Cracking the Safe
Problem Description
There is a safe protected by a password. The password is a sequence of n digits where each digit can be in the range [0, k - 1].
The safe has a peculiar way of checking the password. When you enter in a sequence, it checks the most recent n digits that were entered each time you type a digit.
For example, the correct password is "345" and you enter in "012345":
- After typing
0, the most recent3digits is"0", which is incorrect. - After typing
1, the most recent3digits is"01", which is incorrect. - After typing
2, the most recent3digits is"012", which is incorrect. - After typing
3, the most recent3digits is"123", which is incorrect. - After typing
4, the most recent3digits is"234", which is incorrect. - After typing
5, the most recent3digits is"345", which is correct and the safe unlocks.
Return any string of minimum length that will unlock the safe at some point of entering it.
Example 1:
Input: n = 1, k = 2
Output: "10"
Explanation: The password is a single digit, so enter each digit. "01" would also unlock the safe.
Example 2:
Input: n = 2, k = 2
Output: "01100"
Explanation: For each possible password:
- "00" is typed in starting from the 4th digit.
- "01" is typed in starting from the 1st digit.
- "10" is typed in starting from the 3rd digit.
- "11" is typed in starting from the 2nd digit.Thus "01100" will unlock the safe. "10011", and "11001" would also unlock the safe.
Constraints:
- 1 <= n <= 4
- 1 <= k <= 10
- 1 <= kn <= 4096
Solve it on LeetCode
Approaches
1. Eulerian Circuit (Hierholzer’s Algorithm)
Intuition:
The problem can be seen as finding an Eulerian path in a de Bruijn graph. For a given n and k, the graph is constructed where each vertex is a string of length n-1, and edges are created by appending one more character from 0 to k-1. An Eulerian circuit exists if each vertex in a directed graph has equal in-degree and out-degree, which is the case here.
Here's the step-by-step intuition for this approach:
- Graph Formation:
- Nodes: Every password of length
n-1. - Edges: Each edge represents the possible extensions of these nodes by adding one of
kcharacters (forming the next state). - Hierholzer's Algorithm:
- Choose an arbitrary starting point and follow a trail of unused edges to build the path.
- Continue walking through unused edges until you return to your starting point, creating a cycle.
- If there are still unused edges, start a new cycle from one of the vertices on your current path.
- Connect all cycles to form the Eulerian circuit.
Code:
- Time Complexity: O(k^n). This is because each node can be extended by
kpossibilities and there are approximatelyk^(n-1)nodes to explore. - Space Complexity: O(k^n). To store the visited nodes (each of which is a password).
2. Recursive Backtracking (Brute Force)
Intuition:
In this approach, we attempt every possible combination while ensuring that the solution is of the minimum length. The backtracking will involve generating each possible path and checking if it could complete the cracking of the safe.
- Path Exploration:
- Start with an initial state.
- Explore all potential extensions from each state sequentially.
- Backtrack once each potential path is entirely explored or deemed unviable.
- Cycle Formulation:
- Start forming cycles from smaller sequences while checking if they form a valid sequence.
Code:
- Time Complexity: O(k^n + n*k^n), due to the high cost of exploring each path and verifying possible configurations.
- Space Complexity: O(n*k^n), for storing the paths and visited nodes.