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. Constant Factor Approximation 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.

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 +L T △ T – the maximum generalized degree of T generalized degree of a node = actual degree*switching time L T – the maximum latency λ rv in T from source r to any nodes v in U λ uv = c(u,v) + (p(u) + p(v))/2

26. The approximation solution(I) U 0 = the original multicast group U0U0 U i+1 = core(U i ) U i+1 Multicast scheme U 1 = core(U 0 ) U1U1 Multicast scheme UiUi U k = {r} UkUk Multicast scheme 1)size | U i+1 | <= ¾ * | U i |, r U k 2)a multicast scheme to disseminate the message from U i to U i-1 in time proportional to the minimum multicast time from r to U i-1

27. The approximation solution(II) Core computation core(U i ) 1)Find a set of edge-disjoint paths, each path connects a pair of nodes in U i ; the length of each path is bounded by 2L T*, 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 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. The heuristic solution(II) s v1v1 v2v2 v5v5 v4v4 v3v3 m(v 5 ) V(T) = {s, v 1, v 2 } m(v 3 ) m(v 4 ) d 2,5 d 1,3 d 3,4 d 3,4 + d 1,3 > d 2,5

30. The heuristic solution(II) s v1v1 v2v2 v5v5 v4v4 v3v3 V(T) = {s, v 1, v 2, v 3, v 4, }

31. Simulation

32. Comparison Approx-MDMHeuristic performance (OPT + (p max – p min )). O(log n) OPT group sizesmalllarge & small different network topology undirected, fully connected directed, partial or fully connected multiple sources support

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

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 18 8 17 16 15 4 222120 10 19 9 6 5 RD(N8)=1 RD(N9)=1 RD(N10)=1 Impossible!! RD(N16)=1 RD(N17)=1 Possible!

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

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