This problem requires counting splits of a binary array where each subarray contains exactly one 1. A state transition dynamic programming approach tracks valid partitions efficiently, handling large arrays with modulo operations. The solution considers distances between 1s and combines results from previous states to compute the total number of splits.
Problem Statement
Given a binary array nums, a subarray is considered good if it contains exactly one element with the value 1. Determine how many ways the entire array can be split into consecutive good subarrays.
Return the total number of valid splits modulo 10^9 + 7. If the array contains only zeros, the result should be zero. Each split must maintain the original order and cover the entire array.
Examples
Example 1
Input: nums = [0,1,0,0,1]
Output: 3
There are 3 ways to split nums into good subarrays:
- [0,1] [0,0,1]
- [0,1,0] [0,1]
- [0,1,0,0] [1]
Example 2
Input: nums = [0,1,0]
Output: 1
There is 1 way to split nums into good subarrays:
- [0,1,0]
Constraints
- 1 <= nums.length <= 105
- 0 <= nums[i] <= 1
Solution Approach
Identify positions of 1s
Traverse the array to record indices where nums[i] == 1. These positions determine the boundaries of potential good subarrays and allow calculation of distances between consecutive 1s for dynamic programming transitions.
Dynamic programming state definition
Define dp[i] as the number of ways to split the array up to the i-th 1. Use the distances between 1s to update dp[i] from dp[i-1] by multiplying the number of ways to partition the zeros between them, applying modulo 10^9 + 7.
Compute total splits
Iterate through all recorded 1 positions using the state transition formula to fill the dp array. The final dp value at the last 1 gives the total number of ways to split the array into good subarrays.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
Time complexity is O(n) for a single pass to identify 1 positions and compute dp transitions. Space complexity is O(k) where k is the number of 1s, as we only store dp states and positions of 1s.
What Interviewers Usually Probe
- Check for edge case when the array contains only zeros; answer should be 0.
- Ask about handling large arrays and modulo operations to prevent integer overflow.
- Probe understanding of state transition DP and distances between consecutive 1s.
Common Pitfalls or Variants
Common pitfalls
- Forgetting to return 0 when there are no 1s in the array.
- Incorrectly handling zeros at the beginning or end of the array in the DP calculation.
- Neglecting the modulo operation, which can cause integer overflow for large arrays.
Follow-up variants
- Allow subarrays to contain at most one 1 instead of exactly one 1, changing the DP transition rules.
- Count splits where each good subarray contains exactly k ones, requiring generalized state transitions.
- Consider arrays with more than two distinct values, extending the definition of good subarrays.
How GhostInterview Helps
- GhostInterview guides you through indexing 1s and constructing dynamic programming states step by step.
- It highlights potential pitfalls like all-zero arrays and modulo operations, ensuring correct results.
- It provides immediate verification of DP transitions with test cases, reinforcing the state transition logic.
Topic Pages
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- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
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- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the key pattern used in Ways to Split Array Into Good Subarrays?
The key pattern is state transition dynamic programming, where each dp state depends on distances between consecutive 1s.
How do I handle arrays that contain only zeros?
Return 0 immediately since there are no good subarrays possible if the array has no 1s.
Why do we apply modulo 10^9 + 7 in the solution?
Because the number of ways can be very large, modulo ensures results fit within integer limits.
Can this approach handle arrays up to length 10^5 efficiently?
Yes, identifying 1 positions and using state transition DP ensures an O(n) time solution suitable for large arrays.
What common mistake occurs in computing dp states?
A common mistake is miscounting the number of zeros between 1s or forgetting to multiply correctly for multiple partition options.
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