Algorithm 1: The LGR Algorithm.

Begin

Input: Network topology, data source nodes, sink node and end-to-end delay QoS requirements

Output: Data aggregation tree

Initialize Lagrangean multiplier vectors ui(0) = 0, ∀i = 1,2,…,8.

UB = a very large positive number (e.g., M) and LB = a very small negative number (e.g., −M) //upper and lower bounds, respectively.

quiescence_age = 0, and step_size = 2.

For iteration = 1 to Max_Iteration_Number, perform the following:

Solve Subproblem 1, Subproblem 2, Subproblem 3, Subproblem 4, Subproblem 5 and Subproblem 6.

ComputeZLR in (LR).

If ZLR(u) > LB

LB = ZLR(u) and quiescence_age = 0.

Else quiescence_age = quiescence_age + 1.

If quiescence_age = Quiescence_Threshold

step_size = step_size/2 and quiescence_age = 0.

Run the LGR-Primal algorithm.

Compute the new upper bound ub.

If ub < UB then UB = ub.

Update the step_size.

Update the Lagrangean multiplier vectors.

End For

End