CSE%20550%20Computer%20Network%20Design

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CSE 550Computer Network Design

• Dr. Mohammed H. Sqalli
• COE, KFUPM
• Spring 2007 (Term 062)

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Outline

• Topology Design for Centralized Networks
• Multipoint Line Topology
• Terminal Assignment
• Concentrator Location

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Centralized Network Design

• Centralized network is where all communication
  is to and from a single central site
• The central site is capable of making routing
  decisions
• ? Tree topology provides only one path through
  the center (For reliability, lines between other
  sites can be included)

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Centralized Network Design Problems

• Multipoint line topology selection of links
  connecting terminals to concentrators or directly
  to the center
• Terminal assignment association of terminals
  with specific concentrators
• Concentrator location deciding where to place
  concentrators, and whether or not to use them at
  all

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Multipoint Line Topology
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The Greedy Algorithm

• At each stage, select the shortest edge possible
  (myopic or near-sighted)
• Start with empty solution s
• While elements exist
• Find e, the best element not yet considered
• If adding e to s is feasible, add it if not,
  discard it
• May not find a feasible solution when one exists
• Efficient and simple to implement ? widely used
• Basis of other more complex and effective
  algorithms
• A Minimum Spanning Tree (MST) is a tree of
  minimum total cost
• In the case of MST, the greedy algorithm
  guarantees both optimality and reasonable
  computational complexity

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Constrained/Capacitated MST (CMST)

• CMST Problem
• Given
• A central node N0
• A set of other nodes (N1, N2, , Nn)
• A set of weights (W1, W2, , Wn) for each node
• The capacity of each link Wmax
• A cost matrix Cij Cost(i,j)
• Find a set of trees T1, T2, , Tk such that
• Each Ni belongs to exactly one Tj, and
• Each Tj contains N0

is minimized
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CMST

• Objective Find a tree of minimum cost and which
  satisfies a number of constraints such as
• Flow over a link
• Number of ports
• The CMST problem is NP-hard (i.e., cannot be
  solved in polynomial time)
• ? Resort to heuristics (approximate algorithms)
• These heuristics will attempt to find a good
  feasible solution, not necessarily the best,
  that
• Minimizes the cost
• Satisfies all the constraints
• Well-known heuristics
• Kruskal
• Prim
• Esau-Williams

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Kruskals Algorithm for CMST

• Example Given a network with five nodes,
  labelled 1 to 5, and characterized by the
  following cost matrix
• Node 1 is the central backbone node
• fmax5, f10, f22, f33, f42, f51

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Prims Algorithm for CMST
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Esau-Williams Algorithm for CMST

• Node 1 is the central node.
• tij is the tradeoff of connecting i to j or i
  directly to the root
• If (tij lt 0) ? better to connect i to j
• If (tij 0) ? better to connect i directly to
  the root
• Algorithm

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Terminal Assignment
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Problem Statement

• Terminal Assignment Association of terminals
  with specific concentrators
• Given
• T terminals (stations) i 1, 2, , T
• C concentrators (hubs/switches) j 1, 2, , C
• Cij cost of connecting terminal i to
  concentrator j
• Wj capacity of concentrator j
• Assume that terminal i requires Wi units of a
  concentrator capacity
• Assume that the cost of all concentrators is the
  same
• xij 1 if terminal i is assigned to
  concentrator j
• xij 0 otherwise
• Objective
• Minimize
• Subject to
• i 1, 2, , T (Each terminal associated
  with one Concentrator)
• j 1, 2, , C (Capacity of
  concentrators is not exceeded)

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Assignment Problem

• Given a cost matrix
• One column per concentrator
• One row per terminal
• Assume that
• Weight of each terminal is 1 (i.e., each terminal
  consumes exactly one unit of concentrator
  capacity)
• A concentrator has a capacity of W terminals
  (e.g., number of ports)
• A feasible solution exists iff T W C

C1 C2
T1
T2
T3
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Augmenting Path Algorithm

• It is based on the following observations
• Ideally, every terminal is assigned to the
  nearest concentrator
• Terminals on concentrators that are full are
  moved only to make room for another terminal that
  would cause a higher overall cost if assigned to
  another concentrator
• An optimal partial solution with k1 terminals
  can be found by finding the least expensive way
  of adding the (k1)th terminal to the k terminal
  solution

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Augmenting Path Algorithm

• Initially, try to associate each terminal to its
  nearest concentrator
• If successful in assigning all terminals without
  violating capacity constraints, then stop (i.e.,
  an optimal solution is found)
• Else,
• Repeat
• Build a compressed auxiliary graph
• Find an optimal augmentation
• Until all terminals are assigned

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Building a Compressed Auxiliary Graph

• U set of unassociated terminals
• T(Y) set of terminals associated with Y

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Building a Compressed Auxiliary Graph
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Example

• Cost Matrix
• W 2 (capacity of each concentrator)
• Solution (a, H) (b, I), (c, H), (d, G), (e, I),
  (f, G)
• Cost 15

G H I
a 6 3 8
b 2 9 4
c 3 1 4
d 2 5 9
e 1 6 3
f 2 7 9
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Concentrator Location
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Problem Statement

• Concentrator location deciding where to place
  concentrators, and whether or not to use them at
  all
• Given
• T terminals (stations) i 1, 2, , T
• C concentrators (hubs/switches) j 1, 2, , C
• Cij cost of connecting terminal i to
  concentrator j
• dj cost of placing a concentrator at location j
  (i.e., cost of opening a location j)
• Kj maximum capacity (of terminals) that can be
  handled at possible location j
• Assume that terminal i requires Wi units of a
  concentrator capacity
• xij 1 if terminal i is assigned to
  concentrator j 0, otherwise
• yj 1 if a concentrator is decided to be
  located at site j 0, otherwise
• Objective
• Minimize
• Subject to
• i 1, 2, , T (Each terminal associated
  with one Concentrator)
• j 1, 2, , C (Capacity of
  concentrators is not exceeded)

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Add Algorithm

• Greedy Algorithm
• Start with all terminals connected directly to
  the center
• Evaluate the savings obtainable by adding a
  concentrator at each site
• Greedily select the concentrator which saves the
  most money

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Add Algorithm

• Initialization
• M set of locations
• Select an initial location m
• Assume all terminals are connected to m
• Set L0 m
• Set k 0 (iteration count)
• c'i cim, i 1, 2, , N
• Compute
• For , do where
• Determine a new m such that
• If there is no such m, go to step 4.
• Update and for
• Set and k k1, and go to Step 1.
• No more improvement possible stop.

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Example

• Cost Matrix
• Kj 3, j 1, 2, 3, 4 (capacity of each
  concentrator)
• d1 0, d2 d3 d4 2
• Solution S2 (1, 2, 3), S1 (4), S4 (5, 6)
• Cost 9

S1 S2 S3 S4
1 2 1 2 4
2 1 0 1 2
3 4 1 2 2
4 1 2 1 2
5 2 3 2 0
6 4 4 3 2
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References

• A. Kershenbaum, Telecommunications Network
  Design Algorithms, McGraw-Hill,1993
• M. Pióro and D. Medhi, Routing, Flow, and
  Capacity Design in Communication and Computer
  Networks, Morgan Kaufmann Publishers, Inc., 2004