Dependency Preserving Decomposition

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from typing import Set, List, Tuple, FrozenSet
from dataclasses import dataclass
 
@dataclass(frozen=True)
class FD:
    """Immutable functional dependency representation."""
    lhs: FrozenSet[str]  # Left-hand side (determinant)
    rhs: FrozenSet[str]  # Right-hand side (dependent)
    
    def __str__(self):
        lhs_str = ','.join(sorted(self.lhs))
        rhs_str = ','.join(sorted(self.rhs))
        return f"{lhs_str} → {rhs_str}"
 
 
def compute_canonical_cover(fds: List[FD]) -> List[FD]:
    """
    Compute the canonical cover of a set of functional dependencies.
    
    Algorithm:
    1. Decompose RHS to single attributes
    2. Remove extraneous LHS attributes
    3. Remove extraneous RHS attributes (redundant FDs)
    4. Recombine FDs with same LHS
    
    Time Complexity: O(n³ × m) where n = |attributes|, m = |FDs|
    """
    
    # Step 1: Decompose multi-attribute RHS into single-attribute FDs
    # A → BC becomes A → B and A → C
    working_fds = []
    for fd in fds:
        for attr in fd.rhs:
            working_fds.append(FD(fd.lhs, frozenset([attr])))
    
    # Step 2: Remove extraneous LHS attributes
    # For each FD X → A, check if any attribute in X is extraneous
    changed = True
    while changed:
        changed = False
        new_fds = []
        for fd in working_fds:
            if len(fd.lhs) <= 1:
                new_fds.append(fd)
                continue
            
            # Try removing each LHS attribute
            reduced_fd = fd
            for attr in fd.lhs:
                reduced_lhs = fd.lhs - {attr}
                if reduced_lhs:  # Can't have empty LHS
                    # Check if reduced_lhs → rhs is derivable
                    test_fds = [f for f in working_fds if f != fd]
                    test_fds.append(FD(reduced_lhs, fd.rhs))
                    
                    if is_fd_derivable(fd.lhs, fd.rhs, test_fds):
                        # Attribute is extraneous
                        reduced_fd = FD(reduced_lhs, fd.rhs)
                        changed = True
                        break
            
            new_fds.append(reduced_fd)
        working_fds = new_fds
    
    # Step 3: Remove extraneous RHS / redundant FDs
    # An FD is redundant if it can be derived from the remaining FDs
    final_fds = []
    for fd in working_fds:
        other_fds = [f for f in working_fds if f != fd]
        if not is_fd_derivable(fd.lhs, fd.rhs, other_fds):
            final_fds.append(fd)
    
    # Step 4: Recombine FDs with same LHS
    # A → B and A → C become A → BC
    lhs_to_rhs = {}
    for fd in final_fds:
        if fd.lhs not in lhs_to_rhs:
            lhs_to_rhs[fd.lhs] = set()
        lhs_to_rhs[fd.lhs].update(fd.rhs)
    
    canonical = [
        FD(lhs, frozenset(rhs_set)) 
        for lhs, rhs_set in lhs_to_rhs.items()
    ]
    
    return canonical
 
 
def is_fd_derivable(lhs: FrozenSet, rhs: FrozenSet, fds: List[FD]) -> bool:
    """
    Test if lhs → rhs is derivable from fds using attribute closure.
    """
    closure = compute_closure(lhs, fds)
    return rhs.issubset(closure)
 
 
def compute_closure(attrs: FrozenSet[str], fds: List[FD]) -> Set[str]:
    """
    Compute attribute closure of attrs under fds.
    """
    closure = set(attrs)
    changed = True
    
    while changed:
        changed = False
        for fd in fds:
            if fd.lhs.issubset(closure) and not fd.rhs.issubset(closure):
                closure.update(fd.rhs)
                changed = True
    
    return closure

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@dataclass
class RelationSchema:
    """Schema for a decomposed relation."""
    name: str
    attributes: FrozenSet[str]
    primary_key: FrozenSet[str]
    functional_dependencies: List[FD]
    
    def __str__(self):
        attrs = ', '.join(sorted(self.attributes))
        key = ', '.join(sorted(self.primary_key))
        return f"{self.name}({attrs}) [Key: {key}]"
 
 
def generate_relations_from_canonical(canonical_cover: List[FD]) -> List[RelationSchema]:
    """
    Phase 2: Generate one relation per FD in the canonical cover.
    
    For each FD X → Y in Fc:
    - Create relation with attributes X ∪ Y
    - Primary key is X
    - The FD is preserved in this relation
    
    Time Complexity: O(|Fc|)
    """
    relations = []
    
    for i, fd in enumerate(canonical_cover):
        # Relation attributes = LHS ∪ RHS
        attrs = fd.lhs.union(fd.rhs)
        
        # Primary key = LHS of the FD
        key = fd.lhs
        
        # This relation preserves this FD
        preserved_fds = [fd]
        
        relation = RelationSchema(
            name=f"R{i+1}",
            attributes=attrs,
            primary_key=key,
            functional_dependencies=preserved_fds
        )
        
        relations.append(relation)
    
    return relations
 
 
# Example
def example_phase_2():
    """Demonstrate Phase 2 with a simple example."""
    # Canonical cover: {A → BC, C → D}
    canonical = [
        FD(frozenset(['A']), frozenset(['B', 'C'])),
        FD(frozenset(['C']), frozenset(['D']))
    ]
    
    relations = generate_relations_from_canonical(canonical)
    
    for rel in relations:
        print(rel)
    
    # Output:
    # R1(A, B, C) [Key: A]
    # R2(C, D) [Key: C]

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def find_all_candidate_keys(all_attrs: FrozenSet[str], fds: List[FD]) -> List[FrozenSet[str]]:
    """
    Find all candidate keys of a relation.
    
    A candidate key is a minimal set of attributes whose closure equals all attributes.
    
    Algorithm:
    1. Start with attributes that never appear on RHS (must be in every key)
    2. Add attributes that appear on both sides
    3. Check if current set is a superkey
    4. If not, incrementally add remaining attributes
    5. Minimize to find all candidate keys
    """
    # Partition attributes
    lhs_only = set()  # Must be in every key
    rhs_only = set()  # Never in any minimal key
    both_sides = set()
    
    all_lhs = set()
    all_rhs = set()
    for fd in fds:
        all_lhs.update(fd.lhs)
        all_rhs.update(fd.rhs)
    
    for attr in all_attrs:
        if attr in all_lhs and attr not in all_rhs:
            lhs_only.add(attr)
        elif attr in all_rhs and attr not in all_lhs:
            rhs_only.add(attr)
        else:
            both_sides.add(attr)
    
    # Start with LHS-only attributes (guaranteed in key)
    base = frozenset(lhs_only)
    
    # If base is already a superkey, it's the only candidate key
    if compute_closure(base, fds) >= all_attrs:
        return [base] if is_minimal_key(base, fds, all_attrs) else find_minimal_subsets(base, fds, all_attrs)
    
    # Need to add from both_sides until we get a superkey
    candidate_keys = []
    from itertools import combinations
    
    for size in range(len(both_sides) + 1):
        for combo in combinations(both_sides, size):
            potential_key = base.union(combo)
            if compute_closure(potential_key, fds) >= all_attrs:
                # Check if minimal
                if is_minimal_key(potential_key, fds, all_attrs):
                    candidate_keys.append(potential_key)
        
        # If we found keys at this size, stop (larger ones won't be minimal)
        if candidate_keys:
            break
    
    return candidate_keys if candidate_keys else [all_attrs]  # Worst case: all attrs
 
 
def is_minimal_key(key: FrozenSet[str], fds: List[FD], all_attrs: FrozenSet[str]) -> bool:
    """Check if key is minimal (no proper subset is a superkey)."""
    for attr in key:
        smaller = key - {attr}
        if smaller and compute_closure(smaller, fds) >= all_attrs:
            return False
    return True
 
 
def ensure_lossless_join(
    relations: List[RelationSchema], 
    original_attrs: FrozenSet[str],
    original_fds: List[FD]
) -> List[RelationSchema]:
    """
    Phase 3: Add a key relation if needed for lossless join.
    
    If no existing relation contains a candidate key, add one.
    """
    candidate_keys = find_all_candidate_keys(original_attrs, original_fds)
    
    # Check if any relation contains a candidate key
    for rel in relations:
        for key in candidate_keys:
            if key.issubset(rel.attributes):
                # Already have a relation with a key - lossless guaranteed
                return relations
    
    # No relation contains a key - add one
    key_to_add = candidate_keys[0]  # Pick first candidate key
    
    key_relation = RelationSchema(
        name=f"R_key",
        attributes=key_to_add,
        primary_key=key_to_add,
        functional_dependencies=[]  # No FDs to preserve (just for lossless)
    )
    
    return relations + [key_relation]

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def remove_redundant_relations(relations: List[RelationSchema]) -> List[RelationSchema]:
    """
    Phase 4: Remove relations whose attributes are a subset of another.
    
    If Ri ⊂ Rj, remove Ri (its data is available via projection from Rj).
    
    Time Complexity: O(n² × k) where n = |relations|, k = max |attributes|
    """
    non_redundant = []
    
    for i, rel_i in enumerate(relations):
        is_redundant = False
        
        for j, rel_j in enumerate(relations):
            if i != j:
                # Check if rel_i is a proper subset of rel_j
                if rel_i.attributes < rel_j.attributes:  # Proper subset
                    is_redundant = True
                    break
        
        if not is_redundant:
            non_redundant.append(rel_i)
    
    return non_redundant
 
 
def remove_redundant_optimized(relations: List[RelationSchema]) -> List[RelationSchema]:
    """
    Optimized version using sorting by size.
    Smaller relations can only be subsumed by larger ones.
    """
    # Sort by number of attributes (ascending)
    sorted_rels = sorted(relations, key=lambda r: len(r.attributes))
    
    result = []
    
    for i, rel in enumerate(sorted_rels):
        # Only need to check against larger relations
        is_subsumed = False
        for larger in sorted_rels[i+1:]:
            if rel.attributes < larger.attributes:
                is_subsumed = True
                break
        
        if not is_subsumed:
            result.append(rel)
    
    return result

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"""
Complete Dependency-Preserving 3NF Decomposition Algorithm
 
Guarantees:
- Result is in Third Normal Form (3NF)
- All functional dependencies are preserved
- Decomposition is lossless-join
 
Time Complexity: O(n³ × m²) where n = |attributes|, m = |FDs|
Space Complexity: O(n × m)
"""
 
from typing import List, FrozenSet, Set, Tuple
from dataclasses import dataclass
 
 
@dataclass(frozen=True)
class FD:
    lhs: FrozenSet[str]
    rhs: FrozenSet[str]
    
    def __str__(self):
        return f"{','.join(sorted(self.lhs))} → {','.join(sorted(self.rhs))}"
 
 
@dataclass
class RelationSchema:
    name: str
    attributes: FrozenSet[str]
    primary_key: FrozenSet[str]
    preserved_fds: List[FD]
    
    def __str__(self):
        attrs = ', '.join(sorted(self.attributes))
        key = ', '.join(sorted(self.primary_key))
        return f"{self.name}({attrs}) [Key: {key}]"
 
 
class ThreeNFDecomposer:
    """
    Main class for 3NF decomposition with dependency preservation.
    """
    
    def __init__(self, attributes: Set[str], fds: List[Tuple[Set[str], Set[str]]]):
        """
        Initialize with relation attributes and functional dependencies.
        
        Parameters:
        - attributes: Set of attribute names
        - fds: List of (lhs, rhs) tuples representing FDs
        """
        self.original_attrs = frozenset(attributes)
        self.original_fds = [
            FD(frozenset(lhs), frozenset(rhs)) 
            for lhs, rhs in fds
        ]
    
    def decompose(self) -> List[RelationSchema]:
        """
        Execute the complete 3NF decomposition algorithm.
        
        Returns list of RelationSchema objects representing the decomposition.
        """
        print("=" * 60)
        print("3NF DECOMPOSITION ALGORITHM")
        print("=" * 60)
        
        # Phase 1: Compute canonical cover
        print("\n[Phase 1] Computing canonical cover...")
        canonical = self._compute_canonical_cover()
        print(f"  Canonical cover: {[str(fd) for fd in canonical]}")
        
        # Phase 2: Generate relations
        print("\n[Phase 2] Generating relations...")
        relations = self._generate_relations(canonical)
        for rel in relations:
            print(f"  {rel}")
        
        # Phase 3: Ensure lossless join
        print("\n[Phase 3] Checking lossless join property...")
        relations = self._ensure_lossless(relations)
        
        # Phase 4: Remove redundant relations
        print("\n[Phase 4] Removing redundant relations...")
        relations = self._remove_redundant(relations)
        
        print("\n" + "=" * 60)
        print("FINAL DECOMPOSITION")
        print("=" * 60)
        for rel in relations:
            print(f"  {rel}")
        
        return relations
    
    def _compute_canonical_cover(self) -> List[FD]:
        """Phase 1: Compute canonical cover of FDs."""
        # Step 1: Decompose RHS to singletons
        working = []
        for fd in self.original_fds:
            for attr in fd.rhs:
                working.append(FD(fd.lhs, frozenset([attr])))
        
        # Step 2: Remove extraneous LHS attributes
        working = self._remove_extraneous_lhs(working)
        
        # Step 3: Remove redundant FDs
        working = self._remove_redundant_fds(working)
        
        # Step 4: Recombine by LHS
        lhs_to_rhs = {}
        for fd in working:
            if fd.lhs not in lhs_to_rhs:
                lhs_to_rhs[fd.lhs] = set()
            lhs_to_rhs[fd.lhs].update(fd.rhs)
        
        return [FD(lhs, frozenset(rhs)) for lhs, rhs in lhs_to_rhs.items()]
    
    def _remove_extraneous_lhs(self, fds: List[FD]) -> List[FD]:
        """Remove extraneous attributes from LHS of each FD."""
        result = list(fds)
        changed = True
        
        while changed:
            changed = False
            new_result = []
            
            for fd in result:
                reduced = fd
                
                if len(fd.lhs) > 1:
                    for attr in fd.lhs:
                        test_lhs = fd.lhs - {attr}
                        if test_lhs:
                            # Check if reduced FD still works
                            other_fds = [f for f in result if f != fd]
                            other_fds.append(FD(test_lhs, fd.rhs))
                            
                            closure = self._compute_closure(test_lhs, other_fds)
                            if fd.rhs.issubset(closure):
                                reduced = FD(test_lhs, fd.rhs)
                                changed = True
                                break
                
                new_result.append(reduced)
            
            result = new_result
        
        return result
    
    def _remove_redundant_fds(self, fds: List[FD]) -> List[FD]:
        """Remove FDs that are derivable from others."""
        result = []
        
        for fd in fds:
            others = [f for f in fds if f != fd]
            closure = self._compute_closure(fd.lhs, others)
            
            if not fd.rhs.issubset(closure):
                result.append(fd)
        
        return result
    
    def _compute_closure(self, attrs: FrozenSet[str], fds: List[FD]) -> Set[str]:
        """Compute attribute closure."""
        closure = set(attrs)
        changed = True
        
        while changed:
            changed = False
            for fd in fds:
                if fd.lhs.issubset(closure) and not fd.rhs.issubset(closure):
                    closure.update(fd.rhs)
                    changed = True
        
        return closure
    
    def _generate_relations(self, canonical: List[FD]) -> List[RelationSchema]:
        """Phase 2: Create one relation per FD."""
        relations = []
        
        for i, fd in enumerate(canonical):
            rel = RelationSchema(
                name=f"R{i+1}",
                attributes=fd.lhs.union(fd.rhs),
                primary_key=fd.lhs,
                preserved_fds=[fd]
            )
            relations.append(rel)
        
        return relations
    
    def _ensure_lossless(self, relations: List[RelationSchema]) -> List[RelationSchema]:
        """Phase 3: Add key relation if needed."""
        candidate_keys = self._find_candidate_keys()
        
        # Check if any relation contains a key
        for rel in relations:
            for key in candidate_keys:
                if key.issubset(rel.attributes):
                    print(f"  ✓ Lossless: {rel.name} contains key {set(key)}")
                    return relations
        
        # Add key relation
        key = candidate_keys[0]
        key_rel = RelationSchema(
            name="R_key",
            attributes=key,
            primary_key=key,
            preserved_fds=[]
        )
        print(f"  + Adding key relation: {key_rel}")
        return relations + [key_rel]
    
    def _find_candidate_keys(self) -> List[FrozenSet[str]]:
        """Find candidate keys of original relation."""
        # Find attributes that must be in every key
        all_lhs = set()
        all_rhs = set()
        for fd in self.original_fds:
            all_lhs.update(fd.lhs)
            all_rhs.update(fd.rhs)
        
        must_be_in_key = frozenset(a for a in self.original_attrs if a not in all_rhs)
        
        # Try to close this set
        closure = self._compute_closure(must_be_in_key, self.original_fds)
        
        if closure >= self.original_attrs:
            return [must_be_in_key]
        
        # Need to add more attributes
        from itertools import combinations
        remaining = self.original_attrs - must_be_in_key - frozenset(a for a in self.original_attrs if a in all_rhs and a not in all_lhs)
        
        for size in range(len(remaining) + 1):
            for combo in combinations(remaining, size):
                potential = must_be_in_key.union(combo)
                if self._compute_closure(potential, self.original_fds) >= self.original_attrs:
                    return [potential]
        
        return [self.original_attrs]
    
    def _remove_redundant(self, relations: List[RelationSchema]) -> List[RelationSchema]:
        """Phase 4: Remove subsumed relations."""
        result = []
        
        for rel in relations:
            subsumed = False
            for other in relations:
                if rel != other and rel.attributes < other.attributes:
                    print(f"  - Removing {rel.name} (subsumed by {other.name})")
                    subsumed = True
                    break
            
            if not subsumed:
                result.append(rel)
        
        if len(result) == len(relations):
            print("  No redundant relations found")
        
        return result
 
 
# =====================================================
# USAGE EXAMPLE
# =====================================================
 
if __name__ == "__main__":
    # Example: Employee database
    attributes = {'EmpID', 'Name', 'DeptID', 'DeptName', 'ManagerID'}
    fds = [
        ({'EmpID'}, {'Name', 'DeptID'}),
        ({'DeptID'}, {'DeptName', 'ManagerID'}),
    ]
    
    decomposer = ThreeNFDecomposer(attributes, fds)
    result = decomposer.decompose()