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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Comparing ILP with Greedy SPECTS 2005, July 24-28

Comparing ILP with Greedy SPECTS 2005, July 24-28

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

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

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

Project: Aggregated Multicast Questions ??? Project: Aggregated Multicast http://www.cse.uconn.edu/~jcui SPECTS 2005, July 24-28

Thank you! SPECTS 2005, July 24-28