Hongyu Gong, Lutian Zhao, Kainan Wang, Weijie Wu, Xinbing Wang Shanghai Jiao Tong University A Distributed Algorithm to Construct Multicast Trees in WSNs:

Hongyu Gong, Lutian Zhao, Kainan Wang, Weijie Wu, Xinbing Wang

Shanghai Jiao Tong University

A Distributed Algorithm to Construct Multicast Trees in WSNs:

An Approximate Steiner Tree Approach

Wireless Sensor Networks

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Data dissemination and aggregation are common in wireless sensor networks

What is multicast tree?

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Connect a group of sensors

No redundant links

Support one-to-many or many-to-onedata transmission

Applications

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Smart home

Safety

Environmental monitoring

Illumination control

Irrigation control

Trafficmonitoring

Multicast treeIn WSN

Communication & Computation

Evaluation of the multicast tree

Euclidean tree length is an important concern

Larger tree length might result in Higher probability of transmission failure: wireless

interference More energy consumption: more power to transmit

messages farther Longer delay: more time is needed for messages travel

for a long distance

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Outline

Introduction and Challenges Network Model Distributed algorithm Performance Evaluation

Tree Length Running time Energy Consumption

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Outline



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Our work

An algorithm to construct the minimum-length tree in Wireless Sensor Networks In a distributed manner Time efficiency Energy efficiency

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Challenge: relay selection

Minimum-length tree is formulated as the Steiner Tree Problem, NP-hard in graph theory

relay addition tree length decrease

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Fig. 1 Without relay, tree length is 2 Fig. 2 With relay, tree length is

Challenge: quantitative analysis

Quantitative analysis of the Steiner Tree Famous Gilbert-Pollak conjecture

on the Steiner ratio Prof. Dingzhu Du proved this conjecture

Quantitative analysis in stochastic network Big-O notation of tree length is not

so accurate Consider the general distribution instead of

the uniform distribution Hop count is not enough to describe the tree

performance

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Dingzhu Du

Challenge: practical concern

Time complexity

Wireless sensor networks have dynamic topology, so the tree should be constructed in a short time.

Energy consumption

Sensors are usually battery-powered, so the algorithm should be energy saving.

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Outline



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Network model

Sensor are identically and independently distributed in a unit square.

A group of m members are randomly chosen to participate in multicasting among n nodes in the network.

Density distribution of nodes is f(x), , where x is the position vector, and are positive constants.

The constructed tree is called Toward Source Tree (TST).

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Assumptions

Every sensor knows its geographical location through GPS or signal sensing

Every node is distinguished from each other by their identification numbers

Only the source knows which nodes are chosen as destinations

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Outline



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Distributed algorithm

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Connecting Multicast Group Members

Selection of Relays

Cycle Detection and Elimination

Multicast Tree Constructed


Make full use of local topology

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Connectivity: every multicast group member connects to another member

Acyclic: neighbor is closer to source than it is (B, C in the grey region are potential neighbors of A)

Locality: only the closest member will be chosen (B is closer to A than C, so B is chosen by A) S: source

A,B,C,D: destinations


Construct temporary tree among multicast members


Selection of relays Limited transmission rangeTradeoff between hop count and Euclidean distance

Hop count: determines the delay and energy consumption Euclidean distance: determines tree length

Minimum-hop+shortest path between two adjacent multicast group members Among all paths with fewest hops, the shortest one is

chosen to connect members Nodes on the chosen path are selected as relays

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Relay addition

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Distributed cycle detection and elimination

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Cycle detection: A cycle exists if and only if one node acts as relay for more than one pair of multicast group members.

Cycle elimination: Relays forwarded the Eliminate message along the redundant paths, and wipe them out.

Further reduce the tree length and relay count

Black nodes are multicast membersRed nodes are relaysDotted rectangle shows the existence of cycles

Outline



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Performance: tree length

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Theorem 1 Assume that nodes are uniformly distributed in . The length of the temporary tree among multicast group members is denoted as . The expected tree length is

Proof points• Multicast group members are uniformly distributed• According to the neighbor selection criteria, nodes

with larger distance are chosen with smaller probability

Tree Length in Uniform Case

Divide the unit square into small grids Establish coordinate system with source as origin A receiver is in the cell with coordinate : the probability that connects to another receiver in cell

: the probability that connects to the source

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Tree Length in Uniform Case

: the expected length of temporary tree

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Tree Length in General Case

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Lemma 2 Assume that node distribution satisfy density function . The expected length of the temporary tree among members is

Proof points• Divide the network region into cells small enough• Nodes in each cell can be regard as uniformly

distributed• Inter-cell edges connect nodes in the same cell and

intra-cell edges connect nodes in different cells

Tree Length in General Case

Divide the unit square into square cells, and then partition the cells into smaller grids with edge length of . where , Tree length is the sum of inter-square edge and intra-

square edge length : the probability that length of inter-square edge is

,

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Tree Length in General Case k small squares

Intra-square edge length is:

By Riemann-Stiejies integration, the upper bound is:

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Path Length

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Lemma 3 Let nodes be independently and identically distributed. Suppose that the Euclidean distance between two nodes and is , and the transmission range is . The following properties hold:• The expected number of fewest relays that are needed to

connect and converges to as approaches ;• The expected length of the path connecting and involving

the fewest relays converges to .

Path Length

Hops on the path is chosen one by one Let A be the event that the next hop exists with

distance from last hop Let B be the event that there is a node within last

hop’s transmission region

is expected hop count in minimum-hop shortest path between nodes with Euclidean distance x

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Path Length

Lower bound of

Upper bound of Proof by induction when Assume that ()

holds when

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Path Length

As , , and .

The expected hop count is .

Since the transmission range is , the expected path length approaches the Euclidean distance between two nodes.

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Upper bound of length of the constructed tree

: length of the tree among m nodes, : expenctaion.

. Lower bound of length of the Steiner Tree

Theoretical approximation ratio is smaller than 10.

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Fig. 3 Tree length comparison

Approximation ratio is 1.11 in simulations when nodes are uniformly distributed.

Outline



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Running time

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TST algorithm:

Upper bound of running time

Lower bound of running time

The algorithm with minimal running time:Upper bound of running time

Lower bound of running time

The running time of our algorithm shares the same upper and lower bound as the minimal time to construct the multicast tree. The ratio between the upper and lower bound is only .

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Tradeoff between tree length and time complexity

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Outline



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Energy consumption

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Energy consumption is evaluated from two aspects: during and after the tree construction.

During tree construction: energy consumed during tree construction

After tree construction: energy for data transmission along the constructed tree

Energy Consumption Energy consumption is measured by the quantity of

exchanged messages in our distributed algorithm.

Divide the network into cells : the probability that connects to another receiver in cell : the probability that connects to the source

Five types messages:

: Messages used to wake up all receivers

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Energy Consumption : Messages used to request neighbors.

We have

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Energy Consumption : Messages used to respond the requests from

receivers

We have .

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Energy Consumption : Messages used to connect to the neighbor.

: Messages used to eliminate cycle.

Total message complexity is:

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Energy consumption

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Energy consumed during tree construction

Energy consumption

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Energy for data transmission along the constructed tree – measured by the number of forwarding nodes,

Network size: , multicast group size: : minimal number of forwarding nodes

Energy consumption

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Nodes in TST

When , the number of forwarding nodes in TST is order optimal

Performance

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Algorithm Tree Length

Time Complexity

Exchanged Messages

Assumptions

SPH

Shortest paths are already

known

KSPH

ADH

DAShortest paths are unknownOur

algorithm -TST

Comparisons with other algorithms （ table）

Conclusion

Simple algorithm: limited computation and storage ability of nodes in wireless sensor networks

Tree length: good approximation ratio of the Steiner tree in both theory and simulations

Time efficiency: fast tree construction in dynamic networks.

Energy efficiency: energy-efficient in both tree construction and data transmission.

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Discussion

Good approximation ratio at low cost Distributed algorithm making parallel processing

possible and reducing the time cost Applies to dense networks such as wireless sensor

networks, but might not perform well in sparse networks

The approximation ratio is shown in expectation, but not always ensure the good performance

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The Institute of Wireless Communication Technology,

School of Electronics, Information and Electrical Engineering,

Shanghai Jiao Tong University

A Distributed Algorithm to Construct Multicast Trees in WSNs:

An Approximate Steiner Tree Approach

Documents

Hongyu Gong, Lutian Zhao, Kainan Wang, Weijie Wu, Xinbing Wang Shanghai Jiao Tong University A Distributed Algorithm to Construct Multicast Trees in WSNs: