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

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

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

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

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

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

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

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

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

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

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.

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

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).

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

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

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

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

Distributed algorithm Construct temporary tree among multicast members

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

Distributed algorithm Relay addition

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

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

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

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

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 ) ≤5.622 𝑚

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

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− 1 2 log 2 , 𝑃 𝑘 →1

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

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.

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 2 𝑒 2𝑛 𝜖 1 𝑠𝑟 𝑑𝑠 =1+ 2 log 𝑛 𝑛 2 −1 0 1 𝐸 𝑟 𝑥−𝛼𝑟 𝑛 2𝛼 𝑑𝛼

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 −1 0 1 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

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.

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=5.622 . 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.

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

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

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

Tradeoff between tree length and time complexity

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

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

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)

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 2 𝑥 2 + 𝑦 2 𝑚 𝑟 𝑝 𝑆 𝜋 2 𝑘 𝑟 2 𝜖 2 𝑛 ) We have 𝐸 𝑀𝑠𝑔 2 =𝑂 𝑛 .

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 2 𝑥 2 + 𝑦 2 𝑚 𝑟 We have 𝐸 𝑀𝑠𝑔 3 =𝑂 𝑚 𝑟 .

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)

Energy consumption Energy consumed during tree construction

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 )

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 )

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)

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.

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

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