1. CSE 550 Computer Network Design Dr. Mohammed H. Sqalli COE, KFUPM Spring 2007 (Term 062)

2. CSE-550-T062 Lecture Notes - 52 Outline Topology Design for Centralized Networks Multipoint Line Topology Terminal Assignment Concentrator Location

3. CSE-550-T062 Lecture Notes - 53 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) Center Concentrator Terminal High speed lines Low speed lines

4. CSE-550-T062 Lecture Notes - 54 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

5. Multipoint Line Topology

6. CSE-550-T062 Lecture Notes - 56 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

7. CSE-550-T062 Lecture Notes - 57 Constrained/Capacitated MST (CMST) CMST Problem: Given: A central node N 0 A set of other nodes (N 1, N 2, …, N n ) A set of weights (W 1, W 2, …, W n ) for each node The capacity of each link W max A cost matrix C ij = Cost(i,j) Find a set of trees T 1, T 2, …, T k such that: Each N i belongs to exactly one T j, and Each T j contains N 0 is minimized

8. CSE-550-T062 Lecture Notes - 58 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

9. CSE-550-T062 Lecture Notes - 59 Kruskal’s 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 f max =5, f 1 =0, f 2 =2, f 3 =3, f 4 =2, f 5 =1

10. CSE-550-T062 Lecture Notes - 510 Prim’s Algorithm for CMST

11. CSE-550-T062 Lecture Notes - 511 Esau-Williams Algorithm for CMST Node 1 is the central node. t ij : is the tradeoff of connecting i to j or i directly to the root If (t ij < 0) → better to connect i to j If (t ij ≥ 0) → better to connect i directly to the root Algorithm:

12. Terminal Assignment

13. CSE-550-T062 Lecture Notes - 513 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 C ij : cost of connecting terminal i to concentrator j W j : capacity of concentrator j Assume that terminal i requires W i units of a concentrator capacity Assume that the cost of all concentrators is the same x ij = 1; if terminal i is assigned to concentrator j x ij = 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 )

14. CSE-550-T062 Lecture Notes - 514 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 C1C1 C2C2 T1T1 T2T2 T3T3

15. CSE-550-T062 Lecture Notes - 515 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 k+1 terminals can be found by finding the least expensive way of adding the (k+1) th terminal to the k terminal solution

16. CSE-550-T062 Lecture Notes - 516 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

17. CSE-550-T062 Lecture Notes - 517 Building a Compressed Auxiliary Graph U: set of unassociated terminals T(Y): set of terminals associated with Y

18. CSE-550-T062 Lecture Notes - 518 Building a Compressed Auxiliary Graph

19. CSE-550-T062 Lecture Notes - 519 Example Cost Matrix: W = 2 (capacity of each concentrator) Solution: (a, H) (b, I), (c, H), (d, G), (e, I), (f, G) Cost = 15 GHI a638 b294 c314 d259 e163 f279

20. Concentrator Location

21. CSE-550-T062 Lecture Notes - 521 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 C ij : cost of connecting terminal i to concentrator j d j : cost of placing a concentrator at location j (i.e., cost of opening a location j) K j : maximum capacity (of terminals) that can be handled at possible location j Assume that terminal i requires W i units of a concentrator capacity x ij = 1; if terminal i is assigned to concentrator j; 0, otherwise y j = 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 )

22. CSE-550-T062 Lecture Notes - 522 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

23. CSE-550-T062 Lecture Notes - 523 Add Algorithm Initialization: M: set of locations Select an initial location m Assume all terminals are connected to m Set L 0 = {m} Set k = 0 (iteration count) c' i = c im, i= 1, 2, …, N Compute: 1. For, dowhere 2. Determine a new m such that: If there is no such m, go to step 4. 3. Update:and for Set and k:= k+1, and go to Step 1. 4. No more improvement possible; stop.

24. CSE-550-T062 Lecture Notes - 524 Example Cost Matrix: K j = 3, j = 1, 2, 3, 4 (capacity of each concentrator) d 1 = 0, d 2 = d 3 = d 4 = 2 Solution: S2 (1, 2, 3), S1 (4), S4 (5, 6) Cost = 9 S1S2S3S4 12124 21012 34122 41212 52320 64432

25. CSE-550-T062 Lecture Notes - 525 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