​
Core Implementation

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Basic DSU with Path Compression and Union by Rank

class DisjointSet:
    def __init__(self, n):
        self.parent = list(range(n))
        self.rank = [0] * n
        self.size = [1] * n  # Track size of each set
        self.components = n  # Track number of components
    
    def find(self, x):
        # Path compression
        if self.parent[x] != x:
            self.parent[x] = self.find(self.parent[x])
        return self.parent[x]
    
    def union(self, x, y):
        px, py = self.find(x), self.find(y)
        if px == py:
            return False  # Already in same set
        
        # Union by rank
        if self.rank[px] < self.rank[py]:
            px, py = py, px
        self.parent[py] = px
        self.size[px] += self.size[py]
        
        if self.rank[px] == self.rank[py]:
            self.rank[px] += 1
        
        self.components -= 1
        return True
    
    def connected(self, x, y):
        return self.find(x) == self.find(y)
    
    def get_size(self, x):
        return self.size[self.find(x)]

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Implementation Variations

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1. DSU with Path Splitting

class PathSplittingDSU:
    def __init__(self, n):
        self.parent = list(range(n))
    
    def find(self, x):
        # Path splitting
        while self.parent[x] != x:
            next_parent = self.parent[self.parent[x]]
            self.parent[x] = next_parent
            x = next_parent
        return x

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2. DSU with Path Halving

class PathHalvingDSU:
    def __init__(self, n):
        self.parent = list(range(n))
    
    def find(self, x):
        # Path halving
        while self.parent[x] != x:
            self.parent[x] = self.parent[self.parent[x]]
            x = self.parent[x]
        return x

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3. Weighted Quick Union

class WeightedQuickUnion:
    def __init__(self, n):
        self.parent = list(range(n))
        self.size = [1] * n
    
    def find(self, x):
        while self.parent[x] != x:
            x = self.parent[x]
        return x
    
    def union(self, x, y):
        px, py = self.find(x), self.find(y)
        if px == py:
            return
        
        # Always attach smaller tree under root of larger tree
        if self.size[px] < self.size[py]:
            self.parent[px] = py
            self.size[py] += self.size[px]
        else:
            self.parent[py] = px
            self.size[px] += self.size[py]

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Time Complexities

OperationSimple DSUPath CompressionPath SplittingWeighted QU
Make SetO(1)O(1)O(1)O(1)
FindO(N)O(α(N))*O(log N)O(log N)
UnionO(N)O(α(N))*O(log N)O(log N)
ConnectedO(N)O(α(N))*O(log N)O(log N)
* α(N) is the inverse Ackermann function, which is effectively constant

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Space Complexities

ImplementationSpace ComplexityAdditional Notes
Basic DSUO(N)Parent array only
Ranked DSUO(N)Extra array for ranks
Weighted QUO(N)Extra array for sizes
Path CompressionO(N)No extra space beyond parent array

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Common Data Structure Patterns

  1. Connected Components
    • Graph connectivity
    • Component counting
    • Cycle detection
  2. Dynamic Connectivity
    • Online connectivity queries
    • Real-time updates
    • Network partitioning
  3. Minimum Spanning Tree
    • Kruskal’s algorithm
    • Network design
    • Clustering
  4. Component Properties
    • Size tracking
    • Component merging
    • Set operations
  5. Graph Algorithms
    • Cycle detection
    • Connected components
    • Path compression
  6. Network Flow
    • Network connectivity
    • Cut problems
    • Flow partitioning

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Edge Cases to Consider

  1. Single Element Set
def handle_single(self, x):
    return self.parent[x] == x and self.size[x] == 1
  1. Self-Union
def union_safe(self, x, y):
    if x == y:
        return False  # Cannot union element with itself
    return self.union(x, y)
  1. Invalid Elements
def validate_element(self, x):
    if not 0 <= x < len(self.parent):
        raise ValueError(f"Element {x} out of range")

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Optimization Techniques

  1. Path Compression with Recursion Elimination
def find_iterative(self, x):
    # Find root
    root = x
    while self.parent[root] != root:
        root = self.parent[root]
    
    # Compress path
    while x != root:
        next_parent = self.parent[x]
        self.parent[x] = root
        x = next_parent
    
    return root
  1. Rank Balancing
class RankBalancedDSU:
    def __init__(self, n):
        self.parent = list(range(n))
        self.rank = [0] * n
        self.max_rank = 0
    
    def union(self, x, y):
        px, py = self.find(x), self.find(y)
        if px == py:
            return
        
        if self.rank[px] < self.rank[py]:
            px, py = py, px
        
        self.parent[py] = px
        if self.rank[px] == self.rank[py]:
            self.rank[px] += 1
            self.max_rank = max(self.max_rank, self.rank[px])

​
Memory Management

  1. Compact Representation
class CompactDSU:
    def __init__(self, n):
        # Use a single array for both parent and rank
        # Higher 16 bits for rank, lower 16 bits for parent
        self.data = list(range(n))
    
    def get_parent(self, x):
        return self.data[x] & 0xFFFF
    
    def get_rank(self, x):
        return self.data[x] >> 16
    
    def set_parent(self, x, parent):
        self.data[x] = (self.get_rank(x) << 16) | parent
    
    def set_rank(self, x, rank):
        self.data[x] = (rank << 16) | self.get_parent(x)
  1. Dynamic Allocation
class DynamicDSU:
    def __init__(self):
        self.parent = {}
        self.rank = {}
    
    def make_set(self, x):
        if x not in self.parent:
            self.parent[x] = x
            self.rank[x] = 0
    
    def find(self, x):
        if x not in self.parent:
            self.make_set(x)
        if self.parent[x] != x:
            self.parent[x] = self.find(self.parent[x])
        return self.parent[x]

​
Performance Considerations

  1. Batch Operations
def batch_union(self, pairs):
    # Sort pairs to optimize cache usage
    pairs.sort()
    for x, y in pairs:
        self.union(x, y)
  1. Size-based Optimization
class SizeOptimizedDSU:
    def __init__(self, n):
        self.parent = list(range(n))
        self.size = [1] * n
    
    def union(self, x, y):
        px, py = self.find(x), self.find(y)
        if px == py:
            return
        
        # Always merge smaller component into larger
        if self.size[px] < self.size[py]:
            self.parent[px] = py
            self.size[py] += self.size[px]
        else:
            self.parent[py] = px
            self.size[px] += self.size[py]

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Common Pitfalls

  1. Missing Path Compression
# Wrong
def find_wrong(self, x):
    while self.parent[x] != x:
        x = self.parent[x]
    return x

# Correct
def find_correct(self, x):
    if self.parent[x] != x:
        self.parent[x] = self.find_correct(self.parent[x])
    return self.parent[x]
  1. Incorrect Rank Update
# Wrong
def union_wrong(self, x, y):
    px, py = self.find(x), self.find(y)
    if px == py:
        return
    self.parent[py] = px
    self.rank[px] += 1  # Always incrementing rank

# Correct
def union_correct(self, x, y):
    px, py = self.find(x), self.find(y)
    if px == py:
        return
    if self.rank[px] < self.rank[py]:
        px, py = py, px
    self.parent[py] = px
    if self.rank[px] == self.rank[py]:
        self.rank[px] += 1
  1. Not Maintaining Component Count
# Wrong
def union_no_count(self, x, y):
    px, py = self.find(x), self.find(y)
    if px == py:
        return
    self.parent[py] = px  # Lost track of component count

# Correct
def union_with_count(self, x, y):
    px, py = self.find(x), self.find(y)
    if px == py:
        return
    self.parent[py] = px
    self.components -= 1  # Track components