1. CSE 326 Union Find
Richard Anderson
Text Chapter 8
CSE 326

2. Union Find Problem Fixed collection of items Disjoint subsets
Find(x) – which subset is x in?
Union(x,y) – merge the sets containing x and y
What run times do you expect for implementations of Union and Find?
CSE 326

3. Applications of Union Find
Building connected components
foreach edge (e = (v1, v2)){
if (Find(v1) != Find(v2))
Union(v1, v2); }
CSE 326

4. Equivalence Classes
Fixed collection of items partitioned into equivalence classes
Equivalence classes identified by one of their members
{1, 3, 5}, {2, 8}, {4}, {6, 9, 10, 11}, {7}
CSE 326

5. Draw an in-tree representation of the following equivalence relation
{1,11}, {2, 4, 6, 8, 10, 12}, {3, 5, 7, 9}, {13}, {14}, {15}
CSE 326

6. Array representation Store parents in the array P 3 8 3 4 3 9 7 8 9 96
CSE 326

7. Basic Find operation Write the code for Find(x) using the Array P[ ]
CSE 326

8. Union Union(x, y){ x1 = Find(x); y1 = Find(y); P[x1] = y1; } 7-29-2005
CSE 326

9. Unions create long chains
CSE 326

10. Weighted Union W[x]: number of descendents of x (written wt(x))
Union(x, y){
x1 = Find(x);
y1 = Find(y);
if (W[x1] > W[y1]){
P[y1] = x1;
W[x1] = W[x1] + W[y1]; }
else {
P[x1] = y1;
W[y1] = W[x1] + W[y1];
W[x]: number of descendents of x (written wt(x))
Weighted union: smaller tree points to the bigger tree
CSE 326

11. Binomial Trees, B0, B1, …, Bk B0 Bk = Union(Bk-1, Bk-1) Bk-1 Bk-1 Bk
CSE 326

12. Binomial Trees Draw B4 What is the height of Bk?
What is the weight of Bk?
CSE 326

13. Theorem: Weight balanced unions guarantee O(log n) depth
Show wt(T) >= 2ht(T)
Show this is an invariant maintained by Union and Find operations
Base case – true for singleton node
Trivial case – invariant for Find operations
Interesting case: T = Union(T1, T2)
CSE 326

14. wt(T1) >= 2ht(T1), wt(T2) >= 2ht(T2) Show wt(T) >= 2ht(T)
Assume
wt(T2) >= wt(T1)
Idea, consider cases:
ht(T1) >= ht(T2)
ht(T1) < ht(T2) T1 T T2
CSE 326

15. Case 1 ht(T1) >= ht(T2) T Show 2ht(T) <= wt(T)
2ht(T) = 2ht(T1)+1 <= 2wt(T1) <= wt(T) T1 T T2
Show 2ht(T) <= wt(T)
Hint: what is ht(T)?
Recall:
wt(T2) >= wt(T1)
2ht(T1) <= wt(T1)
2ht(T2) <= wt(T2)
CSE 326

16. Case 2: ht(T1) < ht(T2) T Show 2ht(T) <= wt(T)
2ht(T) = 2ht(T2) <= wt(T2) <= wt(T) T1 T T2
Show 2ht(T) <= wt(T)
Hint: What is ht(T)?
Recall:
wt(T2) >= wt(T1)
2ht(T1) <= wt(T1)
2ht(T2) <= wt(T2)
CSE 326

17. Path Compression
Long chains are only expensive if they are traversed multiple times
Idea: Compress chains when they are traversed
CSE 326

18. Path Compression Algorithms
Find(x) {
y = x;
while (P[x] != x)
x = P[x];
while (P[y] != x){
t = y; P[y] = x; y = t; }
return x;
Find(x){
while (P[x] != x)
x = P[x] = P[P[x]];
return x; }
CSE 326

19. Amortized Complexity Cost of n intermixed Union-Find operations
Basic algorithm O(n2)
Weighted union O(n log n)
Path Compression O(n log n)
Weighted union + Path Compression O(a(n) n)
CSE 326