Problem Statement
Maximum Subarray Finder
You are a financial analyst working for an investment firm. You're analyzing a stock's daily price changes (which can be positive or negative) over a period of time. You want to find the maximum profit you could have made by buying and selling the stock multiple times within a specific timeframe.
However, you suspect that there might be one day with incorrect data that's affecting your analysis. You decide to find the maximum sum of a subarray after removing at most one element from the array.
Given an array of integers arr, find the maximum sum of a subarray with at most one element deletion.
Examples
Example 1:
Input: arr = [1, -2, 0, 3]
Output: 4
Explanation: We can get a sum of 4 by deleting -2 and summing the remaining elements: 1 + 0 + 3 = 4.
Example 2:
Input: arr = [1, -2, -2, 3]
Output: 3
Explanation: We can get a sum of 3 by keeping just the last element, 3.
Example 3:
Input: arr = [-1, -1, -1, -1]
Output: -1
Explanation: We can get a sum of -1 by deleting any one of the elements and summing the remaining three elements.
Constraints
- 1 <= arr.length <= 10^5
- -10^4 <= arr[i] <= 10^4
Problem Breakdown
To solve this problem, we need to:
- This problem is an extension of the classic maximum subarray problem (Kadane's algorithm).
- We need to consider two cases: the maximum subarray sum without deletion, and the maximum subarray sum with one deletion.
- To handle the deletion case efficiently, we can use dynamic programming to keep track of the maximum subarray sum ending at each position, both with and without a deletion.
- The key insight is that when we delete an element, we're essentially connecting two subarrays on either side of the deleted element.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Maximum Subarray Finder
You are a financial analyst working for an investment firm. You're analyzing a stock's daily price changes (which can be positive or negative) over a period of time. You want to find the maximum profit you could have made by buying and selling the stock multiple times within a specific timeframe.
However, you suspect that there might be one day with incorrect data that's affecting your analysis. You decide to find the maximum sum of a subarray after removing at most one element from the array.
Given an array of integers arr, find the maximum sum of a subarray with at most one element deletion.
Examples
Example 1:
We can get a sum of 4 by deleting -2 and summing the remaining elements: 1 + 0 + 3 = 4.
Example 2:
We can get a sum of 3 by keeping just the last element, 3.
Example 3:
We can get a sum of -1 by deleting any one of the elements and summing the remaining three elements.
Problem Breakdown
Constraints:
- 1 <= arr.length <= 10^5
- -10^4 <= arr[i] <= 10^4
Key Insights:
This problem is an extension of the classic maximum subarray problem (Kadane's algorithm).
We need to consider two cases: the maximum subarray sum without deletion, and the maximum subarray sum with one deletion.
To handle the deletion case efficiently, we can use dynamic programming to keep track of the maximum subarray sum ending at each position, both with and without a deletion.
The key insight is that when we delete an element, we're essentially connecting two subarrays on either side of the deleted element.
Real-World Application
This problem has several practical applications:
Financial Analysis
Analyzing stock price movements while accounting for potential data errors.
Signal Processing
Finding the strongest signal segment while filtering out noise or interference.
Data Cleaning
Identifying and removing outliers in sequential data to improve analysis.