Problem Description

In this problem, we are given a string s that consists of lowercase English letters, and a matrix shift consisting of pairs [direction, amount]. We are tasked with performing a series of shift operations on the string according to the instructions provided by each element in the shift matrix. The direction indicates whether the shift should be to the left (0) or to the right (1) and amount tells us how many times the shift should occur.

A left shift (direction = 0) means removing the first character of string s and appending it to the end. Conversely, a right shift (direction = 1) means removing the last character of s and adding it to the beginning. After applying all shifts in the order they appear in the shift matrix, our goal is to return the resultant string.

To illustrate, if we have string s = "abcde":

• A left shift of 1 would change the string to "bcdea".
• A right shift of 1 would change the string to "eabcd".

We need to ensure that we carry out these operations in an efficient manner to determine the final state of the string s.

Intuition

To efficiently perform the shift operations on the string, we notice that shifts in opposite directions counteract each other. That means if we shift a string to the left by 2 positions and then to the right by 2 positions, we end up with the original string. Furthermore, shifting the string by its length or multiples of its length also results in the same string. Therefore, we can simplify the problem by combining all the shifts into a single effective shift.

To achieve this, we iterate over the shift matrix and maintain a variable x to calculate the net effect of all shifts. For left shifts, we can subtract the amount from x, and for right shifts, we can add the amount to x. By doing this for each shift and then simplifying the net shifts (x) modulo the length of the string, we arrive at a single shift value that represents the overall effect of all the given shifts.

Once the effective shift is determined, we can perform the shift in one step. If x is positive, it indicates a right shift; if negative, a left shift. The implementation in Python takes advantage of string slicing to perform this operation:

• s[-x:]: This slice takes the last x characters of s, which is what moves to the start of the string in a right shift.
• s[:-x]: This slice takes all characters of s except the last x, which remain in their original order after the shift.

Combining these two slices, we create the new string post shift, s[-x:] + s[:-x], effectively performing the necessary left or right shift in a single operation. This approach effectively reduces the complexity of the problem from potentially having to do multiple shifts to simply calculating the net shift and applying it once. This not only simplifies the problem but also ensures that the solution is efficient, even when the number of shift operations is large.

Learn more about Math patterns.

Solution Approach

The provided solution follows a direct algorithmic approach to combine all the individual shift commands into one effective shift operation and then apply it to the string s. Here is a step-by-step walkthrough of the implementation:

1. Initialize a variable x to 0. This will keep track of the net number of shifts needed. When we encounter a left shift, we will decrease x, and for a right shift, we will increase x.

2. Loop through each shift operation in the shift matrix. Each operation is a list containing two elements: direction and amount. We check the direction:

   • If the direction is 0 (left shift), we convert the amount to a negative number (by doing -b) and add it to x.
   • If the direction is 1 (right shift), we simply add the amount to x.

   In Python, this can be compactly written as:

   for a, b in shift:
       if a == 0:
           b = -b
       x += b

3. After processing all shift operations, we normalize the net total shifts x by taking the remainder when x is divided by the length of s (x %= len(s)). This is an essential step because shifting the string by its length (or multiples thereof) results in the same string, so shifts beyond that point are redundant.

   In mathematical form, this step is expressed as:

   x %= len(s)

   If x is positive, it represents a net right shift; if negative, it represents a net left shift. In either case, since the remainder operation normalizes x within the range from 0 to len(s) - 1, we have a guaranteed valid index for the slicing operation in the next step.

4. Finally, we apply the effective shift to s using Python's string slicing feature:

   • s[-x:] generates a substring consisting of the last x characters in s.
   • s[:-x] creates a substring of all characters in s except for the last x.

   The result is then concatenated to get the shifted string. As x has been adjusted to be within the proper index range, these slices will always be valid. The Python code for this operation is:

   return s[-x:] + s[:-x]

This method uses the string slicing capability of Python, which is very efficient, to perform the shifting operation in constant time, which is much more efficient than performing each shift individually. The space complexity of the solution is constant since it uses a fixed amount of extra space, regardless of the input size, and the time complexity is linear relative to the number of shift operations due to the single pass through the shift array.

Example Walkthrough

Let's consider a small example to illustrate the solution approach:

Suppose we have a string s = "abcdef" and a shift matrix with the following operations: [[0, 1], [1, 2], [0, 3]].

1. We would initialize our variable x to 0, which will keep track of the net number of shifts.

2. The first operation is a left shift ([0, 1]), so we decrease x by 1. Now x becomes -1.

3. The second operation is a right shift ([1, 2]), so we increase x by 2. Now x becomes 1 (-1 + 2).

4. The final operation is a left shift ([0, 3]), so we decrease x by 3. Now x becomes -2 (1 - 3).

5. Once we've processed all operations, we will normalize x with the length of s which is 6. We compute x %= len(s), which results in x becoming 4 because -2 % 6 is 4. It means we have a net right shift of 4.

6. We now apply this effective shift to string s with a single slicing operation. The substring from the last x characters will be s[-4:], which gives us "cdef". The substring from the start to the point before the last x characters will be s[:-4], giving us "ab".

7. Combining these two substrings to perform the net right shift operation, we get the transformed string "cdefab".

Thus, after applying an algorithmic approach, we were able to reduce multiple shifts to a single operation and efficiently determine the final state of s after all the shifting instructions.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code is O(n + m), where n is the length of the string s, and m is the number of shifts in the shift list. This is because the code loops once over the list of shifts (m operations) and then applies a single string operation that has a complexity of O(n).

Let's break it down:

• The for loop processes each element in the shift array of size m, which means it operates in O(m) time.
• The modulo operation (x %= len(s)) is done in constant time O(1) since the length of the string is calculated just once.
• The string concatenation (s[-x:] + s[:-x]) is O(n) because it involves creating two substrings and then concatenating them together, which requires building a new string of length n.

Therefore, when combined we get O(m + n) for the time complexity of the entire function.

Space Complexity

The space complexity of the code is O(n). This is due to the space needed for the output string in the final concatenation step. While x is calculated in place and does not need additional space proportional to the input, the slicing operation creates new strings which are then concatenated to form the final result, requiring a buffer that can hold the entire string s.

Here's why:

• The integer x is a single counter variable and thus takes up O(1) space.
• The space required to store the input s and the shift instructions does not count towards the space complexity as they are considered input.
• The final line (return s[-x:] + s[:-x]) creates substrings and concatenates them, requiring O(n) space since the resulting string is of the same length as the original string s.

In summary, the space complexity is O(n).

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