#include <MsgBoxConstants.au3> ; to declare the Constants of MsgBox
Local $sRegex = "(?ms)\\\[(.*?)\\\]"
Local $sString = "# Equivalency Lookups: Concepts, Techniques, and Applications " & @CRLF & _
"" & @CRLF & _
"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. " & @CRLF & _
"" & @CRLF & _
"---" & @CRLF & _
"" & @CRLF & _
"## 1. **What Are Equivalency Lookups?** " & @CRLF & _
"" & @CRLF & _
"### Definition " & @CRLF & _
"An **equivalency lookup** is the process of identifying if two elements belong to the same equivalence class based on a defined equivalence relation $( R )$. " & @CRLF & _
"" & @CRLF & _
"### Properties of Equivalence Relations " & @CRLF & _
"An equivalence relation $( R )$ satisfies three properties:" & @CRLF & _
"1. **Reflexivity**: $( a R a )$ (an element is equivalent to itself). " & @CRLF & _
"2. **Symmetry**: $( a R b \implies b R a )$ (if $( a )$ is equivalent to $( b )$, then $( b )$ is equivalent to $( a )$). " & @CRLF & _
"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 $( c )$). " & @CRLF & _
"" & @CRLF & _
"### Real-World Examples " & @CRLF & _
"- **Dictionary Synonyms**: Checking if two words (e.g., "fast" and "quick") are synonyms. " & @CRLF & _
"- **User Identity Matching**: Verifying if two user accounts refer to the same individual. " & @CRLF & _
"- **Canonical Representation**: Mapping equivalent objects to a single representative for efficiency (e.g., hash-based deduplication). " & @CRLF & _
"" & @CRLF & _
"---" & @CRLF & _
"" & @CRLF & _
"## 2. **Techniques for Equivalency Lookups** " & @CRLF & _
"" & @CRLF & _
"### 2.1 Hashing " & @CRLF & _
"- Use **hash functions** to map equivalent objects to the same hash value. " & @CRLF & _
"- Efficient for exact matches (e.g., strings, integers). " & @CRLF & _
"- Example: Checking file equivalency via MD5 or SHA-256 hash comparison. " & @CRLF & _
"" & @CRLF & _
"#### Example: Hash-Based Lookup " & @CRLF & _
"```python" & @CRLF & _
"# Equivalency check for strings using hashes" & @CRLF & _
"import hashlib" & @CRLF & _
"" & @CRLF & _
"def get_hash(value):" & @CRLF & _
" return hashlib.md5(value.encode()).hexdigest()" & @CRLF & _
"" & @CRLF & _
"a = "hello"" & @CRLF & _
"b = "hello"" & @CRLF & _
"" & @CRLF & _
"print(get_hash(a) == get_hash(b)) # Output: True" & @CRLF & _
"```" & @CRLF & _
"" & @CRLF & _
"### 2.2 Union-Find (Disjoint Set) " & @CRLF & _
"- Efficient data structure for handling equivalency relations in dynamic systems. " & @CRLF & _
"- Operations:" & @CRLF & _
" - **Find**: Determine the equivalence class of an element. " & @CRLF & _
" - **Union**: Merge two equivalence classes. " & @CRLF & _
"- Applications: Network connectivity, Kruskal’s algorithm for MST, and clustering. " & @CRLF & _
"" & @CRLF & _
"#### Example: Union-Find Implementation " & @CRLF & _
"```python" & @CRLF & _
"class UnionFind:" & @CRLF & _
" def __init__(self, size):" & @CRLF & _
" self.parent = list(range(size))" & @CRLF & _
" " & @CRLF & _
" def find(self, x):" & @CRLF & _
" if self.parent[x] != x:" & @CRLF & _
" self.parent[x] = self.find(self.parent[x]) # Path compression" & @CRLF & _
" return self.parent[x]" & @CRLF & _
" " & @CRLF & _
" def union(self, x, y):" & @CRLF & _
" rootX = self.find(x)" & @CRLF & _
" rootY = self.find(y)" & @CRLF & _
" if rootX != rootY:" & @CRLF & _
" self.parent[rootX] = rootY" & @CRLF & _
"" & @CRLF & _
"# Usage" & @CRLF & _
"uf = UnionFind(10)" & @CRLF & _
"uf.union(1, 2)" & @CRLF & _
"uf.union(2, 3)" & @CRLF & _
"print(uf.find(1) == uf.find(3)) # Output: True" & @CRLF & _
"```" & @CRLF & _
"" & @CRLF & _
"### 2.3 Canonicalization " & @CRLF & _
"- Transform each object into a canonical form such that equivalent objects are identical. " & @CRLF & _
"- Examples:" & @CRLF & _
" - Sorting strings for anagrams (e.g., "cat" and "tac" → "act"). " & @CRLF & _
" - Reducing fractions to lowest terms. " & @CRLF & _
"" & @CRLF & _
"#### Example: Canonicalizing Anagrams " & @CRLF & _
"```python" & @CRLF & _
"def canonical_form(word):" & @CRLF & _
" return ''.join(sorted(word))" & @CRLF & _
"" & @CRLF & _
"print(canonical_form("listen") == canonical_form("silent")) # Output: True" & @CRLF & _
"```" & @CRLF & _
"" & @CRLF & _
"### 2.4 Database Indexes " & @CRLF & _
"- Use indexes for equivalency lookups in structured data. " & @CRLF & _
"- Example: SQL query to find all users with the same email address. " & @CRLF & _
"" & @CRLF & _
"#### Example: SQL Query " & @CRLF & _
"```sql" & @CRLF & _
"SELECT user_id FROM users WHERE email = 'example@example.com';" & @CRLF & _
"```" & @CRLF & _
"" & @CRLF & _
"---" & @CRLF & _
"" & @CRLF & _
"## 3. **Applications of Equivalency Lookups** " & @CRLF & _
"" & @CRLF & _
"### 3.1 Data Deduplication " & @CRLF & _
"- Identifying and removing duplicate records or files. " & @CRLF & _
"- Technique: Hash-based deduplication or clustering similar records. " & @CRLF & _
"" & @CRLF & _
"### 3.2 Graph Connectivity " & @CRLF & _
"- Check if two nodes are in the same connected component. " & @CRLF & _
"- Technique: Union-Find or BFS/DFS. " & @CRLF & _
"" & @CRLF & _
"### 3.3 Synonym Matching " & @CRLF & _
"- Resolve different words or phrases that refer to the same concept. " & @CRLF & _
"- Technique: Canonicalization or synonym dictionaries. " & @CRLF & _
"" & @CRLF & _
"### 3.4 Distributed Systems " & @CRLF & _
"- Ensure consistency by checking if replicas are equivalent. " & @CRLF & _
"- Technique: Compare hash values of data on different servers. " & @CRLF & _
"" & @CRLF & _
"---" & @CRLF & _
"" & @CRLF & _
"## 4. **Optimizations for Large-Scale Lookups** " & @CRLF & _
"" & @CRLF & _
"### 4.1 Bloom Filters " & @CRLF & _
"- Space-efficient data structure for approximate membership testing. " & @CRLF & _
"- Useful for checking if an element might be equivalent to others in a large dataset. " & @CRLF & _
"" & @CRLF & _
"#### Example: Using Bloom Filter " & @CRLF & _
"```python" & @CRLF & _
"from pybloom_live import BloomFilter" & @CRLF & _
"" & @CRLF & _
"bloom = BloomFilter(capacity=1000, error_rate=0.01)" & @CRLF & _
"bloom.add("hello")" & @CRLF & _
"print("hello" in bloom) # Output: True" & @CRLF & _
"```" & @CRLF & _
"" & @CRLF & _
"### 4.2 Caching " & @CRLF & _
"- Store results of equivalency checks to avoid recomputation. " & @CRLF & _
"- Use LRU (Least Recently Used) or LFU (Least Frequently Used) caches. " & @CRLF & _
"" & @CRLF & _
"#### Example: Caching Results with `functools.lru_cache` " & @CRLF & _
"```python" & @CRLF & _
"from functools import lru_cache" & @CRLF & _
"" & @CRLF & _
"@lru_cache(maxsize=1000)" & @CRLF & _
"def is_equivalent(a, b):" & @CRLF & _
" return sorted(a) == sorted(b)" & @CRLF & _
"" & @CRLF & _
"print(is_equivalent("listen", "silent")) # Output: True" & @CRLF & _
"```" & @CRLF & _
"" & @CRLF & _
"---" & @CRLF & _
"" & @CRLF & _
"## 5. **Challenges in Equivalency Lookups** " & @CRLF & _
"" & @CRLF & _
"1. **Scalability**: " & @CRLF & _
" - Large datasets require efficient data structures and algorithms. " & @CRLF & _
" - Use distributed systems or approximate methods for very large inputs. " & @CRLF & _
"" & @CRLF & _
"2. **Precision vs. Performance**: " & @CRLF & _
" - Approximate methods (e.g., Bloom filters) trade off precision for speed. " & @CRLF & _
"" & @CRLF & _
"3. **Ambiguity**: " & @CRLF & _
" - Defining equivalence relations can be complex for real-world data (e.g., synonym matching may depend on context). " & @CRLF & _
"" & @CRLF & _
"4. **Data Quality**: " & @CRLF & _
" - Inconsistent or noisy data can lead to false equivalences. " & @CRLF & _
"" & @CRLF & _
"---" & @CRLF & _
"" & @CRLF & _
"## 6. **Summary** " & @CRLF & _
"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. "
Local $sSubst = "$$\1$$"
Local $sResult = StringRegExpReplace($sString, $sRegex, $sSubst)
MsgBox($MB_SYSTEMMODAL, "Result", $sResult)
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 AutoIt, please visit: https://www.autoitscript.com/autoit3/docs/functions/StringRegExp.htm