Problem Description

The goal is to determine if it's possible to convert string s into string t by performing several operations. In each operation, you can select a non-empty substring from s and rearrange its characters in ascending order. A substring is defined as a contiguous block of characters within the string. The operation can be applied any number of times to any substring within s. The problem asks to return true if s can be transformed into t using these operations, otherwise return false.

Intuition

To solve the problem, the solution uses the intuition that each character in string s can only move towards the left in each operation. This is due to the fact that sorting a substring in an ascending order will not move any character to a position greater than its original position.

Given that constraint, we check if we can transform s into t by considering the characters in t from left to right. We make use of a data structure pos, which is a dictionary of deques, where the keys are the digits (characters represented as integers) found in s, and the values are deques holding the indices of these digits.

As we iterate through each character in t, we check if the character can be moved to its correct position in s. This involves checking if there is an occurrence of the character left in s (pos[x] is not empty), and ensuring that there are no smaller characters (digits) to the left of the character's current position in s (any(pos[i] and pos[i][0] < pos[x][0] for i in range(x)) returns false). If these conditions are not met, it means that it is not possible to transform s into t and we return false. Otherwise, we "move" the character in s to its next position (which is simulated by popping from the left of the deque corresponding to that character) and continue with the next character in t.

If we successfully go through the entire string t without returning false, it means the transformation is possible and we return true.

Learn more about Greedy and Sorting patterns.

Solution Approach

The implementation of the solution takes a strategic approach using the following algorithmic concepts and data structures:

• Deque Data Structure: The solution uses a defaultdict to create a mapping of deques which are essentially double-ended queues to store the indices of corresponding characters. The deque allows efficient removal of elements from both ends which is essential since we are trying to simulate the movement of characters within the string.

• Index Tracking: Each character's position in s is tracked by storing its indices in order in the corresponding deque. This helps to determine the possibility of moving a character to a certain position.

• Greedy Approach: A greedy approach is applied by attempting to match characters from t with characters in s from left to right. This checks the feasibility of reaching the desired pattern.

To understand the step-by-step algorithm:

1. Initialize pos: A defaultdict(deque) called pos is created to store the indices of characters in s.

2. Store Indices: Iterating through s, the indices are stored in pos. For example, if s = '3421' then pos[1] will be deque([3]), pos[2] will be deque([2]), and so on.

3. Transformation Check: The algorithm runs through each character c in t and performs the following:

   • It converts c to an integer x.
   • It checks if pos[x] is not empty, which means there's still an occurrence of that character in s.
   • It uses a combination of list comprehension and the any function to determine if there's any smaller character to the left of the current one in s. This is done with any(pos[i] and pos[i][0] < pos[x][0] for i in range(x)). If true, then it's not possible to move x to the desired position without violating the sorting constraint (since a smaller number cannot be passed).

4. Simulate Character Movement: If the checks pass, the character's index is moved (deque is popped from the left) as if the character has been sorted to its place in s. This is effectively simulating the operation defined in the problem.

5. Return the Result: If no inconsistencies are found during the whole iteration, the transformation is possible, and therefore we return true. If at any point an inconsistency arises, the function returns false immediately.

The algorithm's correctness hinges on the property that you can only sort substrings in ascending order, which consequently only allows characters in s to move leftward (smaller index) and not rightward.

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have s = "4321" and t = "1234". We want to find out if it's possible to convert s into t using the defined operations.

Following the solution steps:

1. Initialize pos: We create a defaultdict(deque) to hold the indices of characters from string s. The pos will look like this after we've finished:

   pos[4] = deque([0])
   pos[3] = deque([1])
   pos[2] = deque([2])
   pos[1] = deque([3]) 

2. Store Indices: As we've initialized pos, each character from s has its index stored in the corresponding deque.

3. Transformation Check: Now, we check if we can transform s into t character by character.

   • We start with t[0] which is '1', convert it into an integer x = 1, and see if pos[x] is not empty (which means '1' is still in s). pos[1] is deque([3]), so it's not empty.
   • We check if there's a smaller character to the left of the '1' in s using the any statement. Since '1' is the smallest character, no smaller character can be to the left, so we pass this check.
   • We then "move" the '1' by popping from pos[1], simulating the sorting operation. pos now looks like this:

   pos[4] = deque([0])
   pos[3] = deque([1])
   pos[2] = deque([2])
   pos[1] = deque([])  // '1' has been moved

4. Repeat for the Next Character: We repeat the same check for the next character in t, which is '2'.

   • x becomes 2 and pos[2] is deque([2]), so we have a '2' to work with.
   • With the any statement, we check for smaller characters. Since '1' has already been moved, and there is no '0' in s, the check passes.
   • We "move" the '2' by popping from pos[2], and pos now looks like:

   pos[4] = deque([0])
   pos[3] = deque([1])
   pos[2] = deque([]) // '2' has been moved
   pos[1] = deque([])

5. Continue With Remaining Characters: We continue this process for '3' and '4'. Each character passes the condition checks and gets moved.

6. Return the Result: Since we were able to move all characters from s to match the sorted order of t without encountering any blocking condition, we conclude that it's possible to transform s into t. The function would return true.

This walkthrough demonstrates that given these operations and their constraints, we can determine the possibility of transforming one string into another algorithmically, using deques for index tracking and a series of checks to simulate the transformation.

Solution Implementation

Time and Space Complexity

The given Python function isTransformable checks if it's possible to transform the string s into the string t by repeatedly moving a digit to the leftmost position if there are no smaller digits to its left.

Time Complexity:

To determine the time complexity, let us analyze the various operations being performed in the function.

1. Initializing pos: Building the pos dictionary takes O(n) time, where n is the length of s, as we iterate over all characters in s once.

2. Checking Transformability: We iterate over each character in t, and for each character, perform a check that could iterate over a maximum of 10 possible values (since the digits range from 0 to 9) - this is the any() function call.

   The inner check pos[i] and pos[i][0] < pos[x][0] is O(1) for every digit i for each character of t, because it's merely indexing and comparison.

   Therefore, the total time for transforming, in the worst case, would be O(10 * n) which simplifies to O(n), since we don't count constants in Big O notation.

3. Popping from pos: The popleft() operation is O(1) for a deque.

Therefore, the overall time complexity of the function is O(n).

Space Complexity:

1. Space for pos: The pos dictionary stores deque objects for each unique digit found in s, and each deque can grow to the size of the number of occurrences of that digit. The total size of all deques combined will not exceed n. Therefore, the space complexity contributed by pos is O(n).

2. Miscellaneous Space: Constant additional space used by iterators and temporary variables.

Thus, the total space complexity is also O(n).

In conclusion, the time complexity is O(n) and the space complexity is O(n).

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