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Disjoint Set Neil Tang 02/26/2008. Class Overview. Disjoint Set and An Application Basic Operations Linked-list Implementation Array Implementation Union-by-Size and Union-by-Height(Rank) Find with Path Compression Worst-Case Time Complexity. Disjoint Set. - PowerPoint PPT Presentation
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CS223 Advanced Data Structures and Algorithms 1
Disjoint Set Disjoint Set
Neil TangNeil Tang02/26/200802/26/2008
CS223 Advanced Data Structures and Algorithms 2
Class OverviewClass Overview
Disjoint Set and An Application
Basic Operations
Linked-list Implementation
Array Implementation
Union-by-Size and Union-by-Height(Rank)
Find with Path Compression
Worst-Case Time Complexity
CS223 Advanced Data Structures and Algorithms 3
Disjoint SetDisjoint Set
Given a set of elements, we can have a collection S = {S1, S
2, ... Sk} of disjoint dynamic (sub) sets.
Representative of a set: We choose one element of a set to identify the set, e.g., we use the root of a tree to identify a tree, or the head element of a linked list to access the linked list.
Usually, we want to find out if two elements belong to the same set.
CS223 Advanced Data Structures and Algorithms 4
An ApplicationAn Application
Given an undirected graph G = (V, E)
We may want to find all connected components, whether the graph is connected or whether two given nodes belong to the same connected component.
a
b
c
d
ge
f h
i
CS223 Advanced Data Structures and Algorithms 5
Basic OperationsBasic Operations
find(x): find which disjoint set x belongs to
Union(x,y): Union set x and set y.
CS223 Advanced Data Structures and Algorithms 6
Linked-list ImplementationLinked-list Implementation
union(f, b)
fnil
head
taila b c
nilfind(b) tail
a b c fnil tail
CS223 Advanced Data Structures and Algorithms 7
Array ImplementationArray Implementation
Assume that all the elements are numbered sequentially from 0 to N-1.
CS223 Advanced Data Structures and Algorithms 8
Array ImplementationArray Implementation
CS223 Advanced Data Structures and Algorithms 9
Array ImplementationArray Implementation
CS223 Advanced Data Structures and Algorithms 10
Union OperationUnion Operation
Time complexity: O(1)
CS223 Advanced Data Structures and Algorithms 11
Find OperationFind Operation
Time complexity: O(N)
CS223 Advanced Data Structures and Algorithms 12
Union-by-SizeUnion-by-Size
Make the smaller tree a subtree of the larger and break ties arbitrarily.
CS223 Advanced Data Structures and Algorithms 13
Union-by-Height (Rank)Union-by-Height (Rank)
Make the shallow tree a subtree of the deeper and break ties arbitrarily.
CS223 Advanced Data Structures and Algorithms 14
Size and HeightSize and Height
-1 -1 -1 4 -5 4 4 6
-1 -1 -1 4 -3 4 4 6
0 1 2 3 4 5 6 7
CS223 Advanced Data Structures and Algorithms 15
Union-by-Height (Rank)Union-by-Height (Rank)
Time complexity: O(logN)
CS223 Advanced Data Structures and Algorithms 16
Worst-Case TreeWorst-Case Tree
CS223 Advanced Data Structures and Algorithms 17
Find with Path CompressionFind with Path Compression
CS223 Advanced Data Structures and Algorithms 18
Find with Path CompressionFind with Path Compression
CS223 Advanced Data Structures and Algorithms 19
Find with Path CompressionFind with Path Compression
Fully compatible with union-by-size.
Not compatible with union-by-height.
Union-by-size is usually as efficient as union-by-height.
CS223 Advanced Data Structures and Algorithms 20
Worst-Case Time ComplexityWorst-Case Time Complexity
If both union-by-rank and path compression heuristics are used, the worst-case running time for any sequence of M union/find operations is O(M * (M,N)), where (M, N) is the inverse Ackermann function which grows even slower than logN.
For any practical purposes, (M, N) < 4.