains the element at k, adhering to the problem’s constraints.
2. Expanding Outwards: As we move the pointers, we’re essentially exploring all possible subarrays containing the element at k. By expanding outwards, we ensure we don’t miss any potential subarray configurations.
3. Making Optimal Choices: At each step, we choose to expand towards the larger of the two boundary values. This is because our subarray’s score is determined by its minimum value. Including a larger value might not immediately increase our score, but it gives us the potential to expand further without reducing the minimum value too quickly.
4. Comprehensive Exploration: Even though we’re making greedy choices by expanding towards the larger value, we’re not missing out on any possibilities. This is because both pointers will eventually explore the entirety of the array’s length. Regardless of our immediate choices, we will consider all viable subarray configurations by the end of the algorithm.
5. Constant Score Updates: At each step, we calculate and update the maximum score. This ensures that we’re always tracking the best possible score at any given point in our traversal.

In essence, the two-pointer technique, in this context, systematically and efficiently explores all possible “good subarrays.” By always starting at k and expanding outwards while making optimal choices at each step, we ensure that we’re considering all potential solutions. This comprehensive exploration guarantees that we’ll find the maximum possible score for a good subarray.