1. Minimum Spanning Tree Section 7.3: Examples {1,2,3,4}

2. Section 7.3: Examples

7. Paths and cycles
A path is a sequence of nodes v1, v2, …, vN such that (vi,vi+1)E for 0<i<N
The length of the path is N-1.
Simple path: all vi are distinct, 0<i<N
A cycle is a path such that v1=vN
An acyclic graph has no cycles 7

8. Cycles
BOS
DTW
SFO
PIT
JFK
LAX 8

9. More useful definitions
In a directed graph:
The indegree of a node v is the number of distinct edges (w,v)E.
The outdegree of a node v is the number of distinct edges (v,w)E.
A node with indegree 0 is a root. 9

10. Trees are graphs A dag is a directed acyclic graph.
A tree is a connected acyclic undirected graph.
A forest is an acyclic undirected graph (not necessarily connected), i.e., each connected component is a tree. 10

11. Graph Traversals

12. Graph Traversals 12

13. Use of a stack It is very common to use a stack to keep track of:
nodes to be visited next, or
nodes that we have already visited.
Typically, use of a stack leads to a depth-first visit order.
Depth-first visit order is “aggressive” in the sense that it examines complete paths. 13

14. Use of a queue It is very common to use a queue to keep track of:
nodes to be visited next, or
nodes that we have already visited.
Typically, use of a queue leads to a breadth-first visit order.
Breadth-first visit order is “cautious” in the sense that it examines every path of length i before going on to paths of length i+1. 14

15. Minimum Spanning Trees15

16. Problem: Laying Telephone Wire
Central office 16

17. Wiring: Naïve Approach
Central office
Expensive! 17

18. Wiring: Better Approach
Central office
Minimize the total length of wire connecting the customers 18

19. Understanding the terms
“Minimal cost” = Obvious
“Spanning” = No isolated vertices
“Tree” = A graph with no loops
At step 3 draw the actual tree formed in example 1
6.5 Trees

20. Definition of MST Let G=(V,E) be a connected, undirected graph.
For each edge (u,v) in E, we have a weight w(u,v) specifying the cost (length of edge) to connect u and v.
We wish to find a (acyclic) subset T of E that connects all of the vertices in V and whose total weight is minimized.
Since the total weight is minimized, the subset T must be acyclic (no circuit).
Thus, T is a tree. We call it a spanning tree.
The problem of determining the tree T is called the minimum-spanning-tree problem.

21. Application of MST: an example
In the design of electronic circuitry, it is often necessary to make a set of pins electrically equivalent by wiring them together.
Running cable TV to a set of houses. What’s the least amount of cable needed to still connect all the houses?

22. Where might this be useful? Can also be used to approximate some
Spanning Tree
Definition
A spanning tree of a graph G is a tree (acyclic) that connects all the vertices of G once
i.e. the tree “spans” every vertex in G
A Minimum Spanning Tree (MST) is a spanning tree on a weighted graph that has the minimum total weight
Where might this be useful? Can also be used to approximate some
NP-Complete problems

23. Here is an example of a connected graph and its minimum spanning tree:b h c d e f g i 4 8 7 9 10 14 2 6 1 11
Notice that the tree is not unique:
replacing (b,c) with (a,h) yields another spanning tree
with the same minimum weight.

24. Minimum Spanning Trees
1/2/2019 6:45 AM
Minimum Spanning Trees
2704
867
BOS
849
PVD
ORD
187
740
144
1846
621
JFK
184
1258
802
SFO
1391
BWI
1464
337
1090
DFW
946
LAX
1235
1121
MIA
2342

25. Minimum Spanning Trees
Spanning subgraph
Subgraph of a graph G containing all the vertices of G
Spanning tree
Spanning subgraph that is itself a (free) tree
Minimum spanning tree (MST)
Spanning tree of a weighted graph with minimum total edge weight
Applications
Communications networks
Transportation networks
ORD 10 1
PIT
DEN 6 7 9 3
DCA
STL 4 8 5 2
DFW
ATL
Minimum Spanning Trees

26. Minimum Spanning Trees
Cycle Property 8 4 2 3 6 7 9 e C f
Cycle Property:
Let T be a minimum spanning tree of a weighted graph G
Let e be an edge of G that is not in T and C let be the cycle formed by e with T
For every edge f of C, weight(f)  weight(e)
Proof:
By contradiction
If weight(f) > weight(e) we can get a spanning tree of smaller weight by replacing e with f
Replacing f with e yields a better spanning tree 8 4 2 3 6 7 9 C e f
Minimum Spanning Trees

27. Minimum Spanning Trees
Partition Property U V 7
Partition Property:
Consider a partition of the vertices of G into subsets U and V
Let e be an edge of minimum weight across the partition
There is a minimum spanning tree of G containing edge e
Proof:
Let T be an MST of G
If T does not contain e, consider the cycle C formed by e with T and let f be an edge of C across the partition
By the cycle property, weight(f)  weight(e)
Thus, weight(f) = weight(e)
We obtain another MST by replacing f with e f 4 9 5 2 8 8 3 e 7
Replacing f with e yields another MST U V 7 f 4 9 5 2 8 8 3 e 7
Minimum Spanning Trees

28. Minimum Spanning Trees
Example  7 7 D 7 D 2 2 B 4 B 4 8 9  5 9  5 5 2 F 2 F C C 8 8 3 3 8 8 E E A 7 A 7 7 7 7 7 7 D 2 7 D 2 B 4 B 4 5 9  5 5 9 4 2 F 5 C 2 F 8 C 8 3 8 3 8 E A E 7 7 A 7 7
Minimum Spanning Trees

29. Minimum Spanning Trees
Example (contd.) 7 7 D 2 B 4 9 4 5 5 2 F C 8 3 8 E A 3 7 7 7 D 2 B 4 5 9 4 5 2 F C 8 3 8 E A 3 7
Minimum Spanning Trees

30. Minimum Spanning Trees
Example
JFK
BOS
MIA
ORD
LAX
DFW
SFO
BWI
PVD
867
2704
187
1258
849
740
144
1846
621
184
802
1391
1464
337
1090
946
1235
1121
2342
Minimum Spanning Trees

31. Prim's algorithm (simple)
MST_PRIM(G,w,r){
A={}
S:={r} (r is an arbitrary node in V)
Q=V-{r};
while Q is not empty
do {
take an edge (u, v) such that
uS and vQ (vS ) and (u,v) is the shortest edge
add (u, v) to A,
add v to S and delete v from Q }

32. Prim's algorithm Initialization
for each vertex in graph
set min_distance of vertex to ∞
set parent of vertex to null
set minimum_adjacency_list of vertex to empty list
set is_in_Q of vertex to true
set min_distance of initial vertex to zero
add to minimum-heap Q all vertices in graph, keyed by min_distance
inputs: A graph, a function returning edge weights weight-function, and an initial vertex
Initial placement of all vertices in the 'not yet seen' set, set initial vertex to be added to the tree, and place all vertices in a min-heap to allow for removal of the min distance from the minimum graph.
Wikipedia

33. Prim's algorithm

34. Prim's algorithm

35. Prim's algorithm

36. Prim's algorithm

37. Prim's algorithm

38. Prim's algorithm

39. Prim's algorithm

40. Prim's algorithm

41. Prim's algorithm

42. Prim's algorithm

43. Prim's algorithm

44. Prim's algorithm

45. Prim's algorithm

46. Prim's algorithm

47. Prim's algorithm

48. Prim's algorithm

49. Prim's algorithm

50. Prim's algorithm

51. Another example

52. Prim’s algorithm d b c a  e b a d d b c a 4 5  e c Vertex Parent e -9 b a 2 6 d d b c a 4 5 
Vertex Parent
e -
b e
c e
d e 4 5 5 4 5 e c
The MST initially consists of the vertex e, and we update
the distances and parent for its adjacent vertices 52

53. Prim’s algorithm d b c a 4 5  b a d a c b 2 4 5 e c Vertex Parent e -
c e
d e 9 b a 2 6 d 4 5 5 4 a c b 2 4 5
Vertex Parent
e -
b e
c d
d e
a d 5 e c 53

54. Prim’s algorithm a c b 2 4 5 b a d e c b 4 5 c Vertex Parent e - b e
c d
d e
a d 9 b a 2 6 d 4 5 5 4 5 e c b 4 5
Vertex Parent
e -
b e
c d
d e
a d c 54

55. Prim’s algorithm c b 4 5 b a d e b 5 c Vertex Parent e - b e c d d e9 b a 2 6 d 4 5 5 4 5 e b 5
Vertex Parent
e -
b e
c d
d e
a d c 55

56. Prim’s algorithm b 5 b a d e c The final minimum spanning tree
Vertex Parent
e -
b e
c d
d e
a d 9 b a 2 6 d 4 5 5 4 5 e
Vertex Parent
e -
b e
c d
d e
a d c
The final minimum spanning tree 56