Problem Statement
Dynamic Window Statistics
You're analyzing a time series dataset and need to calculate the median value for each window of consecutive data points.
Median is the middle value in an ordered integer list. If the size of the list is even, there is no middle value. So the median is the mean of the two middle values.
For example:
[2,3,4], the median is3[2,3], the median is(2 + 3) / 2 = 2.5
Given an array nums, there is a sliding window of size k which is moving from the very left of the array to the very right. You can only see the k numbers in the window. Each time the sliding window moves right by one position.
Your task is to output the median array for each window in the original array.
Examples
Example 1:
Input: nums = [1,3,-1,-3,5,3,6,7], k = 3
Output: [1,-1,-1,3,5,6]
Explanation: Window position Median
--------------- -----
[1 3 -1] -3 5 3 6 7 1
1 [3 -1 -3] 5 3 6 7 -1
1 3 [-1 -3 5] 3 6 7 -1
1 3 -1 [-3 5 3] 6 7 3
1 3 -1 -3 [5 3 6] 7 5
1 3 -1 -3 5 [3 6 7] 6
Constraints
- 1 <= k <= nums.length <= 10^5
- -2^31 <= nums[i] <= 2^31 - 1
Problem Breakdown
To solve this problem, we need to:
- The median of a window can be efficiently maintained using two heaps: a max heap for the smaller half and a min heap for the larger half
- As the window slides, we need to add a new element and remove an old element
- Removing elements from a heap is not straightforward, so we need to handle 'lazy deletion'
- We can also use a self-balancing binary search tree (like a multiset in C++) to maintain the sorted order of elements in the window
- The time complexity is dominated by the operations to maintain the sorted order of elements in the window
- For even-sized windows, the median is the average of the two middle elements
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Dynamic Window Statistics
You're analyzing a time series dataset and need to calculate the median value for each window of consecutive data points.
Median is the middle value in an ordered integer list. If the size of the list is even, there is no middle value. So the median is the mean of the two middle values.
For example:
[2,3,4], the median is3[2,3], the median is(2 + 3) / 2 = 2.5
Given an array nums, there is a sliding window of size k which is moving from the very left of the array to the very right. You can only see the k numbers in the window. Each time the sliding window moves right by one position.
Your task is to output the median array for each window in the original array.
Examples
Example 1:
Window position Median --------------- ----- [1 3 -1] -3 5 3 6 7 1 1 [3 -1 -3] 5 3 6 7 -1 1 3 [-1 -3 5] 3 6 7 -1 1 3 -1 [-3 5 3] 6 7 3 1 3 -1 -3 [5 3 6] 7 5 1 3 -1 -3 5 [3 6 7] 6
Problem Breakdown
Constraints:
- 1 <= k <= nums.length <= 10^5
- -2^31 <= nums[i] <= 2^31 - 1
Key Insights:
The median of a window can be efficiently maintained using two heaps: a max heap for the smaller half and a min heap for the larger half
As the window slides, we need to add a new element and remove an old element
Removing elements from a heap is not straightforward, so we need to handle 'lazy deletion'
We can also use a self-balancing binary search tree (like a multiset in C++) to maintain the sorted order of elements in the window
The time complexity is dominated by the operations to maintain the sorted order of elements in the window
For even-sized windows, the median is the average of the two middle elements
Real-World Application
This problem has several practical applications:
Time Series Analysis
Calculating moving medians for financial data, weather patterns, or sensor readings.
Signal Processing
Filtering noise from signals by using median filters that are less sensitive to outliers than mean filters.
Data Monitoring
Monitoring system metrics and detecting anomalies based on median values in recent time windows.
Statistical Analysis
Computing rolling medians for robust statistical analysis of sequential data.
Database Systems
Implementing moving median queries for analytical database functions.