De-Nian Young Ming-Syan Chen IEEE Transactions on Mobile Computing Slide content thanks in part to Yu-Hsun Chen, University of Taiwan

 Environment:  Wireless Multicast Networks  Heterogeneous Devices and Cells  Differing “Costs” per Cell  Problem:  Given a Heterogeneous Network, Select the Lowest Cost Distribution Tree  NOT STATED: From the perspective of the network owner!

 Current mobile devices have multiple radios  Can connect via:  Wi-Fi  WiMax  3G  EVDO  Satellite  Bluetooth (presumably tethered)

 Devices (mobile hosts) can choose which radio and which “cell” to connect to with that radio to get Mobile-IP multicast messages  Different cells have different costs to both the distributor and mobile host  By aggregating individual mobile hosts appropriately, the provider can reduce overall bandwidth costs for multicasting

 SPT  Easy to build (Dijkstra’s algorithm)  Not necessarily the most efficient in bandwidth usage

 MCT  Finds the minimum cost tree for a given graph  NP-hard!

 CTSP – reformulation of Minimum Cost tree problem.  Contributions:  For each technology – Clusters mobile hosts and reduces the number of cells in the SPT.  Takes into account bandwidth costs of links (weighted edges).  Transparent to the IP multicast protocols  Supports dynamic group membership (necessary for moving hosts)

 All wireless cells are multicast capable  Paths from root to host are pre-given by the multicast protocol  Unwritten:  The root bears the bandwidth costs (questionable in practice)  The individual nodes have multiple cells and multiple technologies to choose from (again, questionable AND irrelevant – different technologies are the same as different cells when weighted!)

 Objective function for ILP formulation   Constraints  Minimum bandwidth Each mobile host selects one cell A cell is used in the shortest path tree if it is selected by any mobile host A link is used in the shortest path tree if it is on the path from any selected cell to the root of the tree

 Modification to ILP  Relaxes a constraint to reduce complexity (relaxation just sound better than cheating by approximation)

 Relax the second constraint ( ) in the ILP  New objective function  Lagrange multiplier : the cost of cell c for mobile host m  Constraints

 Properties  For any feasible solution to the LRP that contradicts the relaxed constraints ( ), the objective value is larger  Any feasible solution to CTSP is a feasible solution to the LRP  When adopting the optimal solution to CTSP, [the objective value of LRP] <= [the objective value of CTSP]  The objective value of the optimal solution to the LRP provides a lower bound to CTSP

 Objective function of the subproblem 1   Constraint   The runtime is  The cost for cell c is stored in each mobile host m Find the cell with the minimum cost for each mobile host m

 Objective Function   Constraint  Minimize the net cost of all selected Cells in the shortest path tree

 To find the minimum net cost of the whole shortest path tree, we consider each link in the bottom-up manner  : the minimum net cost of the subtree that includes link and the subtree rooted at v

 All cells in the subtree corresponding to a link are not selected if net cost is not negative  Each candidate cell c is selected in the second subproblem if the net cost of every link in the shortest path from c to the root of the tree is negative

 The selected cells may not be feasible to CTSP  Each mobile host is not guaranteed to be covered by a cell that is selected in the second subproblem  Each member m in the LAGRANGE algorithm selects the cell c according to the cost in the first subproblem  Adjust the cost iteratively with the subgradient algorithm and the solutions to the two subproblems of the LRP  : the objective function of the LRP  The subgradient of the LRP:

 The subgradient indicates the direction of adjusting to find the better feasible solution to CTSP  : increase  : decrease  The second subproblem tends to  Select the cells cover more mobile hosts to save wireless bandwidth  Select the cells such that the shortest path from the cells to the root share more common wireline links

 A distributed protocol based on the LAGRANGE algorithm  Data tree: the shortest path tree for data delivery  Control tree: to solve the second subproblem in a distributed manner  Initially the control tree spans every candidate cell  Incrementally prune the control tree to reduce the protocol overhead  Each router and base station in the control tree maintains a node agent and cell agent

 Each node agent stores the following states  Multicast group address  The address of the parent node agent in the control tree  The bandwidth cost of the link with the parent node agent  The address of the child agent and a Join timer  Each cell agent stores the following states  The bandwidth cost of the cell  Control Flag (whether the cell is selected)  Data Flag (whether the base station is in the data tree)  The address of the mobile host  The cost of the cell for the mobile host (Lagrange multiplier)  Join timer

 Join  Mobile hosts or node agents send Join to join the control tree  Join_Ack  Confirm the Join message  Contain the Data Flag and the cost of the cell for the mobile host (sent by cell agent)  Leave  Sent by mobile hosts, cell agents, and node agents  Request, Reply, and Inform  Update the cost of each cell in a distributed manner

 Join a multicast group  Mobile host sends a Join message to the cell agent of each cell that covers the mobile host  Handover to a new cell  Mobile host sends a Join message to the new cell and a Leave message to the original cell  Leave the multicast group  Mobile host sends a Leave message to cell agent

 Update the cost of each cell  Root periodically sends a Request message  Cell agent first calculates the net cost → Set Control Flag → send Reply message  Node agent first calculates the net cost → send Reply message to parent node agent  If net cost = 0, send Inform message to child node agent Inform

 Prune the control tree  Cell agent or node agent obtains a zero net cost for a period of time  A node agent leaves the control tree if it receives a Leave message from every child agent

 25 km × 25 km, 36 hexagon cells Simulation results of small wireless networks. (a) total bandwidth cost. (b) number of cells in the tree.

Simulation results of large wireless networks (a) original scenario (b) larger transmission range

Simulation results of large wireless networks. (c) (d) zero bandwidth cost for each link.

Transient behavior of the LAGRANGE algorithm with different mobility (a) Probability = 0 percent (b) 0.1 percent (c) 0.5 percent (d) 2 percent

 LAGRANGE provides a solution to the lowest cost spanning tree problem  The solution uses an iterative approximation approach  Problems:  It really doesn’t address heterogeneous networks  The comparison choices in the experimental results are dubious  It assumes the root bears the cost (not likely) or that it can be somehow transferred to the client

assign a unit cost to each cell for each member find the solution to the first subproblem initial topology every cell is selected in the first subproblem

1+1-2= = =-1 find the solution to the second subproblem

1+(-1)=0 no cell is selected in the second subproblem

optimal threshold

after the second iteration

H 3 handovers from C 4 to C 2 H 5 moves out C 4 H 7 leaves the multicast group

adjustment after the mobility