#3281

Medium

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Maximize Score of Numbers in Ranges

Maximize Score of Numbers in Ranges asks to find the maximum score by selecting integers within given intervals with a focus on minimizing differences.

array binary search greedy sorting

Problem Statement

You are given an array 'start' of integers and an integer 'd'. Each entry in 'start' defines a range [start[i], start[i] + d]. You need to pick exactly one integer from each range. The score of your selection is defined as the minimum absolute difference between any two of the selected integers.

Your goal is to maximize the score by selecting integers in such a way that the minimum absolute difference between the selected integers is as large as possible. You should return this maximum score.

Examples

Example 1

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

Output: 4

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

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

Solution Approach

Binary Search over the Valid Score Space

The optimal solution requires finding the maximum score through binary search. The key idea is to search over a range of possible scores and validate each score by checking if it is feasible to select integers from the intervals that satisfy the score condition. This approach allows us to efficiently narrow down the solution space.

Greedy Approach to Validation

For each candidate score, the greedy strategy helps by attempting to select the integers from each interval such that the score condition is met. This involves selecting the largest possible integer that still maintains the valid score and ensuring it doesn't conflict with previous selections.

Efficient Check Using Sorting

To quickly check the feasibility of a score, sorting the intervals by their start values enables a simple traversal through them. This ensures that each selection is optimized and meets the score condition efficiently, making the validation step faster.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the binary search over the possible scores, typically O(log(maxDifference)) where maxDifference is the range between the smallest and largest possible values of the score. Each score check involves sorting the intervals, making the overall time complexity O(n log n log(maxDifference)). Space complexity is mainly driven by the sorting step, O(n).

What Interviewers Usually Probe

Candidate understands binary search and can apply it in unusual contexts.
Candidate can break down complex problems into manageable parts with clear strategies.
Candidate demonstrates proficiency with greedy algorithms and interval-based problems.

Common Pitfalls or Variants

Common pitfalls

Failure to recognize the binary search over possible scores as the most efficient approach.
Improper validation logic when checking if a score is feasible, leading to incorrect answers.
Ignoring sorting as a key step for efficiently handling the interval selections.

Follow-up variants

Increasing the number of intervals or the size of d to test scalability.
Introducing constraints where the ranges are no longer uniformly spaced or have overlaps.
Modifying the score condition to include other factors such as maximum instead of minimum differences.

FAQ

How do you maximize the score in Maximize Score of Numbers in Ranges?

To maximize the score, use binary search over the possible scores. For each candidate score, validate it by trying to select integers from the given intervals using a greedy approach.

What is the time complexity of the Maximize Score of Numbers in Ranges problem?

The time complexity is O(n log n log(maxDifference)), where n is the number of intervals and maxDifference is the range between the smallest and largest possible score values.

What pattern is primarily used in solving the Maximize Score of Numbers in Ranges problem?

The main pattern used is binary search over the valid answer space, combined with a greedy approach for validating candidate scores.

How does sorting affect the solution to Maximize Score of Numbers in Ranges?

Sorting the intervals helps in efficiently selecting integers from them by ensuring the greedy approach works optimally, as it allows quick traversal and decision-making for each interval.

What are common pitfalls when solving the Maximize Score of Numbers in Ranges?

Common pitfalls include failing to apply binary search correctly, making mistakes in the validation logic for feasible scores, and neglecting the importance of sorting the intervals.

terminal

Solution

Solution 1: Sorting + Binary Search

We can first sort the $\textit{start}$ array. Then, we consider selecting integers from left to right, where the score is equal to the minimum difference between any two adjacent selected integers.

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class Solution:
    def maxPossibleScore(self, start: List[int], d: int) -> int:
        def check(mi: int) -> bool:
            last = -inf
            for st in start:
                if last + mi > st + d:
                    return False
                last = max(st, last + mi)
            return True

        start.sort()
        l, r = 0, start[-1] + d - start[0]
        while l < r:
            mid = (l + r + 1) >> 1
            if check(mid):
                l = mid
            else:
                r = mid - 1
        return l

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