Shortest path problem

Jun 16, 2014Download as PPTX, PDF8 likes9,998 views

This document presents information about the shortest path problem in graphs. It defines key graph terms like vertices, edges, and discusses weighted, directed, and undirected graphs. It provides an example of finding the shortest path between two vertices in a graph using Dijkstra's algorithm and walks through the steps of running the algorithm on a sample graph to find the shortest path between vertices 1 and 9.

PRESENTED TO:
MAM MERYAM
DEPARTMENT OF
COMPUTER SCIENCE

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IFRA ILYAS (470)
AQSA SHAUKAT (586)
PAZEER ZARA (452)
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GRAPH:
 A graph is a representation of a set of objects
where some pairs of objects are connected by links.
 Vertices:
The interconnected objects are represented
by mathematical abstractions called vertices.
 Edges:
The links that connect some pairs of vertices
are called edges.

SHORTEST PATH
PROBLEM :
 The shortest path problem is the problem of finding
a path between two vertices in a graph such that
the sum of the weights of its constituent edges is
minimized.


TYPES OF GRAPH:
1. Weighted graph
2. Un-weighted graph
3. Directed graph
4. Un-directed graph

TYPES OF GRAPHS:
 Weighted:
A graph is a
weighted graph if a
number (weight) is
assigned to each edge.
Example:
Such weights might
represent, for example,
costs, lengths or
capacities, etc.
 Un-weighted:
A graph is a un-
weighted graph if a
number(weight) is not
assigned to each edge.

TYPES OF GRAPHS:
 Directed:
An directed
graph is one in which
edges have orientation.
 Un-directed:
An undirected
graph is one in which
edges have no
orientation.

SHORTEST PATH PROBLEM:
 Given the graph below, suppose we wish to
find the shortest path from vertex 1 to vertex
13.

EXAMPLE:
 After some consideration, we may determine
that the shortest path is as follows, with length
14
 Other paths exists, but they are longer

NEGATIVE CYCLES:
 Clearly, if we have negative vertices, it may be
possible to end up in a cycle whereby each pass
through the cycle decreases the total length
 Thus, a shortest length would be undefined for such
a graph
 Consider the shortest path
from vertex 1 to 4...
 We will only consider non-
negative weights.

SHORTEST PATH EXAMPLE:
 Given:
 Weighted Directed graph G = (V, E).
 Source s, destination t.
 Find shortest directed path from s to t.
s
3
t
2
6
7
4
5
23
18
2
9
14
15
5
30
20
44
16
11
6
19
6
Cost of path s-2-3-5-t
= 9 + 23 + 2 + 16
= 48.

DISCUSSION ITEMS
 How many possible paths are there from s
to t?
 Can we safely ignore cycles? If so, how?
 Any suggestions on how to reduce the set of
possibilities?
s
3
t
2
6
7
4
5
23
18
2
9
14
15
5
30
20
44
16
11
6
19
6

KEY OBSERVATION:
 A key observation is that if the shortest path contains
the node v, then:
 It will only contain v once, as any cycles will only add to the
length.
 The path from s to v must be the shortest path to v from s.
 The path from v to t must be the shortest path to t from v.

DIJKSTRA’S ALGORITHM:
 Works when all of the weights are positive.
 Provides the shortest paths from a source to
all other vertices in the graph.
 Can be terminated early once the shortest
path to t is found if desired.

EXAMPLE:
 Consider the graph:
 the distances are appropriately initialized
 all vertices are marked as being unvisited

EXAMPLE:
 Visit vertex 1 and update its neighbours,
marking it as visited
 the shortest paths to 2, 4, and 5 are
updated

EXAMPLE:
 The next vertex we visit is vertex 4
 vertex 5 1 + 11 ≥ 8 don’t update
 vertex 7 1 + 9 < ∞ update
 vertex 8 1 + 8 < ∞ update

EXAMPLE:
 Next, visit vertex 2
 vertex 3 4 + 1 < ∞ update
 vertex 4 already
visited
 vertex 5 4 + 6 ≥ 8 don’t update
 vertex 6 4 + 1 < ∞ update

EXAMPLE:
 Next, we have a choice of either 3 or 6
 We will choose to visit 3
 vertex 5 5 + 2 < 8 update
 vertex 6 5 + 5 ≥ 5 don’t update

EXAMPLE:
 We then visit 6
 vertex 8 5 + 7 ≥ 9 don’t update
 vertex 9 5 + 8 < ∞ update

EXAMPLE:
 Next, we finally visit vertex 5:
 vertices 4 and 6 have already been visited
 vertex 7 7 + 1 < 10 update
 vertex 8 7 + 1 < 9 update
 vertex 9 7 + 8 ≥ 13 don’t update

EXAMPLE:
 Given a choice between vertices 7 and 8, we
choose vertex 7
 vertices 5 has already been visited
 vertex 8 8 + 2 ≥ 8 don’t update

EXAMPLE:
 Next, we visit vertex 8:
 vertex 9 8 + 3 < 13 update

EXAMPLE:
 Finally, we visit the end vertex
 Therefore, the shortest path from 1 to 9 has length
11

EXAMPLE:
 We can find the shortest path by working back from
the final vertex:
 9, 8, 5, 3, 2, 1
 Thus, the shortest path is (1, 2, 3, 5, 8, 9)

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Shortest path problem

2. PRESENTED TO: MAM MERYAM DEPARTMENT OF COMPUTER SCIENCE

3. GROUP MEMBERS: IFRA ILYAS (470) AQSA SHAUKAT (586) PAZEER ZARA (452) SAMRA ASLAM (427) AQSA ANWAR (453)

4. SHORTEST PATH PROBLEM

5. GRAPH:  A graph is a representation of a set of objects where some pairs of objects are connected by links.  Vertices: The interconnected objects are represented by mathematical abstractions called vertices.  Edges: The links that connect some pairs of vertices are called edges.

6. SHORTEST PATH PROBLEM :  The shortest path problem is the problem of finding a path between two vertices in a graph such that the sum of the weights of its constituent edges is minimized. 

7. TYPES OF GRAPH: 1. Weighted graph 2. Un-weighted graph 3. Directed graph 4. Un-directed graph

8. TYPES OF GRAPHS:  Weighted: A graph is a weighted graph if a number (weight) is assigned to each edge. Example: Such weights might represent, for example, costs, lengths or capacities, etc.  Un-weighted: A graph is a unweighted graph if a number(weight) is not assigned to each edge.

9. TYPES OF GRAPHS:  Directed: An directed graph is one in which edges have orientation.  Un-directed: An undirected graph is one in which edges have no orientation.

10. SHORTEST PATH PROBLEM:  Given the graph below, suppose we wish to find the shortest path from vertex 1 to vertex 13.

11. EXAMPLE:  After some consideration, we may determine that the shortest path is as follows, with length 14  Other paths exists, but they are longer

12. NEGATIVE CYCLES:  Clearly, if we have negative vertices, it may be possible to end up in a cycle whereby each pass through the cycle decreases the total length  Thus, a shortest length would be undefined for such a graph  Consider the shortest path from vertex 1 to 4...  We will only consider non- negative weights.

13. SHORTEST PATH EXAMPLE:  Given:  Weighted Directed graph G = (V, E).  Source s, destination t.  Find shortest directed path from s to t. s 3 t 2 6 7 4 5 23 18 2 9 14 15 5 30 20 44 16 11 6 19 6 Cost of path s-2-3-5-t = 9 + 23 + 2 + 16 = 48.

14. DISCUSSION ITEMS  How many possible paths are there from s to t?  Can we safely ignore cycles? If so, how?  Any suggestions on how to reduce the set of possibilities? s 3 t 2 6 7 4 5 23 18 2 9 14 15 5 30 20 44 16 11 6 19 6

15. KEY OBSERVATION:  A key observation is that if the shortest path contains the node v, then:  It will only contain v once, as any cycles will only add to the length.  The path from s to v must be the shortest path to v from s.  The path from v to t must be the shortest path to t from v.

16. DIJKSTRA’S ALGORITHM:  Works when all of the weights are positive.  Provides the shortest paths from a source to all other vertices in the graph.  Can be terminated early once the shortest path to t is found if desired.

17. EXAMPLE:  Consider the graph:  the distances are appropriately initialized  all vertices are marked as being unvisited

18. EXAMPLE:  Visit vertex 1 and update its neighbours, marking it as visited  the shortest paths to 2, 4, and 5 are updated

19. EXAMPLE:  The next vertex we visit is vertex 4  vertex 5 1 + 11 ≥ 8 don’t update  vertex 7 1 + 9 < ∞ update  vertex 8 1 + 8 < ∞ update

20. EXAMPLE:  Next, visit vertex 2  vertex 3 4 + 1 < ∞ update  vertex 4 already visited  vertex 5 4 + 6 ≥ 8 don’t update  vertex 6 4 + 1 < ∞ update

21. EXAMPLE:  Next, we have a choice of either 3 or 6  We will choose to visit 3  vertex 5 5 + 2 < 8 update  vertex 6 5 + 5 ≥ 5 don’t update

22. EXAMPLE:  We then visit 6  vertex 8 5 + 7 ≥ 9 don’t update  vertex 9 5 + 8 < ∞ update

23. EXAMPLE:  Next, we finally visit vertex 5:  vertices 4 and 6 have already been visited  vertex 7 7 + 1 < 10 update  vertex 8 7 + 1 < 9 update  vertex 9 7 + 8 ≥ 13 don’t update

24. EXAMPLE:  Given a choice between vertices 7 and 8, we choose vertex 7  vertices 5 has already been visited  vertex 8 8 + 2 ≥ 8 don’t update

25. EXAMPLE:  Next, we visit vertex 8:  vertex 9 8 + 3 < 13 update

26. EXAMPLE:  Finally, we visit the end vertex  Therefore, the shortest path from 1 to 9 has length 11

27. EXAMPLE:  We can find the shortest path by working back from the final vertex:  9, 8, 5, 3, 2, 1  Thus, the shortest path is (1, 2, 3, 5, 8, 9)

Shortest path problem

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Shortest path problem