TCOM 541

1
TCOM 541

• Session 3

2
MENTOR-II

• Last session, we talked about routing and how the
  limitations of routing algorithms introduce
  complications
• They may not be smart enough to find a feasible
  routing for our beautiful network design
• MENTOR-II improves the design algorithm to
  account for the limitations of routing algorithms
• Also increases the complexity of the design
  process

3
Incremental Shortest Path (ISP)

• Goal of ISP algorithm is to identify all pairs
  that could use a link in place of the current
  path
• Initially all paths pass through the tree
• As direct links are added, the situation becomes
  more complex

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ISP (2)

• Maintain two nxn matrices
• Shortest-path distances (sp_dist)nxn
• Matrix of node pointers (sp_pred) nxn
• Maintains all shortest paths through the network
  simultaneously
• Update matrices after each link addition

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sp_pred

• sp_pred(i,j) m contains the next-to-last node
  on the shortest path from i to j
• Can trace back the shortest path from i to j by
  next looking at sp_pred(i,m), etc.

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ISP in MENTOR II

• Each link must be assigned a length greater than
  its cost
• Do not want to give links artificially short
  lengths
• Consider a source node s, and a destination node d

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ISP in MENTOR II (2)

• ISP algorithm builds s_list and d_list
• For a proposed link of proposed length L
• Node added to s_list if
• sp_dist nodes L lt sp_distnoded
• Node added to d_list if
• sp_distnoded L lt sp_distnodes

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ISP in MENTOR II (3)

• Now work through all pairs of ni in s_list and nj
  in d_list
• If sp_distnis L sp_distdnj lt
  sp_distninj
• Then (ni,nj) traffic will shift to the proposed
  link
• Maximum permissible L for (ni,nj) traffic to
  shift is
• maxL sp_distninj sp_distdnj
  sp_distnis

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ISP in MENTOR II (4)

• We can sort pairs by maxL and get a sequence
  for example
• maxL(P1) 2000
• maxL(P2) 1800
• maxL(P3) 1800
• maxL(P4) 1700
• Do not want to set L 2000 or 1800 or 1700
  creates equal length paths which will probably
  confuse the routers

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Commodity Links

• MENTOR-II has a weakness when the clustering
  algorithm chooses too few backbone sites
• Recall in last weeks example that best costs
  were achieved with a high number of backbone
  nodes
• Backbone node choice determined by parameters
  WPARM and RPARM

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Recap Threshold Clustering

• Weight of a site is sum of all traffic into and
  out of the site
• Normalized weight of site i is
• NW(i) W(i)/C
• Sites with NW(i) gt WPARM are made into backbone
  sites
• Where WPARM is a parameter

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Recap -Threshold Clustering (2)

• All sites that do not meet the weight criterion
  and are close to a backbone site are made into
  end sites
• Close is defined as when the link cost from the
  end site e to the backbone site is less than a
  predefined fraction of the maximum link cost
  MAXCOST maxi,jcost(Ni,Nj)
• cost(e,Ni) lt MAXCOSTRPARM

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Wrong Parameter Choices

• Wrong choice of parameters can lead to problems
• E.g., WPARM 100, RPARM 1
• Most likely, no site will be chosen in the
  initial pass as a backbone node because WPARM is
  so high
• Computation of merit will choose a first backbone
  site

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Recap Merit

• Define
• dcn (xn-xctr)2 (yn-yctr)20.5
• maxdc max(dcn)
• maxW max(Wn)
• Then
• meritn 0.5(maxdcdcn)/maxdc 0.5(Wn/maxW)
• That is, merit gives equal value to a nodes
  proximity to the center and to its weight

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Wrong Parameter Choices (2)

• Then no other nodes will will be chosen as
  backbone nodes because of RPARM
• With only one backbone node, no direct links will
  be added
• End result is a star network

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Fixing the Problem

• Define a new class of links called 1-commodity
  links that link endpoints
• Carry only traffic between the endpoints linked
• Insert a new step into the MENTOR algorithm to
  add 1-commodity links where appropriate

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1-Commodity Links - Example
H
T(A,H) 1.8 T(H,I) 2.4 T(C,I) 1.4
Links have capacity 2 We want load lt 1
A
G
B
E
F
D
I
C
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Recap Adding Links

• For each pair (N1,N2), execute the following
  algorithm
• If capacity of a link is C, compute
• n ceilT(N1,N2)/C
• Compute utilization
• u T(N1,N2)/(nC)
• Add link if u gt umin, otherwise move traffic 1
  hop through the network
• I.e., add T(N1,N2) to both T(N1,H) and T(H,N2)
• And do same for T(N2,N1)
• Note there is a special case when (N1,N2)
  belongs to the original tree
• In this case just add the link (N1,N2) to the
  design
• Added note Define slack 1 - umin

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1-Commodity Links Example (2)
Links have nominal capacity 64 We want lt 32
H
T(A,H) 38 T(H,I) 77 T(C,I) 45
A
G
B
E
F
D
I
If slack 0.1, we add 2 parallel links between A
and H If slack 0.2, also add 3 links between H
and I If slack 0.3, also add 2 links between C
and I
C
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1-Commodity Links Example (3)

• Adding the A-to-H links, etc., may or may not be
  a good idea, depending on the rest of the traffic
  matrix
• Also, must not add these requirements to the
  backbone, since they are carried separately

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Setting 1-Commodity Backbone Link Lengths

• Need to set 1-commodity link lengths so that they
  only carry traffic between the two end nodes
• Assume all links have length gt 1
• With shortest-path routing, set 1-commodity link
  lengths sp_disttree - 1

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MENTOR II Overview

• Divide sites into backbone and edge sites
• Select the median
• Build a Prim-Dijkstra tree rooted at median, link
  length cost
• Compute distance through the tree between each
  node pair put information in sp_dist and sp_pred

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MENTOR II Overview (2)

• 5. Add 1-commodity links between pairs not
  considered for direct-link addition if traffic on
  the link will exceed umin
• 6. Collapse requirements onto backbone
• 7. Sequence backbone node pairs in decreasing
  order of shortest-path distance
• 8. Consider each pair using ISP algorithm, add
  link if desired, choosing appropriate length

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MENTOR II Overview (3)

• Set 1-commodity links lengths to carry only
  traffic between edge nodes, if possible
• Do final link re-sizing

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MENTOR-II Difficulty

• It is not always possible to add 1-commodity
  links with lengths that will only carry the
  desired traffic
• Then there are three possibilities
• Do not add 1-commodity links
• Extend direct-link addition from backbone pairs
  to all pairs
• Add links and check routing

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MENTOR-II Difficulty (2)

• Without the 1-commodity links, the algorithm is
  too dependent upon the initial choice of backbone
  thats why we started adding them
• Extending link addition to all node pairs is too
  expensive raises algorithm complexity to O(n4)
• Adding links and checking routing is only
  reasonable possibility not ideal, but better
  than using OSPF routers on original MENTOR design

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We Are Still Not Done With MENTOR

• Backbone link addition algorithm in MENTOR can go
  astray example

Assume 64 kbps links, slack set to add if a
link attracts between 25 and 32 kbps
A
B
Y
Traf(A,Z) 5 kbps Traf(B,Y) 25 kbps
Z
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MENTOR Goes Astray
Assume 64 kbps links, slack set to add if a
link attracts between 25 and 32 kbps
A
B
Y
Traf(A,Z) 5 kbps Traf(B,Y) 25 kbps
Z
sp_dist(A,Z) gt sp_dist(B,Y), so consider adding
(A,Z) first. This will attract 30 kbps but
adding it would detour traffic from the larger
pair through the smaller pair.
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Improving MENTOR

• To avoid this error, we reorder as follows
• For pair P, create the requirement list (P1,
  P2, Pk)
• Check if any pairs Pi in this list
• Have not yet been processed by the direct-link
  addition algorithm
• Have more than 2x traffic between Pi than between
  P
• If so, exchange order of processing P and Pi

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Multispeed MENTOR

• Initial MENTOR assumption was that only one link
  speed was available
• Now generalize to multispeed links

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Multispeed MENTOR (2)

• In considering whether to replace multiple lower
  speed links with one higher speed link, there are
  three cases
• Cost of multiple lower speed links gt cost of
  higher speed link
• Cost of multiple lower speed links cost of
  higher speed link
• Cost of multiple lower speed links lt cost of
  higher speed link

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Multispeed MENTOR (3)

• In case 1, its clearly advantageous to use the
  higher speed link
• In case 2, its still advantageous still
  improves performance and allows for growth
• In case 3, we have to balance the extra cost now
  against the improved performance now and cost
  savings as traffic grows (future)
• Avoid termination charges/installation
  charges/higher rates in future
• Decision depends on expected growth rate

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Multispeed MENTOR (4)

• We will not add time-dependencies to MENTOR
• Will merely find the cheapest set of homogenous
  parallel circuits that will carry the traffic
  with acceptable utilization
• Homogeneous because routing algorithms can do
  stupid things with inhomogeneous circuit
  combinations

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Link Sizing Steps

• For each link capacity CAPi, compute ni, the
  number of circuits needed to carry the traffic
  with utilization lt umax
• If njCostj is smallest, configure as nj circuits
  of type j
• Break ties in favor of configuration with higher
  total capacity

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Complexity of MENTOR-II

• MENTOR-II has same complexity as MENTOR, except
  for direct-link addition
• That is, O(n2)
• Direct-link addition step
• Incremental shortest path step is O(b2), where b
  is number of backbone nodes
• Number of backbone node pairs is O(b2)
• Overall complexity of direct-link addition is
  O(b4)
• MENTOR-II complexity is O(n2) O(b4)

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Complexity of MENTOR-II (2)

• O(n2) O(b4) is viable so long as b ltlt n
• For instance, if b O(n0.5) then complexity of
  MENTOR-II is O(n2) O(n2) O(n2)
• Often reasonable to cluster many nodes to
  concentrators in large networks
• If b is large relative to n, the algorithm slows
  down a lot

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Comparing MENTOR and MENTOR-II

• Compare performance of MENTOR and MENTOR-II
• Network with 20 sites (Cahn figs 8.23.and 8.24)
• MENTOR produces infeasible design
• Some links have 118 utilization

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MENTOR-II Design

• MENTOR-II initial design with
• a 0
• Slack 0
• WPARM 5
• RPARM 0.2
• MENTOR-II design is a feasible tree costing 102k
• Alternate mesh network design costs 122k

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MENTOR-II Design (2)

• Mesh increases cost by 20
• Increases reliability from 0.939 to 0.987
• Reduces average hops from 3.905 to 2.826

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Controlling the Algorithm

• Changing the design parameters changes the
  characteristics of the resulting design
• To make network more dense, increase slack, or
  increase number of backbone nodes
• To make network more reliable, increase density,
  or use lower-speed links
• To make network less dense, decrease slack, or
  use only higher-speed links

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Controlling the Algorithm (2)

• To decrease number of hops
• Decrease number of backbone nodes
• Increase slack
• Increase a
• To increase performance
• Use higher-speed links
• Decrease utilization
• Increase a

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Controlling the Algorithm (3)

• To minimize cost no easy answer
• Best approach is an thorough search of the design
  space
• I.e., vary the values of a, slack, RPARM and
  WPARM in a systematic fashion

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Parameter Search

• Brute force method would simply assign a range of
  k possible values for a, slack, RPARM and WPARM,
  and compute all possible designs
• This is O(k4)
• Better approaches are possible

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MENTOR Parameter Search

• Recall the main steps of the MENTOR algorithm
• Backbone selection, governed by WPARM and RPARM
• Tree building, governed by a
• Direct-link addition, governed by slack

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MENTOR Parameter Search (2)

• Parameter search in is, again, a heuristic
  process
• Search can be shortened
• E., g., if new values of WPARM and RPARM give
  rise to the same set of backbone nodes
• Then there is no need to reiterate on a, because
  we will just get the same set of trees

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MENTOR Parameter Search (3)

• Alternatively, can perform one-dimensional
  searches on the parameters (fast search)
• Start with values (a0, slack0, RPARM0, WPARM0)
• Search along a axis for best (cheapest?) design
  say at a1
• Alternatively, search in direction of positive
  gradient until a local maximum is attained
• Then search along the slack axis starting at (a1,
  slack0, RPARM0, WPARM0) find new best design at
  (a1, slack1, RPARM0, WPARM0)

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MENTOR Parameter Search (4)

• Continue through RPARM and WPARM to design (a1,
  slack1, RPARM1, WPARM1)
• Can repeat iterations until a satisfactory design
  (an, slackn, RPARMn, WPARMn) is produced
• Note there is no guarantee that this
  hill-climbing process will work, but it seems
  effective in practice

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Example of Fast-Search Process

• Network of 20 cities and 2.048 Mbps traffic
• File ment23.gen on FTP site

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Example of Fast-Search Process (2)

• Exhaustive search with 4000 network designs gave
  best cost of 165k with parameters (0.0, 0.1,
  0.2, 5.5)

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Other Clustering Procedures

• Threshold clustering fails when the network does
  not have natural centers (backbone nodes)
• Also insensitive individual sites access costs
• Various alternatives possible

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Other Clustering Procedures - Pre-Select Types

• Pre-define certain sites as backbone or end
  before using the clustering algorithm
• Can take account of cost variations, user
  population

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Other Clustering Procedures K-Means Clustering

• We want to choose K backbone sites from among a
  set si of sites with coordinates (xi, yi) and
  weight Wi
• Choose K random centers cj
• N the set of indices of the subset of points in
  si that are closer to cj than any other center
• Set cj (xav, yav) where
• xav (SneN Wnxn)/(SneN Wn) etc.

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Other Clustering Procedures K-Means Clustering
(2)

• If ci ci then stop, else set ci ci and
  repeat
• This algorithm enables the analyst to choose
  explicitly the number of backbone nodes
• Variations allow for splitting clusters,
  predefining backbone nodes etc.

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Other Clustering Procedures - Hybrid

• Combines threshold and K-means
• Parametrized by (nclst, n, seed)
• Selects first sites that pass weight threshold of
  ncap say m of them
• If m gt nclst, stop
• Else select remaining (nclst m) sites by K-means

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Further Elaborations of MENTOR

• Access design algorithm is limited to stars
• Can use access design algorithms such as
  Esau-Williams or MSLA to design better access
  networks

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Assignment

• Cahn, 8.1and 8.9