import java.util.regex.Matcher;
import java.util.regex.Pattern;
public class Example {
public static void main(String[] args) {
final String regex = "\\\\\\[(.*?)\\\\\\]";
final String string = "# Equivalency Lookups: Concepts, Techniques, and Applications \n\n"
+ "Equivalency lookups are a common computational task where you determine whether two or more entities are equivalent based on certain criteria. This concept underpins a wide range of problems, from database joins to synonym matching, hash-based comparisons, and distributed system consistency checks. \n\n"
+ "---\n\n"
+ "## 1. **What Are Equivalency Lookups?** \n\n"
+ "### Definition \n"
+ "An **equivalency lookup** is the process of identifying if two elements belong to the same equivalence class based on a defined equivalence relation $( R )$. \n\n"
+ "### Properties of Equivalence Relations \n"
+ "An equivalence relation $( R )$ satisfies three properties:\n"
+ "1. **Reflexivity**: $( a R a )$ (an element is equivalent to itself). \n"
+ "2. **Symmetry**: $( a R b \\implies b R a )$ (if $( a )$ is equivalent to $( b )$, then $( b )$ is equivalent to $( a )$). \n"
+ "3. **Transitivity**: $( a R b )$ and $( b R c \\implies a R c )$ (if $( a )$ is equivalent to $( b )$, and $( b )$ is equivalent to $( c )$, then $( a )$ is equivalent to $( c )$). \n\n"
+ "### Real-World Examples \n"
+ "- **Dictionary Synonyms**: Checking if two words (e.g., \"fast\" and \"quick\") are synonyms. \n"
+ "- **User Identity Matching**: Verifying if two user accounts refer to the same individual. \n"
+ "- **Canonical Representation**: Mapping equivalent objects to a single representative for efficiency (e.g., hash-based deduplication). \n\n"
+ "---\n\n"
+ "## 2. **Techniques for Equivalency Lookups** \n\n"
+ "### 2.1 Hashing \n"
+ "- Use **hash functions** to map equivalent objects to the same hash value. \n"
+ "- Efficient for exact matches (e.g., strings, integers). \n"
+ "- Example: Checking file equivalency via MD5 or SHA-256 hash comparison. \n\n"
+ "#### Example: Hash-Based Lookup \n"
+ "```python\n"
+ "# Equivalency check for strings using hashes\n"
+ "import hashlib\n\n"
+ "def get_hash(value):\n"
+ " return hashlib.md5(value.encode()).hexdigest()\n\n"
+ "a = \"hello\"\n"
+ "b = \"hello\"\n\n"
+ "print(get_hash(a) == get_hash(b)) # Output: True\n"
+ "```\n\n"
+ "### 2.2 Union-Find (Disjoint Set) \n"
+ "- Efficient data structure for handling equivalency relations in dynamic systems. \n"
+ "- Operations:\n"
+ " - **Find**: Determine the equivalence class of an element. \n"
+ " - **Union**: Merge two equivalence classes. \n"
+ "- Applications: Network connectivity, Kruskal’s algorithm for MST, and clustering. \n\n"
+ "#### Example: Union-Find Implementation \n"
+ "```python\n"
+ "class UnionFind:\n"
+ " def __init__(self, size):\n"
+ " self.parent = list(range(size))\n"
+ " \n"
+ " def find(self, x):\n"
+ " if self.parent[x] != x:\n"
+ " self.parent[x] = self.find(self.parent[x]) # Path compression\n"
+ " return self.parent[x]\n"
+ " \n"
+ " def union(self, x, y):\n"
+ " rootX = self.find(x)\n"
+ " rootY = self.find(y)\n"
+ " if rootX != rootY:\n"
+ " self.parent[rootX] = rootY\n\n"
+ "# Usage\n"
+ "uf = UnionFind(10)\n"
+ "uf.union(1, 2)\n"
+ "uf.union(2, 3)\n"
+ "print(uf.find(1) == uf.find(3)) # Output: True\n"
+ "```\n\n"
+ "### 2.3 Canonicalization \n"
+ "- Transform each object into a canonical form such that equivalent objects are identical. \n"
+ "- Examples:\n"
+ " - Sorting strings for anagrams (e.g., \"cat\" and \"tac\" → \"act\"). \n"
+ " - Reducing fractions to lowest terms. \n\n"
+ "#### Example: Canonicalizing Anagrams \n"
+ "```python\n"
+ "def canonical_form(word):\n"
+ " return ''.join(sorted(word))\n\n"
+ "print(canonical_form(\"listen\") == canonical_form(\"silent\")) # Output: True\n"
+ "```\n\n"
+ "### 2.4 Database Indexes \n"
+ "- Use indexes for equivalency lookups in structured data. \n"
+ "- Example: SQL query to find all users with the same email address. \n\n"
+ "#### Example: SQL Query \n"
+ "```sql\n"
+ "SELECT user_id FROM users WHERE email = 'example@example.com';\n"
+ "```\n\n"
+ "---\n\n"
+ "## 3. **Applications of Equivalency Lookups** \n\n"
+ "### 3.1 Data Deduplication \n"
+ "- Identifying and removing duplicate records or files. \n"
+ "- Technique: Hash-based deduplication or clustering similar records. \n\n"
+ "### 3.2 Graph Connectivity \n"
+ "- Check if two nodes are in the same connected component. \n"
+ "- Technique: Union-Find or BFS/DFS. \n\n"
+ "### 3.3 Synonym Matching \n"
+ "- Resolve different words or phrases that refer to the same concept. \n"
+ "- Technique: Canonicalization or synonym dictionaries. \n\n"
+ "### 3.4 Distributed Systems \n"
+ "- Ensure consistency by checking if replicas are equivalent. \n"
+ "- Technique: Compare hash values of data on different servers. \n\n"
+ "---\n\n"
+ "## 4. **Optimizations for Large-Scale Lookups** \n\n"
+ "### 4.1 Bloom Filters \n"
+ "- Space-efficient data structure for approximate membership testing. \n"
+ "- Useful for checking if an element might be equivalent to others in a large dataset. \n\n"
+ "#### Example: Using Bloom Filter \n"
+ "```python\n"
+ "from pybloom_live import BloomFilter\n\n"
+ "bloom = BloomFilter(capacity=1000, error_rate=0.01)\n"
+ "bloom.add(\"hello\")\n"
+ "print(\"hello\" in bloom) # Output: True\n"
+ "```\n\n"
+ "### 4.2 Caching \n"
+ "- Store results of equivalency checks to avoid recomputation. \n"
+ "- Use LRU (Least Recently Used) or LFU (Least Frequently Used) caches. \n\n"
+ "#### Example: Caching Results with `functools.lru_cache` \n"
+ "```python\n"
+ "from functools import lru_cache\n\n"
+ "@lru_cache(maxsize=1000)\n"
+ "def is_equivalent(a, b):\n"
+ " return sorted(a) == sorted(b)\n\n"
+ "print(is_equivalent(\"listen\", \"silent\")) # Output: True\n"
+ "```\n\n"
+ "---\n\n"
+ "## 5. **Challenges in Equivalency Lookups** \n\n"
+ "1. **Scalability**: \n"
+ " - Large datasets require efficient data structures and algorithms. \n"
+ " - Use distributed systems or approximate methods for very large inputs. \n\n"
+ "2. **Precision vs. Performance**: \n"
+ " - Approximate methods (e.g., Bloom filters) trade off precision for speed. \n\n"
+ "3. **Ambiguity**: \n"
+ " - Defining equivalence relations can be complex for real-world data (e.g., synonym matching may depend on context). \n\n"
+ "4. **Data Quality**: \n"
+ " - Inconsistent or noisy data can lead to false equivalences. \n\n"
+ "---\n\n"
+ "## 6. **Summary** \n"
+ "Equivalency lookups are essential across fields like data processing, graph theory, and distributed systems. Techniques like hashing, union-find, canonicalization, and Bloom filters provide robust solutions depending on the use case. By balancing accuracy and performance, equivalency lookups can be optimized for scalability and reliability in real-world applications. ";
final String subst = "$$\\1$$";
final Pattern pattern = Pattern.compile(regex, Pattern.MULTILINE | Pattern.DOTALL);
final Matcher matcher = pattern.matcher(string);
// The substituted value will be contained in the result variable
final String result = matcher.replaceAll(subst);
System.out.println("Substitution result: " + result);
}
}
Please keep in mind that these code samples are automatically generated and are not guaranteed to work. If you find any syntax errors, feel free to submit a bug report. For a full regex reference for Java, please visit: https://docs.oracle.com/javase/7/docs/api/java/util/regex/Pattern.html