Tree Design for Networks Cahn ch'3

1
Tree Design for Networks (Cahn ch.3)

• Tree reviewed
• 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
Network in Matrix Format

• Early example 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 (MST)

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

4

• 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.
• Using Delite http//eie507.eie.polyu.edu.hk/handou
  ts/delitehttp//eie507.eie.polyu.edu.hk/handouts/
  delite/examples/instructions.htm

5
Squreworld Counter-example

• 1000 miles x 1000 miles.
• One type of transmission lines? 1Mbps.
• dsqrt( dv2dh2/10 ), dvV1-V2,dhH1-H2
• Cost for two sites with locations (V1, H1) and
  (V2, H2) with distance d (10006.1728d)/month
  .
• Consider problems with 5, 10, 20, 50, and 100
  sites, traffics are normalized with 1kbps from
  one site to another. The traffic volumes grow
  quadratically

6
Generate Files for Squareworld Counter-example

• Compile \Delite\Source\gen.c add macro define
  min(X, Y) (((X)lt(Y))?(X)(Y) )
• Run gen ltnumber of nodesgt ltfilename.gengtto
  produce the 5, 20, 50, 100 nodes files, e.g., gen
  5 5out.gen.
• Edit NG_TOT_REQ of .gen file according to total
  traffic (slide 5).
• 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.). bandwidth in
  .REQ file should be 1000.
• Invoke

7
MST for 5 Node Network
8
MST for 10 Node Network
9
Leggy-ness in Network

• Traffic in the above 10 node MST takes a
  circuitous route between source and destination.
• To quantify the leggy-ness 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).

10
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 delay related to of
  hops.
• The sum of the traffic on all links Total
  traffic x hops

11
MST for 20 Node Network
12
MST for 50 Node Network
Delite demo 5out1.inp
13
MST for 100 Node Network
14
Traffic Volume and Costs

• The traffic of 100 node MST grows about 500
  times 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.

15
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.
• 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 n1 and n2.

16
SPT for 20 Node Network
Generated using Prim-Dijkstra with centerN14
alpha1.0
MST SPT Avg(HOPS)
4.4158 1.9000 Max_Util 0.099 0.019
17
SPT vs. MST

• Queuing plus transmission 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!

18
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 re-labeling 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.

19
Choosing Prim-Dijkstra Trees (100 nodes obtained
by Interpid network design tool)
?0.3 and ?0.4 give attractive trees.
20
TSP Tour provides more reliability

• TSP tours 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.