Dijkstra's Outline
1. Algorithm
CSC/ECE 791B Survivable Networks Algorithms for Diverse Routing
George N. Rouskas
Department of Computer Science North Carolina State University
2. Modified Dijkstra's Algorithm 3. Algorithms for Shortest Pair of Disjoint Paths
CSC/ECE 791B, Spring 2007: Algorithms for Diverse Routing
Copyright c 2007 by George N. Rouskas p.1
CSC/ECE 791B, Spring 2007: Algorithms for Diverse Routing
Copyright c 2007 by George N. Rouskas p.2
Dijkstra's Algorithm
Input: weighted graph G start vertex A
Dijkstra's Algorithm: Notation
i : set of neighbors of vertex i
= (V, E)
lij : length of arc from i to j ( if it does not exist) d(i): distance of shortest path from source A to vertex i Pi : predecessor of vertex i on shortest path from A to i S : set of vertices for which the shortest path from A has been found
V destination vertex Z V
Output: shortest path (and its length) from A to Z Greedy algorithm: always selects the best local solution local choices lead to optimal overall solution
Dijkstra's Algorithm: Description
1. Initialization (a)
Dijkstra Algorithm: Example
B 1 3 C 2 5 4 2 E D 2 3 1 1 Z
d(A) 0 (b) d(i) = lAi , i A (, otherwise) (c) S = V - {A} (d) Pi = A i S
2. Select a new vertex and label it permanently (a) find j
S such that d(j) = miniS d(i) (b) S S - {j} (c) if j = Z then stop; else go to Step 3
3. Update distances (a)
A
(i j i S): if d(j) + lij < d(i) then d(i) d(j) + lij , Pi j
(b) go to Step 2
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Dijkstra Algorithm: Example (2)
Dijkstra Algorithm: Example (3)
1 1 B 3 2 C 2 1 4
1 1 3
B 3 1 3 C 2 1 8 Z 2 E 5
5 A
1 3
Z 2 8 E
A
4
5
4
5 S
D
2
S
D
2
8
Dijkstra Algorithm: Example (4)
1 1 3 B 3 2 C 2 5 4 S 4 2 8 E
S 5 1 1
Dijkstra Algorithm: Example (5)
B 3 2 C 2 4 2 8 4 D 2
CSC/ECE 791B, Spring 2007: Algorithms for Diverse Routing
1
1 3
A
1
3
Z
A
1
3
Z
E
D
2
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Copyright c 2007 by George N. Rouskas p.9
Copyright c 2007 by George N. Rouskas p.10
Graphs with Negative Arcs
Special class of graphs: some arcs have negative weights but: no negative cycles Encountered in the construction of disjoint paths Dijkstra's algorithm fails on such graphs
Graph with Negative Arcs: Example
B 5 1 A -1 C 2 7 6 -6 E -2 Z 8
D
Graph with Negative Arcs: Example (2)
Graph with Negative Arcs: Example (3)
5 5 B 8 1 A -1 6 C 2 7 6 7 -6 -2 E 13 Z 8
5 5
B 8 1
A
-1
-6 C 2
-2 E
8
8
8
Z
7 7
6
D
D
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Copyright c 2007 by George N. Rouskas p.13
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Copyright c 2007 by George N. Rouskas p.14
Graph with Negative Arcs: Example (4)
5 5 1 A -1 6 C 2 6 7 -6 -2 E 13 Z B 8
Graph with Negative Arcs: Example (5)
5 5 1 A -1 6 C 2 7 7 6 -6 9 E -2 13 Z B 8
7
D
8
D
Graph with Negative Arcs: Example (6)
5 5 1 A -1 6 C 2 7 7 6 -6 9 E -2 13 Z B 8
Modified Dijkstra's Algorithm
1. Initialization (a)
d(A) 0 (b) d(i) = lAi , i A (, otherwise) (c) S = V - {A} (d) Pi = A i S
2. Select a new vertex and label it (a) find j
S such that d(j) = miniS d(i) (b) S S - {j} (c) if j = Z then stop; else go to Step 3
3. Update distances and re-enter vertices into set S , if necessary (a)
D
(i j ): if d(j) + lij < d(i) then d(i) d(j) + lij , Pi j S S {i}
(b) go to Step 2
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Modified Dijkstra Algorithm: Example
B 5 1 A -1 C 2 7 6 -6 E -2 Z 8
Modified Dijkstra Algorithm: Example (2)
5 5
B 8 1
A
-1
-6 C 2
-2 E
8
8
8
Z
7 7
6
D
D
Modified Dijkstra Algorithm: Example (3)
5 5 1 A -1 6 C 2 7 7 6 -6 -2 E 13 Z
A
Modified Dijkstra Algorithm: Example (4)
5 B 8 1 -1 6 C 2 6 7 -6 -2 E 13 Z
B 8
5
8
7
D
D
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Copyright c 2007 by George N. Rouskas p.21
8
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Modified Dijkstra Algorithm: Example (5)
5 5 1 A -1 3 C 2 7 7 6 -6 9 E -2 13 Z
A
Modified Dijkstra Algorithm: Example (6)
B 8 1 -1 3 C 2 7 7 6 -6 9 E -2 13 Z
B
4
8
5
D
D
Modified Dijkstra Algorithm: Example (7)
4 5 1 A -1 3 C 2 7 7 6 -6 9 E -2 12 Z B 8
Shortest Pair of Disjoint Paths
Objective: find a pair of paths between a source a destination that are: edge-disjoint, or vertex-disjoint and the sum of their lengths is minimum
D
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Copyright c 2007 by George N. Rouskas p.25
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Copyright c 2007 by George N. Rouskas p.26
Shortest Pair of Disjoint Paths
Objective: find a pair of paths between a source a destination that are: edge-disjoint, or vertex-disjoint and the sum of their lengths is minimum Observations: vertex-disjoint path pair edge-disjoint path pair graph of vertices with degrees 3: edge-disjoint path pair vertex-disjoint path pair length of vertex-disjoint pair length of edge-disjoint pair
Simple Two-Step Algorithms
Two-step Edge-disjoint (TE) algorithm: 1. find the shortest path between two nodes 2. find the shortest path in the same graph, after removing the edges of the first shortest path
Simple Two-Step Algorithms
Two-step Edge-disjoint (TE) algorithm: 1. find the shortest path between two nodes 2. find the shortest path in the same graph, after removing the edges of the first shortest path Two-step Vertex-disjoint (TV) algorithm: 1. find the shortest path between two nodes 2. find the shortest path in the same graph, after removing the edges incident on the vertices (except the endpoints) of the first shortest path
Example Graph: Shortest Path
G
8 B C
1 D
2
A
1 1 1
1
1 1
1
Z
2
4 E 4
F
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Copyright c 2007 by George N. Rouskas p.27
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Example Graph: TE Algorithm
G
Example Graph: TV Algorithm
G
8 B C
1 D
2
8 Z B C D
2
A
A
Z
1 1
2
1 4 1 4 E 4
E
4
F
F
Shortcomings of Two-Step Algorithms
1. Suboptimality may yield suboptimal disjoint paths 2. False alarms may fail to find disjoint paths when such paths actually exist
Example Graph: Optimal Edge-Disjoint Paths
G
8 B C
1 D
2
A
1 1 1
1
1 1
1
Z
2
4 E 4
F
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Copyright c 2007 by George N. Rouskas p.31
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Copyright c 2007 by George N. Rouskas p.32
Example Graph: Optimal Vertex-Disjoint Paths
G
Example Graph: TE and TV Algorithms Fail
D
8 B C
1 D
2
2
1 Z
1 B 1 C 1
A
1 1 1
1
1 1
A
1
Z
2
4 E 4
2
1
F
E
Example Graph: TV Algorithm Fails
D 2 A 1 B 1 C 2 1 1 1
Edge-Disjoint Shortest Pair Algorithm
Input: weighted graph G start vertex A
= (V, E)
V destination vertex Z V
Output: shortest pair of edge-disjoint paths from A to Z
Z
E
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Edge-Disjoint Shortest Pair Algorithm: Description
1. Find the shortest path from A to Z with modified the Dijkstra algorithm 2. Replace each edge of the shortest path by a single arc directed towards the source vertex 3. Make the length of each of the above arcs negative 4. Find the shortest path from A to Z in the modified graph with the modified Dijkstra algorithm 5. Transform to the original graph, and erase any interlacing edges of the two paths found 6. Group the remaining edges to obtain the shortest pair of edge-disjoint paths
A
Example: Edge-Disjoint Shortest Pair Algorithm
G H
8 2 5 1 C 2 D 3 1 E 3
1
2 B
2 F
4
1
1
1
1
Z
7
5
I
Example: Edge-Disjoint Shortest Pair Algorithm (2)
G H
Example: Edge-Disjoint Shortest Pair Algorithm (3)
G H
8 2 5 -1 C 2 D 3 -1 E 3
8 2 5 -1 C 2 D 3 -1 E 3
1
2 B
2 F
4
1
2 B
2 F
4
A
-1
-1
-1
-1
Z
A
-1
-1
-1
-1
Z
7
5
7
5
I
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I
Copyright c 2007 by George N. Rouskas p.40
Example: Edge-Disjoint Shortest Pair Algorithm (4)
G H
Example: Edge-Disjoint Shortest Pair Algorithm (5)
G H
8 2 5 1 C 2 D 3 1 E 3
8 2 5 1 C 2 D 3 1 E 3
1
2 B
2 F
4
1
2 B
2 F
4
A
1
1
1
1
Z
A
1
1
1
1
Z
7
5
7
5
I
I
Generalization:
k(> 2) Edge-Disjoint Paths
Example: Edge-Disjoint Shortest Triplet Algorithm
F
Edge-disjoint shortest triplet algorithm: 1. Find the shortest edge-disjoint pair of paths between A and Z 2. Replace the edges of the disjoint paths by arcs directed towards the source 3. Make the length of the arcs negative 4. Find the shortest path from A to Z in the modified graph using the modified Dijkstra algorithm 5. Transform to the original graph, and erase any edges of this shortest path interlacing with the shortest edge-disjoint pair of paths Straightforward generalization to
1 3 D 3
2 E 1 1 C 9
1 A 1 B
1
1
1 1 Z
k>3
1 7 G
4
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Copyright c 2007 by George N. Rouskas p.44
Example: Edge-Disjoint Shortest Triplet Algorithm (2)
F
Example: Edge-Disjoint Shortest Triplet Algorithm (3)
F
1 3 -1 1 A -1 B -1 D 3
2 E -1 9 3 -1 -1 -1 C Z A -1 B 1 D
1 3
2 E -1 9
-1 -1
-1 -1 C Z
-1 1
7 G
4
1 7 G
4
Example: Edge-Disjoint Shortest Triplet Algorithm (4)
F
Example: Edge-Disjoint Shortest Triplet Algorithm (5)
F
1 3 -1 1 A -1 B -1 D 3
2 E -1 9 3 D
1 3
2 E 1 1 C 9
-1 -1 C Z A 1
1 B
1
1
1 1 Z
-1 1
7 G
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1 7 G
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Copyright c 2007 by George N. Rouskas p.48
Example: Edge-Disjoint Shortest Triplet Algorithm (6)
F
Vertex-Disjoint Shortest Pair Algorithm
Input:
1 3 D 3
2 E 1 1 C 9
weighted graph G start vertex A
= (V, E)
V destination vertex Z V
Output: shortest pair of vertex-disjoint paths from A to Z
1 A 1 B
1
1
1 1 Z
1 7 G
4
Vertex-Disjoint Shortest Pair Algorithm: Description
1. Find the shortest path from A to Z with the modified Dijkstra algorithm 2. Replace each edge on the shortest path by an arc directed towards the destination vertex 3. Split each vertex on the shortest path into two co-located subvertices joined by an arc of length zero direct the arc of length zero towards the destination replace each external edge connected to a vertex on the shortest path by its two component arcs (of the same length) let one arc terminate on one subvertex, the other emanate from the other subvertex cycle along zero-length arc
Vertex-Disjoint Shortest Pair Algorithm: Description (2)
4. Reverse the direction of the arcs on the shortest path, and make their length negative 5. Find the shortest path from A to Z in the modified graph with the modified Dijkstra algorithm 6. Remove zero-length arcs; coalesce subvertices into parent vertices; replace arcs on first shortest path with original edges of positive length; remove interlacing edges of the two paths
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Copyright c 2007 by George N. Rouskas p.51
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Copyright c 2007 by George N. Rouskas p.52
Example: Vertex-Disjoint Shortest Pair Algorithm
E
Example: Vertex-Disjoint Shortest Pair Algorithm (2)
E
3 1 3
11
3 1 3
11
1 A B
1 C 4
1 D 1
1 Z A
1 B
1 C 4
1 D 1
1 Z
1
1
F
F
Example: Vertex-Disjoint Shortest Pair Algo...