ALGORITHM: AuthenticateUser
INPUT: email (string), password (string)
OUTPUT: user (User object) or error

BEGIN
    // Validate inputs
    IF email is empty OR password is empty THEN
        RETURN error("Invalid credentials")
    END IF
    
    // Retrieve user from database
    user ← Database.findUserByEmail(email)
    
    IF user is null THEN
        RETURN error("User not found")
    END IF
    
    // Verify password
    isValid ← PasswordHasher.verify(password, user.passwordHash)
    
    IF NOT isValid THEN
        // Log failed attempt
        SecurityLog.logFailedLogin(email)
        RETURN error("Invalid credentials")
    END IF
    
    // Create session
    session ← CreateUserSession(user)
    
    RETURN {user: user, session: session}
END

DATA STRUCTURES:

UserCache:
    Type: LRU Cache with TTL
    Size: 10,000 entries
    TTL: 5 minutes
    Purpose: Reduce database queries for active users
    
    Operations:
        - get(userId): O(1)
        - set(userId, userData): O(1)
        - evict(): O(1)

PermissionTree:
    Type: Trie (Prefix Tree)
    Purpose: Efficient permission checking
    
    Structure:
        root
        ├── users
        │   ├── read
        │   ├── write
        │   └── delete
        └── admin
            ├── system
            └── users
    
    Operations:
        - hasPermission(path): O(m) where m = path length
        - addPermission(path): O(m)
        - removePermission(path): O(m)

PATTERN: Rate Limiting (Token Bucket)

ALGORITHM: CheckRateLimit
INPUT: userId (string), action (string)
OUTPUT: allowed (boolean)

CONSTANTS:
    BUCKET_SIZE = 100
    REFILL_RATE = 10 per second

BEGIN
    bucket ← RateLimitBuckets.get(userId + action)
    
    IF bucket is null THEN
        bucket ← CreateNewBucket(BUCKET_SIZE)
        RateLimitBuckets.set(userId + action, bucket)
    END IF
    
    // Refill tokens based on time elapsed
    currentTime ← GetCurrentTime()
    elapsed ← currentTime - bucket.lastRefill
    tokensToAdd ← elapsed * REFILL_RATE
    
    bucket.tokens ← MIN(bucket.tokens + tokensToAdd, BUCKET_SIZE)
    bucket.lastRefill ← currentTime
    
    // Check if request allowed
    IF bucket.tokens >= 1 THEN
        bucket.tokens ← bucket.tokens - 1
        RETURN true
    ELSE
        RETURN false
    END IF
END

ALGORITHM: OptimizedSearch
INPUT: query (string), filters (object), limit (integer)
OUTPUT: results (array of items)

SUBROUTINES:
    BuildSearchIndex()
    ScoreResult(item, query)
    ApplyFilters(items, filters)

BEGIN
    // Phase 1: Query preprocessing
    normalizedQuery ← NormalizeText(query)
    queryTokens ← Tokenize(normalizedQuery)
    
    // Phase 2: Index lookup
    candidates ← SET()
    FOR EACH token IN queryTokens DO
        matches ← SearchIndex.get(token)
        candidates ← candidates UNION matches
    END FOR
    
    // Phase 3: Scoring and ranking
    scoredResults ← []
    FOR EACH item IN candidates DO
        IF PassesPrefilter(item, filters) THEN
            score ← ScoreResult(item, queryTokens)
            scoredResults.append({item: item, score: score})
        END IF
    END FOR
    
    // Phase 4: Sort and filter
    scoredResults.sortByDescending(score)
    finalResults ← ApplyFilters(scoredResults, filters)
    
    // Phase 5: Pagination
    RETURN finalResults.slice(0, limit)
END

SUBROUTINE: ScoreResult
INPUT: item, queryTokens
OUTPUT: score (float)

BEGIN
    score ← 0
    
    // Title match (highest weight)
    titleMatches ← CountTokenMatches(item.title, queryTokens)
    score ← score + (titleMatches * 10)
    
    // Description match (medium weight)
    descMatches ← CountTokenMatches(item.description, queryTokens)
    score ← score + (descMatches * 5)
    
    // Tag match (lower weight)
    tagMatches ← CountTokenMatches(item.tags, queryTokens)
    score ← score + (tagMatches * 2)
    
    // Boost by recency
    daysSinceUpdate ← (CurrentDate - item.updatedAt).days
    recencyBoost ← 1 / (1 + daysSinceUpdate * 0.1)
    score ← score * recencyBoost
    
    RETURN score
END

ANALYSIS: User Authentication Flow

Time Complexity:
    - Email validation: O(1)
    - Database lookup: O(log n) with index
    - Password verification: O(1) - fixed bcrypt rounds
    - Session creation: O(1)
    - Total: O(log n)

Space Complexity:
    - Input storage: O(1)
    - User object: O(1)
    - Session data: O(1)
    - Total: O(1)

ANALYSIS: Search Algorithm

Time Complexity:
    - Query preprocessing: O(m) where m = query length
    - Index lookup: O(k * log n) where k = token count
    - Scoring: O(p) where p = candidate count
    - Sorting: O(p log p)
    - Filtering: O(p)
    - Total: O(p log p) dominated by sorting

Space Complexity:
    - Token storage: O(k)
    - Candidate set: O(p)
    - Scored results: O(p)
    - Total: O(p)

Optimization Notes:
    - Use inverted index for O(1) token lookup
    - Implement early termination for large result sets
    - Consider approximate algorithms for >10k results

INTERFACE: AuthenticationStrategy
    authenticate(credentials): User or Error

CLASS: EmailPasswordStrategy IMPLEMENTS AuthenticationStrategy
    authenticate(credentials):
        // Email$password logic
        
CLASS: OAuthStrategy IMPLEMENTS AuthenticationStrategy
    authenticate(credentials):
        // OAuth logic
        
CLASS: AuthenticationContext
    strategy: AuthenticationStrategy
    
    executeAuthentication(credentials):
        RETURN strategy.authenticate(credentials)

CLASS: EventEmitter
    listeners: Map<eventName, List<callback>>
    
    on(eventName, callback):
        IF NOT listeners.has(eventName) THEN
            listeners.set(eventName, [])
        END IF
        listeners.get(eventName).append(callback)
    
    emit(eventName, data):
        IF listeners.has(eventName) THEN
            FOR EACH callback IN listeners.get(eventName) DO
                callback(data)
            END FOR
        END IF