tcom 540/11 tcom 540 session 2. tcom 540/12 web page om540541.htm

TCOM 540/1 1

TCOM 540

Session 2

TCOM 540/1 2

Web Page

• http://teal.gmu.edu/ececourses/tcom540/TCOM540541.htm

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Set Theory

• A set is a collection of (mathematical) objects– E.g., {A, B, C}– E.g., {1,2,3, …, 99}

• sS means “s is a member of S”• s ~ S means “s is not a member of S”• T is a subset of S if every member of T is also a

member of S– It’s a proper subset if there is at least one member of S

that is not a member of T

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Set Theory (2)

• The set of all subsets of S is denoted as 2s

• If S has n members, then 2s has 2n members

• Union of two sets is the set of all their members

• Intersection of two sets is the set of common members

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Set Theory (3)

Intersection

Union

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Set Theory (4)

• Cartesian Product of two sets, S and T, is the set SxT – Elements are (s,t) where s S and t T

• The graph of a function f:S T is the subset of SxT that consists of

{(s,t) f(s) = t)}

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Graphs

• A graph consists of a set of vertices (or nodes) V and a set of edges E

A

D

I

ZC

B

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Some Definitions for Graphs

• Each edge connects two vertices (may be same – then it’s called a loop)

• Two edges are called parallel if they connect the same vertices

• A graph is simple if it has no loops or parallel edges

• The degree of a vertex is the number of edges it has

• Two nodes are adjacent if there is an edge that has them as endpoints

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Some Definitions for Graphs (2)

• A path between vertices v1 and vn is a set of edges (e1, e2, …, en) such that ei and ei+1 have a common endpoint, and v1 is an endpoint of e1 and vn is an endpoint of en

• A cycle is a path from a vertex to itself• A graph is connected if for any two nodes there is

a path between them

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A Small (But Not Simple) Graph

A

D

I

ZC

B

Parallel edges

LoopNode degree 3

Adjacent nodes

(DZ), (ZB), (BI), (ID) is a cycle.

This graph is connected

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Definitions (3)• A subgraph G* of a graph G with vertices V and edges E

is a pair (V*, E*) where – V* is a subset of V – E* is a subset of E– If an edge belongs to E* then both its endpoints must belong to

V*

• A component of a graph is a maximal connected subgraph• Two graphs G1, G2 are isomorphic if there is a 1-to-1

mapping f:V1 V2 such that (v1, v2) is a member of E1 iff (f(v1), f(v2)) is a member of E2

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Another Graph

A

D

I

ZC

N

K

M

J

e1

e2

e3

e4

e5

e6

e7

e8

( (D, Z, B, M, J), (e2, e8, e4)) is a subgraph

It is not a component.

The two right-hand components are isomorphic

X

Q

R H

e11

e99

Pe15

e16

e12B

e13

e9

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Definitions (4)

• A tree is a connected simple graph without cycles

• A star is a tree in which exactly one node has degree >1

• A chain is a tree in which no node has degree greater than 2

• Define N(G) = number of nodes in G

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Tree, Star, Chain

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V&H Coordinates• V&H coordinate grid covers U.S.

• Approx 10,000 x 10,000

• Distance between points (v1, h1) and (v2, h2) for tariff calculations is sometimes defined as:

• A simpler formulation is:

• Note V&H assumes earth is flat …

Dist = 1+int{[(dv2+9)/10+(dh2 +9 )/10]0.5}

Where dv = v1-v2 and dh = h1-h2

Dist = 1 + int[(dv2+dh2)/10]

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Latitude and Longitude

• V&H used primarily in North America

• Most of world uses latitude and longitude (L&L)

• Distances between points in L&L coordinates are computed using spherical geometry

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Node Locations

0

2000

4000

6000

8000

10000

0 2000 4000 6000 8000 10000

TCOM 540/1

A Real Network (Backbone Not Shown)

Vendor A Cost: $1.159MVendor B Cost: $1.213M164 Hosts

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Real Network Detail (Atlanta, GA)

Stand Alone

Hosts

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Weighted Graph• A weighted graph is a graph G where each edge e

has a weight w(e)– Denoted by (G, w)– Generally w(e) > 0– Weight of a subgraph G* is sum of weights of edges in

G*

• Real networks are weighted graphs– Weight may be cost, delay, or other parameter

• Minimum spanning tree (MST) is a connected subgraph with minimum weight

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Kruskal’s Algorithm for MST

Is G connected?yes no stop

Sort edgesin ascending

order ofweight

Mark each node as Separate

component

Loop on edgesLet e be candidate edgeIf ends of e are in different components, accept e

Stop when number of edges = N(G) - 1

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Kruskal’s Algorithm for MST (2)

27

30

25

26

2129

31

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27

30

25

26

21

1st add

21

29

31

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27

30

25

26

21

1st add

21

2nd add

2521

29

31

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27

30

25

26

21

1st add

21

2nd add 3rd add

2521

27

2521

29

31

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27

30

25

26

21

1st add

21

2nd add 3rd add

2521

27

2521

29

4th add

27

2521

29

31

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Prim’s Algorithm for MSTStart with all nodesunconnected and

Label = infinity

Select rootnode

Scan neighbors, update Labels =min edge to tree

Add closest neighbor

(smallest Label)

Stop whenN(G) –1added

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Prim’s Algorithm for MST (2)

27

30

25

26

2129

31

Choose as root

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27

30

25

26

2129

31

Choose as root

27

1st Add

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27

30

25

26

2129

31

Choose as root

27

1st Add

27

21

2nd Add

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27

30

25

26

2129

31

Choose as root

27

1st Add

27

21

2nd Add

27

2521

3rd Add

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27

30

25

26

2129

31

Choose as root

27

1st Add

27

21

2nd Add

27

2521

3rd Add

27

2521

29

4th Add

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Limitations of MSTs

• No redundancy– One link failure separates the network into two

disconnected components– Big problem for large networks

• May involve very long paths in large networks

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MSTs Do Not Scale

• Number of hops between nodes n1 and n2 is the number of edges in the path chosen by the routing algorithm

• Average number of hops is traffic-weighted

= (n1,n2traffic(n1,n2)*hops(n1,n2))/n1,n2traffic(n1,n2)

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MSTs Do Not Scale (2)

0

2

4

6

8

10

12

14

16

0 20 40 60 80 100 120

Number of Nodes

Avera

ge H

op

s

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MSTs Do Not Scale (3)

0

20000

40000

60000

80000

100000

120000

140000

160000

0 50 100 150

Number of Nodes

Tra

ffic Total Traffic

Summed Traffic

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Definitions

• For a weighted graph (G,w), and nodes n1 and n2, the shortest path P from n1 to n2 minimizes ePw(e)

• The shortest-path tree (SPT) rooted at node n1 is a tree T such that for any other node n2 the path from n1 to n2 is a shortest path

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Dijkstra’s Algorithm for Shortest-Path Trees

1. Mark each node unscanned, assign label infinity2. Set label of root to 0, and predecessor to self3. Loop through nodes

• Find node n with smallest label• Mark as scanned• Examine all adjacent nodes m, see if distance through n < label

• If so, update label, update predecessor(m) = n

Note that a link may drop out of the tree if a shorter route is found

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Dijkstra ExampleChoose as root 1

2

5

3

27

30

25

1

214

29

31

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Dijkstra ExampleChoose as root

1 & 2. Nodes adjacent to root

1

2

527

30

Label = 27

Label = 30

1

2

5

3

27

30

25

1

214

29

31

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Dijkstra ExampleChoose as root

1

2

527

30

Label = 27

Label = 30

3. Nodes adjacent to 2

1

2

527Label = 28

1

2

5

3

27

30

25

1

214

29

31

1


321Label = 48

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Dijkstra Example

1

2

5

3

27 1

4

31

Choose as root

4&5. Nodes adjacent to 5

1

2

5

3

27

30

25

26

214

29

31

Label = 59

1

2

527

30

Label = 27

Label = 30

3&4. Nodes adjacent to 2

1

2

527Label = 28

1


321Label = 48

25

Label = 53

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Characteristics of SPTs

• In a complete graph, SPT is a star*– High performance and reliability– But likely implies low link utilization, high

expense

* Unless triangle inequality does not hold

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Prim-Dijkstra Trees

• We play with the definition of the Label

• Prim’s Label

= minneighborsdist(node, neighbor)

• Dijkstra’s Label

= minneighbors[dist(root, neighbor) + dist(neighbor, node)]

• Prim-Dijkstra Label =

= minneighbors[*dist(root, neighbor) + dist(neighbor, node)]

• Now is a parameter that we choose, between 0 and 1

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Varying Alpha

0.000

0.200

0.400

0.600

0.800

1.000

1.200

0 0.5 1

Alpha

No

rmali

zed

Valu

e

Normalized Avg Hops

Normalized delay

Normalized Cost

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Tours

• A tree design may be unreliable

• A tour adds one link to significantly increase reliability

• A tour of a set of vertices (v1, v2, …, vn) is a set of n edges such that each vertex has degree 2 and the graph is connected

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Tours (2)

• Leads to the (in)famous Traveling Salesman Problem (TSP)– Given a set of vertices (v1, v2, …, vn) and a

distance function d(vi,vj) between vertices, find the tour T(vti) such that d(vti,ti+1) is minimized

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Reliability of a Tree

• Reliability = probability that functioning nodes are connected by working links

• For a tree, reliability = (1-p)n-1, where– p = probability of a link failing– n = number of nodes

• P(failure) = 1- reliability = 1 - (1-p)n-1

(n-1)*p

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Reliability of a Ring

• A ring can tolerate one failure

• For a ring,

P(failure) = 1- (1-p)n – n*p*(1-p)n-1

0.5*n*(n-1)*p2 if p is small

X

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A Simple Algorithm for Building a Tour

• Denote a root node, set current node = root

• Loop through nodes– Find closest node (not in tour) to current node– Add an edge to it– Reset current node to be this node just added

• Create an edge between last node and root

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Creditable Solutions and Creditability Tests

• A solution is creditable if it is a local optimum– I.e., it is not creditable if, by some method, we

can manipulate the solution to a better one

• Cahn uses a crossing test to determine creditability of the simple tour-building algorithm– It does not do well ….

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A Better Tour-Building Algorithm

• Look for closest neighbor to any node in the partial tour (not just the last one added)

• Insert between two adjacent nodes in tour in “best” place– Minimum increase in partial tour length

• Also “farthest neighbor” heuristic – Avoids stranding distant nodes

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A Difficulty

• TSP tours do not scale– Similar to trees in this respect– Average number of hops increases O(n)

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2-Connectivity

• A vertex v of a connected graph G = (V, E) is an articulation point if removing the vertex and all attached edges disconnects the graph

• If a connected graph has no articulation points, it is said to be 2-connected

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Connecting 2-Connected Graphs

• Suppose G1 = (V1, E1) and G2 = (V2, E2) are two disjoint 2-connected graphs. Take v1 and v2 from G1 and v3 and v4 from G2 and add the edges (v1,v3) and (v2,v4). The resulting graph is 2-connected

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Connecting 2-Connected Graphs

• Suppose G1 = (V1, E1) and G2 = (V2, E2) are two disjoint 2-connected graphs. Take v1 and v2 from G1 and v3 and v4 from G2 and add the edges (v1,v3) and (v2,v4). The resulting graph is 2-connected

• Roughly, if you connect 2 pairs of vertices from two 2-connected graphs, the resulting graph is 2-connected

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Heuristic Based on Partitioning

• Divide set of nodes into multiple “clusters”

• Use nearest-neighbor algorithm to build TSP tour on each cluster

• Connect clusters, ensuring no connectors have a common vertex

• Resulting graph is 2-connected

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Select clusters

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Select clustersDevelop TSPs

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Select clustersDevelop TSP toursJoin clusters

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Homework

• Read Chapter 4 of Cahn

• Do exercises 2.8, 2.9, 3.1, 3.8 (note: there seem to be typos here – use figure 3.9, and extend table 3.15)

tcom 540/11 tcom 540 session 2. tcom 540/12 web page om540541.htm

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