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Find All Possible Stable Binary Arrays I
Find all stable binary arrays of a given number of 0's, 1's, and limit using dynamic programming and state transitions.
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Practice Focus
Medium · State transition dynamic programming
Answer-first summary
Find all stable binary arrays of a given number of 0's, 1's, and limit using dynamic programming and state transitions.
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The problem asks to find all stable binary arrays with a given number of 0's, 1's, and a limit on consecutive identical elements. The solution can be approached by dynamic programming and state transitions to efficiently count valid arrays without violating the limit condition. This can be framed with a 4D DP array to track the count of stable arrays based on specific counts and consecutive values.
Problem Statement
You are given three positive integers zero, one, and limit, which represent the number of 0's, 1's, and the maximum allowed consecutive identical digits in a binary array, respectively.
A binary array is called stable if it does not contain any subarray with consecutive identical elements exceeding the limit. Return the total number of stable binary arrays that can be formed with exactly zero 0's and one 1's under this constraint.
Examples
Example 1
Input: zero = 1, one = 1, limit = 2
Output: 2
The two possible stable binary arrays are [1,0] and [0,1] , as both arrays have a single 0 and a single 1, and no subarray has a length greater than 2.
Example 2
Input: zero = 1, one = 2, limit = 1
Output: 1
The only possible stable binary array is [1,0,1] . Note that the binary arrays [1,1,0] and [0,1,1] have subarrays of length 2 with identical elements, hence, they are not stable.
Example 3
Input: zero = 3, one = 3, limit = 2
Output: 14
All the possible stable binary arrays are [0,0,1,0,1,1] , [0,0,1,1,0,1] , [0,1,0,0,1,1] , [0,1,0,1,0,1] , [0,1,0,1,1,0] , [0,1,1,0,0,1] , [0,1,1,0,1,0] , [1,0,0,1,0,1] , [1,0,0,1,1,0] , [1,0,1,0,0,1] , [1,0,1,0,1,0] , [1,0,1,1,0,0] , [1,1,0,0,1,0] , and [1,1,0,1,0,0] .
Constraints
- 1 <= zero, one, limit <= 200
Solution Approach
Dynamic Programming with State Transitions
Use dynamic programming (DP) to track the number of stable arrays for each possible count of 0's and 1's, along with the current consecutive value. Define a DP state dp[a][b][c][d] where a is the count of 0's, b is the count of 1's, c is the current digit (0 or 1), and d is the number of consecutive identical digits. Transition between states while ensuring the subarray stability condition is met.
Prefix Sum Optimization
To optimize the solution, employ prefix sums to quickly compute and transition between DP states without recalculating from scratch each time. This allows for efficient counting of valid transitions, reducing the time complexity of the approach.
Space Complexity Reduction
To reduce space complexity, use only a sliding window approach on the DP states, keeping track of only the previous and current states. This optimization reduces the overall space requirement and maintains efficiency while processing the transitions.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time and space complexity depend on the approach used for dynamic programming, with the worst-case scenario requiring a 4D DP table. Optimizations like prefix sums and space reduction techniques can significantly improve the overall complexity. A well-optimized solution could have a time complexity of O(zero * one * limit) and space complexity of O(zero * one * limit).
What Interviewers Usually Probe
- Look for the candidate's understanding of dynamic programming and state transitions.
- Evaluate the candidate's ability to optimize both time and space complexity in a DP-based approach.
- Assess the candidate's ability to apply the prefix sum technique to optimize counting and state transitions.
Common Pitfalls or Variants
Common pitfalls
- Forgetting to account for the consecutive identical digit limit while transitioning between states.
- Not using the sliding window technique to optimize space complexity in larger test cases.
- Inefficient transitions or recalculations leading to excessive time complexity.
Follow-up variants
- Different limit values could be tested to assess the solution's ability to handle various stability constraints.
- The number of 0's and 1's could be varied to test the candidate's ability to handle different binary array sizes.
- Testing with edge cases, such as limit being 1 or 0, will assess the robustness of the candidate's solution.
FAQ
What is the primary pattern for solving the Find All Possible Stable Binary Arrays I problem?
The primary pattern is state transition dynamic programming, where each state represents the number of stable arrays with specific counts of 0's, 1's, and consecutive digits.
How do I optimize the space complexity for this problem?
You can reduce space complexity by using a sliding window technique to store only the previous and current DP states, instead of maintaining a full 4D DP table.
What is the time complexity of the best solution for this problem?
The time complexity of the best solution depends on the number of zeros, ones, and the limit, with an optimized approach achieving O(zero * one * limit).
How does the prefix sum optimization improve the solution?
Prefix sum optimization allows for faster state transitions by precomputing the counts, reducing the need for repeated calculations and improving overall time complexity.
What are the edge cases I should consider for this problem?
Consider edge cases such as a limit of 1, a limit of 0, or when the number of 0's or 1's is at its minimum or maximum allowed value.
Solution
Solution 1: Memoized Search
We define a function $\textit{dfs}(i, j, k)$ to represent the number of stable binary arrays that satisfy the problem conditions when there are $i$ zeros and $j$ ones remaining to place, and the next digit to fill is $k$. Then the answer is $\textit{dfs}(\textit{zero}, \textit{one}, 0) + \textit{dfs}(\textit{zero}, \textit{one}, 1)$.
class Solution:
def numberOfStableArrays(self, zero: int, one: int, limit: int) -> int:
@cache
def dfs(i: int, j: int, k: int) -> int:
if i == 0:
return int(k == 1 and j <= limit)
if j == 0:
return int(k == 0 and i <= limit)
if k == 0:
return (
dfs(i - 1, j, 0)
+ dfs(i - 1, j, 1)
- (0 if i - limit - 1 < 0 else dfs(i - limit - 1, j, 1))
)
return (
dfs(i, j - 1, 0)
+ dfs(i, j - 1, 1)
- (0 if j - limit - 1 < 0 else dfs(i, j - limit - 1, 0))
)
mod = 10**9 + 7
ans = (dfs(zero, one, 0) + dfs(zero, one, 1)) % mod
dfs.cache_clear()
return ansSolution 2: Dynamic Programming
We can also convert the memoized search in Solution 1 into dynamic programming.
class Solution:
def numberOfStableArrays(self, zero: int, one: int, limit: int) -> int:
@cache
def dfs(i: int, j: int, k: int) -> int:
if i == 0:
return int(k == 1 and j <= limit)
if j == 0:
return int(k == 0 and i <= limit)
if k == 0:
return (
dfs(i - 1, j, 0)
+ dfs(i - 1, j, 1)
- (0 if i - limit - 1 < 0 else dfs(i - limit - 1, j, 1))
)
return (
dfs(i, j - 1, 0)
+ dfs(i, j - 1, 1)
- (0 if j - limit - 1 < 0 else dfs(i, j - limit - 1, 0))
)
mod = 10**9 + 7
ans = (dfs(zero, one, 0) + dfs(zero, one, 1)) % mod
dfs.cache_clear()
return ansContinue Topic
dynamic programming
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State transition dynamic programming
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Medium
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