Get the Maximum Score

Hard

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Topics:

ArraysDynamic ProgrammingGreedy AlgorithmsTwo Pointers

You are given two sorted arrays of distinct integers nums1 and nums2.

A valid path is defined as follows:

Choose array nums1 or nums2 to traverse (from index-0).
Traverse the current array from left to right.
If you are reading any value that is present in nums1 and nums2 you are allowed to change your path to the other array. (Only one repeated value is considered in the valid path).

The score is defined as the sum of unique values in a valid path.

Return the maximum score you can obtain of all possible valid paths. Since the answer may be too large, return it modulo 10⁹ + 7.

Example 1:

Input: nums1 = [2,4,5,8,10], nums2 = [4,6,8,9]
Output: 30
Explanation: Valid paths:
[2,4,5,8,10], [2,4,5,8,9], [2,4,6,8,9], [2,4,6,8,10],  (starting from nums1)
[4,6,8,9], [4,5,8,10], [4,5,8,9], [4,6,8,10]    (starting from nums2)
The maximum is obtained with the path in green [2,4,6,8,10].

Example 2:

Input: nums1 = [1,3,5,7,9], nums2 = [3,5,100]
Output: 109
Explanation: Maximum sum is obtained with the path [1,3,5,100].

Example 3:

Input: nums1 = [1,2,3,4,5], nums2 = [6,7,8,9,10]
Output: 40
Explanation: There are no common elements between nums1 and nums2.
Maximum sum is obtained with the path [6,7,8,9,10].

Constraints:

1 <= nums1.length, nums2.length <= 10⁵
1 <= nums1[i], nums2[i] <= 10⁷
nums1 and nums2 are strictly increasing.

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the possible ranges for the numbers in the input array, and can the array contain negative numbers, zeros, or floating-point numbers?
Is the input array guaranteed to be non-empty, and what should I return if the array is null or empty?
If there are multiple ways to achieve the maximum score, should I return one of them or is there a specific optimal solution to return?
Could you please clarify the rules for calculating the score; specifically, are there any constraints or dependencies between the different elements of the array when computing the score?
What is the expected data type for the score, and is there a maximum possible score that I should consider for potential overflow issues?

Brute Force Solution

Approach

The brute force method tackles this problem by exploring every single possible combination. It's like trying out every single arrangement to see which one gives us the highest score. We generate all possibilities and then pick the best one.

Here's how the algorithm would work step-by-step:

Imagine we have a way to pick a few starting items.
We then try every possible combination of items we could pick at the start.
For each of these starting possibilities, we need to figure out what we can do next.
We again try all possible actions we can take after this initial selection.
We continue this process, always trying everything, until we reach the end in every branch.
As we explore each path, we calculate a score for that entire arrangement.
Finally, we compare all of the scores from all the different possibilities, and pick the highest one.

Code Implementation

def get_maximum_score_brute_force(items, scorer):    number_of_items = len(items)
    maximum_score = float('-inf')

    def explore_combinations(current_combination, current_score):
        nonlocal maximum_score

        # Base case: all items have been considered
        if not items:
            maximum_score = max(maximum_score, current_score)
            return

        # Recursive step: explore all combinations by either
        # including or excluding the current item.
        for index in range(len(items)): # Iterate through each item
            chosen_item = items[index]
            remaining_items = items[:index] + items[index+1:]
            new_score = current_score + scorer(current_combination, chosen_item)

            explore_combinations(current_combination + [chosen_item], new_score)

    # Start the exploration with an empty combination and score
    explore_combinations([], 0)

    return maximum_score

Big(O) Analysis

Time Complexity

O(2^n) – The described brute force method explores every possible combination of items. In the worst case, for each of the 'n' items, we either include it or exclude it in a combination. This results in 2 options for each item, leading to 2 * 2 * ... * 2 (n times) or 2^n possible combinations to evaluate. Therefore, the time complexity is O(2^n) as we are generating and checking all possible subsets of the input.

Space Complexity

O(N^K) – The described brute force approach explores every possible combination of actions. This implies a decision tree where each level represents a choice and each path represents a complete combination. The branching factor at each level depends on the number of possible actions, and if we assume there are K steps or decisions to be made, and on average N choices at each step, the number of paths grows exponentially. To store each of these complete paths to calculate the scores, we need space to store the sequence of actions (or the items picked). In the worst-case, we might need to store all paths simultaneously, leading to a space complexity of O(N^K) where N is the average number of choices and K is the number of decisions.

Optimal Solution

Approach

The optimal strategy involves deciding at each step whether to include the current element in our selection or not. We avoid recomputing answers to subproblems by remembering the best result we've found so far for each possible state, building up to the final answer efficiently.

Here's how the algorithm would work step-by-step:

Think of the problem as making a decision at each step: should I include this item, or skip it?
Instead of recomputing whether to include a particular item, store that best decision as a number in a chart.
The rows of the chart represent the element we are considering, and the columns are the limit we are given.
To fill each cell, check to see if the current element improves the best value we have so far.
If we decide to include it, we add the element's value to the best value we can get from the items before it with the updated limit.
If not, then the chart just shows the same best value that we could get without this element.
The bottom right corner of the chart will have the maximum possible score.

Code Implementation

def get_maximum_score(items, limit):
    number_of_items = len(items)

    # Initialize the dynamic programming table
    maximum_score_table = [[0] * (limit + 1) for _ in range(number_of_items + 1)]

    for item_index in range(1, number_of_items + 1):
        item_value, item_weight = items[item_index - 1]
        for current_limit in range(1, limit + 1):
            # Cannot include item if it exceeds the current limit.
            if item_weight > current_limit:
                maximum_score_table[item_index][current_limit] = maximum_score_table[item_index - 1][current_limit]
            else:
                # Determine max score by including or excluding current item.
                maximum_score_table[item_index][current_limit] = max(
                    maximum_score_table[item_index - 1][current_limit],
                    item_value + maximum_score_table[item_index - 1][current_limit - item_weight]
                )

    # Final answer is in the bottom right corner
    return maximum_score_table[number_of_items][limit]

Big(O) Analysis

Time Complexity

O(n*limit) – The described dynamic programming approach involves creating a chart (or table) where rows represent the n elements and columns represent the limit. Filling each cell in this chart requires a constant amount of work to compare the current score with and without including the current element. Since there are n rows and 'limit' columns, the total number of cells is n * limit, and each cell requires constant time processing. Therefore, the overall time complexity is O(n*limit).

Space Complexity

O(N*Limit) – The described approach uses a chart (likely a 2D array) to store the maximum score achievable for each element considered and each possible limit. The rows of the chart represent the element being considered (up to N, where N is the number of elements), and the columns represent the limit. Therefore, the auxiliary space required is proportional to the product of the number of elements (N) and the limit. This leads to a space complexity of O(N*Limit).

Edge Cases

Case	How to Handle
Null or empty input array	Return 0 or throw an exception based on requirements since no score is possible.
Array with only one element	Return 0 or throw an exception because a pair is needed.
Array containing duplicate values	The algorithm should correctly account for duplicates when calculating the score; ensure no double-counting of the same index.
Array with all elements equal	The algorithm's scoring logic should handle the case where all numbers are the same, potentially returning 0 or a valid score based on the definition.
Input array contains negative numbers	The algorithm should correctly handle negative values in the array, as they might contribute to a higher score depending on the rules.
Input array contains zero	Zeroes could introduce edge cases, especially in multiplication-based scoring, requiring careful consideration.
Integer overflow during score calculation	Use a data type that can accommodate large scores (e.g., long) to avoid overflow issues if the problem involves products or sums of large numbers.
Maximum input array size approaching memory limits	Optimize for space efficiency and consider using streaming or divide-and-conquer approaches if the input array can be extremely large.

Null or empty input array

How to Handle:

Return 0 or throw an exception based on requirements since no score is possible.

Array with only one element

How to Handle:

Return 0 or throw an exception because a pair is needed.

Array containing duplicate values

How to Handle:

The algorithm should correctly account for duplicates when calculating the score; ensure no double-counting of the same index.

Array with all elements equal

How to Handle:

The algorithm's scoring logic should handle the case where all numbers are the same, potentially returning 0 or a valid score based on the definition.

Input array contains negative numbers

How to Handle:

The algorithm should correctly handle negative values in the array, as they might contribute to a higher score depending on the rules.

Input array contains zero

How to Handle:

Zeroes could introduce edge cases, especially in multiplication-based scoring, requiring careful consideration.

Integer overflow during score calculation

How to Handle:

Use a data type that can accommodate large scores (e.g., long) to avoid overflow issues if the problem involves products or sums of large numbers.

Maximum input array size approaching memory limits

How to Handle:

Optimize for space efficiency and consider using streaming or divide-and-conquer approaches if the input array can be extremely large.