aim: graph theory - trees course: math literacy do now: aim: what’s a tree?
TRANSCRIPT
Aim: Graph Theory - Trees Course: Math Literacy
A Tree
Tree – a graph with the smallest number of edges that allow all vertices to be reached from all other vertices.
A
D E
GB
C F
CD
E
A
B
FA
B C
G HFED
all connectedno circuits
no longer trees
Aim: Graph Theory - Trees Course: Math Literacy
What’s in a tree?
A tree is a graph that is connected and has no circuits. All trees have the following properties:
a. There is one and only one path joining any two vertices.
b. Every edge is a bridge.
c. A tree with n vertices must have n – 1 edges.
a tree with 5 vertices must have 4 edges.
Aim: Graph Theory - Trees Course: Math Literacy
Model Problem
Which graph is a tree?A
B
EC
D
A
B
C
D
E
A
B
C
D
E
Aim: Graph Theory - Trees Course: Math Literacy
Spanning Trees
Spanning tree – a subgraph of a connected graph that contains no circuits.
A
G
D
B C
E Fnot a tree7 vertices7 edges
A
G
D
B C
E F
a tree with removal of BC
7 vertices, 6 edges
A
G
D
B C
FE
a tree with removal of EG7 vertices, 6 edges
Aim: Graph Theory - Trees Course: Math Literacy
Model Problem
Find a spanning tree for the graph below.
A B
D C
E F
H G
8 vertices12 edges
5 gotta go
A B
D C
E F
H G
A B
D C
E F
H G
A B
D C
E F
H G
A B
D C
E F
H G
A B
D C
E F
H G
8 vertices, 7 edges - a tree
Aim: Graph Theory - Trees Course: Math Literacy
Model Problem
Find a spanning tree for the graph below.
6 vertices8 edges
3 gotta go
C
E F
B D
A
Aim: Graph Theory - Trees Course: Math Literacy
Efficiency!
Minimum spanning tree - a spanning tree with the smallest possible total weight on a weight graph.A
G
D
B C
E F
8
20
3517 15
12
24
original weightedgraph =
131
A
G
D
B C
E F
8
20
3517 15
24
35+24+20+8+17+15 =
119A
G
D
B C
FE
8
20
3517 15
12
35+17+12+15+20+8 = 107
Aim: Graph Theory - Trees Course: Math Literacy
Kruskal’s Algorithm
Kruskal’s Algorithm: finding the minimum spanning tree from a weighted graph:
1. Find the edge with the smallest weight in the graph. If there is more than one, pick one at random and mark it.
2. Find the next smallest edge in the graph. If there is more than one, pick at random. Mark it.
3. Find the next-smallest unmarked edge that does not create a red circuit.
4. Repeat step 3 until all vertices are included. The marked edges are the desired minimum spanning tree.
Aim: Graph Theory - Trees Course: Math Literacy
Model Problem
Seven building on a college campus are connected by the sidewalks show in the figure below. The weight graph represents building as vertices sidewalks as edges and sidewalk lengths as weights. A heavy snow has fallen and the sidewalks need to be cleared quickly. Determine the shortest series of sidewalks to clear. What is the total length of the sidewalks that need to be cleared?
264’256’ 262’
242’
259’255’
251’253’
251’241’
245’
274’
Aim: Graph Theory - Trees Course: Math Literacy
Model Problem
264’256’ 262’
242’
259’255’
251’253’
251’241’
245’
274’G
E
D
B
A
C
F
245253
264
256
249
251
242274
251
255 259
262
B C E
F
GA
D
Aim: Graph Theory - Trees Course: Math Literacy
245253
264
256
249
251
242274
251
255 259
262
B C E
F
GA
D
Model Problem
Kruskal’s Algorithm: finding the minimum spanning tree from a weighted graph:
1. Find the edge with the smallest weight in the graph. If there is more than one, pick one at random and mark it.
2. Find the next smallest edge in the graph. If there is more than one, pick at random. Mark it.
3. Find the next-smallest unmarked edge that does not create a red circuit.
4. Repeat step 3 until all vertices are included. The marked edges are the desired minimum spanning tree.
242'
GF
245'
BD
249'
AD
251'
DG
253'
CD
259'
CE+ + + + + = 1499’
Aim: Graph Theory - Trees Course: Math Literacy
Model Problem
Use Kruskal’s Algorithm to find the minimum spanning tree for the graph below. Give the total weight of the minimum spanning tree.
3531
23
28
21
12
14
26
2224