# LC010 - Regular Expression Matching

## Problem

Given an input string `s` and a pattern `p`, implement regular expression matching with support for `'.'` and `'*'` where:

* `'.'` Matches any single character.​​​​
* `'*'` Matches zero or more of the preceding element.

The matching should cover the **entire** input string (not partial).

### Example

**Input:** s = "aa", p = "a"

**Output:** false

**Explanation:** "a" does not match the entire string "aa".

## Solution

#### DFS

The naive approach to traverse the entire parsing decision tree is $$O(2^n)$$ in time complexity.

#### DFS Cache

With a cache, we can get it down to $$O(m\cdot n)$$, at the cost of extra space.

```python
def isMatch(self, s: str, p: str) -> bool:
	@cache
	def dfs(i, j):
		if i >= len(s) and j >= len(p):
			return True
		if j >= len(p):
			return False

		match = i < len(s) and (s[i] == p[j] or p[j] == ".")
		if (j+1) < len(p) and p[j+1] == "*":
			return (dfs(i, j+2) or (match and dfs(i+1, j)))
		if match:
			return dfs(i+1, j+1)
		return False
	return dfs(0,0)
```


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://www.vjssn.dev/practice/24-06-09/lc010-regular-expression-matching.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.