Problem Statement
Special Subsequence Counter
You are a data analyst working on a pattern recognition system. The system needs to identify special patterns in binary data streams. A special pattern is defined as a subsequence consisting of only 0s, followed by only 1s, followed by only 2s.
Given an array of integers nums consisting of only 0s, 1s, and 2s, count the number of special subsequences in the array.
A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.
Since the answer may be very large, return it modulo 10^9 + 7.
Examples
Example 1:
Input: nums = [0,1,2,2]
Output: 3
Explanation: The special subsequences are [0,1,2], [0,1,2,2], and [0,2,2]. Note that [0,2] is not a special subsequence because there is no 1 between 0 and 2.
Example 2:
Input: nums = [2,2,0,0]
Output: 0
Explanation: There are no special subsequences in [2,2,0,0].
Example 3:
Input: nums = [0,1,2,0,1,2]
Output: 7
Explanation: The special subsequences are [0,1,2], [0,1,2,2], [0,1,1,2], [0,1,1,2,2], [0,0,1,2], [0,0,1,2,2], and [0,0,1,1,2].
Constraints
- 1 <= nums.length <= 10^5
- 0 <= nums[i] <= 2
Problem Breakdown
To solve this problem, we need to:
- A special subsequence must have at least one 0, one 1, and one 2, in that order.
- We can use dynamic programming to count the number of special subsequences.
- For each position, we need to keep track of the number of subsequences ending with 0, 1, and 2.
- When we encounter a new element, we update the counts based on the previous counts.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Special Subsequence Counter
You are a data analyst working on a pattern recognition system. The system needs to identify special patterns in binary data streams. A special pattern is defined as a subsequence consisting of only 0s, followed by only 1s, followed by only 2s.
Given an array of integers nums consisting of only 0s, 1s, and 2s, count the number of special subsequences in the array.
A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.
Since the answer may be very large, return it modulo 10^9 + 7.
Examples
Example 1:
The special subsequences are [0,1,2], [0,1,2,2], and [0,2,2]. Note that [0,2] is not a special subsequence because there is no 1 between 0 and 2.
Example 2:
There are no special subsequences in [2,2,0,0].
Example 3:
The special subsequences are [0,1,2], [0,1,2,2], [0,1,1,2], [0,1,1,2,2], [0,0,1,2], [0,0,1,2,2], and [0,0,1,1,2].
Problem Breakdown
Constraints:
- 1 <= nums.length <= 10^5
- 0 <= nums[i] <= 2
Key Insights:
A special subsequence must have at least one 0, one 1, and one 2, in that order.
We can use dynamic programming to count the number of special subsequences.
For each position, we need to keep track of the number of subsequences ending with 0, 1, and 2.
When we encounter a new element, we update the counts based on the previous counts.
Real-World Application
This problem has several practical applications:
Pattern Recognition
Identifying specific patterns in data streams for anomaly detection or trend analysis.
Sequence Analysis
Analyzing genetic sequences or time-series data for specific patterns.
Text Processing
Finding specific patterns in text for natural language processing or information retrieval.