Diverse Routing Algorithms CSC/ECE 772: Survivable Networks Spring, 2009, Rudra Dutta.

Diverse Routing AlgorithmsDiverse Routing Algorithms

CSC/ECE 772: Survivable Networks

Spring, 2009, Rudra Dutta

Copyright Rudra Dutta, NCSU, Spring, 2009 2

Disjoint Paths AlgorithmsDisjoint Paths Algorithms Simple approach (two-step) fails with trap

topologies Remedy - “remember” first path found by link

reversal– As in flow problems

Must be able to find shortest paths in presence of negative-cost arcs

Remedy - modify Dijkstra’s algorithm– Can also use general purpose shortest path

algorithm for negative-cost arcs– Can also use graph transformation to eliminate

negative-cost arcs


Dijkstra’s AlgorithmDijkstra’s Algorithm Given:

– Undirected graph G with vertices V,– Source node A,– Destination node Z– Edge costs l(ij)

Use:– N(i) = set of neighbors of i,– d(i) = distance from A to i,– S = set of nodes to which shortest distance has NOT

been found,– P(i) = predecessor of i in path from A


Dijkstra’s AlgorithmDijkstra’s Algorithm1. Initialize

d(A) = 0; d(i) = l(Ai) for all i in N(A), for all other i S = all vertices except A P(i) = A for all i in S

l Select new vertex to label permanently Choose j from S such that d(j) = min d(i), for i in S Remove j from S If j = Z, then stop; otherwise go to Step 3

l Update distances For all i in N(j) such that i is also in S

If d(j) + l(ij) < d(i), update d(i) to d(j) + l(ij) and set P(i) to j Back to Step 2

l(ij) cost of edge ij

N(i) = neighbors of i

d(i) = distance A to i

S = nodes not found

P(i) = predecessor of i


Dijkstra’s Algorithm - ExampleDijkstra’s Algorithm - Example


Negative Cost ArcsNegative Cost Arcs Many reasonable cost metrics are positive-only However, sometimes negatives costs can turn

up– Sometimes introduced by specific desired

transformations of graphs Graph may have cycles

– Cycles with negative total costs existConcept of shortest path?

– If negative cost arcs, but not cycles? Specific family of such graphs generated by

useful diverse routing algorithms Dijkstra’s algorithm fails with negative arc costs


Failure of Dijkstra - ExampleFailure of Dijkstra - Example


Dijkstra’s Algorithm - ModifiedDijkstra’s Algorithm - Modified1. Initialize

d(A) = 0; d(i) = l(Ai) for all i in N(A), for all other i S = all vertices except A P(i) = A for all i in S

l Select new vertex to label permanently Choose j from S such that d(j) = min d(i), for i in S Remove j from S If j = Z, then stop; otherwise go to Step 3

l Update distances For all i in N(j) such that i is also in S

If d(j) + l(ij) < d(i), update d(i) to d(j) + l(ij) and set P(i) to j

Back to Step 2

-----------------------------

Also add i back to S

S =

l(ij) cost of edge ij

N(i) = neighbors of i

d(i) = distance A to i

S = nodes not found

P(i) = predecessor of i


Modified Dijkstra - ExampleModified Dijkstra - Example


Shortest Disjoint Path PairShortest Disjoint Path Pair Objective: find a pair of paths between a source

a destination that are:– Edge-disjoint, or Vertex-disjoint– Sum of their lengths is minimum

Observations:– Vertex-disjointness implies edge-disjointedness– For graph of vertices with degrees 3 or less:

Edge-disjointedness implies Vertex-disjointedness

Length of vertex-disjoint pair ≥ length of edge-disjoint pair


Simple Two-step Approach FailsSimple Two-step Approach Fails Obvious approach - find one, then other

– May not find shortest pair– May not find pair at all even when one exists– Trap topology

A

B

D

Z

E

C

1 1 1

1

12

2


Shortest Disjoint Path AlgorithmsShortest Disjoint Path Algorithms Various types

– Most based on flow canceling approaches (like flow maximization)

– One early approach by Bhandari– Other early approach by Suurballe (and with Tarjan)

Bhandari’s algorithm– Simpler to state on undirected graphs– Assumes original graph has non-negative edge costs– Creates intermediate mixed graphs with negative directed arc

costs– Only specific family of such graphs - arcs form s-d path– Modified Dijkstra’s algorithm works correctly on such graphs– Can be used in lieu of more complicated negative cost

algorithms


Bhandari’s AlgorithmBhandari’s Algorithm1. Find the shortest path from A to Z with the modified

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


Bhandari’s Algorithm - ExampleBhandari’s Algorithm - Example


kk Disjoint Paths Disjoint Paths Can be generalized to obtain more disjoint paths 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 k > 3


Edge Disjoint Triple - ExampleEdge Disjoint Triple - Example


Vertex Disjoint PathsVertex Disjoint Paths Builds on the edge disjoint path algorithm Simple graph transformation

– Separate “source” nature of “destination” nature of each vertex into two fictitious vertices

– Directed arc from “destination” to “source” for each, zero cost

– Path, flow etc. preserved in magnitude and cost Use of fictitious vertices/arcs can be limited to

“as needed” basis Edge disjoint paths in transformed graph are

clearly vertex disjoint in original Can be generalized to k > 2 as before


Vertex Disjoint Pair AlgorithmVertex Disjoint Pair Algorithm1. Find the shortest path from A to Z with the modified Dijkstra

algorithm2. Replace each edge on the shortest path by an arc directed towards

the destination vertex3. 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 - forming cycle along zero-length arc

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


Vertex Disjoint Pair - ExampleVertex Disjoint Pair - Example


Use for Multiple Source/DestinationsUse for Multiple Source/Destinations

A

Z1

Z2

ZA

Z1

Z2

ZA

Z1

Z2

A1

A2

ZA

Z1

Z2

A1

A2

Diverse Routing Algorithms CSC/ECE 772: Survivable Networks Spring, 2009, Rudra Dutta.

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