1. CS541 Advanced Networking 1 Routing and Shortest Path Algorithms Neil Tang 2/18/2009

2. CS541 Advanced Networking 2 Outline  Network Model  Link and Path Costs  Routing Problems  Dijkstra’s Shortest Path Algorithm  Constrained Shortest Path Algorithms

3. CS541 Advanced Networking 3 Network Model  A network is usually modeled as a graph such that the networking problems, such as routing and scheduling, can be transformed to the corresponding problems in a graph and solved by algorithms in graph theory.  A vertex in the graph represents a router.  In a wired network, there is an edge (A,B) between a pair of vertices A and B if there is a physical link in between.  Disk graph: In a wireless network, there is an edge (A,B) between a pair of vertices A and B if their Euclidean distance is no more than the transmission range of node A.

4. CS541 Advanced Networking 4 Link and Path Cost  There is usually one or multiple cost parameters associated with each link, which indicate its transmission cost, bandwidth, delay and so on.  Some of parameters are bottleneck parameters (e.g. bandwidth), i.e., the cost of the path in terms of this parameter depends on the minimum link cost along the path.  Some of parameters are additive parameters (e.g. delay), i.e., the cost of the path in terms of this parameter is equal to the summation of the costs of all links along the path.

5. CS541 Advanced Networking 5 Routing Problems  Basic routing problem: Given a source and a destination, the basic routing problem seeks a source-destination route such that it has the minimum cost among all possible source-destination routes.  QoS routing problem: Given a source and destination along with one or multiple QoS constrains (e.g, bandwidth, delay), a QoS routing problem seeks a source-destination route such that it has the minimum cost among all source-destination routes which can satisfy all QoS constraints.

6. CS541 Advanced Networking 6 Dijkstra’s Shortest Path Algorithm 10 5  0 5 2 1 9 3 6  8 5 14 0 10 5 2 1 9 3 6 7 8 5 13 0 10 5 2 1 9 3 6 7 8 5 9 0 5 2 1 9 3 6 7 8 5 9 0 5 2 1 9 3 6 7 6 1    0 5 2 9 3  A B C D E Time complexity: |E|+|V|log|V|

7. CS541 Advanced Networking 7 Typical QoS Routing Problems  Find a minimum cost path subject to a bottleneck (e.g., bandwidth) constraint.  Find a minimum cost path subject to an additive (e.g. delay) constraint. This problem is NP-hard.

8. CS541 Advanced Networking 8 Constrained Shortest Path Algorithms Find a shortest path subject to a bottleneck (e.g., bandwidth) constraint: ρ= (A,D), Threshold=3 A B C D (2,4)(2,4) (4,4)(4,4) (3,2)(3,2) (3,4)(3,4) (5,2)(5,2)

9. CS541 Advanced Networking 9 Constrained Shortest Path Algorithms Find a shortest path subject to an additive (e.g., delay) constraint: ρ= (A,D), Threshold=4 A B C D (2,4)(2,4) (4,4)(4,4) (3,2)(3,2) (3,4)(3,4) (5,2)(5,2) A0A0 A1A1 A2A2 A3A3 A4A4 B0B0 B1B1 B2B2 B3B3 B4B4 C0C0 C1C1 C2C2 C3C3 C4C4 D0D0 D1D1 D2D2 D3D3 D4D4 2 2 3 4 3 4 3 3 3 5 5 5 5 5 5 3 3 3 0 0 0 0