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Shortest Path Algorithm

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 AVL Trees / Slide 2

2

Shortest Path Algorithm

Definition:Shortest path between two vertices u

and v of a graph G is a path between u and v

with minimum number of edges in the path for

an unweighted graph.

For a weighted graph, it is the path

 between u and v for which the sum of the

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 AVL Trees / Slide 3

3

Shortest Path Algorithm

Versions of Shortest Path Algorithm:

.!nwighted shortest path problem.

".#eighted shortest problem with no

negative edges.

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 AVL Trees / Slide 4

!nwighted shortest path problem.

Example:

Find the shortest path from S to ever$ other

Verte% in the graph G.

v1 v2

v3 v4 v5

v6 v7

 V K dv pv

 v1 0 0

 v2 0 0

 v3 0 0 0

 v4 0 0

 v5 0 0

 v6 0 0

 v7 0 0&nitiali'e a table with following values:

(nown Field for all vertices are set to 'ero)*+.Distance vector dv for all vertices are set to infinit$.

&nitial onfiguration -able

S

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 AVL Trees / Slide 5

!nwighted shortest path problem.

Step:/ar0 the start node as reachable in 'ero edges

v1 v2

v3 v4 v5

v6 v7

 V K dv pv

 v1 0 0

 v2 0 0

 v3 1 0 0

 v4 0 0

 v5 0 0

 v6 0 0

 v7 0 0

0

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 AVL Trees / Slide 6

!nwighted shortest path problem.

Step":

Find all the vertices that are at a distance awa$ from s./ar0 them reachable with distance

i.e Vertices V and V1 )onl$ v and v1 are ad2acent to v3+

v1 v2

v3 v4 v5

v6 v7

 V K Dv pv

 v1 1 1 v3

 v2 0 0

 v3 1 0 0

 v4 0 0

 v5 0 0

 v6 1 1 v3

 v7 0 0

0

1

1

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 AVL Trees / Slide 7

!nwighted shortest path problem.

v1 v2

v3 v4 v5

v6 v7

0

1

1

2

2

 V K Dv pv

 v1 1 1 v3

 v2 1 2 v1

 v3 1 0 0

 v4 1 2 v1

 v5 0 0

 v6 1 1 v3

 v7 0 0

Step3:

Find all the vertices that are at a distance " awa$ from s./ar0 them reachable with distance ".

i.e Vertices V" and V4 )onl$ v" and v" are ad2acent to v3+

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 AVL Trees / Slide 8

!nwighted shortest path problem.

v1 v2

v3 v4 v5

v6 v7

0

1

1

2

2 V K Dv pv

 v1 1 1 v3

 v2 1 2 v1

 v3 1 0 0

 v4 1 2 v1

 v5 1 3 v2

 v6 1 1 v3

 v7 1 3 v4

3

3

Step4:

Find all the vertices that are at a distance 3 awa$ from s./ar0 them reachable with distance 3.

i.e Vertices V5 and V6 are ad2acent to v" 7 v4 so update

the dv and pv in the table and mar0 it as 0nown.

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 AVL Trees / Slide 9

Di2i0stra8s Algorithm

Definition:

-he general method to solve the single

source shortest path problem is 0nown as Di2i0stra8s

algorithm.

-he algorithm is an e%ample of greed$ algorithm.

Greed$ algorithms generall$ solve a problem in stages

 b$ doing what appears to be the best thing at each

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 AVL Trees / Slide 10

Di2i0stra8s Algorithm

At each stage, Di2i0stra8s algorithm selects a verte% 9v8, which has

the smallest distance dv among all the un0nown vertices, and

declares that the shortst path from s to v is 0nown.

-he remainder of a stage consists of updating the values of dw.

&n unweighted graph, we set dwdv ; if dw infinit$.

&n weighted case, we set dw dv ; v,w if this new value of dw

would be an improvement.

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 AVL Trees / Slide 11

Di2i0stra8s Algorithm

Find the shortest path from v to all the other vertices

 V K dv pv

 v1 0 0 0

 v2 0 0

 v3 0 0 0

 v4 0 0

 v5 0 0

 v6 0 0

 v7 0 0

&nitial configuration -able

v1 v2

v3 v4 v5

v6 v7

4

2

1 3 10

22

5

1

6

48

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 AVL Trees / Slide 12

Di2i0stra8s Algorithm

 

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 AVL Trees / Slide 13

Di2i0stra8s Algorithm

 

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 AVL Trees / Slide 14

Di2i0stra8s Algorithm

Find the ad2acent vertices of v". v4 7 v5 are ad2acent

vertices of v". Since the verte% v4 has alread$ set to

0nown to , no need to consider v4.

=ut v58s 0nown field is still *. So find dw for v5.V K dv pv

 v1 1 0 0

 v2 1 2 V1

 v3 0 3 v4

 v4 1 1 v1

 v5 0 3 v4

 v6 0 9 v4

 v7 0 5 v4

So, dw dv ; v,w

" ; *

  "

=ut in the table to v5 there is alread$ an entr$

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 AVL Trees / Slide 15

Di2i0stra8s Algorithm

 ence mar0 it as

0nown.

 V K dv pv

 v1 1 0 0

 v2 1 2 V1

 v3 1 3 v4

 v4 1 1 v1 v5 1 3 v4

 v6 0 9 v4

 v7 0 5 v4

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 AVL Trees / Slide 16

Di2i0stra8s Algorithm

Find the ad2acent vertices of v3 7 v5. v 7 v1 are

ad2acent to v3. Since v is alread$ mar0ed 0nown no

need to consider v. alculate dw for v1.

So, dw dv ; v,w

3 ; 5

  ?

=ut in the table to v1 there is alread$ an entr$

V K dv pv

 v1 1 0 0

 v2 1 2 V1

 v3 1 3 v4

 v4 1 1 v1

 v5 1 3 v4

 v6 0 9, 8 v4, v3

 v7 0 5 v4

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 AVL Trees / Slide 17

Di2i0stra8s Algorithm

For V5 the ad2acent verte% is v6.

alculate dw for v6.

So, dw dv ; v,w

3 ; 1

  @=ut in the table to v6 there is alread$ an entr$

from v3 with dv 5 which is shortest when

compared to the calculated dw, >ence no

need to alter the table entries.

 V K dv pv

 v1 1 0 0

 v2 1 2 V1

 v3 1 3 v4

 v4 1 1 v1

 v5 1 3 v4

 v6 0 8 v3

 v7 0 5 v4

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 AVL Trees / Slide 18

Di2i0stra8s Algorithm

 

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 AVL Trees / Slide 19

Di2i0stra8s Algorithm

 

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 AVL Trees / Slide 20

Di2i0stra8s Algorithm

-he left out entr$ is v1. v1 is finall$ mar0ed 0nown.

 V K dv pv

 v1 1 0 0

 v2 1 2 V1

 v3 1 3 v4

 v4 1 1 v1

 v5 1 3 v4

 v6 1 6 v7

 v7 1 5 v4

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 AVL Trees / Slide 21Di2i0stra8s Algorithm

-he left out entr$ is v1. v1 is finall$ mar0ed 0nown. V K dv pv

 v1 1 0 0

 v2 1 2 V1

 v3 1 3 v4

 v4 1 1 v1

 v5 1 3 v4

 v6 1 6 v7

 v7 1 5 v4

v1 v2

v3 v4 v5

v6 v7

42

1 3 10

22

5

1

648

0

1

3

65

2

3

Distance and path from v1 to all the other vertices are:

v1 to v2 = v1 -> v2 = 2

v1 to v3 = v1 -> v4 -> v3 = 31 t 4 1 4 3