Bellman Ford Algorithm

Design Analysis of Algorithm
Topic:
Bellman Ford
Algorithm

Contents
• Introduction to Bellman Ford
• Which data structure is used?
• Bellman Ford Algorithm
• Bellman Ford Working
• Example
• Time-complexity
• Applications
• Limitations

Introduction to Bellman Ford Algorithm
• In a directed graph, we have to select one vertex as source vertex
and find out the shortest path to all other vertex.
• Before doing this, we already have Dijkstra algorithm, that will
solve the shortest path problem, but Dijkstra algorithm will not
work if there are negative edges. It may or may not give correct
result
• So we use Bellman ford Algorithm.

Introduction to Bellman Ford Algorithm
• Solve the single-source shortest-paths problem
• The algorithm returns a boolean value indicating whether or
not there is a negative-weight cycle that is reachable from the
source.
• If there are no negative weight cycles, then the algorithm
produces the shortest paths and their weights.

What data structure is used?
• Graph Data Structure is Used Because It Solve the Problem of
Weighted Graph

Bellman-Ford algorithm
d[s] ← 0
for each v ∈ V – {s}
do d[v] ← ∞
for i ← 1 to |V| – 1
initialization
do for each edge (u, v) ∈ E
do if d[v] > d[u] + w(u, v)
then d[v] ← d[u] + w(u, v)
for each edge (u, v) ∈ E
do if d[v] > d[u] + w(u, v)
relaxation
step
then report that a negative-weight cycle exists.

Bellman Ford Working
• The algorithm says to prepare list of all edges,
• This algorithm says to relax all edges
• We repeatedly relax them n-1 times, n is the number of vertices
|v| = n
So we relax them |v|-1

1 3
4
5
6
7
6
-1
5
5 -2
1
-1
3
3
2
-2
• There are total 10 edges, edge-list is
(1,2), (1,3), (1,4), (2,5),(3,2), (3,5), (4,3), (4,6),(5,7),(6,7)
• Now to relax all edges for 6 times. As, |v|-1 =6
• Initially we mark the distance of all edges
• Mark 1st as 0 and others as infinity
0
 




Example

1 3
4
5
6
7
6
-1
5
5 -2
1
-1
3
3
2
-2
• First, we take (1,2)
• If (d[u]+c(u,v)<d[v])
• Then d[v]= d[u]+c(u,v)
0
 



 • As, (0+6<)
• Then d[2]= 6

1 3
4
5
6
7
6
-1
5
5 -2
1
-1
3
3
2
-2
• Now, we take (1,3)
• Then, we take (1,4)
• If (d[u]+c(u,v)<d[v])
0
6 



 • As, (0+5<)
• Then d[3]= 5
• As, (0+5<)
• And d[4] =5

1 3
4
5
6
7
6
-1
5
5 -2
1
-1
3
3
2
-2
Now, we take (2,5)
We take (3,2)
• If (d[u]+c(u,v)<d[v])
0
6 


5
5 • As, (6+(-1)<)
• Then d[5]= 5
• As, (5+(-2)< 6)
• So, d[2]=3

1 3
4
5
6
7
6
-1
5
5 -2
1
-1
3
3
2
-2
Now, we take (3,5)
we take (4,3)
• If (d[u]+c(u,v)<d[v])
0
3 5


5
5 • As, (5+1>5)
• So, we don’t modify it
• As, (5+(-2)< 5)
• So, d[3]=3

1 3
4
5
6
7
6
-1
5
5 -2
1
-1
3
3
2
-2
Now, we take (4,6)
then, we take(5,7)
then, we take(6,7)
• If (d[u]+c(u,v)<d[v])
0
3 5


3
5
• As, (5-1<)
• So, d[6]= 4
• As, (5+3< )
• So, d[7]=8
• As, (4+3<8)
• So d[7] = 7

1 3
4
5
6
7
6
-1
5
5 -2
1
-1
3
3
2
-2
Now, do it again and again for 5
more times
0
3 5
7
4
3
5

• We see that after third time it
had stopped changing
• It will run 6 times, when we are
doing it in algorithm
• So finally, these are the shortest
path
1 2 3 4 5 6 7
0 0      
1 0 3 3 5 5 4 7
2 0 1 3 5 2 4 5
3 0 1 3 5 0 4 3
4 0 1 3 5 0 4 3
5 0 1 3 5 0 4 3
6 0 1 3 5 0 4 3

Time-Complexity
• It relaxes all edges, edges are |E|
• How many time it is relaxing|v-1|
• So this is order of v into e
O(|V| |E|)
If we say V and E are n,n
O(n2 )
This depends on number of edges.

REAL LIFE APPLICATIONS
• A Google Map may uses Bellman-Ford for finding Shortest Path
b/w to locations :-
• consider map as a GRAPH
• locations in the map are our VERTICES
• roads and road lengths between locations are EDGES and
WEIGHTS

REAL LIFE APPLICATIONS
• In a telephone network the lines have bandwidth(BW) We want to
route the phone call via the highest BW and consider the
transmition lines with highest BW as Shortest Path
• consider telephone network as a GRAPH
• switching station in the network are our VERTICES
• transmission lines in network are EDGES
• Bandwidths in network are WEIGHTS

Limitations of Bellman Ford’s Algorithm
• If there is a negative weight cycle then the graph cannot be
solved.
• Negative cycles. A negative cycle is a directed cycle whose total
weight (sum of the weights of its edges) is negative. The concept
of a shortest path is meaningless if there is a negative cycle.
-10
5
3
1
2 3
3+5-10 = -2

Limitations of Bellman Ford’s Algorithm
• Does not scale well
• Changes in network topology are not reflected quickly since
updates are spread node-by-node.

Bellman Ford Algorithm

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