Single sourceshortestpath by emad

Presentation
On
Single Source Shortest Path.

Presenter:
Kazi Emad
ID no. 191902025
01

Agenda
1. Overview of Single Source Shortest
Path
2. Types of Single Source Shortest Path
Algorithm
3. Representation of Single Source
Shortest Path
4. Initialization
5. Relaxation
6. Implementation of Dijkstra's Algorithm
7. Does Dijkstra’s Algorithm Always Work?
8. Implementation of Bellman-Ford
Algorithm
9. Negative Weight Cycles in Bellman-
Ford Algorithm

Overview of Single Source Shortest
Path:
 In a shortest-paths problem, we are given a weighted, directed graph G = (V, E), with weight
function w : E -> R mapping edges to real-valued weights. The weight w(p) of path p = (V0,
V1….., Vk) ki is the sum of the weights of its constituent edges:
w(p)  w(vi ,vi1)
i1
 The single-source shortest path problem, in which we have to find shortest paths from a source
vertex v to all other vertices in the graph.

Types of Single Source Shortest Path Algorithm:
There are two types of single source shortest path algorithm. And they are:
1. Dijkstra’s Algorithm.
2. Bellman-Ford Algorithm

Representation of Single Source Shortest Path:
For each vertex v  V:
 d[v] = δ(s, v): a shortest-path estimate
 Initially, d[v]=∞
 Reduces as algorithms progress
 [v] = predecessor of v on a shortest path from s
 If no predecessor, [v] = NIL
  induces a tree—shortest-path tree

Initialization:
INITIALIZE-SINGLE-SOURCE(V, s)
1. for each v  V
2. do d[v] ← 
3. [v] ← NIL
4. d[s] ← 0

Relaxation:
Relaxing an edge (u, v) = testing whether we can improve the shortest path to v found so far by
going through u.
Example:
If d[v] > d[u] + w(u, v)
 update d[v] and [v]
S 8
2
4
u
v
2 6
u v
4
RELAX(u,v,w)

Implementation of Dijkstra's Algorithm:
Given a graph and a source vertex in the graph, find shortest paths from
source to all vertices in the given graph.
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distance label

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decrease key
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S 2 3 4 5 6 7 t
S 0 9 inf inf inf 14 15 inf
2 0 9 33 inf inf 14 15 inf
6 0 9 32 inf 44 14 15 inf
7 0 9 32 inf 35 14 15 59
3 0 9 32 inf 34 14 15 51
5 0 9 32 45 34 14 15 50
4 0 9 32 45 34 14 15 50
t 0 9 32 45 34 14 15 50
Tabular Solution of the Graph for Shortest Path

Does Dijkstra’s Algorithm Always Work?
No. Dijkstra’s Algorithm doesn’t always work in case of negative weight
edges.
So, we have a new algorithm named “Bellman-Ford Algorithm” which can
also perform in negative weight edges.

Implementation of Bellman-Ford Algorithm:

Initialize all the
distances

iterate over all
edges/vertices
and apply update
rule

check for negative
cycles

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A:

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A: 3

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A: 3
B:

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A: 3
B: 5

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A: 3
B: 5
D:

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A: 3
B: 5
D: 7

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 





Iteration: 0

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 10




8
Iteration: 1

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 10


12
9
8
Iteration: 2

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 5
10

8
9
8
Iteration: 3
A has the correct
distance and path

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 5
6
11
7
9
8
Iteration: 4

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 5
5
7
147
9
8
Iteration: 5
B has the correct
distance and path

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 5
5
6
107
9
8
Iteration: 6

G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 5
5
6
97
9
8
Iteration: 7
D (and all other
nodes) have the
correct distance
and path

Negative Weight Cycles in Bellman-Ford
Algorithm:
• Negative edges are OK, as long as there are no negative weight cycles (otherwise paths with
arbitrary small “lengths” would be possible)
• Shortest-paths can have no cycles (otherwise we could improve them by removing cycles)
– Any shortest-path in graph G can be no longer than n – 1 edges, where n is the number of
vertices.

Single sourceshortestpath by emad

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