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

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

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

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

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

Dijkstra’s Shortest Path Algorithm 6 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

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

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

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

Constrained Shortest Path Algorithms B0 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

Maximum Flow Problem - LP CS223 Advanced Data Structures and Algorithms

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

Shortest Path Problem - ILP CS223 Advanced Data Structures and Algorithms

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