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In wireless sensor networks, data aggregation routing could reduce the number of data transmissions so as to achieve energy efficient transmission. However, data aggregation introduces data retransmission that is caused by co-channel interference from neighboring sensor nodes. This kind of co-channel interference could result in extra energy consumption and significant latency from retransmission. This will jeopardize the benefits of data aggregation. One possible solution to circumvent data retransmission caused by co-channel interference is to assign different channels to every sensor node that is within each other's interference range on the data aggregation tree. By associating each radio with a different channel, a sensor node could receive data from all the children nodes on the data aggregation tree simultaneously. This could reduce the latency from the data source nodes back to the sink so as to meet the user's delay QoS. Since the number of radios on each sensor node and the number of non-overlapping channels are all limited resources in wireless sensor networks, a challenging question here is to minimize the total transmission cost under limited number of non-overlapping channels in multi-radio wireless sensor networks. This channel constrained data aggregation routing problem in multi-radio wireless sensor networks is an NP-hard problem. I first model this problem as a mixed integer and linear programming problem where the objective is to minimize the total transmission subject to the data aggregation routing, channel and radio resources constraints. The solution approach is based on the Lagrangean relaxation technique to relax some constraints into the objective function and then to derive a set of independent subproblems. By optimally solving these subproblems, it can not only calculate the lower bound of the original primal problem but also provide useful information to get the primal feasible solutions. By incorporating these Lagrangean multipliers as the link arc weight, the optimization-based heuristics are proposed to get energy-efficient data aggregation tree with better resource (channel and radio) utilization. From the computational experiments, the proposed optimization-based approach is superior to existing heuristics under all tested cases.

With the capability of sensing, computing and communication embedded on the sensor node, the wireless sensor network (WSN) is a promising technology to probe and collect environmental information. Without the necessity of expensive wiring cost for constructing the sensor network, the WSN could deploy sensor nodes at any location more efficiently [

The application scenario described above is called

Power efficient communication in the WSN is an interesting and blooming research area [

For data aggregation routing, raw data from multiple children sensor nodes are collected and processed before transmission. Data aggregation can minimize the number of transmission by eliminating redundant data from different source nodes [

Although data aggregation in the WSN could reduce the number of transmissions to save transmission cost, it could introduce additional MAC layer retransmission energy loss. Based on the CSMA/CA protocol, data transmission from multiple sensor nodes to the same sensor node for data aggregation will incur _{5}_{5}

By assigning different

Besides limited number of available non-overlapping channels, _{1}_{3}_{2}_{5}_{3}_{4}_{3}_{4}'s_{5}_{3}_{4}_{5}

To perform Channel and Radio Constrained Data Aggregation Routing (CRDAR) in the WSN is even more challenging than pure data aggregation routing in the WSN. The channel assignment in wireless network could be modeled as a graph coloring problem in graph theory where adjacent nodes could not be assigned with the same color. This graph coloring problem is proven to be a NP-hard problem [

Note that this integrated channel assignment and routing problem is also an important issue in multi-radio wireless networks. Hence, the proposed optimization-based algorithm could also be applied to general wireless networks. The reason that I specifically focus on the WSN is because the nature of “many-to-one” communication from multiple data source nodes to one sink in the WSN is different from “one-to-many” communication in the wireless networks. This many-to-one communication will increase the probability of co-channel interference so as to make the integrated channel assignment and routing problem in the multi-radio WSN more challenging than the traditional wireless network.

The remainder of this paper is organized as follows. In Section 2, existing literature on data aggregation routing and channel assignment problem in wireless networks is surveyed. In Section 3, I formulate the CRDAR problem as the MILP mathematical problem. In Section 4, Lagrangean relaxation scheme is applied to relax some constraints and algorithms are proposed to solve the Lagrangean dual problem optimally. In Section 5, the novel optimization-based heuristics are devised to get the primal feasible solution. In Section 6, the numerical results and performance comparisons are demonstrated. Finally, concluding remarks are summarized in Section 7.

In wireless network, if the transmission radius of a node is ^{α}

Pure data aggregation routing problem in the WSN has been studied by several works. In [

In [

In [

In [

In [

In [

The CRDAR is formulated as a MILP problem. The objective function is to minimize the total transmission cost. The constraints include the data aggregation tree, co-channel interference constraint, and channel and radio resource constraint. I consider the case where multiple events occur simultaneously and send back to the sink node via different data aggregation trees and each event is modeled as one multicast group. Each event carries different data such that data aggregation could not be performed between different data aggregation trees. If any sensor node is on two different data aggregation tree, then this sensor node needs two channels/radios to transmit the data on different tree respectively. Hence, more channels/radios will be needed as compared to single data aggregation tree.

The notations used in the formulation are as follows.

Input variables:

_{g}

_{l}

_{gd}

_{g}

δ_{pl} : = 1, if link

ε_{jk} : = 1, if a sensor node

_{lk}

_{lk}

_{j}

Decision variables:

_{gpd}

_{gl}

_{l}

_{ij}

_{i}

Basically, the size of _{gd}_{gd}_{gpd}_{gd}, P_{gd}

Another important parameter is _{jk}_{jk}_{kj}

Objective function:

The objective function is to minimize the total transmission cost. For instance, in _{7}_{7}_{8}, C_{l}_{5}_{7}, C_{l}_{5}_{l}^{α}

Note that in considering the total power consumption, energy consumption in the idle mode is significant such that the sleep/awake mechanism for sensor nodes plays an important role to minimize the total power consumption. In this paper, I only addresses the transmission cost instead of total power consumption for the CRDAR problem. Therefore, the sleep/awake mechanism is outside the scope of this paper.

Constraints _{8}→n_{7}_{n}_{8→}_{n}_{7} must be at least two. Since the objective function is to minimize the _{l}_{n}_{8→}_{n}_{7} will be equal to two at the optimal solution. Hence, it should be an equality at Constraint

In Constraint _{g}_{g}_{g}_{g}_{1}| are the legitimate lower bound. For example, in _{g}_{1}| = 3. The minimum hop count for each data source node could be obtained by running the Bellman-ford algorithm. Then minimum hop count for _{1}, n_{2}_{3}_{g}_{1} = 5. Then the number of selected links (i.e.,

On the left hand side of Constraint _{g}_{9}→n_{8}→ n_{7}→n_{1}, S→n_{9}→n_{8}→ n_{7}→n_{1}→n_{2}, S→n_{9}→n_{8}→ n_{7}→n_{3}

In Constraint _{2}_{1}→ n_{2}_{2}_{1}_{7}_{8}→ n_{7}_{8}→ n_{7}_{7}_{7}_{7}→ n_{5}, n_{7}→ n_{1}, n_{7}→ n_{3}_{1}, n_{3}_{5}_{7}_{7}

Constraint _{jk}_{ij}_{ik}_{jk}_{4}_{6}_{5}

In problem (P), there is total number of

Constraints

Subproblem 1: for _{l}

subject to

Subproblem 2: for _{gl}

subject to

Subproblem 3: for _{gpd}

subject to

Subproblem 4: for _{ij}

subject to

Subproblem 5: for _{i}

subject to

Subproblem 1 is to determine decision variable _{l}_{l}_{l}_{l}_{l}

Sub-problem 2 is to determine decision variable _{gl}

The algorithm to optimally solve

_{gl}_{gl}

_{g}_{g}_{gl}_{gi}

_{g}_{g}_{gl}_{g}_{g}_{gl}_{gl}

The computational complexity of above algorithm is _{g}

Subproblem 3 is to determine decision variable _{gpd}^{2}) for each data source node of the multicast group.

Sub-problem 4 is to determine decision variable _{ij}

In the objective function of _{ij}_{j}_{j}_{ij}_{j}_{ij}_{j}_{ij}

Sub-problem 5 is to determine decision variable _{i}_{i}

Applying the above algorithms, we can solve the Lagrangean dual problem (LR) optimally. According to weak duality theorem, that is, the objective value of LR is a legitimate lower bound to the original problem (P). One can calculate the tightest lower bound and solve the dual problem by using the subgradient method [

The basic idea of getting primal feasible solution (LGR-Primal) is first to identify the energy efficient data aggregation and then adjust the routing path to meet the channel/radio resource constraint. This is a kind of iterative algorithm which is particularly useful under stringent resource constraint (i.e., limited channels and radios). In order to facilitate this idea, the LGR-Primal algorithm is first to identify efficient data aggregation tree by using the GIT algorithm. Then perform the channel and radio assignment algorithm. If the channel and radio constraint is satisfied, then report the data aggregation tree. Otherwise, identify another data aggregation routing path such that the channel and radio constraint could be satisfied, this process is repeated until feasible data aggregation tree is identified.

In

At Step 5, it specifies that if we could not assign feasible channel orradio to any sensor node along the routing path of the data source node, then we identify another routing path which bypasses the sensor nodes that violate the channel or radio constraint for this data source node. By assigning the very large arc weight (Z) to the links incident to these sensor nodes, the data source node could bypass these sensor nodes by running GIT algorithm again. If the total arc weight for the data source node exceeds Z, then we could conclude that there is no feasible channel/radio constrained routing path for this data source node. Note that, by assigning the very large arc weight (Z) to the links incident to the sensor nodes that violate the channel/radio constraint, the maximum number of traversed routing paths for any data source node is limited to be |

At step 1 of

The first two terms are from the subproblem 3 where the physical meaning for these two multipliers is the tree constraint violation cost for multicast group

The fifth term is the cost of violating the co-channel interference for the termination node of link _{j}_{″}_{k}

An illustrative example is given in

The computational complexity for the GIT algorithm is ^{2}) for each data source node. The channel assignment (i.e., step 3 of ^{2}) for each data source node.

In the following, I show the complete algorithm (denoted as LGR) to solve Problem (P). The computational complexity of the above LGR algorithm is

^{i}_{LR}_{LR}_{LR}

The sensor nodes are randomly placed in a 1 × 1 area. Sensor nodes are uniformly distributed on the deployment area. If there are 100 sensor nodes, the deployment area will be partitioned into 100 grids with the same size and each sensor node is placed in the center of the grid. The most top left node is selected as the sink node such that we could have a data aggregation tree with larger depth. The transmission cost is equal to the square of the Euclidean distance of the link (i.e., signal attenuation constant

For LGR,

In

In

In

Institutively, in the dense WSN (i.e., large network size), the number of feasible data aggregation trees will also be increasing. In

When network size is above a certain threshold, we could not get feasible solutions due to the co-channel interference constraint will not be satisfied. LGR and ICADAR algorithms could identify feasible solution even when the number of sensor nodes is 196. In addition, LGR could locate lowest cost aggregation tree. This reveals that with capturing the penalty cost of co-channel interference to construct the multicast tree, LGR algorithm is superior to the other four heuristics under variety of network size. From the comparison between event-driven and random-source model, it indicates that it is not easy to identify another routing path for a randomly distributed data source node under stringent channel constraint. As compared to

In

From

Data aggregation that could eliminate redundant data transmission is particularly useful in the limited power WSN. However, data aggregation also incurs collisions such that it produces extra energy loss from retransmission. By assigning different channels to sensor nodes that are within each other's interference range, it could eliminate the problem of retransmission. This requires sophisticated data aggregation routing and the channel assignment strategies. Besides channel assignment, sensor equipped with multi-radios could transmit/receive from multiple sensor nodes simultaneously to minimize the latency. However, channel and radio are limited resources in the WSN that need to be planned carefully to minimize the total transmission cost without violating the co-channel interference constraint. This paper studies the channel and radio constrained data aggregation routing problem in the WSN. I model this CRDAR problem as a mixed integer and linear programming problem and proposed Lagrangean relaxation technique (LGR) to tackle this problem. Unlike the existing heuristics (SPT, GIT, CAGIT and ICADAR), the data aggregation tree is based on the transmission power. The proposed optimization-based heuristics could identify the data aggregation tree from the perspective of transmission power and channel/radio resources simultaneously. The proposed LGR outperforms the other four heuristics in terms of total transmission cost with respect to all kinds of traffic load, available channels/radio resources constraints, network size, and communication radius.

This work is supported (in part) by the National Science Council, Taiwan, under Grant No. NSC 94-2213-E-128-002.

Typical wireless sensor networks.

Data aggregation in data aggregation routing and address centric routing.

Channel, radio and rata aggregation routing.

Data aggregation trees and the corresponding multicast trees in the WSN.

LGR-Primal Algorithm.

Illustrative example for LGR-Primal algorithm.

Performance comparison with respect to traffic loads.

Performance comparison with respect to the channel and radio in event-driven.

Performance comparison with respect to the channel and radio in random-source.

Performance comparison with respect to the network size.

Performance comparison with respect to communication radius.

Performance comparison between LGR and the other four heuristics.

SPT | (200%, 200%) |
(175%, 150%) | (400%, 350%) | (300%, 142%) | (180%, 175%) |

GIT | (200%, 125%) | (175%, 100%) | (400%, 250%) | (16%, 96%) | (180%, 83%) |

CAGIT | (50%, 125%) | (125%, 100%) | (300%, 250%) | (16%, 142%) | (100%, 175%) |

ICADCR | (31%, 13%) | (17%, 25%) | (50%, 100%) | (7%, 7%) | (61.4%, 92%) |

(Event-driven, Random-source)