1. ICOM 6115©Manuel Rodriguez-Martinez ICOM 6115 – Computer Networks and the WWW Manuel Rodriguez-Martinez, Ph.D. Lecture 20

2. ICOM 6115©Manuel Rodriguez-Martinez Lecture Objectives Mathematical background on Graph Theory –Basic Definitions –Representations –Fundamental Algorithms Introduction to Global Internetworking –Layer 3 – The Network Layer –IP Protocol

3. ICOM 6115©Manuel Rodriguez-Martinez Directed Graph Directed Graph G consists of –a set of vertices V –a set of edges E –Formally, G = (V,E) Vertices are used to represent some object –Like nodes in a tree Edges represent a connection between pairs of vertices –e = (v1, v2) we say v1 is adjacent to v2 e is an ordered pair

4. ICOM 6115©Manuel Rodriguez-Martinez Example Directed Graphs V = {Jil, Mel, Ron, Apu} E = {(Jil, Mel), (Mel, Apu), (Jil,Ron), (Ron, Jil), (Mel, Ron), (Ron, Apu)} Apu Ron Jil Mel G=(V,E)

5. ICOM 6115©Manuel Rodriguez-Martinez Path and Cycles A path is a sequence of vertices v1, v2, v3, …, vn, such that vi is adjacent to vi + 1 –(v1,v2), (v2, v3), …(v(n-1), vn) Simple path: v1, v2, …, vn –Where v1 and vn are distinct Cycle: v1, v2, …, vn –Where v1 and vn are the same Path Length –number of edges on the path

6. ICOM 6115©Manuel Rodriguez-Martinez Example Directed Graph Example Path –P1: Jil, Ron, Apu; P2: Mel, Ron, Jil, Ron –P1: length = 2; P2: length = 3 Cycle –C1: Jil, Ron, Jil; C2: Mel, Ron, Jil, Mel Apu Ron Jil Mel G=(V,E)

7. ICOM 6115©Manuel Rodriguez-Martinez Undirected Graph Undirected Graph G consists of –a set of vertices V –a set of edges E –Formally, G = (V,E) Vertices are node used to represent some object –Like in tree Edges represent a connection between pairs of vertices –e = (v1, v2) we say v1 is adjacent to v2 and vice-versa e is not ordered pair, e = (v1,v2) = (v2,v1)

8. ICOM 6115©Manuel Rodriguez-Martinez Example Undirected Graphs V = {Jil, Mel, Ron, Apu} E = {(Jil, Mel), (Mel, Apu), (Jil,Ron), (Ron, Jil), (Mel, Ron), (Ron, Apu)} Apu Ron Jil Mel G=(V,E)

9. ICOM 6115©Manuel Rodriguez-Martinez Labels and Weights Labels –Information associated with vertices or edges Weight –Often used to denote the label of edges –Most often, weight are numbers Cost function –Establish a mapping between the labels of edges and/or nodes and a numeric value

10. ICOM 6115©Manuel Rodriguez-Martinez Example of Cost: Distance between cities DC NY SJU LA G=(V,E) 10 20 110 100 90 Shortest Path Between: SJ U and DC: 30 NY and LA: 100 LA and SJU: 120

11. ICOM 6115©Manuel Rodriguez-Martinez Minimum-Cost Spanning Tree Spanning Tree –Tree that connects all nodes in the graph Minimum-Cost Spanning Tree (MST) –Graph might have several spanning tree –MST is the one that has minimal cost amongst all spanning trees DC NY SJU LA G=(V,E) 10 20 110 100 90

12. ICOM 6115©Manuel Rodriguez-Martinez Internetworking What is this? –Process of connecting independent LANs into a larger network Again: Independent LANs –Each LAN might have a different Layer 2 Ethernet Token Ring 802.11b What Layer takes care of this service? –Network Layer (Layer 3)

13. ICOM 6115©Manuel Rodriguez-Martinez Example internetwork Network 3 (FDDI) H1H2R1 Network 1 (Ethernet) H3H4R2 Network 2 (802.11b) H5H6H7H8R3R4 Network 4 (Ethernet) Network 5 (Point-to-Point)

14. ICOM 6115©Manuel Rodriguez-Martinez Layer 3: Store-and-Forward Service