Dijkstra’s Algorithm Notation: c(x,y): link cost from node x to y; = ∞ if not direct neighbors D(v): current value of cost of path from source to dest

Dijkstra’s Algorithm

Notation: c(x,y): link cost from

node x to y; = ∞ if not direct neighbors

D(v): current value of cost of path from source to dest. v

p(v): predecessor node along path from source to v

N': set of nodes whose least cost path definitively known

Data structures: Path

An ordered sequence of nodes

a cost to reach the end of the path

• The sum of each lonk cost along the path

Candidate paths (CP)

Shortest paths (SP)

Dijsktra’s Algorithm

Let u be the router running this algorithm 1 Initialization: 2 Candidate paths CP = [0, {u}]3 Least cost paths = empty

4 Loop 5 find path [a, {u, …, w}] in CP with the least

cost6 Move [a, {u, …, w}] to shortest paths7 Extend w and place extensions in CP: For

each neighbor v of w make new candidate path [a+c(w,v), {u,…, w,v}]

8 Clean up candidate paths: if there are multiple entries in the list of candidate paths with the same destination, remove the ones with higher cost. If two have te same cost, remove one with more hops, if two have the same number of hops, remove the one with higher next hop IP address

9 Remove candidate paths that have a destination that is already in the list of shortest paths

10 If candidate paths is empty, then stop

Dijsktra’s Algorithm from Router A

A

B

E

C

D

F

1

2

2

16

121

6

Link CostsC(A,B)=1C(A,D)=12C(B,E)=6C(B,C)=2C(C,F)=1C(C,D)=6C(D,E)=1C(E,F)=2

Least costs paths Cost Candidate paths Cost

A 0

Initialization (Steps 2-3)








A

B

E

C

D

F

1

2

2

16

121

6









Least costs pathsCost Candidate paths Cost

A 0

Step 5

Entry with least cost


A

B

E

C

D

F

1

2

2

16

121

6










Step 6

Move entryA 0 A 0


A

B

E

C

D

F

1

2

2

16

12

1

6










Step 7: Extend newly found least cost pathA’s neighbors are B and D

A 0


A

B

E

C

D

F

1

2

2

16

12

1

6











A 0 A,B 0+C(A,B)


A

B

E

C

D

F

1

2

2

16

12

1

6











A 0 A,B 0+C(A,B)

A,D 0+C(A,D)


A

B

E

C

D

F

1

2

2

16

12

1

6











A 0 A,B 0+1=1

A,D 0+12=12

0+C(A,B)

0+C(A,D)


A

B

E

C

D

F

1

2

2

16

12

1

6










Step 8: Nothing to do

A 0 A,B 1

A,D 12


A

B

E

C

D

F

1

2

2

16

12

1

6










Step 9: Nothing to do

A 0 A,B 1

A,D 12


A

B

E

C

D

F

1

2

2

16

12

1

6










Step 10: Keep on going

A 0 A,B 1

A,D 12


A

B

E

C

D

F

1

2

2

16

12

1

6










Step 5: A,B has the least cost

A 0 A,B 1

A,D 12


A

B

E

C

D

F

1

2

2

16

12

1

6










Step 6: Move to shortest paths

A 0 A,D 12A,B 1


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1

Step 7: Extend newly found least cost pathB’s neighbors are A, C, D, and E


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1

Step 7: Extend newly found least cost pathB’s neighbors are A, C, and E

A,B,A 1+C(B,A)


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1


A,B,A 1+C(B,A)

A,B,C 1+C(B,C)

Dijsktra’s Algorithm from A

A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1


A,B,A 1+C(B,A)

A,B,C 1+C(B,C)

A,B,E 1+C(B,E)


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1


A,B,A 1+C(B,A)

A,B,C 1+C(B,C)

A,B,E 1+C(B,E)

2

3

7


A

B

E

C

D

F

1

2

2

16

12

1

6






8 Clean up candidate paths 1: if there are multiple entries in the list of candidate paths with the same destination, remove the ones with higher cost. If two have the same cost, remove one with more hops, if two have the same number of hops, remove the one with higher next hop IP address

9 Clean up candidate paths 2: Remove candidate paths that have a destination that is already in the list of shortest paths

10 If candidate paths is empty, then stop.


A 0 A,D 12A,B 1

Step 8: Clean up candidate list 1: remove candidate paths with the same end

Nothing to do

A,B,A 2

A,B,C 3

A,B,E 7


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1

Step 9: Clean up candidate list 2: remove candidate paths with the same ends already in least cost paths

Remove A,B,A

A,B,A 2

A,B,C 3

A,B,E 7


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1


Remove A,B,A

A,B,C 3

A,B,E 7


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12

A,B 1 A,B,C 3

A,B,E 7



A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1 A,B,C 3

A,B,E 7

Step 5: A,B,C has the lowest costc


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1 A,B,C 3

A,B,E 7

Step 6: Move path to least costs path

A,B,C 3


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1 A,B,E 7

Step 6: Move path to least costs path

A,B,C 3


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1 A,B,E 7

A,B,C 3

Step 7: Extend newly found least cost pathC’s neighbors are B, D, and F

A,B,C, B 3+C(C,B)


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1 A,B,E 7

A,B,C 3


A,B,C, B 3+C(C,B)

A,B,C, D 3+C(C,D)


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1 A,B,E 7

A,B,C 3


A,B,C, B 3+C(C,B)

A,B,C, D 3+C(C,D)A,B,C, F 3+C(C,F)


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1 A,B,E 7

A,B,C 3


A,B,C, B 3+C(C,B)

A,B,C, D 3+C(C,D)A,B,C, F 3+C(C,F)

5

9

4


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0 A,D 12A,B 1 A,B,E 7

A,B,C 3 A,B,C, B

A,B,C, D

A,B,C, F

5

9

4


A,B,C,D has lower cost than A,D


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1

A,B,E 7

A,B,C 3

A,B,C, B

A,B,C, D

A,B,C, F

5

9

4


A,B,C,D has lower cost than A,D

A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1

A,B,E 7

A,B,C 3

A,B,C, B

A,B,C, D

A,B,C, F

5

9

4


Remove A,B,C,B


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0A,B 1

A,B,E 7

A,B,C 3A,B,C, D

A,B,C, F9

4


Remove A,B,C,B


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1

A,B,E 7

A,B,C 3

A,B,C, D

A,B,C, F9

4



A

B

E

C

D

F

1

2

2

16

12

1

6










A 0A,B 1

A,B,E 7

A,B,C 3

A,B,C, D

A,B,C, F9

4

Step 5: A,B,C,F has the least cost


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1

A,B,E 7

A,B,C 3

A,B,C, D

A,B,C, F9

4


A,B,C, F 4


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1

A,B,E 7

A,B,C 3

A,B,C, D 9


A,B,C, F 4


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1

A,B,E 7

A,B,C 3

A,B,C, D 9

A,B,C, F 4

Step 7: Extend newly found least cost pathF’s neighbors are C, and E


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1

A,B,E 7

A,B,C 3

A,B,C, D 9

A,B,C, F 4


A,B,C, F, C 4+C(F,C)


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1

A,B,E 7

A,B,C 3

A,B,C, D 9

A,B,C, F 4


A,B,C, F, C 4+C(F,C)

A,B,C, F, E 4+C(F,E)


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1

A,B,E 7

A,B,C 3

A,B,C, D 9

A,B,C, F 4


A,B,C, F, C 4+C(F,C)A,B,C, F, E 4+C(F,E)

56


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1

A,B,E 7

A,B,C 3

A,B,C, D 9

A,B,C, F 4A,B,C, F, CA,B,C, F, E

56


A,B,C,F,E has lower cost than A,B,E


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4

A,B,C, F, CA,B,C, F, E

56


A,B,C,F,E has lower cost than A,B,E


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4

A,B,C, F, CA,B,C, F, E

56


Remove A,B,C,F,C


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4

A,B,C, F, E 6


Remove A,B,C,F,C


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4

A,B,C, F, E 6



A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4

A,B,C, F, E 6

Step 5: A, B, C, F, E has the least cost


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4

A,B,C, F, E 6

Step 6: Move the newly found least cost path

A,B,C, F, E 6


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4

Step 6: Move the newly found least cost path

A,B,C, F, E 6


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4

A,B,C, F, E 6

Step 7: Extend newly found least cost pathF’s neighbors are B, D and F


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4A,B,C, F, E 6


A,B,C, F, E, B 6+C(E,B)


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4A,B,C, F, E 6


A,B,C, F, E, B 6+C(E,B)A,B,C, F, E, D6+C(E,D)


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4

A,B,C, F, E 6


A,B,C, F, E, B 6+C(E,B)

A,B,C, F, E, D6+C(E,D)

A,B,C, F, E, F 6+C(E,F)


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4A,B,C, F, E 6


A,B,C, F, E, B 6+C(E,B)

A,B,C, F, E, D6+C(E,D)

A,B,C, F, E, F 6+C(E,F)

12

7

8


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, D 9

A,B,C, F 4A,B,C, F, E 6

A,B,C, F, E, B

A,B,C, F, E, D

A,B,C, F, E, F

12

7

8


A,B,C,F,E,D has lower cost than A,B, C, D


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4A,B,C, F, E 6

A,B,C, F, E, B

A,B,C, F, E, D

A,B,C, F, E, F

12

7

8


A,B,C,F,E,D has lower cost than A,B, C, D


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4A,B,C, F, E 6

A,B,C, F, E, B

A,B,C, F, E, D

A,B,C, F, E, F

12

7

8


Remove A,B,C,F,E,B and A, B, C, F,E,F


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4A,B,C, F, E 6

A,B,C, F, E, D 7


Remove A,B,C,F,E,B and A, B, C, F,E,F


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4

A,B,C, F, E 6

A,B,C, F, E, D 7

Step 10: keep on going


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4

A,B,C, F, E 6

A,B,C, F, E, D 7

Step 5: A,B,C,F,E,D has the least cost


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4

A,B,C, F, E 6

A,B,C, F, E, D 7

Step 6: move newly found least cost path


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4

A,B,C, F, E 6

A,B,C, F, E, D 7


A,B,C, F, E, D 7


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4

A,B,C, F, E 6


A,B,C, F, E, D 7


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4

A,B,C, F, E 6

A,B,C, F, E, D 7

Step 7: Extend newly found least cost pathD’s neighbors are A, C and E

A,B,C, F, E, D,A 19A,B,C, F, E, D,C 13

A,B,C, F, E, D,E 8


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4

A,B,C, F, E 6

A,B,C, F, E, D 7

A,B,C, F, E, D,A 19A,B,C, F, E, D,C 13

A,B,C, F, E, D,E 8

Step 8: Clean up candidate list 1: Nothing to do


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4

A,B,C, F, E 6

A,B,C, F, E, D 7

A,B,C, F, E, D,A 19A,B,C, F, E, D,C 13

A,B,C, F, E, D,E 8


Remove A,B,C,F,E,D,A, A, B, C, F,E,D,C and A,B,C,F,E,D,E


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4

A,B,C, F, E 6

A,B,C, F, E, D 7


Remove A,B,C,F,E,D,A, A, B, C, F,E,D,C and A,B,C,F,E,D,E


A

B

E

C

D

F

1

2

2

16

12

1

6










A 0

A,B 1A,B,C 3

A,B,C, F 4

A,B,C, F, E 6

A,B,C, F, E, D 7

Step 10: DONE!


Dijkstra’s algorithm, discussionAlgorithm complexity: n nodes each iteration:

Need to check all entries in candidate list, at most N comparisons Need to extend best path, at most N extensions Need to compare each entry to all entries in the candidate list (at most N) Need to compare each entry in candidate list to each entry in shortest paths

(N/2) 3.5N each iteration, so 3.5N^2 ~ O(N^2) more efficient implementations possible: O(nlogn)

Oscillations possible: e.g., link cost = amount of carried traffic

A

D

C

B1 1+e

e0

e

1 1

0 0

A

D

C

B2+e 0

001+e1

A

D

C

B0 2+e

1+e10 0

A

D

C

B2+e 0

e01+e1

initially… recompute

routing… recompute … recompute

Documents

Dijkstra’s Algorithm Notation: c(x,y): link cost from node x to y; = ∞ if not direct neighbors D(v): current value of cost of path from source to dest