Remarks

  • Union Find is a data structure keeps track of a set of elements that are partitioned into a number of disjoint subsets.
  • It has two operations union(p, q) and find(p).
    • the find/search finds the subset element p belongs to.
    • and union/merge merges the subsets containing p and q.
  • It memory usage is O(N), and each find and union operations are near O(1) in time.
  • The algorithm can be used to find all connected components in a network.
  • It is also used in kruskal’s algorithm to find the minimal spanning tree for a graph.

Implementation

Simplest implementation to use the core of union find.

parents = list(range(n))

def find(i):
    if i != parents[i]:
        parents[i] = find(parents[i])
        i = parents[i]
    return parents[i]

def union(x, y):
    x, y = find(x), find(y)
    if x == y: return
    parents[x] = y
    return y

Define a class for UF

class UnionFind:
    def __init__(self, n):
        self.parents = list(range(n))
        self.sizes = [1] * n

    def _find_recursive(self, i):
        while i != self.parents[i]:
            # path compression, have i points to the cluster centroid
            self.parents[i] = self._find_recursive(self.parents[i])  
            i = self.parents[i]
        return i

    def _find_iterative(self, i):
        # 1 path to find the root
        root = i
        while self.parents[root] != root:
            root = self.parents[root]
        # 2 path to assign every node in the path to points at root
        while self.parents[i] != root:
            parent = self.parents[i]
            self.parents[i] = root
            i = parent
        return root

    find = _find_recursive

    def union(self, p, q):
        """Attempt to union two elements

        Args:
            p, q: two elements to be attempted for union

        Returns:
            Boolean indicates whether a union operation is performed. 
        """
        root_p, root_q = map(self.find, (p, q))
        if root_p == root_q: return False
        small, big = sorted([root_p, root_q], key=lambda x: self.sizes[x])
        self.parents[small] = big
        self.sizes[big] += self.sizes[small]    
        return True

Use Union Find to find connected components in undirected graph.

connections = [[0, 1], [1, 2], [2, 3], [3, 4], [5, 6], [6, 8], [7, 9]]

uf = UnionFind(10)
for p, q in connections: uf.union(p, q)
num_components = len(set(uf.find(i) for i in range(10)))
print(num_components)

union find animation

  • To find minimum spanning tree in the graph where there are weights on the edges.
    1. Sort edges by edge weights ascendingly.
    2. Iteratve over the edges unify nodes when two nodes don’t belong to same cluster.
    3. Repeat 2 until no nodes or no edges.

An Implementation without pre-allocation

class UnionFind:
    def __init__(self):
        self.parents = dict()
        self.sizes = dict()
        self.n_sets = 0

    def __contains__(self, i):
        return i in self.parents

    def insert(self, i):
        if self.__contains__(i): return
        self.parents[i] = i
        self.sizes[i] = 1
        self.n_sets += 1

    def find(self, i):
        while i != self.parents[i]:
            self.parents[i] = self.find(self.parents[i])  
            i = self.parents[i]
        return i

    def union(self, p, q):
        """Attempt to union two elements
        
        Returns:
            Boolean indicates whether a union operation is performed. 
        """
        root_p, root_q = map(self.find, (p, q))
        if root_p == root_q: return False
        small, big = sorted([root_p, root_q], key=lambda x: self.sizes[x])
        self.parents[small] = big
        self.sizes[big] += self.sizes[small]    
        self.n_sets -= 1
        return True

Sample Questions

  1. 990. Satisfiability of Equality Equations