Problem Description

The task is to rearrange the characters of a given string s in a zigzag pattern on a specified number of rows and then read off the characters line by line to create a new string. The zigzag pattern means that the characters of s are placed diagonally in a zigzag manner in downwards and upwards directions successively. After placing all the characters in this pattern, we would read the characters in each row horizontally and concatenate to form the final string.

To visualize the process, you can think of writing down the characters in the following way:

• Starting from the top row, write characters downwards.
• Upon reaching the last row, switch direction and start writing characters upwards, forming a diagonal line until you reach the top row again.
• Alternate the process until every character is written in this zigzag fashion.

For example, given a string "PAYPALISHIRING" and 3 rows, the characters should be placed like this:

P   A   H   N
A P L S I I G
Y   I   R

After arranging the characters, the string "PAHNAPLSIIGYIR" is obtained by reading each row sequentially. To form this new string programmatically, the code should simulate the reading process.

Intuition

The intuitive approach to simulate the zigzag conversion process involves creating an imaginary 2D grid that represents rows and columns of the pattern. A group is defined which denotes the cycle length of the zigzag pattern, calculated as 2 * numRows - 2. If numRows is 1, then the zigzag pattern is not possible, and the original string is returned.

The solution uses a loop to construct the new string by considering each row individually, assembling the characters that would appear in that row in the zigzag pattern. The interval between characters of the same row varies depending on their position (at the top or bottom, it’s equal to one cycle; in between, it alternates between moving vertically down and obliquely up) and is used to calculate the index of the next character to be added.

1. Initialize an empty list ans to hold characters in the order they should be added to form the zigzag string.
2. Loop through each row. For the top and bottom rows, characters occur after every full cycle (the group), so the interval is constant. For intermediate rows, the interval alternates. We calculate the initial interval based on the current row.
3. Within each row, loop to add characters to ans, incrementing the index by the interval each time. After each addition, update the interval to alternate between moving vertically and diagonally, accounting for edge cases to avoid a zero interval.
4. After processing all rows, join the list of characters ans into a string to get the final result.

The core of the solution relies on finding the pattern of intervals between the positions within the zigzag alignment and using it to construct the final string efficiently without simulating the entire grid.

Solution Approach

The implementation for the given problem follows a direct approach that calculates the next index in the original string for each character in the transformed zigzag string. It does this by determining the appropriate index intervals without needing to simulate the entire 2D zigzag structure.

Here's how the implementation works:

1. Early Return for Single Row: If numRows is 1, the zigzag pattern is trivial and the original string can be returned immediately, as there is no alteration in the order of characters.

   if numRows == 1:
       return s

2. Calculating the Group Cycle: The group cycle (group) is calculated as the number of characters that form the complete vertical and diagonal cycle in the zigzag pattern, given by 2 * numRows - 2. This number is central to the implementation as it helps to understand the symmetry in the pattern, which repeats every group characters.

   group = 2 * numRows - 2

3. Initializing the Answer List: An empty list ans is initialized to store the characters in their new order.

4. Populating Each Row: The algorithm iterates over each row and identifies the characters that would appear in that row in a zigzag pattern:

   • Determine the initial interval for characters in the current row. For the first and last rows, it remains constant and equal to group. For intermediate rows, the interval alternates between two values.
   • Use a while loop to continue adding characters to the ans list until the end of the string is reached.

   for i in range(1, numRows + 1):
       interval = group if i == numRows else 2 * numRows - 2 * i
       idx = i - 1
       while idx < len(s):
           ans.append(s[idx])
           idx += interval
           interval = group - interval if interval != group else interval

   Here, i is the row index, idx is the index in the original string for the current character, and the interval determines the number of steps to the next character in the current row.

   • For intermediate rows (rows between the first and last), after each character is added, the interval for the next character is updated to the complementary interval (the difference between the group and the current interval), thus simulating the zigzag pattern without explicitly constructing a 2D grid structure.

5. Concatenating the Results: After the rows are processed and all characters are placed in the ans list, the list is joined into a string to form the final zigzag transformed string.

   return ''.join(ans)

This implementation optimizes space by avoiding the creation of a 2D grid and computes the string in O(n) time by analyzing the inherent pattern in the positions of characters in the zigzag arrangement.

Example Walkthrough

Let's consider a small example to illustrate the solution approach. Suppose we have the string "HELLO" and the number of rows numRows is 3. The task is to arrange the characters in a zigzag pattern and then read them line by line.

Step 1: Early Return for Single Row

Since numRows is not 1, we don't return the string as is, and we proceed with the rest of the algorithm.

Step 2: Calculating the Group Cycle

We calculate the group cycle (group = 2 * numRows - 2) which in this case is 2 * 3 - 2 = 4. This means that the vertical and diagonal pattern will repeat every 4 characters.

Step 3: Initializing the Answer List

We initialize an empty ans list where we will store the characters.

Step 4: Populating Each Row

We iterate over each row and add the relevant characters to the ans list:

• For the first row (i = 1), the interval will remain constant at 4. The first character is 'H' at index 0. The next character in the zigzag pattern for the first row would be 4 characters away, but since our string is shorter, we stop here for this row. So we add 'H'.

• For the second row (i = 2), the interval starts at 2 * 3 - 2 * 2 = 2 and will alternate between 2 and (4 - 2) = 2 for subsequent characters. The first character for this row is 'E' at index 1. The next character in the zigzag pattern for the second row is 2 characters away, which is 'L' at index 3. There are no more characters 2 or 4 steps away, so we stop for this row. We add 'E' and 'L'.

• For the third row (i = 3), the interval is constant at 4, the same as for the first row. The first character is 'L' at index 2, and there are no more characters 4 steps away. So, we add 'L'.

Step 5: Concatenating the Results

We concatenate the characters in the ans list to form the final string which is 'HEL'.

Following the steps above, the zigzag arrangement for the string "HELLO" with 3 rows would look like this:

H   .
. E .
L . .
. L .
. . O

When reading off each line, the new string formed is "HEL", which matches the characters collected in our answer list.

This walkthrough of a smaller example demonstrates how the designed algorithm would simulate the placement of characters in the zigzag pattern and construct the transformed string accordingly.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code is O(n). This is determined by the observation that we iterate over each character of the string s, which has length n. Every character is visited once during the construction of the ans list. The nested while loop does not increase the complexity because the inner loop's increments are such that the total number of iterations remains linearly dependent on the length of the s.

Space Complexity

The space complexity of the code is O(n). This is because we use an additional list ans to store the rearranged characters. In the worst case, the ans list can contain all characters of the input string s, which requires space that is linearly proportional to the length of the input string.

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