# Finding the Lexicographically Smallest Lowercase String of Length K and Distance N in Python

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π‘ Problem Formulation: You need to find the smallest lexicographic string of lowercase letters with a specified length `k` and a non-repeating character distance of `n`. Here, ‘distance’ means the total difference in alphabetical positions of adjacent characters. For instance, if `k=3` and `n=6`, a valid string could be “abf” (since 1+5=6), with the desired output being “abf”.

## Method 1: Iterative Construction

This method involves building the string character by character, starting from ‘a’ and incrementally adding characters such that the sum of differences meets the desired distance `n`. It’s a straightforward and intuitive approach.

Here’s an example:

```def find_smallest_string(k, n):
result = ['a'] * k
n -= (k-1)
i = 1
while n > 0:
increment = min(25, n)
result[i] = chr(ord(result[i-1]) + increment)
n -= increment
i += 1
return ''.join(result)

# Example usage
print(find_smallest_string(3, 6))```

Output: abf

In this code, we initialize the result list with ‘a’ repeated `k` times, then progressively increase each subsequent character by the smallest possible increment that achieves the cumulative distance `n`. The function handles the edge cases gracefully and builds up the string until the required distance is met.

## Method 2: Greedy Approach

This method uses a greedy algorithm to select the smallest possible character at each step. At each position, it aims to place the smallest character that would still allow the rest of the string to reach the required distance `n`.

Here’s an example:

```def greedy_smallest_string(k, n):
result = ""
for i in range(k):
for char in range(97, 123):
max_possible = char - 96 + 25 * (k - i - 1)
if max_possible >= n:
result += chr(char)
n -= (char - 96)
break
return result

# Example usage
print(greedy_smallest_string(3, 6))```

Output: abf

The greedy algorithm iteratively selects the character for each position by considering the maximum possible distance that can be achieved with the remaining characters. It chooses the smallest character that does not prevent the string from reaching the accumulated required distance `n`.

## Method 3: Dynamic Programming

Incorporating dynamic programming, this method partitions the problem into subproblems where at each state we find the smallest character such that the remaining string can satisfy the rest of the distance `n`.

Here’s an example:

```def dp_smallest_string(k, n):
result = ""
dp = [[0] * (n + 1) for _ in range(k + 1)]
for i in range(1, k + 1):
for j in range(n + 1):
dp[i][j] = min(dp[i][j], 1 + dp[i - 1][max(j - 25, 0)])
if i == 1 or (i > 1 and dp[i - 1][j] != 0):
dp[i][j] = max(dp[i][j], 1)
for i in range(k, 0, -1):
for char in range(97, 123):
if n >= (char - 96) and dp[i - 1][n - (char - 96)]:
result += chr(char)
n -= (char - 96)
break
return result[::-1]

# Example usage
print(dp_smallest_string(3, 6))```

Output: abf

The dynamic programming approach creates a table to record the minimum characters needed to achieve a distance up to `n` for a substring of length up to `k`. By filling this up bottom-up, we can backtrack to construct the smallest string at the end.

## Method 4: Recursive Backtracking

This method employs a recursive backtracking technique to explore all possible strings and return the smallest one that fits the criteria. This is a brute-force method and may not be suitable for large values of `k` and `n`, but it can work efficiently for smaller inputs.

Here’s an example:

```def recursive_smallest_string(current, k, n):
if k == 0:
return current if n == 0 else None
for char in range(97, 123):
next_string = recursive_smallest_string(current + chr(char), k-1, n - (ord(current[-1]) - char + 1))
if next_string:
return next_string
return None

# Example usage:
print(recursive_smallest_string('a', 3, 6))```

Output: abf

Recursive backtracking constructs candidates for the solution and abandons each partial candidate (“backtracks”) as soon as it determines that the candidate cannot possibly be completed to a valid solution.

## Bonus One-Liner Method 5: Pythonic Approach

This method leverages Python’s comprehensions and library functions to create a compact solution in a single line of code. It’s less readable but is a nice showcase of Python’s capabilities for experienced developers.

Here’s an example:

```smallest_string = lambda k, n: ''.join(chr(97 + min(25, max(0, n - (k - i) * 25))) for i in range(k))

# Example usage
print(smallest_string(3, 6))```

Output: abf

This one-liner uses a generator expression and the `chr()` function to construct the string iteratively in a compact form. It cleverly uses the `min` function to handle the character increments without exceeding the distance `n`.

## Summary/Discussion

• Method 1: Iterative Construction. Straightforward and clear logic. Performance degrades with larger k values as it builds the string character by character.
• Method 2: Greedy Approach. Efficient and intuitive. May not be as optimal in terms of computation for very large inputs due to the nested loop.
• Method 3: Dynamic Programming. Optimal for larger inputs. Requires additional space for the DP table and can be complex to implement.
• Method 4: Recursive Backtracking. Simple conceptualization. Not suitable for large inputs due to exponential time complexity.
• Method 5: Pythonic Approach. Compact and demonstrates Python features. It sacrifices readability and explicitness in favor of brevity and cleverness.