1. A Distributed Algorithm to Construct Multicast Trees in WSNs: An Approximate Steiner Tree Approach
Hongyu Gong, Lutian Zhao, Kainan Wang,
Weijie Wu, Xinbing Wang
Shanghai Jiao Tong University

2. Wireless Sensor Networks
Data dissemination and aggregation are common in wireless sensor networks

3. What is multicast tree? Connect a group of sensors No redundant links
Support one-to-
many or many-to-one
data transmission

4. Applications Multicast tree In WSN Traffic monitoring Smart home
Safety
Irrigation control
Environmental monitoring
Illumination control

5. Communication & Computation

6. 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

7. Outline Introduction and Challenges Network Model
Distributed algorithm
Performance Evaluation
Tree Length
Running time
Energy Consumption

8. Outline Introduction and Challenges Network Model
Distributed algorithm
Performance Evaluation
Tree Length
Running time
Energy Consumption

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

10. Challenge: relay selection
Minimum-length tree is formulated as the Steiner Tree Problem, NP-hard in graph theory
relay addition tree length decrease
Fig. 1 Without relay, tree length is 2
Fig. 2 With relay, tree length is 3

11. 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
Dingzhu Du
Hop count is not enough to describe the tree
performance
需要明确transmission range情况下，进行比较

12. 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.

13. Outline Introduction and Challenges Network Model
Distributed algorithm
Performance Evaluation
Tree Length
Running time
Energy Consumption

14. 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), 𝜖 1 ≤𝑓(𝑥)≤ 𝜖 2 , where x is the position vector, 𝜖 1 and 𝜖 2 are positive constants.
The constructed tree is called Toward Source Tree (TST).

15. 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

16. Outline Introduction and Challenges Network Model
Distributed algorithm
Performance Evaluation
Tree Length
Running time
Energy Consumption

17. Distributed algorithm
Connecting Multicast Group Members
Selection of Relays
Multicast Tree Constructed
Cycle Detection and Elimination

18. Distributed algorithm
Make full use of local topology
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)
Acyclic:意思是，这一步可以保证，在destination之间不会形成环
S: source
A,B,C,D: destinations

19. Distributed algorithm
Construct temporary tree among multicast members

20. Distributed algorithm
Selection of relays
Limited transmission range
Tradeoff 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
Hop count:是指，hop数多，可能是最短的distantce，但是hop多，引起的delay往往长。
因此需要在delay和 距离之间做均衡，也就是在hop count和distance里面做均衡。这里distance是指整个的tree length

21. Distributed algorithm
Relay addition

22. Distributed algorithm
Distributed cycle detection and elimination
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 members
Red nodes are relays
Dotted rectangle shows the existence of cycles

23. Outline Introduction and Challenges Network Model
Distributed algorithm
Performance Evaluation
Tree Length
Running time
Energy Consumption

24. Performance: tree length
Theorem 1 Assume that nodes are uniformly distributed in 0,1 × 0,1 . The length of the temporary tree among 𝑚 multicast group members is denoted as 𝐿 𝑉 . The expected tree length is
𝐸 𝐿 𝑉 ≤𝑐 𝑚 , 𝑐=5.622
Proof points
Multicast group members are uniformly distributed
According to the neighbor selection criteria, nodes with larger distance are chosen with smaller probability

25. Tree Length in Uniform Case
Divide the unit square into 1 𝑛 × 1 𝑛 small grids
Establish coordinate system with source as origin
A receiver 𝑅 is in the cell with coordinate (𝑥,𝑦)
𝑝 𝑖,𝑗 : the probability that 𝑅 connects to another receiver in cell (𝑖,𝑗)
𝑝 𝑖,𝑗 ≤ 1− 𝑥 +1−𝑖 𝑦 +1−𝑗 −1 𝑚 𝑚−1 − (1− 𝑥 +1−𝑖 ( 𝑦 +1−𝑗) 𝑚 ) 𝑚−1
𝑝 𝑠 : the probability that 𝑅 connects to the source
𝑝 𝑠 ≤ (1− 𝑥 𝑦 𝑚 ) 𝑚−1

26. Tree Length in Uniform Case
𝐸( 𝐿 𝑉 ): the expected length of temporary tree
𝐸( 𝐿 𝑉 )≤
1 𝑚 𝑘=1 𝑚 𝐸( 𝑖=1 𝑥 𝑗=1 𝑦 𝑝 𝑖,𝑗 (𝑥−𝑖+1) 2 + (𝑦−𝑗+1) 2
+ 𝑝 𝑠 𝑥 2 + 𝑦 2 )
≤ 𝑚

27. Tree Length in General Case
Lemma 2 Assume that node distribution satisfy density function 𝑓 𝑥 . The expected length of the temporary tree among 𝑚 members is
𝐸( 𝐿 𝑉 ) ≤𝑐 𝑚 𝑓 𝑥 𝑑𝑥, 𝑐=5.622
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

28. Tree Length in General Case
Divide the unit square into 1 𝑘 × 1 𝑘 square cells, and then partition the cells into smaller grids with edge length of 1 𝑘 𝛼 .
𝑚= 𝑘 1+𝛾 where 0<𝛾<1, 1 2 <𝛼<𝛾+ 1 2
Tree length is the sum of inter-square edge and intra-square edge length
𝑃: the probability that length of inter-square edge is o( 1 𝑘 )
𝑃=1−(1− 1− 1− 1 𝑘 2𝛼−1 𝑚 𝜖 1 𝑘 2 ) 𝑘 𝛼−1/2
𝑃~1− 𝑒 −2 𝜖 1 𝑘 𝛼− 1 2 − 𝜖 1 𝑘 𝛼− log 2 , 𝑃 𝑘 →1

29. Tree Length in General Case
k small squares { 𝑆𝑞 1 , 𝑆𝑞 2 ,…, 𝑆𝑞 𝑘 }
Intra-square edge length is:
𝐸 𝐼𝑛𝑡𝑒𝑟−𝑠𝑞𝑢𝑎𝑟𝑒 = 𝑖=1 𝑘 𝑐 𝑘 𝑚 𝑥∈𝑆𝑞 𝑖 𝑓 𝑥 𝑑𝑥
By Riemann-Stiejies integration, the upper bound is:
𝐸 𝐼𝑛𝑡𝑒𝑟−𝑠𝑞𝑢𝑎𝑟𝑒 ≤𝑐 𝑚 𝑥∈ [0,1] 2 𝑓 𝑥 𝑑𝑥

30. Path Length
Lemma 3 Let 𝑛 nodes be independently and identically distributed. Suppose that the Euclidean distance between two nodes 𝑢 and 𝑣 is 𝑥, and the transmission range is 𝑟. The following properties hold:
The expected number of fewest relays that are needed to connect 𝑢 and 𝑣 converges to 𝑥 𝑟 as 𝑛 approaches ∞;
The expected length of the path connecting 𝑢𝑣 and involving the fewest relays converges to 𝑥.

31. Path Length Hops on the path is chosen one by one
Let A be the event that the next hop exists with distance 𝑠 from last hop
Let B be the event that there is a node within last hop’s transmission region
𝑃 𝐴 𝐵 = 𝑒 −2𝑛 𝜖 1 𝑟−𝑠 𝑟 (1− 𝑒 −2𝑛 𝜖 1 𝑠𝑟 ) 1− 𝑒 −2𝑛 𝜖 1 𝑟 2
𝐸 𝑟 (𝑥) is expected hop count in minimum-hop shortest path between nodes with Euclidean distance x
𝐸 𝑟 𝑥 = 0 𝑟 (1+ 𝐸 𝑟 𝑥−𝑠 ) 2𝑛 𝜖 1 𝑟 𝑒 −2𝑛 𝜖 1 𝑟 2 1− 𝑒 −2𝑛 𝜖 1 𝑟 𝑒 2𝑛 𝜖 1 𝑠𝑟 𝑑𝑠
=1+ 2 log 𝑛 𝑛 2 − 𝐸 𝑟 𝑥−𝛼𝑟 𝑛 2𝛼 𝑑𝛼

32. 𝐸 𝑟 𝑥 ≤ 𝑑𝑥 𝑟 +1 holds when 𝑑≥2 log 𝑛 𝑛 2 −1 𝑛 2 2 log 𝑛 −1 +1
Path Length
Lower bound of 𝐸 𝑟 (𝑥)
𝐸 𝑟 𝑥 ≥ 𝑥 𝑟
Upper bound of 𝐸 𝑟 (𝑥)
Proof by induction
𝐸 𝑟 𝑥 =1 when 𝑥∈(0, 𝑟)
Assume that 𝐸 𝑟 𝑥−𝛼𝑟 ≤𝑑 𝑥−𝛼𝑟 𝑟 +1 (𝑑<1)
𝐸 𝑟 𝑥 ≤2+ 2 log 𝑛 𝑛 2 − 𝑑 𝑥−𝛼𝑟 𝑟 𝑛 2𝛼 𝑑𝛼
=2+ 𝑑𝑥 𝑟 − 𝑑 𝑛 2 𝑛 2 −1 + 𝑑 2 log 𝑛
𝐸 𝑟 𝑥 ≤ 𝑑𝑥 𝑟 +1 holds when 𝑑≥2 log 𝑛 𝑛 2 −1 𝑛 2 2 log 𝑛 −1 +1

33. Path Length As n→∞, 𝑑→1, and 𝐸 𝑟 𝑥 → 𝑥 𝑟 .
The expected hop count is 𝑥 𝑟 .
Since the transmission range is 𝑟, the expected path length approaches the Euclidean distance between two nodes.

34. Performance: tree length
Upper bound of length of the constructed tree
𝐿 𝑚 : length of the tree among m nodes, 𝐸(∙): expenctaion.
𝐸(𝐿 𝑚 ) ≤𝑐 𝑚 𝑓 𝑥 𝑑𝑥, 𝑐=
Lower bound of length of the Steiner Tree
𝐸( 𝐿 𝑚𝑖𝑛 𝑚 ) ≥𝑐′ 𝑚 𝑓 𝑥 𝑑𝑥, 𝑐′=0.568.
Theoretical approximation ratio is smaller than 10.

35. Performance: tree length
Fig. 3 Tree length comparison
Approximation ratio is 1.11 in simulations when nodes are uniformly distributed.

36. Outline Introduction and Challenges Network Model
Distributed algorithm
Performance Evaluation
Tree Length
Running time
Energy Consumption

37. Running time TST algorithm: Upper bound of running time 𝑇≤𝑂( 𝑛 log 𝑛 )
Lower bound of running time
𝑇≥𝑂( 𝑛 log 𝑛 )
The algorithm with minimal running time:
Upper bound of running time
𝑇 𝑚𝑖𝑛 ≤𝑂( 𝑛 log 𝑛 )
Lower bound of running time
𝑇 𝑚𝑖𝑛 ≥𝑂( 𝑛 log 𝑛 )
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 𝑂 log 𝑛 . 22

38. Tradeoff between tree length and time complexity

39. Outline Introduction and Challenges Network Model
Distributed algorithm
Performance Evaluation
Tree Length
Running time
Energy Consumption

40. Energy consumption
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

41. Energy Consumption 𝑀𝑠𝑔 1 : Messages used to wake up all receivers
Energy consumption is measured by the quantity of exchanged messages in our distributed algorithm.
Divide the network into 1 𝑛 × 1 𝑛 cells
𝑝 𝑖,𝑗 : the probability that 𝑅 connects to another receiver in cell (𝑖,𝑗)
𝑝 𝑠 : the probability that 𝑅 connects to the source
Five types messages: 𝑀𝑠𝑔 𝑖 (𝑖= 1,2,3,4,5 )
𝑀𝑠𝑔 1 : Messages used to wake up all receivers
𝐸 𝑀𝑠𝑔 1 =𝑂(𝑛)

42. Energy Consumption 𝑀𝑠𝑔 2 : Messages used to request neighbors.
𝐸 𝑀𝑠𝑔 2 =𝑚( 𝑖=1 𝑥 𝑦 𝑝 𝑖 𝑘=0 log 2 𝑖+3 𝑚 𝑟 𝜋 2 𝑘 𝑟 2 𝜖 2 𝑛
+ 𝑘=0 log 𝑥 2 + 𝑦 𝑚 𝑟 𝑝 𝑆 𝜋 2 𝑘 𝑟 2 𝜖 2 𝑛 )
We have 𝐸 𝑀𝑠𝑔 2 =𝑂 𝑛 .

43. Energy Consumption
𝑀𝑠𝑔 3 : Messages used to respond the requests from receivers
𝐸 𝑀𝑠𝑔 3 =𝑂 𝑖=1 𝑥 𝑦 𝑚 𝑝 𝑖 𝜋 2 𝑘 𝑟 2 𝜖 2 𝑚 2 𝑘 | 𝑘= log 2 𝑖+3 𝑚 𝑟 𝑂 𝑚 𝑝 𝑆 𝜋 2 𝑘 𝑟 2 𝜖 2 𝑚 2 𝑘 | 𝑘= log 𝑥 2 + 𝑦 𝑚 𝑟
We have 𝐸 𝑀𝑠𝑔 3 =𝑂 𝑚 𝑟 .

44. Energy Consumption 𝑀𝑠𝑔 4 : Messages used to connect to the neighbor.
𝑀𝑠𝑔 5 : Messages used to eliminate cycle.
E(𝑀𝑠𝑔 5 )=𝑂(𝑛)
Total message complexity is:
𝐸 𝑀𝑠𝑔 =𝑂(𝑛)

45. Energy consumption
Energy consumed during tree construction

46. N min = Θ( mn log n ), m=Ο( n log n ) Ω( n log n ), m=𝜔( n log n )
Energy consumption
Energy for data transmission along the constructed tree – measured by the number of forwarding nodes, N TST
Network size: 𝑛,
multicast group size: 𝑚
N min : minimal number of forwarding nodes
N min = Θ( mn log n ), m=Ο( n log n ) Ω( n log n ), m=𝜔( n log n )

47. N TST = Θ( mn log n ), m=Ο( n log n ) Θ(m), m=𝜔( n log n )
Energy consumption
When 𝑚=𝑂 𝑛 log 𝑛 , the number of forwarding nodes in TST is order optimal
Nodes in TST
N TST = Θ( mn log n ), m=Ο( n log n ) Θ(m), m=𝜔( n log n )

48. Performance Comparisons with other algorithms （table） Algorithm
Tree Length
Time Complexity
Exchanged Messages
Assumptions
SPH
O( 𝑚 )
𝑂(𝑚 𝑛 log 𝑛 )
𝑂(𝑚𝑛)
Shortest paths are already known
KSPH
ADH
𝑂(𝑛 log 𝑛 + 𝑚𝑛) DA
𝑂( 𝑛 2 )
𝑂( 𝑚𝑛 2 )
Shortest paths are unknown
Our algorithm -TST
𝑂( 𝑛 log 𝑛 )
𝑂(𝑛)

49. 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.

50. 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

51. A Distributed Algorithm to Construct Multicast Trees in WSNs: An Approximate Steiner Tree Approach
The Institute of Wireless Communication Technology,
School of Electronics, Information and Electrical Engineering,
Shanghai Jiao Tong University