1. Winter 2019 Lecture 10 Minimum Spanning Trees
CSE 421 Algorithms
Winter 2019
Lecture 10
Minimum Spanning Trees

2. Announcement CSE 421 Midterm Wednesday, February 13
In class, closed book, no notes
All material covered in lecture
KT 1.1 – KT 5.5

3. Dijkstra’s Algorithm Implementation and Runtime
Edge costs are assumed to be non-negative
Dijkstra’s Algorithm Implementation and Runtime
S = { }; d[s] = 0; d[v] = infinity for v != s
While S != V
Choose v in V-S with minimum d[v]
Add v to S
For each w in the neighborhood of v
d[w] = min(d[w], d[v] + c(v, w)) y u a
HEAP OPERATIONS
n Extract Mins
m Heap Updates s x v b z

4. Shortest Paths Negative Cost Edges
Dijkstra’s algorithm assumes positive cost edges
For some applications, negative cost edges make sense
Shortest path not well defined if a graph has a negative cost cycle a 6 4 -4 4 e -3 c s -2 3 3 2 6 g b f 4 7

5. Negative Cost Edge Preview
Topological Sort can be used for solving the shortest path problem in directed acyclic graphs
Bellman-Ford algorithm finds shortest paths in a graph with negative cost edges (or reports the existence of a negative cost cycle).

6. Bottleneck Shortest Path
Define the bottleneck distance for a path to be the maximum cost edge along the path u 5 4 s x 4 6 5 3 v

7. Compute the bottleneck shortest paths6 d d 6 a a 5 4 4 4 e e -3 c c s -2 s 3 3 2 5 g g b b 7 7 f f

8. Dijkstra’s Algorithm for Bottleneck Shortest Paths
S = { }; d[s] = negative infinity; d[v] = infinity for v != s
While S != V
Choose v in V-S with minimum d[v]
Add v to S
For each w in the neighborhood of v
d[w] = min(d[w], max(d[v], c(v, w))) 3 y 4 u a 1 2 4 3 s x 1 1 2 v 5 3 b 4 z

9. Minimum Spanning Tree Introduce Problem
Demonstrate three different greedy algorithms
Provide proofs that the algorithms work

10. Minimum Spanning Tree 15 t a 6 9 14 3 4 e 10 13 c s 11 20 5 17 2 7 g b8 f 22 u 12 1 16 v

11. Greedy Algorithms for Minimum Spanning Tree
Extend a tree by including the cheapest out going edge
Add the cheapest edge that joins disjoint components
Delete the most expensive edge that does not disconnect the graph 4 b c 20 5 11 8 a d 7 22 e

12. Greedy Algorithm 1 Prim’s Algorithm
Extend a tree by including the cheapest out going edge 15 t a 6 14 3 9 4 e 10 13 c s 11 20 5
Construct the MST with Prim’s algorithm starting from vertex a
Label the edges in order of insertion 17 2 7 g b 8 f 22 u 12 1 16 v

13. Greedy Algorithm 2 Kruskal’s Algorithm
Add the cheapest edge that joins disjoint components 15 t a 6 14 3 9 4 e 10 13 c s 11 20 5 17 2 7
Construct the MST with Kruskal’s algorithm
Label the edges in order of insertion g b 8 f 22 u 12 1 16 v

14. Greedy Algorithm 3 Reverse-Delete Algorithm
Delete the most expensive edge that does not disconnect the graph 15 t a 6 14 3 9 4 e 10 13 c s 11 20 5 17 2 7
Construct the MST with the reverse-delete algorithm
Label the edges in order of removal g b 8 f 22 u 12 1 16 v

15. Dijkstra’s Algorithm for Minimum Spanning Trees
S = { }; d[s] = 0; d[v] = infinity for v != s
While S != V
Choose v in V-S with minimum d[v]
Add v to S
For each w in the neighborhood of v
d[w] = min(d[w], c(v, w)) 3 y 4 u a 1 2 4 3 s x 1 1 2 v 5 3 b 4 z

16. Minimum Spanning Tree Undirected Graph G=(V,E) with edge weights 15 t6 9 14 3 4 e 10 13 c s 11 20 5 17 2 7 g b 8 f 22 u 12 1 16 v

17. Greedy Algorithms for Minimum Spanning Tree
[Prim] Extend a tree by including the cheapest out going edge
[Kruskal] Add the cheapest edge that joins disjoint components
[ReverseDelete] Delete the most expensive edge that does not disconnect the graph 4 b c 20 5 11 8 a d 7 22 e

18. Why do the greedy algorithms work?
For simplicity, assume all edge costs are distinct

19. Edge inclusion lemma
Let S be a subset of V, and suppose e = (u, v) is the minimum cost edge of E, with u in S and v in V-S
e is in every minimum spanning tree of G
Or equivalently, if e is not in T, then T is not a minimum spanning tree e S
V - S

20. Proof Suppose T is a spanning tree that does not contain e
e is the minimum cost edge between S and V-S
Proof
Suppose T is a spanning tree that does not contain e
Add e to T, this creates a cycle
The cycle must have some edge e1 = (u1, v1) with u1 in S and v1 in V-S
T1 = T – {e1} + {e} is a spanning tree with lower cost
Hence, T is not a minimum spanning tree S
V - S e