Spanning Trees and Minimum Spaning Trees spanning tree: (for connected, undirected graph) minimal set of edges that connect all vertices (no cycles) Minimum spanning tree: (for connected, undirected and weighted graph) minimal set of edges that connect all vertices such that the sum of weights is minimum.
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Spanning Trees and Minimum Spaning Treesspanning tree: (for connected, undirected graph)
minimal set of edges that connect all vertices (no cycles)
Minimum spanning tree: (for connected, undirected and weighted graph)
minimal set of edges that connect all vertices such that the sum of weights is minimum.
Prim’s Algorithm
Similar to Dijkstra’s Algorithm except that dv records edge weights, not path lengths
Prim's Algorithm
MST=NULL;Select an edge of min weight and add it to MST
Iteration:repeat till n-1 edges are added to MST1.select an edge (v1,v2) such that v1 is in MST and v2 is not in MST2.add it to MST
Initialization:
Prims AlgorithmInput:
A connected weighted graph G = {V, E}Initialization:
VMST = EMST = nullSelect an aribitrary vertex, x, from Vadd x to VMST
Iteration:for i = 1 to |V|-1
select an edge v1,v2 with minimum weight such that v1 V∈ MST and v2 V \ V∈ MST
Add v1 to VMST
Add (v1,v2) to EMST
return EMST
Walk-ThroughInitialize array
K dv pvA F
B F
C F
D F
E F
F F
G F
H F
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Start with any node, say D
K dv pvA
B
C
D T 0
E
F
G
H
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Update distances of adjacent, unselected nodes
K dv pvA
B
C 3 D
D T 0
E 25 D
F 18 D
G 2 D
H
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Select node with minimum distance
K dv pvA
B
C 3 D
D T 0
E 25 D
F 18 D
G T 2 D
H
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Update distances of adjacent, unselected nodes
K dv pvA
B
C 3 D
D T 0
E 7 G
F 18 D
G T 2 D
H 3 G
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Select node with minimum distance
K dv pvA
B
C T 3 D
D T 0
E 7 G
F 18 D
G T 2 D
H 3 G
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Update distances of adjacent, unselected nodes
K dv pvA
B 4 C
C T 3 D
D T 0
E 7 G
F 3 C
G T 2 D
H 3 G
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Select node with minimum distance
K dv pvA
B 4 C
C T 3 D
D T 0
E 7 G
F T 3 C
G T 2 D
H 3 G
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Update distances of adjacent, unselected nodes
K dv pvA 10 F
B 4 C
C T 3 D
D T 0
E 2 F
F T 3 C
G T 2 D
H 3 G
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Select node with minimum distance
K dv pvA 10 F
B 4 C
C T 3 D
D T 0
E T 2 F
F T 3 C
G T 2 D
H 3 G
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Update distances of adjacent, unselected nodes
K dv pvA 10 F
B 4 C
C T 3 D
D T 0
E T 2 F
F T 3 C
G T 2 D
H 3 G
2
Table entries unchanged
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Select node with minimum distance
K dv pvA 10 F
B 4 C
C T 3 D
D T 0
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Update distances of adjacent, unselected nodes
K dv pvA 4 H
B 4 C
C T 3 D
D T 0
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Select node with minimum distance
K dv pvA T 4 H
B 4 C
C T 3 D
D T 0
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Update distances of adjacent, unselected nodes
K dv pvA T 4 H
B 4 C
C T 3 D
D T 0
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2
Table entries unchanged
4
25
A
HB
F
E
D
C
G 7
2
10
18
34
3
78
9
3
10
Select node with minimum distance
K dv pvA T 4 H
B T 4 C
C T 3 D
D T 0
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2
4
A
HB
F
E
D
C
G2
34
3
3
Cost of Minimum Spanning Tree = dv = 21
K dv pvA T 4 H
B T 4 C
C T 3 D
D T 0
E T 2 F
F T 3 C
G T 2 D
H T 3 G
2
Done
How many squares can you create in this figure by connecting any 4 dots (the corners of a square must lie upon a grid dot?
TRIANGLES:
How many triangles are located in the image below?
There are 11 squares total; 5 small, 4 medium, and 2 large.
27 triangles. There are 16 one-cell triangles, 7 four-cell triangles, 3 nine-cell triangles, and 1 sixteen-cell triangle.
GUIDED READING
ASSESSMENT
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