#3113

Hard

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Find the Number of Subarrays Where Boundary Elements Are Maximum

Count the subarrays where the first and last elements are the largest in the subarray, utilizing binary search over valid answer space.

array binary search stack monotonic stack

Problem Statement

Given an array of positive integers, you need to return the number of subarrays where the first and last elements of the subarray are the largest element in that subarray.

For example, if nums = [1,4,3,3,2], the solution should count how many subarrays have the first and last elements as the largest value, which in this case is 6.

Examples

Example 1

Input: nums = [1,4,3,3,2]

Output: 6

There are 6 subarrays which have the first and the last elements equal to the largest element of the subarray: Hence, we return 6.

Example 2

Input: nums = [3,3,3]

Output: 6

There are 6 subarrays which have the first and the last elements equal to the largest element of the subarray: Hence, we return 6.

Example 3

Input: nums = [1]

Output: 1

There is a single subarray of nums which is [ 1 ] , with its largest element 1. The first element is 1 and the last element is also 1. Hence, we return 1.

Constraints

1 <= nums.length <= 105
1 <= nums[i] <= 109

Solution Approach

Binary Search over the Answer Space

To optimize the solution, binary search can be used to find the number of valid subarrays in a specified range. This method narrows down the solution space and helps to efficiently count subarrays where the boundary elements are the largest.

Stack-based Counting

A monotonic stack is used to track valid subarrays. For each element, we calculate how many subarrays end with it, ensuring that the first and last elements are equal to the largest element in the subarray.

Dynamic Sliding Window

A sliding window approach can also be used, dynamically adjusting the boundaries of subarrays while counting only the ones that satisfy the condition where both boundary elements are the largest.

Complexity Analysis

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

The time and space complexities of this problem depend on the final approach. A binary search solution typically offers better performance than brute force methods, especially when combined with a stack or sliding window to track subarrays.

What Interviewers Usually Probe

Ability to optimize brute force solutions using binary search and stacks.
Understanding of monotonic stacks for tracking subarray boundaries.
Skill in reducing time complexity by focusing on valid subarray counts.

Common Pitfalls or Variants

Common pitfalls

Ignoring the need to check both the first and last elements of the subarray during counting.
Relying on brute force counting without optimizing using binary search or stack methods.
Misunderstanding the boundary conditions when using sliding window techniques.

Follow-up variants

Variation with larger subarrays or larger input sizes.
Handling arrays with repeated values efficiently.
Different ways to approach the binary search or stack-based optimization.

FAQ

How do I optimize the solution for the problem 'Find the Number of Subarrays Where Boundary Elements Are Maximum'?

You can use binary search over the valid answer space combined with stack-based techniques to efficiently count valid subarrays.

What is the time complexity for this problem?

The time complexity depends on the approach used. Binary search combined with a stack can offer a more efficient solution than brute force.

How does a monotonic stack help in solving this problem?

A monotonic stack helps in efficiently counting subarrays by tracking the valid subarrays ending at each element while ensuring the boundaries are the largest.

What are the common mistakes in solving this problem?

Common mistakes include missing boundary element checks, using brute force without optimization, and misunderstanding the sliding window or stack approach.

Can I apply this problem's solution to other types of array-based subarray counting problems?

Yes, the binary search and stack-based techniques can be applied to similar problems that involve counting subarrays with specific properties.

terminal

Solution

Solution 1: Monotonic Stack

We consider each element $x$ in the array $nums$ as the boundary element and the maximum value of the subarray.

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class Solution:
    def numberOfSubarrays(self, nums: List[int]) -> int:
        stk = []
        ans = 0
        for x in nums:
            while stk and stk[-1][0] < x:
                stk.pop()
            if not stk or stk[-1][0] > x:
                stk.append([x, 1])
            else:
                stk[-1][1] += 1
            ans += stk[-1][1]
        return ans

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