1. A Study of Group-Tree Matching in Large Scale Group Communications
Jun-Hong Cui (UCONN)
Li Lao, Mario Gerla (UCLA)

2. Outline Motivation Problem Formulation Algorithms for Static Version
Group-Tree Matching
Static vs. Dynamic
Algorithms for Static Version
ILP vs. Greedy
A Pseudo-Dynamic Algorithm
Performance Evaluation
Conclusions
SPECTS 2005, July 24-28

3. Background Multicast Multicast State Scalability Problem Solution:
Efficient for multi-user applications
Utilize a tree structure (or mesh)
Multicast State Scalability Problem
Two issues:
Multicast forwarding state information in routers
Control overhead due to multicast tree maintenance
Aggravated when there are a large number of groups
Solution:
Aggregated Multicast: solve both issues
SPECTS 2005, July 24-28

4. Aggregated Multicast The Key Idea:
Force multiple groups to share one aggregated tree
Multiplex and de-multiplex packets on edge routers
Perfect match vs. leaky match:
Perfect match: group and tree are same
Leaky match: tree is bigger than group
Trade-off
aggregation vs. bandwidth waste
SPECTS 2005, July 24-28

5. Aggregated Multicast Reduce multicast state and control overhead
g0, g1: perfect match
g2: leaky match B C E A
(g0, g1, g2)
(g0, g1, g2)
Multicast Groups
Aggregated Trees
Group ID
Members g0
A, D, E g1 g2
A, E
...
Tree ID
Tree Links T0
A-B, B-C, B-E, C-D
...
Reduce multicast state and control overhead
SPECTS 2005, July 24-28

6. How to match groups to trees?
The Key Problem
How to match groups to trees?
Given a set of groups, find a “good” set of trees to cover them
Theoretically: Maximize aggregation and Minimize bandwidth waste
In practice: Maximize aggregation under bandwidth waste threshold
SPECTS 2005, July 24-28

7. Group-Tree Matching Notations Objective
Network: G(V, E)
Multicast Group: g, Native Tree: tn(g)
Bandwidth waste of using t for data delivery
Aggregation Degree:
Objective
Maximize aggregation degree AD (i.e., minimize number of trees) for a given bandwidth waste threshold bth
The network is modeled as an undirected graph
We assume a multicast routing algorithm is given. The native tree is computed by the given routing algorithm, bandwidth waste is the percentage of bandwidth wasted on using an aggregated tree t instead of the native tree. C(t) is the cost of delivering a unit of data on the tree t. without loss of generality, we assume all the link has the same cost in this work; thus, C(t) is proportional to the number of links in the tree
Aggregation degree is the average number of groups aggregated onto the same tree; it is the goal of our group-tree matching algorithm
SPECTS 2005, July 24-28

8. Problem Formulation (I)
Static Pre-defined Trees (our focus)
Assume all the groups are known in advance
Input: G(V, E), Grps, and bth
Output: a minimum number of trees T such that every group is covered by at least one tree without violating bth and AD is maximized
Useful for multicast pre-provisioning based on long-term traffic measurement
IP multicast, Overlay multicast, etc.
Proved to be NP-complete
SPECTS 2005, July 24-28

9. Problem Formulation (II)
Dynamic On-Line Trees
Groups dynamically join and leave
Establish, modify and tear down a set of trees and assign a group to a tree (without violating bth), such that the number of trees is minimized
Useful for on-line systems
Existing protocols:
BEAM, ASSM, MTBF, etc.
A formal study with upper-bound analysis
A generic dynamic on-line algorithm: GDOA
Presented at ISCC 2005
SPECTS 2005, July 24-28

10. Algorithms for Static Problem
The Basic Idea:
Find the candidate trees
Select minimum number of trees for groups
Minimum Set Cover Problem
Map each group to an appropriate tree
Thus three sub-problems
Candidate Tree Generation
Tree Selection
Group-Tree Mapping (straightforward)
SPECTS 2005, July 24-28

11. Candidate Tree Generation
For each group g
Compute its native tree tn(g)
Extend tn(g) by adding more links
Controlled by bandwidth waste threshold bth
Extended trees can surely cover group g
Using shortest path trees
Obtain candidate trees for all groups
SPECTS 2005, July 24-28

12. Tree Selection Find minimum number of trees to cover all groups
Same as Minimum Set Cover Problem
ILP and Greedy g1 g2 …
gNg t1 t2 …
tNt
SPECTS 2005, July 24-28

13. Complexity Analysis ILP Algorithm: non-polynomial Greedy Algorithm:
When tree selection uses ILP, named as ILP Alg.
When tree selection uses greedy, named as Greedy Alg.
ILP Algorithm: non-polynomial
Greedy Algorithm:
Consider three steps
When bth is small
Additional links less than 3, complexity is: O(Ng2 m5)
Ng is number of groups, m is number of nodes
When bth is large?
SPECTS 2005, July 24-28

14. Pseudo-Dynamic Algorithm
The basic idea:
Randomly pick groups to join network
Use dynamic on-line algorithms
Group join: set up a new tree, or extend existing trees
Only consider the joined groups and established trees
No global view of all groups, thus no-optimal
Complexity is: O(Ng2 m2)
Compared with complexity of : O(Ng2 m5), bth is small
SPECTS 2005, July 24-28

15. Performance Evaluation
Network topology
AT&T backbone network with 54 nodes
Group membership
Random Node Weighted Model (NGC 2001)
Performance metrics
Aggregation Degree
State Reduction Ratio
Program Execution Time
In the AT&T network topology, there are 18 core routers and 36 edge routers;
In random node weighted model, we assign a weight to each edge router, which represent the probability that an edge router will have attached group members
Multicast group arrives as a Poisson process and group’s lifetime follows an exponential distribution. We can control the number of concurrent groups by varying the average group arrival rate
Metrics: aggregation degree is defined in previous slides; state reduction ratio is the percentage of multicast forwarding entries reduced by using aggregated multicast; tree control overhead is the number of control messages for tree setup and removal, tree extension and shrinking (in GDOA), and filter setup and removal in MTBF.
SPECTS 2005, July 24-28

16. Comparing ILP with Greedy
SPECTS 2005, July 24-28

17. Comparing ILP with Greedy
SPECTS 2005, July 24-28

18. Greedy vs. Pseudo-Dynamic
SPECTS 2005, July 24-28

19. Greedy vs. Pseudo-Dynamic
SPECTS 2005, July 24-28

20. Conclusions & Future Work
Formulated two versions of group-tree matching problem (static & dynamic)
Proposed algorithms for static version
ILP, Greedy, and Pseudo-Dynamic
Simulation study shows:
Greedy is effective solution when bth is small
Pseudo-Dynamic is more efficient when bth is big, but introduces some performance penalty
Study of the dynamic version: ISCC 05
Future work: the other version of the problem
Given number of trees, maximize aggregation
SPECTS 2005, July 24-28

21. Project: Aggregated Multicast
Questions ???
Project: Aggregated Multicast
SPECTS 2005, July 24-28

22. Thank you!
SPECTS 2005, July 24-28