1. 1 Union-find

2. 2 Maintain a collection of disjoint sets under the following two operations S 3 = Union(S 1,S 2 ) Find(x) : returns the set containing x

3. 3 Union-find We assume there are n fixed elements We start with n sets each containing a single element Each element has a pointer to its representation in the set containing it

4. 4 S1 = {e1} S2 = {e2} S3={e3} S4={e4} …… S1 = {e1} A= {e2,e3} S4={e4} …… A = Union(S3,S4) Find(e2)  A B = Union(A,S7) S1 = {e1} B= {e2,e3,e7} S4={e4} ……

5. 5 Why ? Suppose we want to maintain an equivalence relation: y z t x b v s a y ≡ z

6. 6 y, z t x b v s a y ≡ s

7. 7 y, z, s t x b v a y ≡ s

8. 8 Queries Equivalent?(y,a)

9. 9 Can solve this with union-find Each equivalence class is a set y ≡ s  union(find(y),find(s)) Equivalent?(y,a)  return yes if find(y) = find(a)

10. 10 e2 Representation Represent each set by a tree, each node represents an item, the root also represents the set e3e7 B

11. 11 Concretely e3 e7 e2 B

12. 12 Find(e10) e6 e9 e11 e8e10 C

13. 13 Find(e10) e6 e9 e11 e8e10 Find(x) while (x.parent ≠null) do x ← x.parent return (x) C

14. 14 D=Union(B,C) B e2 e3 e7 C e6 e9 e11 e8e10

15. 15 D=Union(B,C) B e2 e3 e7 C e6 e9 e11 e8e10

16. 16 D=Union(B,C) B e2 e3 e7 D e6 e9 e11 e8e10

17. 17 D=Union(B,C) B e2 e3 e7 C e6 e9 e11 e8e10

18. 18 D=Union(B,C) B e2 e3 e7 C e6 e9 e11 e8e10

19. 19 D=Union(B,C) D e2 e3 e7 C e6 e9 e11 e8e10

20. 20 Link by size For the find’s sake its better to hang the smaller on the larger

21. 21 D=Union(B,C) B e2 e3 e7 C e6 e9 e11 e8e10 5 3 If (C.size > B.size) then B.Parent ← C; C.size = C.size + B.size return (C) Else C.parent ← B; B.size = B.size + C.size return (B)

22. 22 D=Union(B,C) B e2 e3 e7 C e6 e9 e11 e8e10 8 3 If (C.size > B.size) then B.Parent ← C; C.size = C.size + B.size return (C) Else C.parent ← B; B.size = B.size + C.size return (B) D

23. 23 Analysis Union: O(1) time Find: O(log(n)) time The depth of a tree is at most log(n)

24. 24 Proof By induction on the sequence of operations: For every x Depth(x) ≤ log |T(x)| x

25. 25 Path compression