Problem Description

The task is to determine how many times we need to repeat string a such that string b becomes a substring of the repeated string a.

Here are important things to note from the problem:

• If it is not possible for b to be a substring of a no matter how many times a is repeated, we should return -1.
• A string repeated 0 times is an empty string, repeated once remains the same, and so on.

The challenge lies in finding the minimum number of repetitions needed.

Intuition

Let's say a = "abcd" and b = "cdabcdab". The string b isn't immediately a substring of a, but if we repeat a a certain number of times, it can become one.

To find out how many repetitions are needed:

1. We first calculate the minimum number of times a must be repeated such that the length of the resulting string is equal to or just exceeds the length of b. This is because, for b to be a substring of a, the repeated a must be at least as long as b.

2. We start by repeating string a this minimum number of times and check if b is a substring of the resultant string. If not, we increment the number of repetitions by one and check again.

3. We only need to check up to two more repetitions of a beyond the initial calculated number of times. The reasoning is as follows:

   • If b is not a substring of a repeated ans times (where ans is the initial calculated number), b must start near the end of a repeated ans times for it to possibly be included in a further repeated a.
   • If by adding one more a, b is still not a substring, then adding one more repetition on top of that (making it two more repetitions beyond the initial ans) will cover any possible overlap of b as a substring.

4. If after three attempts (ans, ans+1, and ans+2) b is not a substring, it's concluded that b cannot be made a substring by repeating a.

Thus, the solution lies in trying at most three different numbers of repetitions and checking for the substring condition. If none meet the condition, then we return -1.

Solution Approach

The implementation closely follows the intuition:

1. First, we measure the lengths of a and b using len(), storing the lengths in variables m and n respectively.

2. We calculate the initial number of times a needs to be repeated, which we refer to as ans. This is done as follows:

   ans = ceil(n / m)

   Here, ceil is a mathematical function from the math module that takes a float and rounds it up to the nearest integer. This makes sure that if b is not completely covered by multiples of a, we round up to ensure complete coverage.

3. Then we initialize a list t that will contain the repeated string a. We multiply the list [a] by ans to repeat the string a that many times initially:

   t = [a] * ans

4. Next, we begin testing if b is a substring of the repeatedly joined string a. We use a for loop that runs three times, representing the maximum number of additional repeats we decided would be necessary:

   for _ in range(3):
       if b in ''.join(t):
           return ans

   In this loop, we join the elements of t into one string using ''.join(t) and check if b is a substring of this string with b in ''.join(t). If we find b, we immediately return the current number of times a has been repeated (ans).

5. If b was not found to be a substring, before going for the next loop iteration, we add one more a to our list of strings t, effectively repeating a one more time:

       ans += 1
       t.append(a)

   We increment ans by 1 for each additional repeat. Continuously checking every time after appending a.

6. Finally, if three attempts don't result in b becoming a substring, we return -1:

   return -1

   This is outside our loop and is our default case if b was never found within the repeated a.

Remember, the code doesn't explicitly check ans+2 repetitions within the loop because appending a to t inside the for loop happens at the end of the iteration, which means when the loop exits, we would have already checked up to ans+2 repetitions.

The above code leverages the in operator in Python for substring checking, the join method for concatenation of strings, and a simple list to handle the repetitions. It's a straightforward implementation with the focus on optimizing the number of repetition checks.

Example Walkthrough

Let's illustrate the solution approach with a small example:

Suppose we have strings a = "xyz" and b = "xyzxyzx". We want to determine how many repetitions of a we need so that b becomes a substring of the repeated a.

Following the solution approach:

1. We measure the lengths of both strings. For a, the length, m, is 3 and for b, the length, n, is 7.

2. We calculate the minimum number of times a must be repeated to at least cover the length of b. Using ans = ceil(n / m):

   ans = ceil(7 / 3) = ceil(2.33) = 3

   The initial number of repetitions of a required is 3.

3. We prepare our list t to contain the repeated string a. Multiplying the list [a] by ans we get:

   t = ["xyz", "xyz", "xyz"]

4. We concatenate strings in t and check if b is a substring.

   After concatenating, we have: ''.join(t) = "xyzxyzxyz"

   Now, we check if b is a substring of this. Since "xyzxyzx" in "xyzxyzxyz" returns True, we've found that b is indeed a substring after the initial number of repetitions.

5. As a result, we don't need to add more repetitions. The minimum number of repetitions of a needed is 3. We return ans:

   return 3

6. If b had not been a substring, we would add another a to t and check again. In this case, it would become ["xyz", "xyz", "xyz", "xyz"] with the concatenated string being "xyzxyzxyzxyz". We would increment ans by 1 and check again.

Since in this example b becomes a substring after the initial number of repetitions, the algorithm finishes early and returns 3.

This process efficiently checks the minimum and only necessary additional repetitions of a to determine if b can become a substring of the repeated a. If the maximum considered repetitions (ans + 2) do not satisfy the condition, the method will conclude with returning -1.

Solution Implementation

Time and Space Complexity

The time complexity of the given code can be analyzed based on the operations it performs:

• The variable assignments and the ceil operation take constant time, hence O(1).
• Creating t, which is a list of copies of string a, takes O(ans) time in the worst case, as it depends on the initial value of ans, which is ceil(n / m).
• The for loop will run at most 3 times, with each iteration including a ''.join(t) and testing if b in ''.join(t). The join operation takes O(len(t) * m) because it concatenates len(t) strings of length m. Following this, the in operation has a worst-case time complexity of O((len(t) * m) + n) as it needs to check substring b in the concatenated string.
• The append operation inside the loop takes O(1) time. However, as the string concatenation inside the loop occurs in each iteration, it increases the total length of the concatenated string by m every time. So, by the third iteration, the join could be operating on a string of length up to 3 * m + (ceil(n / m) * m).

Given these considerations, we estimate the time complexity as:

• Worst-case time complexity is O((ans + 2) * m + n), reflecting the last iteration of the loop where ans could be incremented twice.

The space complexity is determined by:

• The space needed by the list t and the strings created by ''.join(t). The maximum length of the joined string can go up to 3 * m + (ceil(n / m) * m).
• Hence, the worst-case space complexity is O(3 * m + (ceil(n / m) * m)).

Time Complexity: O((ans + 2) * m + n)

Space Complexity: O(3 * m + (ceil(n / m) * m))

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