Problem Description

You are given a string s and an integer k. The task is to create a function distance(s1, s2) which calculates the sum of the minimum distances between characters at the same positions in two strings s1 and s2 of equal length. These characters are in cyclic order from 'a' to 'z'.

For example:

• distance("ab", "cd") results in 4.
• distance("a", "z") results in 1.

Your goal is to modify the string s to form a new string t such that the distance between s and t is at most k and t is the lexicographically smallest string possible.

Intuition

To solve the problem, we start by iterating through each character in the string s. For each character, we attempt to change it to any smaller letter ranging from 'a' up to just before the current character. The distance d for changing from character c1 to a candidate character c2 is calculated as the minimum of moving forward and moving backward in the cyclic order.

If changing to c2 requires a distance d that can be covered by the remaining k, we perform the change, subtract d from k, and move to the next character in the string. This greedy approach ensures that we always try to make the lexicographically smallest changes while keeping within the allowed distance k.

This method ensures the resulting string is the lexicographically smallest possible while adhering to the given distance constraint.

Learn more about Greedy patterns.

Solution Approach

The solution employs a greedy algorithm to solve the problem. Here is a step-by-step explanation of the implementation:

1. Convert the string s into a list of characters cs so we can modify it easily.

2. Iterate over each character c1 in s, with its index i.

3. For each character c1, we aim to make it as small as possible while keeping the string lexicographically smaller. To achieve this, enumerate all characters c2 from 'a' up to, but not including, c1.

4. Calculate the cyclic distance d between c1 and c2. This is done using:

   • d = min(ord(c1) - ord(c2), 26 - ord(c1) + ord(c2))
   • Here, ord() gives the ASCII value of a character. We consider both the forward and backward paths in the cycle.

5. If d (the cost to change c1 to c2) is less than or equal to k, then:

   • Change the i-th character in cs to c2.
   • Subtract d from k since we've used up d of our allowed distance budget.

6. Break the inner loop once a change is made to move to the next character, as we want to make the smallest possible change first.

7. After completing the traversal of the string, join the modified list cs back into a string to form the result.

8. Return the resulting string, which is the lexicographically smallest string that satisfies the distance constraint.

By following this approach, we ensure that each character's change is minimized first, thus making the overall string the smallest possible while considering the distance constraint k.

Example Walkthrough

Let's walk through an example using the string s = "bdc" and k = 3. The goal is to transform s into the lexicographically smallest string t such that the distance between s and t is at most k.

1. Initialization: Convert s into a list of characters cs = ['b', 'd', 'c'].

2. Iteration Over Each Character in s:

   • Position 0: Character 'b':

     • Attempt to change 'b' to smaller lexicographically characters starting from 'a':
       • c2 = 'a'
         • Calculate the cyclic distance d:
           • d = min(ord('b') - ord('a'), 26 - (ord('b') - ord('a'))) = min(1, 25) = 1
         • Since d = 1 <= k = 3, we change 'b' to 'a'. Now, cs = ['a', 'd', 'c'] and k = 3 - 1 = 2.

   • Position 1: Character 'd':

     • Attempt to change 'd' to smaller lexicographically characters starting from 'a':
       • c2 = 'a'
         • Calculate the cyclic distance d:
           • d = min(ord('d') - ord('a'), 26 - (ord('d') - ord('a'))) = min(3, 23) = 3
         • Since d = 3 <= k = 2 doesn't hold, we can't proceed to change 'd' to 'a'.
       • c2 = 'c'
         • Calculate the cyclic distance d:
           • d = min(ord('d') - ord('c'), 26 - (ord('d') - ord('c'))) = min(1, 25) = 1
         • Since d = 1 <= k = 2, we change 'd' to 'c'. Now, cs = ['a', 'c', 'c'] and k = 2 - 1 = 1.

   • Position 2: Character 'c':

     • Attempt to change 'c' to smaller lexicographical characters starting from 'a':
       • c2 = 'a'
         • Calculate the cyclic distance d:
           • d = min(ord('c') - ord('a'), 26 - (ord('c') - ord('a'))) = min(2, 24) = 2
         • Since d = 2 <= k = 1 doesn't hold, we cannot make a change here.

3. Result Construction: Join the list cs back into a string. The resulting string is t = "acc".

Following this process, we transformed s into "acc", which is the lexicographically smallest string possible given the distance constraint k = 3.

Solution Implementation

Time and Space Complexity

The time complexity of the code is O(n × 26), which simplifies to O(n), where n is the length of the string s. This is because for each character in the string, a loop over a constant set of 26 lowercase letters is performed, resulting in the overall complexity of O(n).

The space complexity is O(n). This arises from storing the modified list of characters cs, which is directly proportional to the size of the input string s.

Learn more about how to find time and space complexity quickly.