**π‘ Problem Formulation:** The problem at hand involves deciphering a message encoded as a string of digits. The mapping follows ‘A’ to 1, ‘B’ to 2, and so on until ‘Z’ with 26. Given an encoded message as a string of digits, we want to count the number of ways it can be decoded. For instance, the input “123” can be decoded as “ABC”, “LC”, or “AW”. So the desired output for “123” is 3, indicating the three possible decodings.

## Method 1: Dynamic Programming

This method uses dynamic programming to efficiently compute the number of ways the message can be decoded. We iterate through the message and dynamically store the number of decodings up to the current position, considering single and double-digit combinations that are valid. The function specification includes a string as input and returns an integer representing the count of possible decodings.

Here’s an example:

def num_decodings(s: str) -> int: if not s or s[0] == '0': return 0 dp = [0] * (len(s) + 1) dp[0:2] = [1, 1] for i in range(2, len(s) + 1): if s[i-1] != '0': dp[i] += dp[i - 1] if 10 <= int(s[i-2:i]) <= 26: dp[i] += dp[i - 2] return dp[len(s)] print(num_decodings("123"))

Output: 3

This code snippet initiates a list `dp`

with length one greater than the message to account for the base case. The first two elements of `dp`

are initialized to 1, representing the count of decodings up to that point. The algorithm then iterates through the message, checking for valid single and double-digit combinations, updating the `dp`

array accordingly to store the results. Finally, the last element of `dp`

gives the total count of decode ways for the entire message.

## Method 2: Recursive Approach

The recursive method breaks down the problem into smaller subproblems. For each digit, we recursively find the number of ways to decode the rest of the message, adding the counts for cases where the current and next digits form a valid character. This method should be used with memoization to avoid repeated calculations. The function takes a string input and returns the number of possible decodings as an integer.

Here’s an example:

def helper(s, k, memo): if k == 0: return 1 start = len(s) - k if s[start] == '0': return 0 if memo[k]: return memo[k] result = helper(s, k-1, memo) if k >= 2 and int(s[start:start+2]) int: memo = [None] * (len(s) + 1) return helper(s, len(s), memo) print(num_decodings("123"))

Output: 3

This code example defines a helper function that uses recursion to calculate the decodings, using a memoization array to store intermediate results and prevent recalculating subproblems. `k`

is the length of the substring to be decoded. When `k`

reaches 0, it signifies all digits have been successfully decoded. The main function `num_decodings`

initializes the memo array and calls the helper function to compute the result.

## Method 3: Iterative Approach with Constant Space

This iterative approach enhances the dynamic programming solution by using constant space instead of a list. The method primarily relies on two variables to store the current and previous number of decodings. It iteratively updates these two variables as it processes each digit β similar to Fibonacci number computation. The function again accepts a string and returns the number of possible decodings as an integer.

Here’s an example:

def num_decodings(s: str) -> int: if not s or s[0] == '0': return 0 prev, cur = 1, 1 for i in range(1, len(s)): tmp = cur if s[i] == '0': if s[i-1] not in ('1', '2'): return 0 cur = prev elif 10 <= int(s[i-1:i+1]) <= 26: cur += prev prev = tmp return cur print(num_decodings("123"))

Output: 3

The code begins by checking if the input is valid and initializes two variables to 1, representing the number of ways for the empty string and the first digit (if not ‘0’). It then loops through each digit, updating `cur`

and `prev`

based on whether a single digit or a pair of digits forms a valid letter. The value of `cur`

at the end of the loop is the total number of decodings.

## Method 4: Using a Stack

While not as intuitive as dynamic programming, a stack-based approach can be employed to count decode ways. We use a stack to keep track of the state of our decoding process, pushing and popping elements according to whether they form valid letters. This approach is more technical and can result in complex code but serves as an example of applying stack structures to dynamic programming problems.

This method will not be laid out in code due to its complexity and lesser relevance than the other methods presented, but it is mentioned to show the versatility of solving this particular problem.

## Bonus One-Liner Method 5: Functional Approach with `reduce()`

If you’re a fan of concise code, Python’s `reduce()`

function from the `functools`

module can be used to apply the dynamic programming logic in a single line. This one-liner nests the logic of the ‘Iterative Approach with Constant Space’ method into a reduction, succinctly and elegantly computing the result.

Here’s an example:

from functools import reduce def num_decodings(s: str) -> int: return reduce(lambda x, y: (x[1], (x[1] if y == '0' else x[0] + x[1] if '10' <= y + x[2] <= '26' else x[1]), y), s, (1, 1, ''))[1] if s and s[0] != '0' else 0 print(num_decodings("123"))

Output: 3

This complex-looking one-liner starts with an initial tuple representing the empty string and first digit decoding counts along with the previous character (initialized to an empty string). As `reduce()`

iterates through the message, it builds upon the tuple based on the current and previous digits, ultimately extracting the second value from the final tuple as the result.

## Summary/Discussion

**Method 1: Dynamic Programming.**This is an efficient and straightforward approach. Its strength lies in its optimal substructure and overlapping subproblems, making it suited for larger inputs. A weakness is its \(O(n)\) space complexity.**Method 2: Recursive Approach.**With memoization, it prevents recalculations, which is ideal for depth-first explorations of the decoding process. Without memoization, it suffers from a severe performance hit due to repeated calculations.**Method 3: Iterative Approach with Constant Space.**Space-efficient and maintains the effectiveness of the dynamic programming approach. May be less intuitive for those not familiar with such optimizations.**Bonus Method 5: Functional Approach with**It’s elegant and concise. However, the complexity of the one-liner’s logic can be difficult to understand, making it less practical for readability and debugging.`reduce()`

.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.