Graph Theory and Topology Design Department of Information Science and Telecommunications

Example: simple Graph (i.e., no loops or parallel edges) Degree of Vertex A = 3, Degree of Vertex E = 2. •Adjacent vertices:[r]

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Graph Theory and Topology Design Graph Theory and Topology Design

David Tipper

Associate Professor

Department of Information Science and Telecommunications

University of Pittsburgh

tipper@tele.pitt.edu

Slides 4

http://www.sis.pitt.edu/~dtipper/2110.html

• Top down network design project approach should follow three phases:

– Conceptual Model

• Objectives, Requirements, Constraints

– Logical Model

• Technology, network graph, node location, link size, etc (where algorithms are used to minimize cost)

– Physical Model

• Specific hardware/software implementations • (e.g., wiring diagram, repeater locations, etc.)

• Focus on Algorithms for Logical Model Design

– Graph Theory – Optimization

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Telcom 2110 Spring 2006 3

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Graphs

V ={A,B,C,D,E,F,G}

E = {(A,B),(A,C), (A,D), (B,C), … , (F,G)}

•Telecommunication and computer networks are naturally represented by graphs

•A graph G = (V, E) is a mathematical structure consisting of two sets V and E

•Elements of V are called vertices (or nodes)

–For example, switches, routers, cross conects

•Elements of E are called edges

–Communication links are edges (wired or wireless) –Each edge has two endpoints

Edge

Vertex V

v v, )∈

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Terminology

• Loop

– an edge where both endpoints are the same vertex Also called a self-loop

• Parallel edges

– a collection of two or more edges having identical end Also called a multi-edge

• A graph is simple if it has no loops or parallel edges. • Focus on simple graphs.

– When considering reliability, we will introduce parallel edges if the network has parallel links

• The degree of a node: the number of edges in the graph that have the node as an endpoint

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Terminology Cont. Example: simple Graph (i.e., no loops or parallel edges) Degree of Vertex A = 3, Degree of Vertex E =

•Adjacent vertices:

Two nodes are adjacent if there is an edge that has them as endpoints Example: A and B are adjacent, A and E are not

Size of graph characterized by Number of edges |E| and number Of vertices |V|

Example |V| = 7, |E| = 10

Paths and Cycles • Path from vertex A to vertex Z:

an alternating sequence of vertices and edges, representing a continuous traversal from vertex A to vertex Z.

Can be represented by sequence of edges or nodes in path

• Trail: a path with no repeated edges.

• Cycle: a path starting and ending on the same node

• Connected graph:

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Terminology Cont.

Example: Path from A to G is given by (A,D),(D,E),(E,G) Cycle at A is given by (A,C), (C,B), (B,A) Example is a connected Graph

Trees

• Tree: a connected, simple graph without cycles.

• Any tree with n nodes has n-1 edges.

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Trees Terminoloy

• Root: One vertex of a tree may be designated as a root (has no parent only childern) • Each vertex (besides root)

has a single parent vertex which is the vertex closest to the root

• Each vertex has zero or more child vertices which are the adjacent vertices farthest from the root

• Leaf: a vertex without a child

IBM MSC BS7 BS5 BS2 BS3 BS4 BS1 BS6 SD Centillion 1400

Bay Ne tworks ETHERRS 232C

PC CARD P* 8x50O OO130A O N6 INSACTALM RSTLINK PWRALMFAN FAN1 PWR0 PWR1ALM

BSC BS7 BS5 BS2 BS3 BS4 BS1 BS6 SD Centillion 1400

Bay Ne tworks ETHERRS 232C

PC CARD P* 8x50O OO130A O N6RSTLINKINSACTALM PWRALMFAN FAN1 PWR0 PWR1ALM

BSC

SD

Centillion 1400

Bay Networks ETHERRS 232C

PC C ARD P* 8x50OOO130A O N6RSTLI NKI NSACT ALM PWRALMFAN0 FAN1 PWR0 PWR 1ALM

BSC

VLR HLR

AUC EIR

Typical Cellular Network

Star

• A tree is a star if only node has degree >1

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Divide and Conquer

• Grouping into groups of 10 nodes Then running the nearest neighbor algorithm gives two rings as below Note that the average hop count is reduced

N20

N13 N6 N2

N7 N15 N9

N14

N10 N1 N5 N9 N12

N16 N18

N17 N4 N8 N11

N3

Divide and Conquer

• Grouping into groups of 10 nodes Then running the nearest neighbor algorithm gives two rings as below Joining the two rings at their closet points results in

N20

N13 N6 N2

N7 N15 N9

N14

N10 N1 N5 N9 N12

N16 N18

N17 N4 N8 N11

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Telcom 2110 Spring 2006 73 Level N American Network

Summary

• If the traffic is small when compared to link size, then the optimal networks are MSTs and TSP tours, depending on the reliability desired. • Both MSTs and TSP tours not scale.

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