Welcome To My

Presentation

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Submitted bySharmin Sultana Jeesa

ID#141-15-3219Sec: C

Dept: CSECourse Code: CSE 221

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ContentBellman Ford

Algorithm With

Negative Cycle3

InventionoBellman Ford algorithm is

named after two of its developers, Richard Bellman and Lester Ford, Jr. , who published it in 1958 and 1956, respectively.

oHowever, Edward F. Moore also published the same algorithm in 1957, and so it is also sometimes called the Bellman–Ford–Moore algorithm.

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Bellman Ford Algorithm

oThis is a single source shortest path algorithm.

oThis algorithm can find shortest path from a single source even if the graph contains negative edges.

oBellman–Ford runs in (|V|.|E|) time, where |V| and |E| are the number of vertices and edges respectively.

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Psuedo Code BELMAN-FORD( G, s ) INIT( G, s ) for i ←1 to |V|-1 do for each edge (u, v) E do RELAX( u, v )

for each edge ( u, v ) E do if d[v] > d[u]+w(u,v) then return FALSE (> neg-weight cycle) return TRUE

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INIT(G, s) for each v V do d[v] ← ∞

π[v] ← NIL

d[s] ← 0 RELAX(u, v) if d[v] >

d[u]+w(u,v) then

d[v] ← d[u]+w(u,v)

π[v] ← u

Example

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s

a

b

c

5 -6

4

-2

-9

1st Step

s a b c

0 ∞ ∞ ∞

0

0

0

0

Example

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s

a

b

c

5 -6

4

-2

-9

1st Step

s a b c

0 ∞ ∞ ∞

0 5 4 ∞

0

0

0

Example

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s

a

b

c

5 -6

4

-2

-9

1st Step

s a b c

0 ∞ ∞ ∞

0 5 4 ∞

0 5 4 -1

0

0

Example

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s

a

b

c

5 -6

4

-2

-9

1st Step

s a b c

0 ∞ ∞ ∞

0 5 4 ∞

0 5 4 -1

0 2 4 -1

0 2 -10 -1

Example

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s

a

b

c

5 -6

4

-2

-9

2nd Step

s a b c0 2 -10 -10 2 -10 -40 -12 -10 -40 -12 -13 -4

Example

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s

a

b

c

5 -6

4

-2

-9

3rd Step

s a b c0 -12 -13 -40 -12 -13 -180 -15 -13 -180 -15 -27 -18

Example

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s

a

b

c

5 -6

4

-2

-9

4th Step

s a b c0 -15 -27 -180 -15 -27 -240 -29 -27 -240 -29 -33 -24

According to the Psuedo Code we continue the loop up to v-1, but here we have continued the loop up to v and the value still changing. So here comes a negative(-ve) cycle.

• Though Bellman-Ford algorithm can handle negative edges but it can’t handle negative cycles. It can only detect negative cycles. Even there is no algorithm which can handle negative cycles.

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Applications• Networks (Routing )• Robot Navigation• Urban Traffic Planning• Telemarketer operator scheduling• Routing of Communication messages• Optimal truck routing through given

traffic congestion pattern• OSPF routing protocol for IP

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Thank You

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