shortest-paths trees

Shortest-Paths Trees

Kun-Mao Chao (趙坤茂 )Department of Computer Science an

d Information EngineeringNational Taiwan University, Taiwan

E-mail: kmchao@csie.ntu.edu.tw

WWW: http://www.csie.ntu.edu.tw/~kmchao

2

Shortest-Paths Trees

• The objective is to find the set of edges connecting all nodes such that the sum of the edge lengths from the source to each node is minimized.

• In order to minimize the total path lengths, the path from the source to each node must be a shortest path connecting them.

3

4 3

4

1

1

5

2

1 1

2

a b c

d

e f

g h

2

4 3

4

1

1

5

2

1 1

2

a b c

d

e f

g h

2

(a) (b)

Shortest-Paths Trees(source: vertex a)

4

Negative edges in an undirected graph

4 3

4

1

1

5

2

-1 1

2

a b c

d

e f

g h

2

5

Directed graphs

4 3

4

1

1

5

2

1 1

1

a b c

d

e f

g h

2

6

Dijkstra's Algorithm

7

Choose a

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 ∞ ∞

∞

∞ ∞

∞∞

8

Relax (a, b) and (a, g)

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

∞

∞

∞ ∞

∞

9

Choose b; Add (a, b) to T

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

∞

∞

∞ ∞

∞

10

Relax (b, c) and (b, d)

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

∞ ∞

∞

11

Choose g; Add (a, g) to T

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

∞ ∞

∞

12

Relax (g, e) and (g, h)

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

7 ∞

6

13

Choose d; Add (b, d) to T

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

7 ∞

6

14

Relax (d, e)

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

6 ∞

6

15

Choose h; Add (g, h) to T

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

6 ∞

6

16

Choose e; Add (d, e) to T

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

6 ∞

6

17

Relax (e, f)

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

6 7

6

18

Choose c; Add (b, c) to T

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

6 7

6

19

Relax (c, h)

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

6 7

6

20

Choose f; Add (e, f) to T

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

6 7

6

21

Relax (f, d) and (f, h)

4 3

4

1

15

2

1

1

1

a b c

d

e f

g h

2

0 4

5

7

5

6 7

6

22

The resulting SPT

4 3

1

1

5 1

1

a b c

d

e f

g h

23

Negative edge

2 2

4

1

1

5

-3

1 1

1

a b c

d

e f

g h

2

24

Choose a

2 2

4

1

15

-3

1

1

1

a b c

d

e f

g h

2

0 2 ∞

∞

∞ ∞

∞5

25

Choose b; Add (a, b) to T

2 2

4

1

15

-3

1

1

1

a b c

d

e f

g h

2

0 2 4

3

∞ ∞

∞5

26

Choose d; Add (b, d) to T

2 2

4

1

15

-3

1

1

1

a b c

d

e f

g h

2

0 2 4

3

4 ∞

∞5

27

Choose c; Add (b, c) to T

2 2

4

1

15

-3

1

1

1

a b c

d

e f

g h

2

0 2 4

3

4 ∞

85

28

Choose e; Add (d, e) to T

2 2

4

1

15

-3

1

1

1

a b c

d

e f

g h

2

0 2 4

3

4 5

85

29

Something went wrong

2 2

4

1

15

-3

1

1

1

a b c

d

e f

g h

2

0 2 4

3

4 5

65

30

A wrong SPT

2 2

1

1

5

1

a b c

d

e f

g h

1

31

A correct SPT

2 2

1

5

-3

1

a b c

d

e f

g h

2

32

The Bellman-Ford Algorithm

33

δ[b] and δ[g] modified

2 2

4

1

15

-3

1

1

1

a b c

d

e f

g h

2

0 2 ∞

∞

∞ ∞

∞5

34

δ[c], δ[d], δ[e] and δ[h] modified

2 2

4

1

15

-3

1

1

1

a b c

d

e f

g h

2

0 2 4

3

2 ∞

65

35

δ[f] modified

2 2

4

1

15

-3

1

1

1

a b c

d

e f

g h

2

0 2 4

3

2 3

65

36

δ[h] modified

2 2

4

1

15

-3

1

1

1

a b c

d

e f

g h

2

0 2 4

3

2 3

55

37

A correct SPT

2 2

4

1

15

-3

1

1

1

a b c

d

e f

g h

2

0 2 4

3

2 3

55

38

Try this in class

4 3

4

1

1

5

2

1 -3

1

a b c

d

e f

g h

2