Problem Statement
Array Increaser
You are a data analyst working with a time series dataset that should be strictly increasing. However, due to measurement errors, some values in the dataset are incorrect. You want to fix the dataset by replacing some values with values from another reference dataset.
Given two arrays of integers arr1 and arr2, your task is to make arr1 strictly increasing by replacing any number of elements in arr1 with any number of elements from arr2.
An array is strictly increasing if and only if arr[i] < arr[i+1] for all 0 <= i < arr.length - 1.
Return the minimum number of replacements needed to make arr1 strictly increasing. If it's impossible to make arr1 strictly increasing, return -1.
Examples
Example 1:
Input: arr1 = [1, 5, 3, 6, 7], arr2 = [1, 3, 2, 4]
Output: 1
Explanation: Replace arr1[2] with arr2[1] = 3. The resulting array is [1, 5, 3, 6, 7] -> [1, 5, 3, 6, 7], which is strictly increasing.
Example 2:
Input: arr1 = [1, 5, 3, 6, 7], arr2 = [4, 3, 1]
Output: 2
Explanation: Replace arr1[0] with arr2[0] = 4 and arr1[1] with arr2[1] = 3. The resulting array is [1, 5, 3, 6, 7] -> [4, 3, 3, 6, 7], which is not strictly increasing. We need to replace arr1[2] with arr2[2] = 1, resulting in [4, 3, 1, 6, 7], which is still not strictly increasing. It's impossible to make arr1 strictly increasing with the given arr2.
Example 3:
Input: arr1 = [1, 5, 3, 6, 7], arr2 = [1, 6, 3, 3]
Output: -1
Explanation: It's impossible to make arr1 strictly increasing with the given arr2.
Constraints
- 1 <= arr1.length, arr2.length <= 2000
- 0 <= arr1[i], arr2[i] <= 10^9
Problem Breakdown
To solve this problem, we need to:
- The key insight is to use dynamic programming to track the minimum number of replacements needed to make the array strictly increasing up to each position.
- For each position, we have two options: keep the original element or replace it with an element from arr2.
- We need to ensure that the element at each position is greater than the element at the previous position.
- Sorting arr2 and removing duplicates can simplify the problem, as we only need to consider the smallest valid replacement.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Array Increaser
You are a data analyst working with a time series dataset that should be strictly increasing. However, due to measurement errors, some values in the dataset are incorrect. You want to fix the dataset by replacing some values with values from another reference dataset.
Given two arrays of integers arr1 and arr2, your task is to make arr1 strictly increasing by replacing any number of elements in arr1 with any number of elements from arr2.
An array is strictly increasing if and only if arr[i] < arr[i+1] for all 0 <= i < arr.length - 1.
Return the minimum number of replacements needed to make arr1 strictly increasing. If it's impossible to make arr1 strictly increasing, return -1.
Examples
Example 1:
Replace arr1[2] with arr2[1] = 3. The resulting array is [1, 5, 3, 6, 7] -> [1, 5, 3, 6, 7], which is strictly increasing.
Example 2:
Replace arr1[0] with arr2[0] = 4 and arr1[1] with arr2[1] = 3. The resulting array is [1, 5, 3, 6, 7] -> [4, 3, 3, 6, 7], which is not strictly increasing. We need to replace arr1[2] with arr2[2] = 1, resulting in [4, 3, 1, 6, 7], which is still not strictly increasing. It's impossible to make arr1 strictly increasing with the given arr2.
Example 3:
It's impossible to make arr1 strictly increasing with the given arr2.
Problem Breakdown
Constraints:
- 1 <= arr1.length, arr2.length <= 2000
- 0 <= arr1[i], arr2[i] <= 10^9
Key Insights:
The key insight is to use dynamic programming to track the minimum number of replacements needed to make the array strictly increasing up to each position.
For each position, we have two options: keep the original element or replace it with an element from arr2.
We need to ensure that the element at each position is greater than the element at the previous position.
Sorting arr2 and removing duplicates can simplify the problem, as we only need to consider the smallest valid replacement.
Real-World Application
This problem has several practical applications:
Data Cleaning
Correcting errors in time series data to ensure monotonicity.
Sequence Alignment
Aligning biological sequences by making minimal substitutions.
Sorting Optimization
Optimizing sorting algorithms by minimizing the number of swaps or replacements.