Example: simple Graph (i.e., no loops or parallel edges) Degree of Vertex A = 3, Degree of Vertex E = 2. •Adjacent vertices:[r]
Trang 1Graph 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
Top Down Network Design Approach
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A
B
C
D
E
F
G
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
(1 2
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
– Number of outgoing links of a node
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B
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D
E
F
G
Terminology Cont.
Example: simple Graph (i.e., no loops or parallel edges)
Degree of Vertex A = 3, Degree of Vertex E = 2
•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:
a graph in which every pair of distinct vertices
has a path between them
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A
B
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D
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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.
A
B
C
D
E
F
G
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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
Centillion 1400
Bay Ne tworks
ETHER RS 232C
FAN 0 FAN1 PWR0 PWR1ALM
BSC
BS2
BS3
BS4
BS1
BS6
Centillion 1400
Bay Ne tworks
ETHER RS 232C
FAN 0 FAN1 PWR0 PWR1
BSC
Centillion 1400
Bay Networks
ETHER RS 232C
P* 8x50OOO130A O N RST ACT ALM
FAN0 FAN1 PWR0 PWR 1
BSC
VLR HLR
Typical Cellular Network
Star
• A tree is a star if only 1 node has degree >1
X Y
Z
P
Q
A
B
D C
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Divide and Conquer
• Grouping into 2 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
N16 N18
N3
Divide and Conquer
• Grouping into 2 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
N16 N18
N3
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Level 3 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 do not scale.
• The growth in the average # of hops is at the
heart of the problem It’s better to build a
Prim-Dijkstra tree or a multi-ring “ring of rings” to
control the length of the routes.