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Chapter 6 Multispeed Access Designs
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One-speed One-Center Design
Review
Problem Connecting sites to a backbone node, all
links with the same capacity
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Capacitated Minimum Spanning Tree Problem(CMST)

• CMST problem Given a central node N0 and a set
  of other nodes (N1, , Nn), a set of
  weights(w1,,wn) for each node, the capacity of a
  link, W, and a cost matrix Cost(i,j), find a set
  of trees T1, , Tk such that each Ni belongs to
  exactly one Tj and each Tj contains N0.

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The Esau-William Algorithm

• Heuristic Algorithm but guarantees the tree meets
  the capacity constraint
• Each node starts off in a tree with 1 node.
• Compute the tradeoff function for each node
• Tradeoff(Nk)minj Cost(Nk, Nj)-Cost(Comp(Nk),Cente
  r)
• If the tradeoff is negative, a merge is
  attractive
• Merge is allowed if
• Tradeoff for merging components A and B computes
  the potential savings of going to a neighbor
  instead of going to the center node.

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Overview

• We need to introduce designs with multiple link
  types, because networks are built of a variety of
  different links.
• The cost of links is set by the market or
  government regulations, not any law of nature.
• Assume we have 3 different sorts of access links

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Multispeed Access Designs

• Assume the weight of the access sites is in the
  range of 2,400 bps to 36 Kbps, what happens if we
  are allowed multiple link types and still use
  either the Esau-Williams or Sharmas algorithm?
• If we use 9.6 Kbps or 56 Kbps links, both
  algorithms fail because its impossible to fit 36
  Kbps of flow on a single line and still keep the
  utilization under 50.
• If we use 128 Kbps links, we will massively
  overdesign the networks for the smaller nodes.

We need an algorithm that builds a tree with
links of different capacity.
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Continued

• Well replace the capacitated MST problem with a
  multispeed MST problem.
• Intuitively, the tree should have small-capacity
  links at the ends and should become fatter as
  we move toward the center of the networks, like
  the structure of a real tree.

To define this type of tree we need a
bit more notation.
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Predecessor Function (definition 6.1)
Pred(Root)Root
Root
Pred(2)Root
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?Pred(2) Pred(Pred(5))Pred2(5)Root
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4
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Pred(5)2

• A tree T rooted at a node Root can be represented
  uniquely by a predecessor function pred V
  V on the set of vertices. The predecessor
  function moves 1 step closer to the root.
• Requirement
• pred(Root) Root
• pred(N) ?N for any other node N
• For any node N, ? n gt 0 such that pred n(N)
  Root

A tree is defined by the set of vertices V and
edges (N, pred(N)) for all N ?Root
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Ancestors (definition 6.2)
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Pred(6)4
6
5
7
8
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Pred2(9)4

• Given a tree T and the associated predecessor
  function, the ancestors of N are all the nodes N
  such that
• pred n (N) N for some
  n gt 0

Node 5 through 9 are ancestors of node 4. A
misnomer?
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Given the following notations

• A set of nodes N0, N1, N2, , Nn.
• A set of weights (w1, w2, , wn) for each node
• A set of link types L1, L2, , Lm
• Capacities W1, W2, , Wm
• A cost matrix C(i, j, k) that gives the cost of a
  link of type Lk between Ni and Nj

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Multispeed CMST (definition 6.3)
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5
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• The multispeed CMST problem is to find the tree
  rooted at N0 and the link assignments such that
• (i)
• (ii)

If N is node 4, then w(4)w(5)w(6)w(7)w(8)w(9)
lt Wlink(4,pred(4))
is a minimum

• A set of link types L1, L2, , Lm
• Clearly if m1, this problem becomes the CMST
  problem

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Multispeed Local Access Algorithm(MSLA)
Assume the center is node 0

• Assign each node the smallest link l possible to
  connect it to the center. For each node n,
  compute spare_capacity(n)Wl -wn and set
  pred(n)0
• Calculate trade-offs (savings from linking site n
  to site i rather than linking directly to the
  center). And upgrade links to carry additional
  traffic
• Tradeoffn(i) c(n,i, l )
  Upgrade(i,wn) c(n,0, l )
• Tradeoff(n) mini Tradeoffn(i)
• Add the edges as long as the tradeoffs are less
  than or equal to zero. Terminate when the tree is
  built and each edge is assigned to its link type.

• - Upgrade(i,wn) is the cost of adding wn units to
  the links that connect i and 0.

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MSLA Example
Table 6.1

• Use the links shown in Table 6.1 at a 50
  utilization
• Define D96 as link type L0
• Define D56 as link type L1
• Define F128 as link type L2

L0 L1 L2
Figure 6.1
L0
L0

• Assume N0 is center.
• We have 4 access nodes and their weights in
  Figure 6.1

L1
L1

• Initial State (Utilization0.5)
• spare_capacity(1)0.556000-200008000
• spare_capacity(2)0.59600-24002400
• spare_capacity(3)0.556000-960018400
• spare_capacity(4)0.59600-48000

Initial state
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L0
Initial state
L0
L0L1L2
L1
L1

• State 2 N2 is furthest away from N0. It is
  closer to N4. For N2 to go through N4, require
  (4,0) to upgrade from 9.6 Kbps to 56 Kbps.
• Upgrade(4,2400) c(4,0,1) c(4,0,0)
• Tradeoff2(4)c(2,4,0)(c(4,0,1)c(4,0,0))
  c(2,0,0)
• Positive, not pick.
• Let N4 goes through N3,no upgrade is needed.
• Its the best tradeoff. w3w3w49600480014400
• spare_capacity(3)18400-480013600

L0
L0
State 2
L1
L1
L0

• State 3 Next, the most attractive tradeoff is
  route N2 through N3. Again no upgrade is needed.
• w3w3 w2 14400 2400 16800
• spare_capacity(3)13600-240011200

L0
State 3
L1
L1
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L0
L0
State 3
L0L1L2
L1

• spare_capacity(3)13600-240011200
• spare_capacity(1)8000 (initial)

L1

• Finally, connect N3 to N1 and increase (1,0) to
  128 Kbps link.
• Spare_capacity(1)0.5128000-16800-2000027200

L0
L0
Final design
L1

• My question why not use state 3 ?

L2

• Reason Final design might make good use of
  the economy of scale offered by the higher speed
  links.

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A realistic example of MSLA Algorithm

• We have 20 nodes in Squareworld and the weights
  of the nodes are generated according to the above
  TABLE TRAFDIST.
• Weights of nodes are shown in the TABLE SITES.
  Note that the weight of N0 is normalized so that
  it sums to the traffic from all the other sites.
• To simplify the mathematics, we assume that every
  line can be used to 100 of capacity.

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Esau Williams 20 nodes with 9.6Kbps links
Cost 26,963 Only 9 sites share links to N0,
more like a star.
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Esau Williams 20 nodes with 56Kbps links
Cost 30,160 A nice tree structure, but the
cost is higher because out on the periphery of
the network there is too much capacity.
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MSLA 20 nodes with multispeed links
Cost 22,760 the best There is a central
D56 tree and a peripheral D96 tree
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Chapter 7MultiCenter Local-Access Design
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What happens if there are multiple centers?
Must build a forest instead of a tree. So what
is a forest?

• Definition 7.1
• A forest F ( V,E ) is a simple graph without
  cycles.
• Note a forest need not be connected.

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Notations

• A set of backbone sites (B0, , Bm) B
• A set of access nodes (N1, , Nn) N
• A set of weights (w1, , wn) for each access
  node
• A upper limit of weight, W.
• A cost matrix Cost(i,j) giving the costs between
  each backbone/access pair of sites.

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MultiCenter Local Access Problem(MCLA)

• Definition 4.2 The multicenter local access
  problem is to find a set of trees T1, , Tk such
  that
• (1) Exactly 1 backbone site belongs to
  each tree
• (2)
• (3)
  is a minimum

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An example

• Circle ? 3 backbone nodes
• X, Y and Z
• Square ? 17 access nodes
• A, B, C and D, etc

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Comments

• This problem is a bit more complex than the
  single-center one
• Suppose we have n access nodes that we want to
  partition into 3 sets. The number of possible
  partitions is
• Each partition of the access sites results in 3
  capacitated MST problems each of which can be
  attacked by the Esau-Williams algorithm

Even for the modest number 17, the complexity is
daunting!
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Nearest-Neighbor Esau-Williams(NNEW)

• For each b in B, let Sb n?N Cost(n,b) lt
  Cost(n,b) ?b?BIf n is equidistant between
  several backbone nodes, add n to one Sb at
  random.
• Use Esau-Williams to construct a capacitated MST
  on each set b?Sb.
• Example
  
• 
  
• 
  

A definitely belongs to X
(since X is not
only the closest backbone node to A it is almost
the closest node to A)
B, C and D not
clear
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Creditability Test

• Test reattach the leaves to a different tree and
  see if it reduces the cost.
• The creditability of NNEW is not good

Let us look at two failed examples.
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Example 1 a 10-site bad design
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Example 2 another 10-site bad design
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What do we deduce from the 2 examples?

• Design Principle 7.1
• In local-access design with multiple centers,
  the location of the other access nodes cannot be
  ignored when deciding which access nodes should
  home to which center.

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MultiCenter Esau-Williams algorithm(MCEW or
Kershenbaum-Chou)

• A variant of the original Esau-Williams algorithm
• Recall that, in Esau-Williams algorithm, we
  calculate the tradeoff as the saving by linking
  Ni to Nj instead of linking it directly to the
  center.
• Tradeoff(Ni)minjCost(Ni, Nj) -
  Cost(Comp(Ni),Center)
• MCEW algorithm replaces the tradeoff
  function as Tradeoff(Ni)minjCost(Ni, Nj) -
  Cost(Comp(Ni),Center(Ni))
• Initially, we set Center(Ni) to be the closest
  center.
• As node Ni is merged with node Nj, update
  Center(Ni)Center(Nj).

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NNEW vs. MCEW

• Only a slight cost advantage of using MCEW as
  opposed to NNEW.
• MCEW is far more creditable than NNEW.

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Practical Suggestions
From GÖdels theorem Given any set of
algorithms, it is always possible to formulate a
problem for which they provide no good solution.

• Design Principle 7.2
• The designer needs to be inventive and agile
  when dealing with unusual constraints.

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Some access trees contain too many nodes.

• EW tests only if the combined weight of the two
  components doesnt exceed the upper bound
  weight_limit.
• Solution add additional size_limit constraints
  that prohibit the merge of two components with
  too many nodes.

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Some access trees contain too many hops.

• Solution add depth checking constraint,
• i.e., depth-limit the
• tree built by EW
• Each site maintains a value depthni
• Initially set to 1, update when we evaluate the
  tradeoff between n1 and n2,
• Depthn2 max (depthn2, depthn1 1)
• and compare against threshold.

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Some site in the access tree has too many links

• Solution add degree constraint, or a valence
  constraint.
• Initialize the valence of each site to 1, when we
  accept the merge from n1 to n2 then we increase
  the valence of n2 by 1. Do not accept merges that
  violates the constraint.

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A central site has too many links

• Solution 1 Modify the tradeoff function as
• Tradeoff(Ni)minj Cost(Ni,Nj)-aCost(Comp(Ni)
  ,Center))
• where a gt 1. By adjusting a we limit the
  links that connect to a center.
• Solution 2 If we have multiple centers and EW
  has overloaded a given site, use NNEW or MCEW
  with initial assignment of centers overridden to
  low utilization center.

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Some site fails too often

• If a given site is known to have availability
  problems, it shouldnt be an interior point.
• Solution Mark it as not being able to be set as
  a predecessor of other sites.

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To overcome such constraints without the ability
to rewrite programs is hopeless!

• Design Principle 7.3
• Designers need to be able to modify
  algorithms to deal with unusual constraints. This
  may necessitate adding a programmer to the design
  team but it will be well worth the effort and
  expense. The programmer needs to know the code
  and to understand the idea that is being carried
  out in the algorithm. This requires a deeper
  understanding than can be obtained from reading
  the comments in the existing code.

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HW 7 Due Date July 04
Design a connected topology that connects nodes
1, , 4 to center 0. The link types are listed
below. Cost tables are listed on the next slide.
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  Link
0
 

Link 1
 



Link 2
Cost Tables
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THE END

• Thank you!