Data Structures and Algorithms
CS245-2009S-17
Shortest Path
Dijkstra’s Algorithm
David Galles
Department of Computer Science
University of San Francisco
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Dijkstra's Algorithm: Finding Shortest Paths in Graphs - Prof. David J. Galles, Study notes of Data Structures and Algorithms
An explanation of dijkstra's algorithm, a popular algorithm used to find the shortest path between two vertices in a weighted graph. The concept of shortest path, the difference between shortest path and path with fewest edges, and the process of implementing dijkstra's algorithm. It also discusses the differences between single source shortest path and all-source shortest path, and introduces floyd's algorithm as an alternative for finding shortest paths.
Typology: Study notes
Pre 2010
1 / 78
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Dijkstra's Algorithm: Finding Shortest Paths in Graphs - Prof. David J. Galles and more Study notes Data Structures and Algorithms in PDF only on Docsity!
Data Structures and Algorithms
CS245-2009S-17Shortest PathDijkstra’s Algorithm^ David GallesDepartment of Computer ScienceUniversity of San Francisco
17-0:^ Computing Shortest Path^ Given a directed weighted graph
G^ (all weights
non-negative) and two vertices
x^ and^ y, find the
least-cost path from
x^ to^ y^ in^ G.
Undirected graph is a special case of a directedgraph, with symmetric edges Least-cost path may not be the path containing thefewest edges “shortest path” == “least cost path” “path containing fewest edges” = “pathcontaining fewest edges”
17-2:^ Shortest Path Example^ Shortest path
6 =^ path containing fewest edges
B A
C D
21 E
44 Shortest Path from A to E: A, B, C, D, E
17-3:^ Single Source Shortest Path^ To find the shortest path from vertex
x^ to vertex^ y
we need (worst case) to find the shortest path from x^ to^ all^ other vertices in the graph^ Why?
17-5:^ Single Source Shortest Path^ If all edges have unit weight ...
17-6:^ Single Source Shortest Path^ If all edges have unit weight,^ We can use Breadth First Search to compute theshortest path^ BFS Spanning Tree contains shortest path to eachnode in the graph^ Need to do some more work to create & saveBFS spanning tree^ When edges have differing weights, this obviouslywill not work
17-8:^ Single Source Shortest Path^ A^
B
C^
D^
E
F^
2 G
Start with the vertex A
17-9:^ Single Source Shortest Path^ A^
B
C^
D^
E
F^
2 G
Node^ Distance^ A^0 B C D E F G
Known vertices are circled in red We can now extend the known set by 1 vertex
17-11:^ Single Source Shortest Path^ A^
B
C^
D^
E
F^
2 G
Node^ Distance^ A^0 B C D^1 E F G
Why is it safe to add D, with cost 1?^ Could we do better with a more roundaboutpath?
17-12:^ Single Source Shortest Path^ A^
B C^
D^ E F^ 2 G (^1 ) 3 10 2 2 (^8 )
Node (^4 ) Distance A^0 B C D^1 E F G
Why is it safe to add D, with cost 1?^ Could we do better with a more roundaboutpath?^ No – to get to any other node will cost at least 1^ No negative edge weights, can’t do better than^1
17-14:^ Single Source Shortest Path^ A^
B
C^
D^
E
F^
2 G
Node^ Distance^ A^0 B^2 C D^1 E F G
How do we know that we could not get to Bcheaper than by going through D?
17-15:^ Single Source Shortest Path^ A^
B C^
D^
E F^ 2 G (^1 ) 3 10 2 2 (^8 ) (^4 )
Node^ Distance^ A^0 B^2 C D^1 E F G
How do we know that we could not get to Bcheaper than by going through D?^ Costs 1 to get to D^ Costs at least 2 to get anywhere from D^ Cost^ at least
(1+2 = 3) to get to B through D
17-17:^ Single Source Shortest Path^ A^
B
C^
D^
E
F^
2 G
Node^ Distance^ A^0 B^2 C^3 D^1 E F G
(We also could have added E for this step) Next vertex to add to Known ...
17-18:^ Single Source Shortest Path^ A
B
C^
D^
E
F^
2 G
Node^ Distance^ A^0 B^2 C^3 D^1 E^3 F G