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]

Graph Theory and Topology Design Graph Theory and Topology Design

David Tipper

Associate Professor

Associate Professor

Department of Information Science and Telecommunications

University of Pittsburgh

tipper@tele.pitt.edu

tipper@tele.pitt.edu

Slides 4

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

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, )∈

(1

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

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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

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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.