Problem Description

In this problem, we're given a pattern, which is a string consisting of characters, and a string s. Our goal is to determine if the string s matches the given pattern.

A string s is considered to match a pattern if we can associate each character in pattern with a non-empty string in s such that the concatenation of the strings, in the order they appear in pattern, exactly makes up s. The mapping must be bijective, which means two conditions must be met: first, no two different characters in pattern can map to the same substring in s, and second, each character must map to exactly one substring (and not multiple different substrings at different instances).

For example, if pattern is "abab" and s is "redblueredblue" then we could map 'a' to "red" and 'b' to "blue", which matches the pattern.

Flowchart Walkthrough

Let's analyze LeetCode 291. Word Pattern II using the Flowchart. Here's the step-by-step process:

Is it a graph?

• No: This problem does not involve a typical graph structure with nodes interconnected by edges.

Need to solve for kth smallest/largest?

• No: The problem is not about finding kth smallest or largest elements, but about finding a mapping from characters to strings that match a given pattern.

Involves Linked Lists?

• No: This problem does not involve manipulating or traversing linked lists.

Does the problem have small constraints?

• Yes: The problem involves matching a pattern and a string, potentially with small constraints given that the pattern length is generally small, facilitating the feasibility of trying out different mappings.

Brute force / Backtracking?

• Yes: The problem requires exploring all potential mappings from pattern characters to parts of the string, checking each mapping to see if it fits the pattern. This exhaustive search is typically implemented using backtracking.

Conclusion: Based on the analysis using the flowchart, backtracking is advised for solving LeetCode 291. Word Pattern II, as it involves trying multiple mappings and seeing which ones correctly map the entire pattern to the string under the defined constraints.

Intuition

Approaching the solution requires a way to test different mappings from pattern characters to substrings of s. Since the problem's constraints don't offer an obvious pattern or formula, we opt for a strategy that tries out different possibilities—this is a typical case for using a backtracking algorithm.

The intuition behind the solution is to recursively attempt to build a mapping between each character in pattern and the substrings of s. We start by considering the first character in the pattern and try to map it to all possible prefixes of the string s. For each mapping we consider, we recursively attempt to extend the mapping to the rest of the characters in the pattern. If at any point, we find a complete and valid mapping, we return true.

If we ever reach a situation where a character is already mapped to a different string, or if we try to associate a substring with a pattern character that is already taken by another substring, we backtrack. We also stop exploring the current recursive branch if we exhaust the characters in pattern or s, or if the remaining length of s is too short to cover the remaining characters in pattern. This process continues until either a full mapping is found or all possibilities have been exhausted, which would lead us to conclude that no valid mapping exists.

Learn more about Backtracking patterns.

Solution Approach

The provided solution uses a recursive backtracking approach to implement the logic described in the intuition section. Let's examine the implementation in detail:

1. Depth-First Search (DFS) with Backtracking:

   • The dfs function is a recursive function that keeps track of the current position i in the pattern and the current position j in the string s.
   • If i equals the length of pattern (m) and j equals the length of s (n), it means we have successfully mapped the entire pattern to the string s, and we return True.
   • If either i equals m or j equals n before the other, or if there aren't enough characters left in s to match the remaining pattern (n - j < m - i), we return False as it signifies that the current mapping does not cover s or pattern correctly.

2. Attempt to Match Substrings:

   • The dfs function iterates over the substring of s starting from index j and ending at various points k within the string. This loop essentially tries to map the current pattern character to various lengths of substrings in s.
   • We extract the current substring of s from index j to k and store it in a variable t.
   • If the current pattern character at index i already has a mapping in the dictionary d and it is equal to t, we recursively call dfs for the next character in the pattern and the next part of the string. If this returns True, then the current mapping is part of a solution, and we propagate this success back up the recursion chain.

3. Adding New Mappings:

   • If the current pattern character at index i is not yet mapped in the dictionary d and the substring t is not already used (checked using set vis), we add the mapping pattern[i] -> t to d and add t to vis.
   • We then recurse to see if this new mapping could lead to a solution.
   • Whether the recursive call returns True or False, we then backtrack by removing the current mapping from d and t from vis, allowing the next iteration to try a different mapping for the same pattern character.

4. Data Structures:

   • A dictionary d is used to keep track of the current mapping from pattern characters to substrings in s.
   • A set vis is used to keep track of the substrings of s that have already been mapped to some pattern characters. This helps ensure the bijective nature of the mapping.

5. Initiating the Recursive Search:

   • Before starting the backtracking process, we initialize the variables m and n to hold the lengths of pattern and s, respectively. The dictionary d and the set vis are initialized to be empty.
   • The dfs function is then initiated with i and j both set to 0, which signifies the start of pattern and s.

In conclusion, the recursive backtracking approach is quite fitting for this problem, as it allows the solution to explore all possible mappings and backtrack as soon as it detects a mapping scenario that cannot lead to a successful pattern match. This approach is particularly powerful for problems related to pattern matching, permutations, and combinations where all potential scenarios need to be considered.

Example Walkthrough

Let's illustrate the solution approach using a small example. Consider the pattern "aba" and the string s "catdogcat".

1. Start with the First Character in Pattern:

   • We examine the first character in the pattern, which is 'a'.
   • We try to map 'a' to each possible prefix of the string s. This means we initially consider 'c', 'ca', 'cat', 'catd', 'catdo', 'catdog', 'catdogc', 'catdogca', and 'catdogcat'.

2. First Recursive Branch - 'a' Mapped to 'c':

   • On the first attempt, we associate 'a' with 'c' and then recurse to the next character in the pattern.
   • Now, our pattern looks like "c_b_c" with 'b' unassigned, and the remaining part of the string is "atdogcat".
   • Next, we attempt to map 'b' to 'a', but this leaves us with "c_ac_c", and the remaining string is "tdogcat", which no longer has a viable place to map the next 'a', as the next 'a' in the pattern would be mapped to 't'. Since 't' is not equal to our existing mapping of 'a', we must backtrack.

3. Second Recursive Branch - 'a' Mapped to 'ca':

   • Backtracking from the previous step, we now try to map 'a' to 'ca'.
   • Following a similar pattern to the step above, we look at the next character 'b' and attempt to map it to 't', and then 'd', and then 'do', and so on until 'tdog'.
   • Every time, we check if the subsequent character in the pattern ('a') can continue with the already established mapping ('ca'). If not, we continue.

4. Successful Recursive Branch - 'a' Mapped to 'cat':

   • Continuing to explore, we finally map 'a' to 'cat'.
   • Our pattern now looks like "cat_b_cat", and the remaining string is "dog".
   • We map 'b' to 'dog', as it's the only substring left that fits.
   • The full string according to our pattern and mapping is "catdogcat", which matches s.
   • Since we've successfully mapped the entire pattern to the string s without any conflicting mappings, the algorithm would return True.

Throughout the process, the algorithm uses a dictionary to maintain the mapping of pattern characters to substrings in s and a set to ensure that no two characters in the pattern map to the same substring. If at any point the mapping is not consistent or a character cannot be mapped to a remaining substring without conflict, the algorithm backtracks and tries a different mapping. In this example, we eventually found a successful mapping that satisfies the requirement of the problem.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the function is dependent on the number of recursive calls made by the dfs function. In each call, the function tries to map pattern[i] to a substring of s starting at index j. The for loop in dfs can run up to O(n) times for each call, where n is the size of string s.

In the worst case, the pattern can be bijectively mapped onto a prefix of s, leading to a situation where each character in pattern maps to a different substring of s. This means that there could be up to n levels in our recursion tree, each having n branches if we assume that the length of s is n.

Therefore, the worst-case time complexity is O(n^m), where m is the length of the pattern and n is the length of the string s. However, since the depth of the recursion is also restricted by the size of m with the condition n - j < m - i, the actual time complexity could potentially be lower, but the upper bound still holds as the worst-case scenario.

Space Complexity

The space complexity is influenced by the storage of the mappings (d) and the visited substrings (vis), in addition to the recursion call stack.

The maximum size of the dictionary d is bounded by the length of the pattern m, since at most each character in pattern can map to a unique substring of s. Therefore, the space taken by d is O(m).

The vis set can potentially contain all possible substrings of s that are mapped to pattern characters. In the worst case, this could result in O(n^2) space complexity since there can be n choices for the starting point and n choices for the ending point of the substring.

The recursion stack can go as deep as the number of characters in pattern, so the maximum depth is O(m).

Combining these considerations, the total space complexity becomes O(m + n^2), with n^2 usually dominating unless m is significantly larger than n.

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