Freedom Trail

The problem is related to finding the minimum number of steps required to align characters from a “ring” string to spell a given “key” string. You can rotate the ring either clockwise or anticlockwise, and after aligning the character, you need to press a button. Both the rotation and pressing count as steps.

We can approach this problem using dynamic programming, where we’ll define a memoization table to keep track of the minimum steps required to align the characters.

Here’s a step-by-step explanation for the solution:

1. Build a Mapping of Characters: Create a dictionary to keep track of the positions of each character in the ring. Since there might be multiple occurrences of a character, you can use lists to store their positions.

2. Create a Memoization Table: Define a 2D DP array dp, where dp[i][j] represents the minimum steps required to align the i-th character of the key at the j-th position of the ring.

3. Initialize the First Row: For the first character in the key, calculate the minimum steps required to align it with the corresponding characters in the ring.

4. Iteratively Build the Solution: For each subsequent character in the key, calculate the minimum steps required to align it based on the previous characters and update the DP array.

5. Return the Final Result: The answer will be in the last row of the DP array. Return the minimum value from that row.

Here’s the code:

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class Solution:
    def findRotateSteps(self, ring: str, key: str) -> int:
        from collections import defaultdict

        # Create a mapping of characters and their positions in the ring
        char_positions = defaultdict(list)
        for i, ch in enumerate(ring):
            char_positions[ch].append(i)

        n, m = len(key), len(ring)

        # Initialize DP array with maximum possible value
        dp = [[float('inf')] * m for _ in range(n)]

        # Initialize the first row
        for pos in char_positions[key[0]]:
            dp[0][pos] = min(pos, m - pos) + 1

        # Build the DP array
        for i in range(1, n):
            for pos in char_positions[key[i]]:
                for prev_pos in char_positions[key[i - 1]]:
                    # Calculate clockwise and anticlockwise distances
                    clockwise = abs(pos - prev_pos)
                    anticlockwise = m - clockwise
                    distance = min(clockwise, anticlockwise)
                    # Update the DP array
                    dp[i][pos] = min(dp[i][pos], dp[i - 1][prev_pos] + distance + 1)

        # Return the minimum value from the last row
        return min(dp[-1])

This approach has a time complexity of (O(n \times m^2)), where (n) is the length of the key, and (m) is the length of the ring, and a space complexity of (O(n \times m)).

10 Prerequisite LeetCode Problems

The “Freedom Trail” problem is a dynamic programming problem that requires understanding of string manipulation and state transitions. Here are 10 problems to prepare for solving it:

1. 72. Edit Distance: This problem is about computing the edit distance between two strings, a fundamental concept for understanding state transitions on strings.

2. 322. Coin Change: This dynamic programming problem helps you understand the concept of achieving a target using minimum steps.

3. 139. Word Break: A problem that involves string manipulation using dynamic programming.

4. 5. Longest Palindromic Substring: Understanding how to approach this problem will help with managing string manipulation in dynamic programming.

5. 300. Longest Increasing Subsequence: This problem will help you understand how to keep track of an optimal solution while iterating over the problem space.

6. 1143. Longest Common Subsequence: This problem requires a similar approach as the Freedom Trail problem, where you are dealing with two strings and looking for a sequence.

7. 516. Longest Palindromic Subsequence: This problem is related to string manipulation and finding a subsequence, which is useful for the Freedom Trail problem.

8. 97. Interleaving String: This problem requires understanding of both strings and dynamic programming.

9. 221. Maximal Square: Though not directly related to strings, the 2D dynamic programming concept in this problem is helpful.

10. 647. Palindromic Substrings: This problem helps with string manipulation and dynamic programming in a 2D problem space.

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class Solution:
    def findRotateSteps(self, ring: str, key: str) -> int:
        memo = {}
        def dp(i: int, j: int) -> int:
            if (i, j) in memo:
                return memo[(i, j)]
            if j == len(key):
                return 0
            ans = float('inf')
            for k in range(len(ring)):
                if ring[k] == key[j]:
                    delta = abs(k - i)
                    steps = min(delta, len(ring) - delta)
                    ans = min(ans, steps + dp(k, j+1))
            memo[(i, j)] = ans
            return ans
        return dp(0, 0) + len(key)

Problem Classification

Language Agnostic Coding Drills

1. Variables and Basic Data Structures: Understand how to create variables to store data. Learn to create and manipulate basic data structures like strings, integers, floats, and arrays. Also, understand the usage of dictionaries, a key-value pair data structure.

2. Control Flow - Loops and Conditionals: Learn to use if-else conditional statements and for and while loops. Practice using these in combination to iterate over data structures and make decisions in your code.

3. Functions and Recursion: Understand how to create functions to encapsulate pieces of code that you want to reuse. Learn about recursion, a method where the solution to a problem depends on solutions to smaller instances of the same problem.

4. Dynamic Programming: Learn the concept of dynamic programming, where you break down a problem into smaller subproblems and use the solutions to these subproblems to build up the solution to the original problem. Practice creating and using memoization to store and reuse solutions to subproblems.

5. Understanding the Problem Domain: The problem is a variant of the classic dynamic programming problem. The aim is to find the minimum number of steps to rotate a ring to get a key. Understand how the problem can be broken down into subproblems - finding the minimum steps to reach each character in the key from the current position in the ring.

6. Designing the Solution: Once you understand the problem, start thinking about the approach to solve it. Here the approach would be to iterate over each character in the ring and check if it matches the current character in the key. For each match, calculate the steps taken to reach the current character in the ring from the previous character and recursively compute the steps needed to reach the next character in the key. Use memoization to store the steps needed to reach each character in the key from each position in the ring to avoid recomputation.

7. Implementing the Solution: Once you have a good grasp of the problem and a clear idea of the solution, the final step would be to implement the solution. The implementation should be a combination of all the concepts learned in the previous steps.

Remember, the key is to understand each concept thoroughly and practice it in isolation. Once you are comfortable with each concept, you can then start combining them to solve complex problems. This method of learning and problem-solving is applicable to almost all modern programming languages.

Targeted Drills in Python

1. Variables and Basic Data Structures:

   • Strings and integers:

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   x = 10 y = "Hello, World!" print(x) print(y)

   • Lists and Dictionaries:

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   my_list = [1, 2, 3, 4, 5] my_dict = {'key1': 'value1', 'key2': 'value2'} print(my_list) print(my_dict)

2. Control Flow - Loops and Conditionals:

   • If-Else statements:

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   a = 5 if a > 0: print("Positive") else: print("Negative or Zero")

   • For and while loops:

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   for i in range(5): print(i) count = 0 while count < 5: print(count) count += 1

3. Functions and Recursion:

   • Creating a function:

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   def say_hello(name): print(f"Hello, {name}!") say_hello("John")

   • Recursion:

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   def factorial(n): if n == 1: return 1 else: return n * factorial(n - 1) print(factorial(5))

4. Dynamic Programming:

   • Basic memoization with Fibonacci sequence:

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   def fib(n, memo = {}): if n in memo: return memo[n] if n <= 2: return 1 memo[n] = fib(n-1, memo) + fib(n-2, memo) return memo[n] print(fib(50))

5. Problem Specific Drill:

   • Create a function to calculate the number of steps to rotate from one position to another on a ring with n positions:

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   def calculate_steps(n, from_pos, to_pos): delta = abs(to_pos - from_pos) steps = min(delta, n - delta) return steps print(calculate_steps(10, 3, 7))

Each of these drills focuses on a specific aspect that is part of the final solution. By mastering these drills, you can then combine them to solve the complex problem at hand.