Overlay Multicast Trees

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

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)

Overlay Multicast Trees of Minimal Delay

Antonio Riabov

Columbia University

Zhen Liu and Li Zhang

IBM T.J. Waston Research Center

Introduction

MotivationFor all of the previous proposed heuristics, the scalability issue remains open with respect to the optimal solution.

Our jobPresent an algorithm for constructing a degree-constraint spanning tree and show that it arrives at asymptotically optimal solution.

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.

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.

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

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

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

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

Extensions

Out-degree 2

Higher Dimensions

Lifting the assumptions

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?

Approximation and Heuristic Algorithms for Minimum Delay Application-Layer Multicast Trees

IEEE INFOCOM 2004

Author: Eli Brosh, Yuval Shavitt

Presenter: Jing(Janet) Liu

Agenda

Research motivation Goal statement An approximation algorithm A heuristic algorithm Evaluation & conclusion

Issues in creating multicast trees

By intuition:

•short latency

•small degreeApplication layer issue:

•sequential message distribution

•Application-centric cost

•processing capacity of end hosts

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]

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 Tgeneralized degree of a node = actual degree*switching time

LT– the maximum latency λrv in T from source r to any nodes v in U

λuv = c(u,v) + (p(u) + p(v))/2

The approximation solution(I)

U0 = the original multicast group

U0

Ui+1 = core(Ui)

Ui+1

Multicast scheme

U1 = core(U0)

U1

Multicast scheme

Ui

Multicast scheme

Uk = {r}

Uk

Multicast scheme

1) size | Ui+1 | <= ¾ * | Ui |, r Uk

2) a multicast scheme to disseminate the message from Ui to Ui-1 in time proportional to the minimum multicast time from r to Ui-1

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

The heuristic solution(I)Note:1. path <v1, …, vk> has cost 2. Shortest path = minimum cost3. t[v] - the minimal time at which the host is free to initiate a non notified hostT - the constructed tree T4. V[T] - set of notified hosts

5. denotes the predecessor of m[v] in between m[v] and w

The heuristic solution(II)

s

v1

v2

v5

v4

v3

m(v5)

V(T) = {s, v1, v2}

m(v3)

m(v4)

d2,5

d1,3 d3,4

d3,4 + d1,3 > d2,5

The heuristic solution(II)

s

v1

v2

v5

v4

v3

V(T) = {s, v1, v2 , v3, v4, }

Simulation

Comparison

Approx-MDM Heuristic

performance (OPT + (pmax – pmin)) . O(log n) OPT

group size small large & small

different network topology

undirected, fully connecteddirected,

partial or fully connected

multiple sources

support support

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

A Proactive Approach to Reconstructing Overlay Multicast Trees

INFOCOM 2004

Mengkun Yang, Zongming Fei

University of Kentucky

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

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

Existing Schemes

Reactive Schemes: Grandfather Root Grandfather-All Root-All

Same drawback Concentration of traffic Long time recovery

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

Pre-computation Algorithm

2

18

8

171615

4

222120

10

19

9

65

RD(N8)=1RD(N9)=1

RD(N10)=1Impossible!!RD(N16)=1

RD(N17)=1Possible!

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

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

Recovery Protocol

Performance Study

Performance Study

Questions

Shortest Path Approach is not optimal! Bandwidth, Processing Power

The quality of tree is not optimal

Conclusion

Faster Recovery Comparable Quality of tree Comparable Amortized cost