1. Minimum Spanning Trees
Given an undirected graph G=(V,E), find a graph G’=(V,E’) such that
E’ is a subset of E
|E’| = |V| - 1
G’ is connected
is minimal
G’ is a minimal spanning tree.
Applications: wiring a house, power grids, Internet connections
Nov 26, 2001
CSE 373, Autumn 2001

2. Prim’s algorithm
Idea: Grow a tree by adding an edge from the “known” vertices to the “unknown” vertices. Pick the edge with the smallest weight. G v
known
Nov 26, 2001
CSE 373, Autumn 2001

3. Example 5 5 v1 v0 5 8 2 2 v3 10 1 v2 v4 10 6 5 3 v5
Nov 26, 2001
CSE 373, Autumn 2001

4. Analysis Running time. Same as Dijkstra’s: O(|E| log |V|) Correctness:
Suppose we have a partially built tree that we know is contained in some minimum spanning tree T. Let (u,v)E, where u is “known” and v is “unknown” and has minimal cost.
Then there is a MST T’ that contains the partially built tree and (u,v) that has as low a cost as T.
Nov 26, 2001
CSE 373, Autumn 2001

5. Union/Find Algorithms
Equivalence Relations
Suppose R is a relation defined on a set S. R is an equivalence relation if:
aRa for all aS (reflexive)
aRb iff bRa for all a,bS (symmetric)
aRb and bRc implies aRc for all a,b,cS (transitive)
Examples:
Yes No
= over the integers
 over the integers
path from u to v in undirected graphs
path from u to v in directed graphs
Nov 26, 2001
CSE 373, Autumn 2001

6. Dynamic Equivalence An equivalence relation defines a partition of S.b a
An equivalence relation defines a partition of S.
operations:
find(a): return the name of the equivalence class of a.
union(a,b): merge the equivalence classes of a and b.
Dynamic: the classes change because of the union operation.
Nov 26, 2001
CSE 373, Autumn 2001

7. Simple Strategy find(i): return A[i]. O(1) union(a,b): Let i = A[a]1 2 3 …
N-1 i
find(i): return A[i] O(1)
union(a,b): Let i = A[a]
Let j = A[b]
Scan the array and change all the i’s to j’s O(N)
Is there a better way? Yes!
Nov 26, 2001
CSE 373, Autumn 2001

8. Better Strategy Make unions “easy” but finds “hard”.
Idea: Keep a tree for each equivalence class.
find(x) returns the root of the tree x is in.
union(x,y): merge the two trees by making one a subtree of the other. 1 2 3 4 5 6 7
after union(1, 2): 1 3 4 5 6 7 2
Nov 26, 2001
CSE 373, Autumn 2001

9. Example, continued after union(3, 4): 1 3 5 6 7 2 4 after union(2, 4):1 3 5 6 7 2 4
after union(2, 4): 1 5 6 7 3 2 4
implicit (array) implementation:
A[i] -1 1 3 i 2 4 5 6 7
Nov 26, 2001
CSE 373, Autumn 2001