#3584

Medium

auto_awesomeTwo-pointer scanning with invariant tracking

LeetCode Problem Workspace

Maximum Product of First and Last Elements of a Subsequence

Maximize the product of the first and last elements of any subsequence of size m in an integer array.

Problem Statement

You are given an integer array nums and an integer m. Your task is to return the maximum product of the first and last elements of any subsequence of nums of size m.

To achieve this, you need to select any subsequence of size m, where the first element is chosen from nums[i], and the last element is chosen from nums[i+m-1] through nums[n-1]. Your goal is to maximize the product of these two elements for any valid subsequence.

Examples

Example 1

Input: nums = [-1,-9,2,3,-2,-3,1], m = 1

Output: 81

The subsequence [-9] has the largest product of the first and last elements: -9 * -9 = 81 . Therefore, the answer is 81.

Example 2

Input: nums = [1,3,-5,5,6,-4], m = 3

Output: 20

The subsequence [-5, 6, -4] has the largest product of the first and last elements.

Example 3

Input: nums = [2,-1,2,-6,5,2,-5,7], m = 2

Output: 35

The subsequence [5, 7] has the largest product of the first and last elements.

Constraints

1 <= nums.length <= 105
-105 <= nums[i] <= 105
1 <= m <= nums.length

Solution Approach

Two-pointer scanning with invariant tracking

Use two pointers to identify the first and last elements of the subsequence. Track the maximum product by considering both positive and negative numbers while iterating through the array.

Optimized scanning using sliding window

Instead of checking every possible subsequence, use the sliding window technique to efficiently explore potential subsequences by adjusting the window size to m. This reduces redundant checks.

Considerations for negative values

When dealing with negative values, remember that two negative numbers can result in a positive product. Adjust your tracking to account for the possible impact of negative values on the product.

Complexity Analysis

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

Time complexity depends on the final approach, specifically the two-pointer scanning or sliding window technique. Space complexity also depends on the method chosen, but both approaches aim to minimize unnecessary space usage.

What Interviewers Usually Probe

Look for candidates who can implement the sliding window technique effectively.
Assess if the candidate handles both positive and negative numbers properly.
Evaluate whether the candidate can optimize the algorithm to avoid unnecessary computations.

Common Pitfalls or Variants

Common pitfalls

Failing to handle the case where negative numbers lead to a positive product.
Incorrectly tracking the subsequence, leading to suboptimal solutions.
Overcomplicating the problem by checking all possible subsequences rather than optimizing the approach.

Follow-up variants

Change the size of the subsequence m and assess the algorithm's adaptability.
Consider an additional constraint such as a maximum number of operations.
Test with arrays where the product results from negative numbers.

FAQ

What is the core approach to solving the Maximum Product of First and Last Elements of a Subsequence problem?

The core approach involves two-pointer scanning with invariant tracking, allowing you to efficiently find the subsequence with the maximum product of the first and last elements.

How can I optimize my solution for the Maximum Product of First and Last Elements of a Subsequence problem?

Use the sliding window technique to avoid checking all subsequences and reduce the time complexity. Additionally, be mindful of negative numbers and their impact on the product.

What are the key pitfalls in solving this problem?

Common pitfalls include failing to properly track the maximum product when negative numbers are involved and overcomplicating the solution by checking every subsequence.

Can I apply this approach to other similar array problems?

Yes, the two-pointer and sliding window approach is widely applicable to array problems that involve selecting subsequences or subarrays with specific properties.

How does GhostInterview help me solve problems like Maximum Product of First and Last Elements of a Subsequence?

GhostInterview provides real-time tips, insights on optimal solutions, and helps break down the problem into manageable steps, ensuring a targeted and efficient approach.

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Solution

Solution 1: Enumeration + Maintaining Prefix Extremes

We can enumerate the last element of the subsequence, assuming it is $\textit{nums}[i]$. Then the first element of the subsequence can be $\textit{nums}[j]$, where $j \leq i - m + 1$. Therefore, we use two variables $\textit{mi}$ and $\textit{mx}$ to maintain the prefix minimum and maximum values respectively. When traversing to $\textit{nums}[i]$, we update $\textit{mi}$ and $\textit{mx}$, then calculate the products of $\textit{nums}[i]$ with $\textit{mi}$ and $\textit{mx}$, taking the maximum value.

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class Solution:
    def maximumProduct(self, nums: List[int], m: int) -> int:
        ans = mx = -inf
        mi = inf
        for i in range(m - 1, len(nums)):
            x = nums[i]
            y = nums[i - m + 1]
            mi = min(mi, y)
            mx = max(mx, y)
            ans = max(ans, x * mi, x * mx)
        return ans

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