Cracking the Safe

Identifying Problem Isomorphism

“All Paths From Source to Target” presents a similar challenge to “Cracking the Safe”.

“Cracking the Safe” involves finding a string such that every possible password of length n can be seen as a substring. This problem can be seen as finding a path in a graph, where each node represents a password of length n and an edge between two nodes represents a password of length n+1 that can be formed by appending a digit at the end. The objective is to find a Hamiltonian path in this graph.

“All Paths From Source to Target” also involves graph traversal. In this problem, we are given a directed, acyclic graph and the task is to find all possible paths from the source (node 0) to the target (the last node).

While the context of the problems is different, both require exploring all paths in a graph. The key difference is that “Cracking the Safe” requires finding a Hamiltonian path (a path that visits each node exactly once), which adds an extra level of complexity compared to “All Paths From Source to Target”.

So, while they share the concept of graph traversal, the “Cracking the Safe” problem is more complex due to the requirement of finding a Hamiltonian path.

“Cracking the Safe” involves depth-first search (DFS) and graph theory. Here are 10 problems for a good preparation:

  1. 200. Number of Islands: A basic problem on depth-first search which is used to traverse a 2D grid.

  2. 79. Word Search: Another application of depth-first search in a 2D grid.

  3. 207. Course Schedule: This problem involves detecting a cycle in a directed graph which is a common pattern in many graph theory problems.

  4. 210. Course Schedule II: A continuation of the previous problem, but this time we are required to return a possible ordering of courses.

  5. 417. Pacific Atlantic Water Flow: This problem has a more complex condition for DFS traversal.

  6. 332. Reconstruct Itinerary: This problem involves creating a path from a graph.

  7. 261. Graph Valid Tree: This problem involves checking the structure of a graph.

  8. 130. Surrounded Regions: A good problem to understand how to traverse and modify a grid using DFS.

  9. 133. Clone Graph: This problem involves graph traversal and deep copying of a graph.

  10. 547. Number of Provinces: A good problem to understand disjoint set union and connected components in a graph.

The concept of DFS is often used in backtracking and graph theory problems, which are useful for solving problems like “Cracking the Safe”. Take the time to understand these concepts and how they are applied in these problems.

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15

class Solution:
    def crackSafe(self, n: int, k: int) -> str:
        seen = set()
        ans = []
        
        def dfs(node):
            for x in map(str, range(k)):
                nei = node + x
                if nei not in seen:
                    seen.add(nei)
                    dfs(nei[1:])
                    ans.append(x)
        
        dfs("0" * (n - 1))
        return "".join(ans) + "0" * (n - 1)

Identifying Problem Isomorphism

Can you help me with finding the isomorphism for this problem?

Which problem does this problem map to the corresponding isomorphic problem on Leetcode ?

How did you identify that this problem is a variation of problem?

Problem Classification

This problem falls under the domain of combinatorics and sequences. The problem involves finding a sequence of digits such that every possible combination of digits of a certain length appears at least once.

What Components:

  1. A safe protected by a password: This is the problem context which provides a scenario for the problem.

  2. The password: This is a sequence of ’n’ digits where each digit can be in the range [0, k - 1].

  3. The way the safe checks the password: When you enter a digit, the safe checks the most recent ’n’ digits entered.

  4. The objective: Find a sequence of minimum length that will unlock the safe at some point.

  5. Type: This is a combinatorial problem since it involves finding all possible combinations of digits of a certain length. It can also be seen as a string problem as it involves creating a sequence (string) of digits.

  6. Difficulty Level: The difficulty level seems to be medium as it requires some knowledge of combinatorics and perhaps graph theory to solve efficiently.

  7. Common Techniques Used for this Category: Depth-First Search (DFS) can be applied on a graph where each node represents a unique sequence of length ’n’ and there is an edge from ‘x’ to ‘y’ if the last ’n-1’ digits of ‘x’ are the same as the first ’n-1’ digits of ‘y’. The problem then becomes finding an Eulerian path or circuit in the graph, which can be found using Hierholzer’s Algorithm. Other combinatorial techniques like backtracking could also be potentially used.

Overall, this problem is a good test of understanding of combinatorics and graph theory, and would require careful construction of the sequence to ensure every possible password is covered.

Thought Process

Step 1: Understanding the problem

This problem is about generating a sequence of digits such that every possible combination of digits of length ’n’ appears at least once in the sequence.

The problem describes a safe that is opened by a password. The password is a sequence of n digits, and each digit can be anything from 0 to k-1. The safe checks the last n digits every time a digit is entered and opens if it matches the password.

Our job is to find a sequence of minimum length that will unlock the safe at some point.

Step 2: Problem Approach

Our first intuition can be that this problem can be solved using combinatorics and sequences. The number of possible combinations of digits of length ’n’ is k^n. So, our sequence needs to be long enough to accommodate all these combinations.

A useful way to view this problem is to consider a graph where each node represents a sequence of length ’n’ and there is a directed edge from node ‘x’ to node ‘y’ if the last ’n-1’ digits of ‘x’ are the same as the first ’n-1’ digits of ‘y’. In this graph, our problem becomes finding a sequence of nodes (an Eulerian path) such that every node is visited at least once.

We can construct this graph and then use Hierholzer’s algorithm to find an Eulerian path in the graph.

Step 3: Algorithm

  1. Construct a graph where each node represents a unique sequence of length ’n’ and there is an edge from ‘x’ to ‘y’ if the last ’n-1’ digits of ‘x’ are the same as the first ’n-1’ digits of ‘y’.
  2. Find an Eulerian path in the graph using Hierholzer’s algorithm. The Eulerian path is a path in the graph which visits every edge exactly once.

Step 4: Coding the solution

Let’s write a Python solution for this problem:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15

class Solution:
    def crackSafe(self, n: int, k: int) -> str:
        seen = set()
        ans = []
        
        def dfs(node):
            for x in map(str, range(k)):
                nei = node + x
                if nei not in seen:
                    seen.add(nei)
                    dfs(nei[1:])
                    ans.append(x)
        
        dfs("0" * (n - 1))
        return "".join(ans) + "0" * (n - 1)

Here, the main function is the crackSafe method. It uses a nested helper function dfs (depth-first search) which is the standard way to traverse through all nodes in a graph, which in this case represents all the possible combinations of the digits for the safe password. The solution implements a variation of Hierholzer’s Algorithm to find an Euler path through this graph of digit combinations.

In this function, we are first initializing an empty set seen to keep track of the visited nodes and an empty list ans to store our answer.

In the dfs function, we iterate over all possible next nodes (node + x) and if a node is not visited, we mark it as visited and perform DFS on it. After DFS, we add x to our answer.

Finally, we call dfs on the initial node (“0” * (n - 1)) and then return our answer as a string. We also append “0” * (n - 1) to our answer to ensure that the last n digits in our sequence form a valid combination.

This solution has a time complexity of O(k^n), which is the total number of possible combinations, and a space complexity of O(k^n) for storing the nodes in the set.

The above Python code follows the Hierholzer’s Algorithm to find the Eulerian path in the graph which guarantees to generate a sequence of minimum length that will unlock the safe at some point.

Language Agnostic Coding Drills

  1. Dissecting the code and identifying distinct concepts:

    • Variables and Data Types: This involves understanding and using different data types including strings, integers, lists, and sets.
    • Functions and Methods: Understanding how to declare and use functions is important. This code includes a main function crackSafe and an internal function dfs.
    • Loops: This code uses a for loop to iterate through a range of numbers.
    • Conditional Statements: An if condition is used to check if a specific node has been seen.
    • Recursion: The internal function dfs calls itself - a clear example of recursion.
    • String Manipulation: This includes concatenating strings and removing characters from strings.
    • Set Operations: The code uses a set to keep track of seen nodes and performs operations like checking membership and adding elements.

  2. Listing the coding concepts in order of increasing difficulty:

    • Variables and Data Types: This is one of the most basic concepts in any programming language.
    • Functions and Methods: Using functions and methods is also a fundamental concept but is a step up in complexity because it involves understanding function definitions and calls, and the flow of data in and out of functions.
    • Loops: Loops are an essential construct in programming for performing repetitive tasks.
    • Conditional Statements: They increase code complexity by adding decision points.
    • String Manipulation: This is slightly more complex as it involves working with indices and substrings.
    • Set Operations: These operations require understanding of how sets work, which is a bit more complex than basic data types.
    • Recursion: This is one of the more advanced concepts as it requires a solid understanding of the call stack and can lead to more complex code paths.

  3. Describing the problem-solving approach:

    • Understanding the problem: The first step is to understand the problem and constraints.
    • Brainstorming the approach: Think about how to tackle the problem. This involves considering the unique password sequence problem as an Eulerian path problem in a graph.
    • Implementing DFS: The Depth-First Search algorithm is a perfect fit for this problem. Implement a recursive DFS function that explores each node and its neighbors.
    • Adding conditions: Add a condition in the DFS to avoid revisiting nodes that have already been seen.
    • Building the answer: Use the DFS function to build the sequence that will unlock the safe.
    • Finishing touches: Concatenate the final password sequence with a string of zeroes to complete the password.

Each of these drills contributes to the overall solution by breaking down the problem into manageable chunks. The key is to build up from the basic concepts like variables and data types, towards more complex ones such as recursion, thereby constructing the solution piece by piece.

Targeted Drills in Python

  1. Python-based coding drills for each identified concept:

    • Variables and Data Types:

      1
2
3
4
5

      my_int = 10
my_str = "Hello, World!"
my_list = [1, 2, 3, 4, 5]
my_set = {1, 2, 3, 3, 4}
print(my_int, my_str, my_list, my_set)

    • Functions and Methods:

      1
2
3
4

      def greet(name):
    return f"Hello, {name}!"

print(greet("Alice"))

    • Loops:

      1
2

      for i in range(5):
    print(i)

    • Conditional Statements:

      1
2
3
4
5

      x = 10
if x > 5:
    print("x is greater than 5")
else:
    print("x is not greater than 5")

    • String Manipulation:

      1
2
3

      s = "Hello, World!"
print(s + " How are you?")
print(s[1:5])

    • Set Operations:

      1
2
3
4
5
6

      s = set()
s.add(1)
s.add(2)
print(s)
if 1 in s:
    print("1 is in the set")

    • Recursion:

      1
2
3
4
5
6
7

      def factorial(n):
    if n == 1:
        return 1
    else:
        return n * factorial(n-1)

print(factorial(5))

  2. Problem-specific concept: Depth-first Search (DFS):

    Depth-first search is a fundamental graph traversal algorithm. Here’s a simple illustration:

     1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19

    graph = {
    'A': ['B', 'C'],
    'B': ['A', 'D', 'E'],
    'C': ['A', 'F'],
    'D': ['B'],
    'E': ['B', 'F'],
    'F': ['C', 'E']
}

visited = set()

def dfs(visited, graph, node):
    if node not in visited:
        print(node)
        visited.add(node)
        for neighbor in graph[node]:
            dfs(visited, graph, neighbor)

dfs(visited, graph, 'A')

    This drill is essential for our problem because DFS is the core algorithm we use to find a path that covers all possible n-length sequences.

  3. Integrating the drills:

    Start by defining the main function crackSafe and initialize the required variables and data structures. Implement the DFS algorithm inside this function to traverse through all the possible sequences. During the DFS, perform string manipulations and set operations to track seen sequences and to generate new sequences. Keep adding the last character of each visited sequence to the answer using list operations. Finally, return the password by joining the characters in the answer and appending ‘0’ repeated (n - 1) times. The order of the drills would be: Variables & Data Types, Functions & Methods, Loops, Conditional Statements, String Manipulations, Set Operations, and finally Recursion (DFS).