1. Disjoint Set Neil Tang 02/26/2008
CS223 Advanced Data Structures and Algorithms

2. CS223 Advanced Data Structures and Algorithms
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
CS223 Advanced Data Structures and Algorithms

3. CS223 Advanced Data Structures and Algorithms
Disjoint Set
Given a set of elements, we can have a collection S = {S1, S2, ... 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. CS223 Advanced Data Structures and Algorithms
An 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 g e f h i
CS223 Advanced Data Structures and Algorithms

5. CS223 Advanced Data Structures and Algorithms
Basic 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 Implementationf
nil
head
tail a b c
find(b) a b c f
nil
tail
union(f, b)
CS223 Advanced Data Structures and Algorithms

7. CS223 Advanced Data Structures and Algorithms
Array Implementation
Assume that all the elements are numbered sequentially from 0 to N-1.
CS223 Advanced Data Structures and Algorithms

8. CS223 Advanced Data Structures and Algorithms
Array Implementation
CS223 Advanced Data Structures and Algorithms

9. CS223 Advanced Data Structures and Algorithms
Array Implementation
CS223 Advanced Data Structures and Algorithms

10. CS223 Advanced Data Structures and Algorithms
Union Operation
Time complexity: O(1)
CS223 Advanced Data Structures and Algorithms

11. CS223 Advanced Data Structures and Algorithms
Find Operation
Time complexity: O(N)
CS223 Advanced Data Structures and Algorithms

12. CS223 Advanced Data Structures and Algorithms
Union-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)
Make the shallow tree a subtree of the deeper and break ties arbitrarily.
CS223 Advanced Data Structures and Algorithms

14. CS223 Advanced Data Structures and Algorithms
Size and Height -1 4 -5 6 1 2 3 4 5 6 7 -1 4 -3 6
CS223 Advanced Data Structures and Algorithms

15. Union-by-Height (Rank)
Time complexity: O(logN)
CS223 Advanced Data Structures and Algorithms

16. CS223 Advanced Data Structures and Algorithms
Worst-Case Tree
CS223 Advanced Data Structures and Algorithms

17. Find with Path Compression
CS223 Advanced Data Structures and Algorithms

18. Find with Path Compression
CS223 Advanced Data Structures and Algorithms

19. Find 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 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.
CS223 Advanced Data Structures and Algorithms