1. Graph Algorithms and Their Applications on Computer Networks Neil Tang 2/6/2010

2. CS223 Advanced Data Structures and Algorithms
Outline
Network Model
Link and Path Costs
Routing Problems
Dijkstra’s Shortest Path Algorithm
Constrained Shortest Path Algorithms
Another Approach: LP and ILP
CS223 Advanced Data Structures and Algorithms

3. CS223 Advanced Data Structures and Algorithms
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.
CS223 Advanced Data Structures and Algorithms

4. CS223 Advanced Data Structures and Algorithms
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.
CS223 Advanced Data Structures and Algorithms

5. CS223 Advanced Data Structures and Algorithms
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.
CS223 Advanced Data Structures and Algorithms

6. Dijkstra’s Shortest Path Algorithm6 1  10 5 2 9 3 A B C D E 8 5 14 10 2 1 9 3 6 7 10 5  2 1 9 3 6 8 5 9 10 2 1 3 6 7 8 5 9 10 2 1 3 6 7 8 5 13 10 2 1 9 3 6 7
Time complexity: |E|+|V|log|V|
CS223 Advanced Data Structures and Algorithms

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.
CS223 Advanced Data Structures and Algorithms

8. Constrained Shortest Path Algorithms
Find a shortest path subject to a bottleneck (e.g., bandwidth) constraint:
ρ= (A,D), Threshold=3
(4,4) D B
(2,4)
(3,4)
(5,2) A
(3,2) C
CS223 Advanced Data Structures and Algorithms

9. Constrained Shortest Path Algorithms
Find a shortest path subject to an additive (e.g., delay) constraint:
ρ= (A,D), Threshold=4
(4,4) D B
(2,4)
(3,4)
(5,2) A
(3,2) C
CS223 Advanced Data Structures and Algorithms

10. Constrained Shortest Path AlgorithmsB0 D0
Find a shortest path subject to an
additive (e.g., delay) constraint:
ρ= (A,D), Threshold=4 2 4 4 5 3 B1 D1 5 A0 C0 5 3 3 2 B2 D2 A1 C1 5 5
(4,4) B D 3
(2,4)
(3,4) B3 D3 3
(5,2) A2 C2 A 5 3
(3,2) 3 C B4 D4 A3 C3 3 A4 C4
CS223 Advanced Data Structures and Algorithms

11. Maximum Flow Problem - LP
CS223 Advanced Data Structures and Algorithms

12. Multi-Commodity Flow Problem - LP
CS223 Advanced Data Structures and Algorithms

13. Shortest Path Problem - ILP
CS223 Advanced Data Structures and Algorithms

14. Constrained Shortest Path Problem - ILP
CS223 Advanced Data Structures and Algorithms