1. Algorithms for Precomputing Constrained Widest Paths and Multicast Trees Paper by Stavroula Siachalou and Leonidas Georgiadis Presented by Jeremy Witmer CS 622 Fall 2007

2. 10.22.2007 Precomputing Multicast Trees - CS622 2 Overview 1.Large Multicast Trees 2.Proposed Solution for adding nodes 3.Network Model 4.Precomputation Algorithms 5.Time and Space Requirements 6.Multicast Tree Algorithm 7.Simulated Results 8.Conclusion

3. 10.22.2007 Precomputing Multicast Trees - CS622 3 Multicast Trees

4. 10.22.2007 Precomputing Multicast Trees - CS622 4 Large Multicast Trees In large networks, adding nodes becomes inefficient Wish to add nodes on a widest-bandwidth path Wish to add nodes with QoS constraints

5. 10.22.2007 Precomputing Multicast Trees - CS622 5 Definitions Constrained Widest Path: the path that provides the most leftover bandwidth, after bandwidth requirements are satisfied and delay constraints are guaranteed to be met

6. 10.22.2007 Precomputing Multicast Trees - CS622 6 Proposed Solution Precompute as many of the links in the tree as possible When adding a node, calculate delay and bandwidth for the links it has to other nodes in the network When a node is added to the multicast tree, choose the path with the highest available bandwidth while obeying QoS delay constraints

7. 10.22.2007 Precomputing Multicast Trees - CS622 7 Proposed Solution Solution defined as solutions to two separate problems First, compute the delay and width data for the links in the network, and when nodes are added to the network Second, selection of a new path from the precomputed data when a new node subscribes to the multicast tree

8. 10.22.2007 Precomputing Multicast Trees - CS622 8 Network Model Given a directed graph G = (V, E) V is the set of nodes in the graph E is the set of edges in the graph N = |V| M = |E|

9. 10.22.2007 Precomputing Multicast Trees - CS622 9 Network Model Each edge in E has a corresponding delay and width, (d,W), where width is the available bandwidth in a link A path from source node s to another node u in the network with delay no greater than d is represented as Pu(d) The optimal path is represented as Pu*(d) A path is defined in terms of the overall delay D(p) and minimum width W(p)

10. 10.22.2007 Precomputing Multicast Trees - CS622 10 Network Model

11. 10.22.2007 Precomputing Multicast Trees - CS622 11 Network Model

12. 10.22.2007 Precomputing Multicast Trees - CS622 12 Problem 1 Definition Find the path Pu*(d) that has the greatest width of all the paths from s to u, meeting the bandwidth requirement W(pu*) > W(p) for all paths Pu(d)

13. 10.22.2007 Precomputing Multicast Trees - CS622 13 Dominated Pairs Pair (D(p1), W(p1)) dominates pair (D(p2), W(p2)) or path p1 dominates path p2 iff W(p1) > W(p2) and D(p1) < D(p2) OR W(p1) > W(p2) and D(p1) < D(p2)

14. 10.22.2007 Precomputing Multicast Trees - CS622 14 Algorithm 1 Create a heap P to store all possible discontinuities For each node u in G, except for the source node s: 1.Initialize queue D’(u) 2.Create all possible successor discontinuities to u 3.Store the discontinuities (d, W, u) in P for each u in V Note: (d, W, u) is actually stored as (d, W, u, prev_link)

15. 10.22.2007 Precomputing Multicast Trees - CS622 15 Algorithm 1 4.Take the discontinuity in the minimum lexicographic order off of P, in the form (d, W, u). 5.If the current discontinuity pair isn’t dominated by any pair currently on D’(u), add the current pair to D’(u), otherwise, discard the pair. 6.Do this for all discontinuities in P

16. 10.22.2007 Precomputing Multicast Trees - CS622 16 Algorithm 1

17. 10.22.2007 Precomputing Multicast Trees - CS622 17 Algorithm 1 This will result in a set of queues D’(u), one for each node u in G. Each queue is then sorted in lexicographical order, so the optimal discontinuity for each node u is at the head of the queue Because each discontinuity except for the source s has a predecessor discontinuity (d, W, v), the path can be found by keeping track of these predecessor discontinuities

18. 10.22.2007 Precomputing Multicast Trees - CS622 18 Algorithm 2 P is an array A[u,k] where 1 < k < K, K < M where k is a function of the width w of a link Initialize H[k] heaps, one for each k Initialize D’(u) queues, one for each node u

19. 10.22.2007 Precomputing Multicast Trees - CS622 19 Algorithm 2 1.For each node u, generate all discontinuities (d, k, u) 2.Add or replace the discontinuity at A[u,k] only if d is < the discontinuity at A[u,k], and add the discontinuity to H[k] 3.Starting at the last column in A (largest width discontinuities), find the first non-empty heap H[k] 4.Find the lexicographically smallest discontinuity (d, k, u) in H[k]. 5.If d is < the smallest discontinuity (dmin, kmin, u) currently on D’(u), add (d, k, u) to D’(u) 6.Set A[u,k] to null 7.Repeat until all entries in A[u,k] are null or all H[k] are empty

20. 10.22.2007 Precomputing Multicast Trees - CS622 20 Algorithm 2

21. 10.22.2007 Precomputing Multicast Trees - CS622 21 Algorithm 2 Since the algorithm works from largest to smallest widths k in A, and only adds links that have delay < the current delay on D’(u), D’(u) is guaranteed to have the best possible links

22. 10.22.2007 Precomputing Multicast Trees - CS622 22 Algorithm 3 Given the same graph G = (V, E) 1.Find the widest-shortest path from s to each u node in G 2.Let W* be the minimum among the widths of the paths from step 1 3.For all nodes, if W(pu) > W* then add (d, w, u) to the appropriate queue D’(u) 4.Remove from G all links with width < W* 5.If s has no more outgoing links, then stop, else repeat

23. 10.22.2007 Precomputing Multicast Trees - CS622 23 Algorithm 3 The widest-shortest paths in step 1 are found by Dijkstra’s Algorithm

24. 10.22.2007 Precomputing Multicast Trees - CS622 24 Time and Space Requirements Worst Case Requirements Running TimeSpace Requirements Algorithm 1O(MNlogN + M 2 logN)Space: O(MN) Algorithm 2O(KNlogN + K 2 )Space: O(KN) Algorithm 3O(MNlogN + M 2 )Space: O(MN)

25. 10.22.2007 Precomputing Multicast Trees - CS622 25 Current Multicast Tree Design The optimization problem to conserve resources is known to be NP complete. Existing multicast tree-calculation protocols do not rely on trees that solely optimize resources Problem aggravated by the need to satisfy QoS restraints

26. 10.22.2007 Precomputing Multicast Trees - CS622 26 Problem 2 Given a source node s, a destination node set U, and delay requirement dmax, find a multicast tree T that satisfies a width requirement Wmin 1.Path width W(p) will be > Wmin 2.Path delay D(p) will be < dmax

27. 10.22.2007 Precomputing Multicast Trees - CS622 27 Computation of Constrained Trees Assume that we need to create a multicast tree T T is a subset U of the nodes V in G Where D(T) < QoS constraint dmax And W(T) is the width of the narrowest link in T

28. 10.22.2007 Precomputing Multicast Trees - CS622 28 Computation of Constrained Trees Any calculated tree T must satisfy Property 1: The delay d of discontinuity (d, W, u) is the smallest one among the delays of the discontinuities in D’(u) whose width is larger than or equal to Wmin

29. 10.22.2007 Precomputing Multicast Trees - CS622 29 Algorithm 4 Assuming that D’(u) is an array 1.For all nodes in U, determine the minimum width Wmin that has a delay still less than dmin 2.For each (d, W, u) in each of the arrays D’, determine the discontinuity having property 1, and mark that discontinuity as belonging to the tree 3.Construct T using the predecessor node information stored in D’(u)

30. 10.22.2007 Precomputing Multicast Trees - CS622 30 Algorithm 4 Performance Running Time: O(max{|U|logN, N}) Space Requirements: O(U)

31. 10.22.2007 Precomputing Multicast Trees - CS622 31 Simulation Results Simulations were run on two different networks Power Law Networks: a network with N nodes and M links, where M=άN, ά > 1 Real Internet Networks: observed internet topologies from 9/20/1998, 1/1/2000, and 2/1/2000

32. 10.22.2007 Precomputing Multicast Trees - CS622 32 Simulation Results The delays of the links in both network types were picked randomly in the interval [1,100]. Width 1 networks: width of each link chosen at random from the interval [1,100] Width 2 networks: link width is a function of link delay, based on w = β(101 – d), where β is random from the interval [1,10]

33. 10.22.2007 Precomputing Multicast Trees - CS622 33 Simulation Results Power Law networks generated with 400, 800, and 1200 nodes and ratios ά = 4, 8, 16 Real networks selected with M = 9360, 16568, 27792 and N = 2107, 4120, 6474

34. 10.22.2007 Precomputing Multicast Trees - CS622 34 Simulation Results

35. 10.22.2007 Precomputing Multicast Trees - CS622 35 Simulation Results

36. 10.22.2007 Precomputing Multicast Trees - CS622 36 Simulation Results

37. 10.22.2007 Precomputing Multicast Trees - CS622 37 Simulation Results

38. 10.22.2007 Precomputing Multicast Trees - CS622 38 Simulation Results Running times are increased using Width 2 method, as there are more available discontinuities Algorithm 2 has the best running time, Algorithm 3 the worst Algorithm 1 takes up to 1.6 times as long as Algorithm 2 Algorithm 3 takes up to 14 times as long as Algorithm 2 Algorithm 2 performs the best, especially on larger networks

39. 10.22.2007 Precomputing Multicast Trees - CS622 39 Simulation Results Algorithm 3 has the smallest memory requirements, followed closely by Algorithm 1. Algorithm 2 requires significantly more space than either of Algorithms 1 and 3, due to the memory requirements of the two-dimensional array A[u, k]

40. 10.22.2007 Precomputing Multicast Trees - CS622 40 Conclusions The performance of all algorithms decreases rapidly as u increases Algorithm 1 presents the best trade-off between time and space requirements for precomputing tree paths. Algorithm 4 is simple by comparison, picking the multicast tree T is fairly quick, no matter the number of subscriber nodes

41. 10.22.2007 Precomputing Multicast Trees - CS622 41 References [1] S. Siachalou and L. Georgiadis. “Algorithms for Precomputing Constrained Widest Paths and Multicast Trees”. IEEE/ACM Transactions on Networking. Vol. 13, No. 5. pp 1174-1187. October 2005. [2] S. Siachalou and L. Georgiadis. “Efficient QoS Routing”. INFOCOM 2003. 22 nd Annual Joint Conference of the IEEE Computer and Communications Societies. Vol. 2. pp 938-947. 30 March-3 April 2003.