Introduction to Multiple-multicast Routing Chu-Fu Wang

The tree packing problem...  Given a network and member node set.  To find a set of multicast tree under the bandwidth constraint, such that, and is minimized.

The mathematical programming formulation The tree packing problem...

The solution approaches... feasibleinfeasible greedy method Steiner tree based heuristic (STH) Cut set based heuristic (CSH)

 To ignore the bandwidth constraint and then find the optimal trees for sessions,respectively. => The lower bound for tree packing problem…

Example. The lower bound for tree packing problem (cont.)

Step 1 Compute Residual network The greedy method…

The greedy method (cont.) Step 2 Compute Residual network Step 3 Compute Residual network

The complete description for greedy method...

The Steiner-tree based heuristic (STH)  Initial step  apply the SPH k times on graph and obtains  Numerical example.

Introduction to the SPH The SPH start with an arbitrary multicast member. It then joins the next close node to the current tree using the shortest path. An numerical example Step 1

Introduction to the SPH (cont.) Step 2 Step 3 Step 4 Notations: : the minimum Steiner tree : the resulting tree for applying SPH

 Step 2  Determine the residual network (R) & the overloaded link set (S). The overloaded link

 Step 3  Chosen any tree that contains an overloaded link for applying the substitution process & the recycling process The overloaded link The shortest path between these subtrees The substitution process

Numerical example (cont). The substitution process Note: In order to compute the shortest path between two subtrees, ones need performed the Dijkstra algorithm times. However, we can obtain this path by only performing the Dijkstra algorithm once. 2

The recycling process

Step 4 Repeat Step 1-3, until no overloaded link exists. Step 5 (The refinement process) (1)Choosing any two multicast tree, say and placing them into the residual network R. (2)Recompute (3)If a better solution found, the replace it; otherwise, remain unchanged.

The complete description for MTPH...

The cut-set based heuristic (CSH)  Assume the member node sets are identical (i.e., ) Step 1. Determine the minimum all-pair cut set with respect to set D. Step 2. Add edge into the multicast tree T, where Step 3. Recursive call.

Simulation results  The STH can find a better approximate solution in a shorter computation time compared to CSH.  CSH has a higher probability than STH to find a solution. |V|=15|V|=20

Simulation results (cont.)

|V|=50 with 20% extra bandwidth percentage |V|=50 with 30% extra bandwidth percentage

 Introduction  The single-multicast routing problem  T he Steiner tree problem  The multiple-multicast routing problem  Tree packing problem  The optimal source gain multicasting problem  Conclusion

The OSGMP  The optimum source gain multicast problem (OSGMP)  Assumptions: Each stream requires one unit of link capacity. Node 0 is the source node and nodes 1,2,…,n-1 are switching nodes. The video server has bid information.

 An example for video distribution scheme The OSGMP (cont.)

The mathematical formulation The OSGMP (cont.)

 The solution approach:  To solve OSGMP on DAG by branch-and-bound method.  To solve OSGMP approximately on general graph by DAGs’ method. Properly choose a directed acyclic subgraph from the given graph, and then solved it by the proposed branch-and-bound algorithm. Distributed video streams over the residual capacities. The OSGMP (cont.)

 The state-space tree The OSGMP (cont.)

The branching rule: The OSGMP (cont.)

The branching rule: The OSGMP (cont.)

 A numerical example

A DAG’s based heuristic algorithm for general graphs

Determine the nodes order

Determine the nodes order (cont.)

 The simulation results

 The simulation results (cont.)

Future works  To solve the Steiner tree problem with optimal reduction The time complexity of our algorithm i.e., to design an algorithm to minimize value c  Note: W.W. Bein et al. “Optimal reduction of two-terminal directed acyclic graphs,” SIAM Journal on Computing, Vol. 21, No. 6, pp , 1992.

Future works (cont.)  W.W. Bein et al. “Optimal reduction of two-terminal directed acyclic graphs,” SIAM Journal on Computing, Vol. 21, No. 6, pp , Interdictive graph

Future works (cont.) (1) Complexity auxiliary graph C(G) (2) Compute the minimum vertex cover in C(G) Property: C(G) is a transitive DAG Equivalent to find a maximum matching in a bipartite graph. So, its is a polynomial time savable problem.