Algorithm 1: LGR Algorithm.

Begin

Input: Network topology, data source nodes

Output: Data aggregation tree

Initialize Lagrangean multiplier vectors ui(0) = 0, ∀i = 1,.…,6.

$\text{UB}=\left|G\right|\times \sum _{l\in L}{a}_l$ and LB = a very large negative number (e.g., −∞) //upper and lower bounds, respectively

quiescence_age = 0, and step_size = 2.

Foriteration = 1 to Max_Iteration_Number, perform the following:

Solve subproblem 1, subproblem 2, subproblem 3, subproblem 4, subproblem 5.

ComputeZLR in (LR).

IfZLR (u) > LB

LB = ZLR(u) and quiescence_age = 0.

Elsequiescence_age = quiescence_age + 1.

Ifquiescence_age = Quiescence_Threshold

step_size = step_size/2 and quiescence_age = 0.

RunLGR-Primal algorithm (Figure 5).

Compute the new upper bound ub.

Ifub < UB then UB = ub.

Update the step_size.

Update the Lagrangean multiplier vectors.

End For

End