#1855

Medium

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Maximum Distance Between a Pair of Values

Find the maximum distance between a pair of values in two non-increasing integer arrays using binary search.

Problem Statement

You are given two non-increasing integer arrays, nums1 and nums2. You need to find the maximum distance between a pair of indices (i, j) such that nums1[i] >= nums2[j] and i < j.

Both arrays are sorted in non-increasing order, meaning that nums1[i] >= nums1[i+1] and nums2[i] >= nums2[i+1] for all valid indices i. You must return the maximum distance between valid pairs of indices.

Examples

Example 1

Input: nums1 = [55,30,5,4,2], nums2 = [100,20,10,10,5]

Output: 2

The valid pairs are (0,0), (2,2), (2,3), (2,4), (3,3), (3,4), and (4,4). The maximum distance is 2 with pair (2,4).

Example 2

Input: nums1 = [2,2,2], nums2 = [10,10,1]

Output: 1

The valid pairs are (0,0), (0,1), and (1,1). The maximum distance is 1 with pair (0,1).

Example 3

Input: nums1 = [30,29,19,5], nums2 = [25,25,25,25,25]

Output: 2

The valid pairs are (2,2), (2,3), (2,4), (3,3), and (3,4). The maximum distance is 2 with pair (2,4).

Constraints

1 <= nums1.length, nums2.length <= 105
1 <= nums1[i], nums2[j] <= 105
Both nums1 and nums2 are non-increasing.

Solution Approach

Binary Search over Valid Pairs

Since both arrays are non-increasing, binary search is ideal to find the farthest valid pair. For each element in nums1, find the furthest element in nums2 that satisfies nums1[i] >= nums2[j]. This can be efficiently done using binary search to minimize time complexity.

Two Pointers Optimization

Two pointers can also help by keeping track of the current position in both arrays. Iterate over nums1 while adjusting a pointer in nums2 to maintain the condition nums1[i] >= nums2[j]. This can also be optimized by moving the pointer only forward, reducing redundant checks.

Edge Case Handling

Consider edge cases such as when nums1 or nums2 has only one element. Ensure that the solution works for the smallest array sizes, as well as for very large arrays up to the constraint limits.

Complexity Analysis

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

The time complexity depends on the approach chosen. Using binary search for each element in nums1 gives O(n log m), where n is the length of nums1 and m is the length of nums2. The space complexity can be O(1) if using only pointers or O(log m) if binary search requires additional space.

What Interviewers Usually Probe

Candidate uses binary search effectively for optimization.
Candidate demonstrates understanding of the relationship between the two arrays.
Candidate can handle edge cases and large inputs.

Common Pitfalls or Variants

Common pitfalls

Forgetting to handle edge cases where the arrays are very small.
Not optimizing the search for the valid pair using binary search.
Incorrectly assuming that the maximum valid pair is always at the ends of the arrays.

Follow-up variants

Modify the problem to handle arrays in increasing order instead of non-increasing.
Consider allowing duplicates in the arrays and adjusting the logic to handle these scenarios.
Extend the problem to find the maximum distance with different constraints, such as allowing negative numbers or non-sorted arrays.

FAQ

How do I apply binary search in the Maximum Distance Between a Pair of Values problem?

Binary search is used to find the furthest valid pair for each element in nums1, leveraging the fact that both arrays are sorted in non-increasing order.

What is the time complexity of the Maximum Distance Between a Pair of Values problem?

The time complexity can be O(n log m) where n is the length of nums1 and m is the length of nums2, if binary search is used.

How do I handle edge cases in this problem?

Ensure to check for small arrays and cases where no valid pairs exist, as these can cause errors in logic or assumptions.

Can I solve the Maximum Distance Between a Pair of Values problem without binary search?

Yes, but binary search significantly optimizes the solution, especially for larger inputs. Without it, the solution might be less efficient.

What is the primary pattern used in the Maximum Distance Between a Pair of Values problem?

The problem utilizes binary search over the valid answer space, which is efficient due to the sorted nature of the arrays.

terminal

Solution

Solution 1: Binary Search

Assume the lengths of $nums1$ and $nums2$ are $m$ and $n$ respectively.

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class Solution:
    def maxDistance(self, nums1: List[int], nums2: List[int]) -> int:
        ans = 0
        nums2 = nums2[::-1]
        for i, v in enumerate(nums1):
            j = len(nums2) - bisect_left(nums2, v) - 1
            ans = max(ans, j - i)
        return ans

Solution 2

#### Python3

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class Solution:
    def maxDistance(self, nums1: List[int], nums2: List[int]) -> int:
        ans = 0
        nums2 = nums2[::-1]
        for i, v in enumerate(nums1):
            j = len(nums2) - bisect_left(nums2, v) - 1
            ans = max(ans, j - i)
        return ans

Continue Practicing

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