CS622 Page 1

1
Graph Theory

• A loop is an edge where both endpoints are the
  same
• Two edges are parallel if they have the same end
  points.
• A graph is simple if there is no loops or
  parallel edges.
• The degree of a node is the number of edges in
  the graph that have the node as an endpoint.The
  degrees of nodes in a graph are important graph
  property .
• A component of a graph is a maximal connected
  subgraph.
• A Tree is a connected, simple graph without
  cycles.
• Any tree with n nodes has n-1 edges.
• Trees are optimal network designs when links have
  very high capacity or enormously expensive, and
  there is no reliability constraints.

2
Minimum Cost Network

• Problem Given the distances among 4 cities,
  build the minimum road to connect them. Assume
  1000/km to build the road.
• Solution Choose the shorter distance links first
  and avoid those links that connect nodes already
  included. This is Kruskals MST Algorithm.

3
Minimum Cost Network

• Select Charmes-Duval, Bregen-Charmes,
• Avoid Bregen-Duval since both them already
  included.
• Choose Anagon-Charmes.
• The results is a star topology.

4
Topology

• A tree is a star if only one node has degree
  greater than 1.
• A tree is a chain if no node has degree greater
  than 2.
• A graph G is weighted if there is a real number
  associated with each edge. It is denoted as
  (G,w).
• The weight of an edge e is denoted as w(e).
• The weighting function w E?R.
• If G is connected, we like to select a connected
  subgraph with minimum weight.

5
Kruskals MST Algorithm

• MST ? Minimum Spanning Tree.
• Check if graph is connected. If not, abort.
• Sort the edges in ascending order by weight.
• Mark each node as separate component.
• Loop on the edges until we accepted V-1
  edges.Let e be the candidate edge.
• If the endpoints are in different components,
  merge the two components and accept the edge and
  increment the of edges accepted by 1.

6
Prims MST Algorithm

• Start with a node v, no edge, T0(v, ?).
• Add least expensive edge.
• 1 Tree Prim(Graph G, Node root)
• 2
• 3 for_each(node, G?nodes)
• 4 node? label INF
• 5 node?intree FALSE
• 6
• 7 root?label 0
• 8 root?pred root
• 9 NodesChosen 0
• 10

7

• 11 while (Nodeschosen lt NumberNodes(G))
• 12 nodeFindMinLabel(G) / Only nodes
  not in the tree are checked. /
• 13 node?intree TRUE
• 14 NodeChosen
• 15 for_each(node2, node?adjacent_nodes)
• 16 if (node2?intree FALSE
• 17 node2?label gt
  weight(edge(node, node2))
• 18 node2-gtprednode
• 19 node2?labelweight(edge(node,
  node2))
• 20 / endif /
• 21 / endfor /
• 22 / endwhile /
• 23
• 24 treecreate_Tree()
• 25 BuildTreeFromPreds(G, tree)
• 26 return(tree)
• 27 / end Prim /

8
Use Delite to Calculate MSTs

• Start Delite. Select Start Programs Delite.
• Select File Read Input File Prim.INP.
• The nodes are displayed.
• Select Design Prim for displaying the result of
  the Prim MST Algorithm.
• The legend window shows the link_cost of the MST.
• Select Display Select Map File USAVH.met for
  overlay Map file.
• The result display is shown next.
• Select Design Set Input Parameters Trace
  Yes to generate .trc trace file.
• Use Notepad/wordpad to view the Prim.trc or
  Prim.INP file.

9
(No Transcript)
10
Tree Designs

• MST is good when
• Links are highly reliable
• Networks can tolerate low reliability
• The number of sites are small.
• Either Prims or Kruskals algorithm gives
  optimal solution.
• MSTs are not good networks to use when the
  number of nodes is large.

11
Squreworld Counter-example

• 1000 miles x 1000 miles.
• One type of transmission lines? 1Mbps.
• Cost for two sites with locations (x1, y1) and
  (x2, y2) with distance d (100010d)/month.
• Consider problems with 5, 10, 20, 50, and 100
  sites, traffic are normalized with 1kbps from one
  site to another. The traffic volumes grows
  quadratically

12
MST for 5 Node Network
13
Generate Files for Squreworld Counter-example

• Compile the c\Program Files\Delite\Source\gen.c
• Run gen ltnumber of nodesgt ltfilename.gengtto
  produce the 5, 20, 50, 100 nodes files.
• Run delite, select File Generate Input
• Select the corresponding .gen file.
• The monitor will show the files are successfully
  generated. They include .REQ (traffic
  requirement), .CST (cost file.).
• Edit the .REQ file, replace the link bandwidth
  with 1000.

14
MST for 10 Node Network
15
Legginess? in Network

• Traffic in the above 10 node MST takes a
  circuitous route between source and destination.
• To quantify the legginess in the network, we
  define
• Definition 3.17 The number of hops (hop count)
  between node n1 and n2 is the number of edges in
  the path chosen by the routing algorithm for the
  traffic flowing from n1 to n2. If only one path
  is chosen or if all paths chosen have the same
  number of edges, then we denote the number by
  hops(n1, n2).

16
Average of Hops

• Although the two routes between a pair of nodes
  can be asymmetric, there is only one path between
  a pair of nodes in a tree.
• Definition 3.18 The average number of hops in a
  network,
• The average number of hops is quite important in
  evaluating MST designs.
• The sum of the traffic on all links Total
  traffic x hops

17
MST for 20 Node Network
18
MST for 50 Node Network
19
MST for 100 Node Network
20
Traffic Volume and Costs

• The traffic of 100 node MST grows 5 orders of
  magnitude from 20kbps of 5 node MST.
• 1.5 order of magnitude comes only from average
  hop count.
• Problem MSTs tend to have very long and
  circuitous paths.

21
Shortest Path Trees

• Definition 3.19 Given a weighted graph (G, w)
  and nodes n1 and n2, the shortest path from n1 to
  n2 is a path such that ?e?P w(e) is a minimum.
• Segments of a shortest path are the shortest
  paths of their end points. (recall the optimality
  principle)
• Definition 3.20 Given a weighted graph (G, w)
  and a node n1, a shortest path tree (SPT) rooted
  at n1 is a tree s.t. for any other node n2 ? G,
  the path from n1 to n2 in the tree T is a
  shortest path between the nodes.

22
SPT for 20 Node Network
Generate using Prim-Dijkstar with centerN14
alpha1.0
MST SPT Avg(HOPS)
5.0316 1.9000 Max_Util 0.010 0.002
23
SPT vs. MST

• Queueing delay and Utilization
• Let
• Then
• Average packet delay
• 20 node networks
• Average packet delay for MSTCost48,686.
• Average packet delay for SPT Cost88,612.
  higher cost, lower delay, N14 failure!

24
Prim-Dijkstra Tree

• Try to find trees falls between MST and SPT.
• Both Prim and Dijkstra algorithm start with
  initial label, looping over nodes to find one
  with smallest label, bringing it into the tree,
  finally relabeling all the neighbors.
• Prim-Dijkstra Tree uses the following
  label.Where 0 ???1 is used to parameterize
  the algorithm.
• ?0, we build MST. ?1, we build an SPT from
  root.

25
Choosing Prim-Dijkstra Trees
?0.3 and ?0.4 give attractive trees.
26
Dominace among Designs

• Impose a partial ordering when picking the
  designs.
• Definition 3.21 Given a set S and an operator ?
  that maps S x S ? TRUE, FALSE, then we call S
  a partial ordered set, or poset, if
• For any s ? S, s ? s is FALSE.
• For any s1,s2 ? S, s1 ? s2, if s1 ? s2 is True,
  then s2 ? s1 is FALSE.
• If s1 ? s2 and s2 ? s3 are TRUE, then s1 ? s3 is
  TRUE.
• Note that there may exist 2 elements s1,s2 ? S,
  s.t. neither s1 ? s2 nor s2 ? s1 is TRUE. They
  are called incomparable.

27
Apply POSET to Pick Designs

• Definition 3.22 Suppose D1 has cost C1 and
  performance P1. Suppose D2 has cost C2 and
  performance P2. We will says D1 dominates D2, or
  D1 ? D2, if C1 lt C2 and P1 gt P2.
• Use the above definition, we compute the
  domination partial order in Table 3.12.

28
Domination Partial Ordering

• N0, N1, N2 are dominated by others. Remove them
  from consideration list.

29
Consider Marginal Cost of Delay

• There are still 6 designs to consider.
• The ratio (C1-C2)/(P2-P1) gives the cost for
  delay.
• 5099 buy 17 ms delay (from N3 to N4)the cost
  for delay5099/0.017305329.
• Between N5 and N6, the cost for delay rises 5
  times.

30
Tours

• Sometime trees are just too unreliable.
• There are designs that are far more reliable, yet
  have only one additional link ? Tours.
• It is a possible solution of the traveling
  salesman problem (TSP).
• Definition 3.24 Given a set of vertices v1, v2,
  , vn, a tour T is a set of n edges E s.t. each
  vertex v has degree 2 and the graph is connected.
• The tour can be represented as a permutation
  (vt1, vt2,, vtn). There are n! such
  permutations.
• Since cyclically permuting them and the reverse
  permutation give the same tour , we have (n-1)!/2
  tours.

31
TSP Problem

• Definition 3.25 Given a set of vertices (v1, v2,
  , vn) and a distance function d V x V?R, the
  traveling salesman problem is to find the tour T
  such that is a minimum. In this we identify
  vt n1 with vt1.

32
Network Reliability

• Definition 3.26 The reliability of a network is
  the probability that the functioning nodes are
  connected by working links.
• Assume the probability of each node working is 1
  and the probability of a link falling is p.
  Typically p is very small. Let q 1 p.
• For a 5-node tree, Ptree(failure)1-(1-p)4?4p
• For a 5-node tour, Ptour(failure)1-((1-p)55p(1-p
  )4)1-(q55pq4) 10p2q310p3q25p4qp5
• substitute with terms from 1(pq)5

33
Ptree(failure) vs. Ptour(failure)

• When p10-6, the reliability difference is 5
  orders of magnitude difference.

34
Nearest Neighbor Algorithm

• Start at a root node. Set current_node root
  node.
• Loop until all nodes in the tour
• Go through the node list, find the node closest
  to the current_node that is not in the tour. Let
  this node be best_node.
• Create an edge between current_node and
  best_node.
• Set current_node to be best_node.
• Finally create an edge between the last node and
  root node.

35
What is wrong with this tour?
36
Uncrossing the Tour

• First time 2nd time

37
Creditable Algorithms

• Definition 3.27 A heuristic optimization
  algorithm produces a creditable result if the
  result is a local optimum for the problem.
  Otherwise, it produces an uncreditable result.
• We are asking absence of stupidity, not
  performance.
• Definition 3.28 A suite of network design
  problems is a set of triples (Locationsi,
  Traffici, Costsi) for i1,,S
• Definition 3.29 A creditability test is a
  program test (net, traffic, cost) that takes a
  network problem as input and return OK or FAIL
  depending on whether or not test() can manipulate
  net into another valid network of lower costs.
• Definition 3.30 Given a suite of network design
  problems S, a design algorithm A, and a
  creditability test t(),then

38
Creditability of Tours built by Nearest-Neighbor
Algorithm

• Test program in appendix B, generate a large set
  of locations.
• Start at location 0, build a tour using N-N
  algorithm.
• Call test_tour() to apply line-crossing test
  cross() O(n2) times.
• Collect the statistics.

39
A More Creditable N-N Heuristic

• Improvement 1 The closest node to the list of
  node in the tour, instead of to the last node in
  the list.
• Improvement 2 The closest node can be added to
  any place in the tour, not just append to the
  end. The insert location minimize the following
  formuladist(Ni, best_node)dist(best_node,Nj)
  dist(Ni,Nj)
• Intuitively, you try to find a place where Ni,
  best_node, Nj is as flat as possible (follow
  straight line).
• O(2n2) complexity instead of O(n2) for basic N-N
  algorithm.

40
Result of MC-NN Algorithm

• Note that it performs much better.
• Instead selecting nearest neighbor, we can select
  the furthest neighbors (they are found isolated
  at the end of N-N algorithms and have poor choice
  to join the tour.

41
TSP Tour Do not Scale

• Theorem 3.3 Given uniform traffic any TSP tour
  of n nodes has avg(hops)(n1)/4 if n is odd, and
  n2/(4(n-1) if n is even.
• At 100 node tour, for the avg hop count, TSP tour
  is twice as bad as MST.

42
2-connectivity Graph

• Tour is a-connected graph. They survive the lost
  of a node.
• Definition 3.31 Given a connected graph G(V,E),
  the vertex v is an articulation point if removing
  the vertex and all the attached edges disconnects
  the graph.
• In tree, any node has degree gt 1 is a
  articulation pt.
• Definition 3.32 If a connected graph G(V,E) has
  no articulation points, then the graph is
  2-connected.

43
Unobvious Articulation Point
Easy to detect
Not easy to detect
44
2-connectivity Theorem

• Theorem 3.4 Suppose G1(V1,E1) and G2(V2,E2)
  are 2-connected graphs with V1 ?V2 ?. Let v1,
  v2 ? V1 and v3, v4 ? V2. Then the graph G with
  vertices V1 ? V2 and Edges E1 ? E2 ? (v1, v3) ?
  (v2, v4) is 2-connected.
• Divide and Conquer Divide the nodes into
  clusters, find the tour for each cluster, treat a
  cluster a node and connect clusters with a ring
  of edges that do not share the same vertex.
• Note that for n clusters, the connected tours
  will have n more edges (more reliable) and
  shorter average hop count.

45
Original 20-node Tour
46
2 Cluster Tour
47
Join the Tours
48
Theorem 3.5

• Suppose that G(V,E) is a 2-connected graph with
  V gt 2. Suppose that each node vi ? V is
  replaced by a 2-connected graph Gi. Suppose each
  edge e(u,w) ? E is replaced by an edge e from
  u ? Gu to v ? Gw. Then if no 2 of these
  replacement edges have a common vertex, the graph
  H(?iVi, ?iEi?E) is a 2-connected graph.

49
4-cluster Design for 50 nodes
Single tour HOP12.755 4-cluster HOP6.8971
50
Exercise 6

• Problem 1. Network Restoration.
• For link A-G cut, show the restoration paths
  (route as sequence of nodes, and itsrestored
  bandwidth.
• Discuss the selection trade-offfor the two
  algorithms at Page5 handout.
• Problem 2. Tree and Tour.
• Problem 3.7.
• Explain why for the 20-node random problem
  avg(hops)SPT1.9000.
• Prove for a start on n nodes,S, that
  avg(hops)S2-2/n.
• Download the g20sw. from cs622/public_html/delit
  e.Run delite with 1. Tour (Nearest Neighbor), 2.
  Tour (Furthest Neighbor), 3. Prim, 4. SPT.
  Compare the cost, avg(hops), and max. utilization
  of these four designs.

51
Solution to EX6

• Problem 1. Network Restoration
• For link A-G cut, show the restoration paths
  (route as sequence of nodes, and its restored
  bandwidth.
• Ans 8 working channels lost.3 restoration paths
  through AF.3 restoration paths through ACEF1
  restoration path through ABDEF
• b. Discuss the selection trade-offfor the two
  algorithms at Page5 handout.
• Ans Depends on the restoration deadline and
  whether restoration paths are needed earlier for
  the priority traffic.If the restoration level is
  the utmost important metric, then the final
  restorationlevel at the restoration deadline
  decides theselection of the algorithm. The
  curve however does not show the spare channel
  usage.

52
Solution to HW5 (2)

• Problem 2. Tree and Tour.
• Problem 3.7.
• Explain why for the 20-node random problem
  avg(hops)SPT1.9000.
• Prove for a star on n nodes,S, that
  avg(hops)S2-2/n.
• Ans 1 There are c(20,2)2019/2190
  connections.Among them, 19 are to the
  star.avg(hops)spt((190-19)2191)/1901.9
• Ans 2 There are c(n,2)n(n-1)/2
  connections.Among them, n-1 are to the
  star.Avg(hops)S(n(n-1)/2-(n-1))2(n-1))/(n(n-
  1)/2)(n(n-1)-(n-1))/(n(n-1)/2)2(n-1)/n2-2/n
  .
• Download the g20sw. from cs622/public_html/delit
  e.Run delite with 1. Tour (Nearest Neighbor), 2.
  Tour (Furthest Neighbor), 3. Prim, 4. SPT.
  Compare the cost, avg(hops), and max. utilization
  of these four designs.

53
Delite Tool Analysis (Tour NN)
54
Tour (Furthest Neighbor)
55
PRIM
56
SPT
57
Network Design Comparison