Binary Strings
Logical Breakdown
- [ ] Subproblem Identification:
- [ ] Optimal Substructure:
- [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
- [ ] Optimization: (Matrix Exponentiation if 'A' is huge)
Mathematical Rigor
State Definition
Let \(dp[i]\) be the state for the \(i\)-th subproblem.
Recurrence Relation
Visualization
Complexity Analysis
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Iterative DP | \(O(N)\) | \(O(1)\) |
Code Reference
package com.dsa.heaps;
/
* ============================================================================================
* PROBLEM: Binary Strings (Minimum Flips of Length B)
* ============================================================================================
*
* ---------------------
* 1. Logical Breakdown & Mathematical Formulation
* ---------------------
* We want to transform string A into all '1's using flips of fixed length B.
* * Step 1: The Greedy Choice
* - Scan from left to right. Whenever we encounter a '0', we MUST flip the window
* starting at that index to turn that '0' into a '1'.
* - There is no other way to fix that '0' without affecting elements to its left
* (which are already '1's).
*
* Step 2: Efficient Range Flipping (Difference Array / Sliding Window)
* - Manually flipping B elements takes O(B), leading to O(NB) total.
* - To achieve O(N), we track the "current flip state" using a variable currentFlips.
* - If currentFlips is odd, the actual value of A[i] is toggled.
* - We use an auxiliary array isFlipped[i] to mark where a flip window ends,
* allowing us to decrement currentFlips when the window slides past.
*
* ---------------------
* 2. Complexity Analysis
* ---------------------
* Time Complexity:
* O(N)
* - We traverse the string once, performing constant time operations at each bit.
*
* Space Complexity:
* O(N)
* - To maintain the isFlipped array to track active flip windows.
*
* ---------------------
* 3. Base Case & Failure Condition
* ---------------------
* - If we encounter a '0' within the last B-1 indices, it's impossible to start
* a full window of length B. In this case, return -1.
* ============================================================================================
/
public class BinaryStrings {
public int solve(String number, int A) {
// TODO: Implement solution
return 0;
}
}