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Maximize Score of Numbers in Ranges

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ArraysBinary Search

You are given an array of integers start and an integer d, representing n intervals [start[i], start[i] + d].

You are asked to choose n integers where the i^th integer must belong to the i^th interval. The score of the chosen integers is defined as the minimum absolute difference between any two integers that have been chosen.

Return the maximum possible score of the chosen integers.

Example 1:

Input: start = [6,0,3], d = 2

Output: 4

Explanation:

The maximum possible score can be obtained by choosing integers: 8, 0, and 4. The score of these chosen integers is min(|8 - 0|, |8 - 4|, |0 - 4|) which equals 4.

Example 2:

Input: start = [2,6,13,13], d = 5

Output: 5

Explanation:

The maximum possible score can be obtained by choosing integers: 2, 7, 13, and 18. The score of these chosen integers is min(|2 - 7|, |2 - 13|, |2 - 18|, |7 - 13|, |7 - 18|, |13 - 18|) which equals 5.

Constraints:

2 <= start.length <= 10⁵
0 <= start[i] <= 10⁹
0 <= d <= 10⁹

Try coding it in LeetCode

Maximize Score of Numbers in Ranges

Medium

Asked by:

18 views

Topics:

ArraysBinary Search

You are given an array of integers start and an integer d, representing n intervals [start[i], start[i] + d].

Return the maximum possible score of the chosen integers.

Example 1:

Input: start = [6,0,3], d = 2

Output: 4

Explanation:

The maximum possible score can be obtained by choosing integers: 8, 0, and 4. The score of these chosen integers is min(|8 - 0|, |8 - 4|, |0 - 4|) which equals 4.

Example 2:

Input: start = [2,6,13,13], d = 5

Output: 5

Explanation:

Constraints:

2 <= start.length <= 10⁵
0 <= start[i] <= 10⁹
0 <= d <= 10⁹

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 themselves and the length of the input array?
Can the start and end indices of the ranges overlap?
If no selection of numbers yields a positive score, what value should I return?
Are the ranges inclusive, meaning both the start and end indices are included in the range?
What data type should I use to represent the score, and is there a possibility of integer overflow?

Brute Force Solution

Approach

The brute force approach to maximizing the score involves exploring every single possible combination of choosing or not choosing each range. We calculate the score for each combination and then pick the best one we find.

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

Consider each range, one at a time.
For the first range, imagine two possibilities: either we include this range or we don't.
If we include the range, mark the numbers within that range as used.
If we don't include the range, we just move on to the next range without marking any numbers.
Now, for the second range, again we have two choices: include it or don't include it. We need to consider both these choices for each of the previous choices regarding the first range. This means we're branching out and exploring all combinations.
Continue this process for every range. At each step, we are either choosing to include the range and marking the corresponding numbers as used, or skipping the range.
Once we have gone through all ranges and made a choice to include or not include each one, we can compute the score for this particular combination. The score is calculated based on the ranges we selected, and the numbers within those ranges were marked.
Repeat this entire process by trying all possible combinations of including or excluding each range. This means we exhaustively try every possibility.
Finally, compare all the scores that were calculated for each combination. Pick the highest score among all the possibilities. That highest score is the maximum score.

Code Implementation

def maximize_score_brute_force(numbers, ranges): 
    max_score = 0

    def calculate_score(selected_ranges): 
        used_numbers = [False] * len(numbers)
        current_score = 0

        for range_index in selected_ranges:
            start, end, score = ranges[range_index]
            for i in range(start, end + 1):
                if not used_numbers[i]:
                    used_numbers[i] = True
                    current_score += score

        return current_score

    def explore_combinations(index, current_combination): 
        nonlocal max_score

        if index == len(ranges):
            # Reached the end of ranges, calculate the score.
            score = calculate_score(current_combination)
            max_score = max(max_score, score)
            return

        # Explore the option of NOT including the current range.
        explore_combinations(index + 1, current_combination)

        # Explore the option of including the current range.
        # Append the current range index to the combination.
        explore_combinations(index + 1, current_combination + [index])

    explore_combinations(0, [])

    return max_score

Big(O) Analysis

Time Complexity

O(2^r) – The brute force approach explores all possible combinations of including or excluding each range. If we have r ranges, each range has two possibilities: either we include it or we don't. Therefore, we are considering 2 * 2 * ... * 2 (r times) possibilities, which equals 2^r. Computing the score for each combination takes O(1) time assuming the cost of marking elements is already factored in. The overall time complexity is dominated by the number of combinations, leading to O(2^r).

Space Complexity

O(N) – The brute force approach uses recursion to explore all possible combinations of including or excluding each range. For each combination, we potentially mark numbers within a range as 'used', requiring a boolean array of size N, where N is the number of unique numbers across all ranges, to keep track of which numbers have already been selected. The depth of the recursion can be, at most, the number of ranges. Each recursive call creates a new copy of this 'used' array, representing the state for that particular combination. Thus the maximum space used is proportional to the size of the boolean array, giving us O(N).

Optimal Solution

Approach

The key to solving this problem quickly is to realize it involves making choices step-by-step and remembering the best score we can achieve at each step. We build up the optimal score by considering each number and each range, making smart decisions based on the previous steps.

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

Think of each number in the list as a potential decision point: Do we include it in a range, or not?
Create a way to remember the best possible score we can have up to each number in the list.
Go through the numbers one at a time. For each number, consider two possibilities: either we don't include it in any range, or we include it in a range that ends at that number.
If we don't include the number, the best score up to that point is just the best score we had *before* reaching that number.
If we *do* include the number, we need to check all the ranges that *end* on that number. For each of these ranges, figure out the score we'd get by *using* that range. This will involve the range's value, plus the best score we had *before* the range even started.
Compare all the scores we calculated (the score from not including the number, and the scores from using each possible range that ends on the number). Take the *highest* of these scores, and remember that as the best score we can achieve up to the current number.
Continue this process for all the numbers in the list. Once we reach the end, the best score we remembered at the very end is the overall best possible score.

Code Implementation

def maximize_score(numbers, ranges):
    number_count = len(numbers)

    # Keep track of the max score
    max_scores = [0] * (number_count + 1)

    for i in range(1, number_count + 1):
        # Option 1: Don't include the current number
        max_scores[i] = max_scores[i - 1]

        # Option 2: Include the current number
        for start, end, score in ranges:
            if end == i:
                # If a range ends at the current number,
                # consider including it.
                max_scores[i] = max(max_scores[i],
                                     max_scores[start - 1] + score)

    return max_scores[number_count]

Big(O) Analysis

Time Complexity

O(n*m) – The algorithm iterates through each of the n numbers in the input list to calculate the maximum score achievable up to that point. Inside this loop, it iterates through a set of m ranges, where m is the total number of ranges. For each number, we consider all ranges that end at that number to determine the best possible score. Therefore, the time complexity is proportional to the product of the number of numbers and the number of ranges to consider, resulting in O(n*m).

Space Complexity

O(N) – The algorithm uses a data structure to remember the best possible score up to each number in the list. This implies creating an array (or similar data structure) of size N, where N is the number of numbers in the input list. Therefore, the auxiliary space required grows linearly with the input size N. No other significant auxiliary space is used, hence the space complexity is O(N).

Edge Cases

Case	How to Handle
Empty ranges array	Return 0 as no scores can be collected.
Null ranges or numbers array	Throw IllegalArgumentException or return appropriate error value (-1) after checking for null inputs.
Empty numbers array	Return 0 if the numbers array is empty as no score can be achieved with no numbers.
Large ranges array that causes memory overflow or performance degradation	Optimize for memory usage and time complexity to ensure efficient execution for large input sizes, possibly using dynamic programming with space optimization.
Ranges overlap significantly causing many possible combinations.	Dynamic programming with memoization can effectively address this by storing and reusing results of subproblems.
Negative numbers in the numbers array	Ensure the solution correctly handles negative numbers; if scores are defined based on value, negative numbers could result in negative score contributions.
Number appearing in no range	That number is automatically skipped because its corresponding dp value in ranges array is 0.
Integer overflow when calculating sum of scores	Use a larger data type (e.g., long) to store the intermediate and final score sums to avoid potential integer overflow issues.

Empty ranges array

How to Handle:

Return 0 as no scores can be collected.

Null ranges or numbers array

How to Handle:

Throw IllegalArgumentException or return appropriate error value (-1) after checking for null inputs.

Empty numbers array

How to Handle:

Return 0 if the numbers array is empty as no score can be achieved with no numbers.

Large ranges array that causes memory overflow or performance degradation

How to Handle:

Optimize for memory usage and time complexity to ensure efficient execution for large input sizes, possibly using dynamic programming with space optimization.

Ranges overlap significantly causing many possible combinations.

How to Handle:

Dynamic programming with memoization can effectively address this by storing and reusing results of subproblems.

Negative numbers in the numbers array

How to Handle:

Ensure the solution correctly handles negative numbers; if scores are defined based on value, negative numbers could result in negative score contributions.

Number appearing in no range

How to Handle:

That number is automatically skipped because its corresponding dp value in ranges array is 0.

Integer overflow when calculating sum of scores

How to Handle:

Use a larger data type (e.g., long) to store the intermediate and final score sums to avoid potential integer overflow issues.

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 themselves and the length of the input array?
Can the start and end indices of the ranges overlap?
If no selection of numbers yields a positive score, what value should I return?
Are the ranges inclusive, meaning both the start and end indices are included in the range?
What data type should I use to represent the score, and is there a possibility of integer overflow?

Brute Force Solution

Approach

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

Consider each range, one at a time.
For the first range, imagine two possibilities: either we include this range or we don't.
If we include the range, mark the numbers within that range as used.
If we don't include the range, we just move on to the next range without marking any numbers.
Now, for the second range, again we have two choices: include it or don't include it. We need to consider both these choices for each of the previous choices regarding the first range. This means we're branching out and exploring all combinations.
Continue this process for every range. At each step, we are either choosing to include the range and marking the corresponding numbers as used, or skipping the range.
Once we have gone through all ranges and made a choice to include or not include each one, we can compute the score for this particular combination. The score is calculated based on the ranges we selected, and the numbers within those ranges were marked.
Repeat this entire process by trying all possible combinations of including or excluding each range. This means we exhaustively try every possibility.
Finally, compare all the scores that were calculated for each combination. Pick the highest score among all the possibilities. That highest score is the maximum score.

Code Implementation

def maximize_score_brute_force(numbers, ranges): 
    max_score = 0

    def calculate_score(selected_ranges): 
        used_numbers = [False] * len(numbers)
        current_score = 0

        for range_index in selected_ranges:
            start, end, score = ranges[range_index]
            for i in range(start, end + 1):
                if not used_numbers[i]:
                    used_numbers[i] = True
                    current_score += score

        return current_score

    def explore_combinations(index, current_combination): 
        nonlocal max_score

        if index == len(ranges):
            # Reached the end of ranges, calculate the score.
            score = calculate_score(current_combination)
            max_score = max(max_score, score)
            return

        # Explore the option of NOT including the current range.
        explore_combinations(index + 1, current_combination)

        # Explore the option of including the current range.
        # Append the current range index to the combination.
        explore_combinations(index + 1, current_combination + [index])

    explore_combinations(0, [])

    return max_score

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

Think of each number in the list as a potential decision point: Do we include it in a range, or not?
Create a way to remember the best possible score we can have up to each number in the list.
Go through the numbers one at a time. For each number, consider two possibilities: either we don't include it in any range, or we include it in a range that ends at that number.
If we don't include the number, the best score up to that point is just the best score we had *before* reaching that number.
If we *do* include the number, we need to check all the ranges that *end* on that number. For each of these ranges, figure out the score we'd get by *using* that range. This will involve the range's value, plus the best score we had *before* the range even started.
Compare all the scores we calculated (the score from not including the number, and the scores from using each possible range that ends on the number). Take the *highest* of these scores, and remember that as the best score we can achieve up to the current number.
Continue this process for all the numbers in the list. Once we reach the end, the best score we remembered at the very end is the overall best possible score.

Code Implementation

def maximize_score(numbers, ranges):
    number_count = len(numbers)

    # Keep track of the max score
    max_scores = [0] * (number_count + 1)

    for i in range(1, number_count + 1):
        # Option 1: Don't include the current number
        max_scores[i] = max_scores[i - 1]

        # Option 2: Include the current number
        for start, end, score in ranges:
            if end == i:
                # If a range ends at the current number,
                # consider including it.
                max_scores[i] = max(max_scores[i],
                                     max_scores[start - 1] + score)

    return max_scores[number_count]

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Empty ranges array	Return 0 as no scores can be collected.
Null ranges or numbers array	Throw IllegalArgumentException or return appropriate error value (-1) after checking for null inputs.
Empty numbers array	Return 0 if the numbers array is empty as no score can be achieved with no numbers.
Large ranges array that causes memory overflow or performance degradation	Optimize for memory usage and time complexity to ensure efficient execution for large input sizes, possibly using dynamic programming with space optimization.
Ranges overlap significantly causing many possible combinations.	Dynamic programming with memoization can effectively address this by storing and reusing results of subproblems.
Negative numbers in the numbers array	Ensure the solution correctly handles negative numbers; if scores are defined based on value, negative numbers could result in negative score contributions.
Number appearing in no range	That number is automatically skipped because its corresponding dp value in ranges array is 0.
Integer overflow when calculating sum of scores	Use a larger data type (e.g., long) to store the intermediate and final score sums to avoid potential integer overflow issues.

Empty ranges array

How to Handle:

Return 0 as no scores can be collected.

Null ranges or numbers array

How to Handle:

Throw IllegalArgumentException or return appropriate error value (-1) after checking for null inputs.

Empty numbers array

How to Handle:

Return 0 if the numbers array is empty as no score can be achieved with no numbers.

Large ranges array that causes memory overflow or performance degradation

How to Handle:

Optimize for memory usage and time complexity to ensure efficient execution for large input sizes, possibly using dynamic programming with space optimization.

Ranges overlap significantly causing many possible combinations.

How to Handle:

Dynamic programming with memoization can effectively address this by storing and reusing results of subproblems.

Negative numbers in the numbers array

How to Handle:

Ensure the solution correctly handles negative numbers; if scores are defined based on value, negative numbers could result in negative score contributions.

Number appearing in no range

How to Handle:

That number is automatically skipped because its corresponding dp value in ranges array is 0.

Integer overflow when calculating sum of scores

How to Handle:

Use a larger data type (e.g., long) to store the intermediate and final score sums to avoid potential integer overflow issues.