Overlay Multicast Trees

1. Overlay Multicast Trees

2. Background • Multicast • IP multicast vs. Application layer multicast • Overlay network • Issues in application layer multicast Construct and maintain efficient distribution trees between the multicast session participants

3. Topics today • Algorithmic solutions for constructing multicast tree • with explicit maximum degree constraints (Fengming Wang) • without explicit maximum degree constraints (Jing Liu) • Multicast Tree Maintenance (Jianming Zhou)

4. Overlay Multicast Trees of Minimal Delay Antonio Riabov Columbia University Zhen Liu and Li Zhang IBM T.J. Waston Research Center

5. Introduction • Motivation For all of the previous proposed heuristics, the scalability issue remains open with respect to the optimal solution. • Our job Present an algorithm for constructing a degree-constraint spanning tree and show that it arrives at asymptotically optimal solution.

6. Assumptions • Each node can be mapped to a point in the Euclidean space, and node-to-node delays can be approximated by Euclidean distances between these points. • Points are uniformly distributed inside a convex region in Euclidean space and at least 2 outgoing links are allowed at each node.

7. Divide the segment into four sub-segments, by splitting it with an arc of radius (R+r)/2 and a ray dividing angle a into two halves. Pick a representative point in each non-empty sub-segment, such that its radius in polar coordinates is closest to the radius of the source node. Connect the source to all the representatives. Repeat the procedure within each non-empty sub-segment, to connect the points inside the sub-segment, using the representative point as a local source. Constant Factor Approximation

12. Argument Length ≤ max (R-q, q-r) + Ra + Ra/2 + … ≤ max (R-q, q-r) + 2Ra • OPT ≥ max (R-q, q-r) • 4 * OPT ≥ 4 * r sin (a) ≥ 2Ra Thus Length ≤ 5 * OPT

13. Main Algorithm • Create a grid of cells with equal area by partitioning the disk. • Connect the cells, using cell representatives, and form a core network • Connect points within the cells, using the constant factor approximation algorithm

17. Analysis • Any path in the constructed spanning tree consists of two parts: the sub-path p connecting cell representatives, and the sub-path q between the points in the last cell. • Length (p) + Length (q) ≤ 1 + 2Ra + S S ≈ 2π / (2^{(k+1)/2}) S is the sum of arc lengths for inner (k-1) circles of k-ring grid

18. Final Statement • For any small ε, δ, which are larger than 0, there exists an N such that with probability greater than 1-δ, when the number of points n is larger than N, the length of the longest path in the tree produced by the algorithm is within ε plus the optimal solution. • This ε denotes the ratio between the length of the longest path in the tree and the optimal solution • N → ∞ implies ε→ 0

19. Extensions • Out-degree 2 • Higher Dimensions • Lifting the assumptions

20. Some questions • The algorithm does not consider the robustness of the multicast tree, what will happen if some point leaves the tree? • The algorithm specifies a minimum degree constraint 2, what if some point does not have this kind of capability or some powerful point wants more degree constraint? • The mapping from real world to Euclidean space is very crucial, how? • Is this solution suitable for existing recovery techniques?

21. Approximation and Heuristic Algorithms for Minimum Delay Application-Layer Multicast Trees IEEE INFOCOM 2004 Author: Eli Brosh, Yuval Shavitt Presenter: Jing(Janet) Liu

22. Agenda • Research motivation • Goal statement • An approximation algorithm • A heuristic algorithm • Evaluation & conclusion

23. Issues in creating multicast trees • By intuition: • short latency • small degree • Application layer issue: • sequential message distribution • Application-centric cost • processing capacity of end hosts

24. Existing solutions • Naive solution: shortest path tree • Other solutions: • Build a minimum height (diameter) tree with fixed degree constraint [Y.-H.Chu et. al. 2000] • Consider processing and communication delays, but assume that each of them are the same for all the nodes [Cidon et. al., 1995] • Considers link delay and switching(sending) time, but assume switching time is always smaller than link delay [Bar-Noy et. al., 2001]

25. Goal statement & Strategy outline • The optimal multicast tree problem (MDM) Given a directed complete graph G = (V,E), a multicast group M V, a source host s M, a non-negative real processing delay p(v) for each vertex v V, and a non-negative real communication cost c(u,v) for each edge (u,v) E, find a multicast scheme which minimizes the delay by which all the hosts in M receive a message from s. • Strategy: find a multicast tree T which minimizes the quantity △T+LT △T – the maximum generalized degree of T generalized degree of a node = actual degree*switching time LT– the maximum latency λrvin T from source r to any nodes v in U λuv= c(u,v) + (p(u) + p(v))/2

26. Multicast scheme Multicast scheme Multicast scheme Multicast scheme Uk Ui+1 Ui U1 U0 U1 = core(U0) U0 =the original multicast group Uk = {r} Ui+1 = core(Ui) • size | Ui+1 | <= ¾ * | Ui |, r Uk • a multicast scheme to disseminate the message from Ui toUi-1 in time proportional to the minimum multicast time from r to Ui-1 The approximation solution(I)

27. The approximation solution(II) • Core computation core(Ui) 1) Find a set of edge-disjoint paths, each path connects a pair of nodes in Ui; the length of each path is bounded by 2LT*, the generalized degree is bounded by 3 △T* 2) Transform the set of paths into a set of spider graphs(graphs in which at most one node has degree more than two) such that each connected component in this subgraph is a spider 3) Arbitrarily select a terminal from each spider to the core and select the nodes not in any spiders to the core Note: the above steps insures that the chosen core members from the spider is able to distribute a message to all the nodes in that spider in the required linear time

28. The heuristic solution(I) Note: 1. path <v1, …, vk> has cost 2. Shortest path = minimum cost 3. t[v] - the minimal time at which the host is free to initiate a non notified host T - the constructed tree T 4. V[T] - set of notified hosts 5. denotes the predecessor of m[v] in between m[v] and w

29. V(T) = {s, v1, v2} s d2,5 v5 m(v5) m(v3) m(v4) v2 v1 v4 d1,3 d3,4 v3 The heuristic solution(II) d3,4 + d1,3 > d2,5

30. The heuristic solution(II) V(T) = {s, v1, v2 , v3, v4, } s v5 v2 v1 v4 v3

31. Simulation

32. Comparison

33. Conclusion • The proposed solutions address the problem of finding minimum multicast tree in a heterogeneous postal model in the application layer • Value: there are some existing solutions, but the proposed one is more realistic • Both the approximation and heuristic solutions are centralized algorithms that could handle the new sender join and multiple senders issue • Critics: fails to consider member join and leave issue, nor other network dynamics such as bandwidth change

34. A Proactive Approach to Reconstructing Overlay Multicast Trees INFOCOM 2004 Mengkun Yang, Zongming Fei University of Kentucky

35. Introduction • Overlay Multicast vs. IP Multicast • Issues of Overlay Multicast • Construction vs. Maintenance • Approaches of maintenance • Reactive vs. Proactive • Challenges of proactive approach • degree constraint, multiple leaves, worst case • Design Principles of proactive approach • responsive, distributed, scalable

36. The Problem • Formalization • Overlay multicast tree => degree constrained spanning tree • Degree-constrained minimum spanning tree problem is NP-hard • The Problem of Recovery • The paper focus • Faster recovery

37. Reactive Schemes: Grandfather Root Grandfather-All Root-All Same drawback Concentration of traffic Long time recovery Existing Schemes

38. The Proactive Approach • Parent-to-be only for child • Solution formalization • Forest => spanning tree • Problem • Tree may not exist • The large distance between root and grandfather • Solution • Pre-computation algorithm • Include grandfather

39. Pre-computation Algorithm 2 RD(N8)=1 RD(N9)=1 RD(N10)=1 Impossible!! 4 6 5 RD(N16)=1 RD(N17)=1 Possible! 8 9 10 15 17 18 19 20 21 22 16

40. The Proactive Approach • Recovery Protocol • Required Information of each node • List of ancestor, from grandfather to root • The parent-to-be, if any • The residual degree of each child • Total residual degree of subtree rooted at each child • Heartbeat and JOIN message • Heartbeat for detection • JOIN for recovery

41. Recovery Protocol • Upon receiving JOIN (parent-to-be) • Accept if residual degree > 1 • Redirect if node who subtree’s residual degree biggest • Reject if no such child • Upon detecting children change (parent) • Re-compute the rescue plan • Upon detecting parent leaving (child) • Join parent-to-be or ancestor

42. Recovery Protocol

43. Performance Study

44. Performance Study

45. Questions • Shortest Path Approach is not optimal! • Bandwidth, Processing Power • The quality of tree is not optimal

46. Conclusion • Faster Recovery • Comparable Quality of tree • Comparable Amortized cost