Distributed Multicast Tree Construction in Wireless Sensor Networks

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Distributed Multicast Tree Construction in Wireless Sensor Networks

Hongyu Gong¹, Luoyi Fu², Xinzhe Fu², Lutian Zhao³, Kainan Wang², and Xinbing Wang ¹

1Dept. of Electronic Engineering, Shanghai Jiao Tong University, China. Email:{ann, xwang8}@sjtu.edu.cn.

2Dept. of Computer Science, Shanghai Jiao Tong University, China.

Email:{yiluofu, fxz0114, sunnywkn}@sjtu.edu.cn

3Dept. of Mathematics, Shanghai Jiao Tong University, China. Email: golbez@sjtu.edu.cn.

Abstract—Multicast tree is a key structure for data dissemina- tion from one source to multiple receivers in wireless networks.

Minimum length multica modeled as the Steiner Tree Problem, and is proven to be NP-hard. In this paper, we explore how to efficiently generate minimum length mult wireless sensor networks (WSNs), where only limited knowledge of network topology is available at each node. We design and analyze a simple algorithm, which we call Toward Source Tree (TST), to build multicast trees in WSNs. We show three metrics of TST algorithm, i.e., running and energy efficiency. We prove that its running time is O(√

n log n), the best among all existing solutions to our best knowledge. We prove that TST tree length is in the same order as Steiner tree, give a theoretical upper bound and use simulations to show the ratio be only 1.114 when nodes are uniformly distributed. We evaluate energy efficiency in terms of message complexity and the number of forwardin prove that they are both order-optimal. We give an efficient way to construct multicast tree in support of transmission of voluminous data.

I. INTRODUCTION

Wireless Sensor Network (WSN) is a network of wireless sensor nodes into which sensing, computation and communication functions are integrated. Sensors are self-organizing and deployed over a geographical region [2]. Multicasting, i.e., one-to-many message transmission, is one of the most common data transmission patterns in WSNs. Tree is the topology for non-redundant data transmission. To enable efficient multicast, multicast tree has been proposed and widely used.

It has not only been used for multicast capacity analysis in wireless networks [3]–[5], but in practice, multicast supports a wide range of applications like distance education, military command and intelligent system [6].

Many researchers have been working on constructing efficient multicast trees [7]–[9], [12]. They have proposed a number of algorithms so as to minimize the routing complexity as well as achieve the time and energy efficiency (for details, please refer to the next section), but most of them did not focus on an important performance measure: the tree length.

This is a critical metric since larger tree length clearly results in longer delay. To enable the messages to be forwarded farther, sensors have to increase their transmission power,

A previous version of this work appears in MobiHoc 2015 [1].

However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org.

causing more energy consumption as well as more serious interference to neighboring nodes. Besides, as the transmission distance increases, the messages suffer from higher probability of transmission failure.

Recently, GEographic Multicast (GEM), inspired by Eu- clidean Steiner Tree, was proposed for routing in dense wireless networks [11]. Formally, given a network G = (V, E), the weight of each edges, and a set of terminals S ⊆ V , the Steiner Tree Problem is to find a tree in G that spans S with the minimum total weight [13]. This problem has been proven to be NP-hard [14], and has not been visited for a long time. Former forms of its approximate implementation were not appropriate for constructing multicast trees in WSNs for various reasons (details will be discussed in the following section.) In GEM, the authors took the first step to utilize the Steiner tree for constructing multicast trees in WSNs, achieving routing scalability and efficiency. This approach can potentially reduce the tree length, but this very simple form of utilization only considers the hop count in an unweighted graph, but not the total length of the multicast tree in a weighted graph. Further, as for the performance analysis, the statistical properties were all under the assumption that all nodes are uniformly distributed, making it difficult to tell its efficiency under a realistic network environment.

In this paper, inspired by taking the advantage of the Steiner tree property, we design a novel distributed algorithm to construct an approximate minimum-length multicast tree for wireless sensor networks, aiming at achieving energy efficiency, ease of implementation and low computational complexity, at an affordable cost on the sub-optimality of tree length. In what follows, we call our design Toward Source Tree Algorithm, or TST for short. We quantitatively evaluate TST algorithm performance under general node distribution, and show that TST has the following satisfactory metrics:

• Its running time is O(√

n log n), the best among all existing solutions for large multicast groups.

• Its tree length is in the same order as Steiner tree, and simulation shows the constant ratio between them is only 1.114 with uniformly distributed nodes.

• Its message complexity (which we will formally define later) and the number of nodes that participate in forwarding are both order-optimal, yielding high energy efficiency for sensor networks.

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• Its theoretical properties and distributive nature render it suitable for sensor network architecture and protocol design for performance improvement.

The rest of paper is organized as follows. Section II states related work. In Section III, we introduce our network model.

In Section IV, we present our Toward Source Tree Algorithm to construct a multicast tree. In Section V, VI and VII, we evaluate the performance of our algorithm mainly from three aspects: multicast tree length, running time and energy efficiency separately. In Section VIII, we use extensive simulations to further evaluate the performance and also illustrate how to apply our TST algorithm to practical applications. We conclude this paper and present discussions of future works in Section IX.

II. RELATEDWORK

We review related works in three categories: the application of minimum multicast tree, multicast routing and the approximate Steiner tree.

Minimum-length Multicast Tree. The most significant advantage of minimum multicast tree in wireless sensor networks is energy efficiency. As sensor nodes are often powered by batteries that drain rather fast and are difficult to replace, energy conserving is extremely crucial in sensor networks.

Furthermore, nodes in sensor network consume most of its energy in communication [18]. Hence, minimum multicast tree-based routing is desirable in many cases. Specifically, it is extensively used in the following two applications in sensor networks - user query and data aggregation.

As wireless sensor networks are mostly data centric [19], users have to query for information and disseminate it in the network. To spread the query in a network as energy-efficient as possible, we need to build a minimum multicast tree and route the data following the trees topology [20]. To achieve this, some existing works apply a Steiner tree-based approach [21].

Data aggregation is to integrate the data from different sources and route for eliminating redundancy. It saves energy by reducing the number of transmissions [22]. Minimum-length tree topology is a widely used technique to solve the implosion problems in data centric routing [20]. In data aggregation, the routing pattern of a sensor network is similar to a reverse multicast tree [20]. Achieving the optimal data aggregation, i.e., constructing the minimum multicast tree, is also treated as a Steiner tree problem [22].

Besides improve energy efficiency and extending network lifetime, routing on a minimum multicast tree also has underlying merits, such as indirectly reducing network delay [23]. Since the total length of tree is minimized, it is obvious that the path won’t be too long between the source and any destination.

Multicase tree construction. Many studies focus on multicast routing in wireless networks, and useful techniques for routing have been proposed in WSN. Sanchez et al. proposed Geo- graphic Multicast Routing (GMR), a heuristic neighborhood selection algorithm based on local geographic information [7].

Later Park et al. [24] combined distributed geographic multicasting with beaconless routing. In Localized Energy-Efficient

Multicast Algorithm (LEMA), forwarding elements apply the MST algorithm locally for routing [8]. Dijkstra-based Lo- calized Energy-Efficient Multicast Algorithm (DLEMA) finds energy shortest paths leading through nodes with maximal geographical advance towards desired destinations [9]. Hier- archical geographic multicast routing (HGMR) tries to combine the advantages of geographic multicast routing (GMR) and hierarchical rendezvous point multicast (HRPM) [10]. It achieves transmission times close to GMR, encoding overhead close to HRPM, and good packet delivery ratio in simulations.

Our concern for delay and the running time of multicast tree construciton is orthogonal to the evaluation metrics in HGMR. In summary, few works have considered minimizing the distance of multicast routing or providing comprehensive quantitative analysis theoretically on the performance of routing policies.

Approximate Steiner Tree. Shortest Path Heuristic (SPH) and Kruskal Shortest Path Heuristic (KSPH) add new nodes to existing subtrees through the shortest path [15]. Average Dis- tance Heuristic (ADH) joins subtrees that contain receivers by a path passing non-receivers with minimal average distance to existing subtrees [16]. Santos et al. pushed forward distributed dual ascent (DA) algorithm, achieving good performance in practice [17]. The comparison of these algorithms with our TST algorithm is shown in Table 2.1. These algorithms were proposed for point-to-point networks. In this paper, we consider the Steiner Tree Problem in wireless sensor networks that are broadcast in nature. In addition, each node has limited computation and storage capability. Devices are usually battery-powered, therefore energy-efficiency is of great importance. Due to these specific features and requirements, existing algorithms for P2P are not suitable for WSN.

To sum up, there have been extensive existing works focus- ing on multicast tree construction or the approximate Steiner tree problems, but we have not found a perfect adoption of Steiner tree into constructing multicast trees.

III. NETWORKMODEL

Let us first use mathematical model to capture a wireless sensor network. We assume the network consists of n nodes in total (or we call the network size is n), distributed independently and identically in a unit square. Each node is assigned with a unique identifier to be distinguished from others. Each time when a source needs to transmit messages, it chooses m receivers randomly. In other words, m is the number of nodes that participate in a multicast transmission, or we call it the multicast group size. For our statistical analysis, we focus on the dense network and large multicast group where m and n are both very large, and m ≤ n. This is particularly suitable to describe a wireless sensor network.

The geographical distribution of nodes is described by a density function f (x) where x is the position vector. Here we allow x to be of any dimension; in the rest of this paper we let it be a two-dimensional vector for ease of presentation, but it does not hurt any generality. We assume f (x) is independent of n and m. We also assume that 0 < 1 ≤ f (x) ≤ 2

where 1 and 2 are both constants, i.e., a node has a positive probability to be located in any region of this area.

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TABLE 2.1: Comparison of Distributed Algorithms for Approximate Steiner Construction Algorithm Expected tree length Expected time Expected messages Assumptions SP H [15] 2-approximation

Steiner tree, O(√

m)

O(m» _n

log n) O(mn)

Known shortest paths;

Point-to-point network.

KSP G [15] O(m» _n

log n) O(mn)

ADH [16] O(m» _n

log n) O(n log n + mn)

DA [17] O(√

m) O(n²) O(mn²) Unknown shortest paths;

Applied in point-to-point network.

T ST O(√

m) O(√

n log n) O(n) Unknown shortest paths;

Applied in wireless network.

To ensure the connectivity of the whole network, we set the transmission range r = Θ»

log n n

[25]. For all nodes, r is the same and fixed. We assume that two nodes u and v can communicate with each other directly if and only if the Euclidean distance between them, duv, is no larger than r. Every node can obtain its own geographical location, e.g., via the Global Position System (GPS). However, nodes do not know the exact location of other nodes until they receive messages containing that piece of information.

TABLE 3.1: Notations and Definitions n the total number of nodes in the network m the number of receivers

r transmission range

rc coverage range of receiver searching LV the length of temporary tree

LM the length of multicast tree IV. ALGORITHM

In this section, we describe our Toward Source Tree algorithm in detail. This algorithm consists of three phases. In the first phase, the source broadcasts a message and wakes up all receivers it chooses. In the second phase, every receiver chooses the closest neighboring receiver that has shorter Euclidean distances to the source node than the receiver node itself, and then a “temporary tree” can be established among all receivers. However “temporary tree” is a virtual topology since multicast group members may not be connected directly to each other given limited transmission range. We select appropriate relays to keep these members connected while controlling the tree length. However, till the end of this stage cycles might exist. Hence we eliminate these cycles to further reduce tree length and avoid redundant transmissions in the third phase. In what follows we describe the process in detail, and we will use an example to illustrate how to generate such a tree at the end of this section.

A. Phase 1: Identifying Receivers

Each node has a label indicating its role in the multicast tree:

“S” stands for the source and “R” for receivers. In this phase, a message containing all receivers’ identifiers is sent from the source so that all nodes in the network can be aware whether they are receivers. Upon receiving this message, receivers then wake up, label themselves with “R” and be ready to participate

Algorithm 1 Neighbor Request from Multicast Members

1: for all receiver R in a multicast group do

2: the number of request session: k ← 0

3: coverage range: rc ← r

4: time out interval: T0← Θ 2^klog n

5: set the node sequence as {R}

6: total hop: H ← 0

7: path length: p ← 0

8: forward the request message to its neighborhood

9: while no response is received when time is out for the k^threquest session do

10: k ← k + 1

11: r_c← 2^kr

12: T_k+1← 2Tk

13: set the node sequence as {R}

14: H ← 0

15: p ← 0

17: end while

18: end for

in the multicast routing. The source will also specify its own location in this message.

This step is necessary for multicast routing since no one except the source knows which nodes the messages are desti- nated for. In this phase, the broadcast information will notify the nodes who are selected into the multicast group, and all receivers will be awakened.

B. Phase 2: Connecting All Receivers

In this phase, we first build a “temporary tree” consisting of only the multicast group members, and then find the minimum- hop shortest path between each pair of members that are directly connected in the “temporary tree”. All multicast group members will be connected with the newly added relays.

Step 1: Searching Receivers in the Neighorhood In this step, each multicast member chooses an appropriate neighbor to connect to. The neighboring member selection criteria is: each member chooses the closest one from the set of members that have shorter Euclidean distances to the source node than this node itself. If no such neighboring member can be found, then this multicast member directly connects to the source.

When a member tries to contact its neighbor members, it is regarded as the sender that sends request message. Its

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form is: <sender id, sender location, location of previous hop, coverage range r_c, node sequence, total hop H, path lengthp>. Sender id is used to identify the multicast members sending the request message, and path length can be updated with the location of previous hop and current hop. The coverage range rc sets the range within which the multicast member searches for its neighboring members. The Euclidean distance between the sender and current node can be calculated with sender location, and messages will be discarded if the distance is larger than rc. Node sequence records in order the nodes through which this message has passed, which acts as a guide for response from neighboring receivers so that the response can be routed via the available path. The hop count H is the number of hops the message has passed through, and p is the path length the message have been through when it reaches the current node.

In each search session, the member broadcasts the request message within search coverage range. The sender sets an appropriate timeout interval. Once the sender receives replies from neighboring nodes, the search session terminates. Then it enters step 2. However, if time runs out and no reply is obtained, it means that no appropriate neighboring members are found. The sender then doubles its search range and initi- ates another search. In Algorithm 1, we show how a multicast group member connects to their neighboring members or the source.

A node may receive more than one request message from the same sender. If it is within coverage range, it will choose the one with the fewest hops among all the messages. If the numbers of hops are the same, it picks out the message with the shortest path length. Then it modifies this message. It adds itself to the node sequence, increases the hop count by 1 and calculates new path length given the location of the previous hop. With these information updated, it forwards the message.

Algorithm 2 describes how nodes deal with request messages in detail.

When a multicast member finds it closer to the source than the sender of the request message, it might be chosen as the neighbor by the sender. Therefore, this member will choose a path to the sender and respond with the respond message.

The form of the respond message is: <sender id, respondent id, node sequence, total hop H, path length p>. The respond message can be routed with the path information provided by the node sequence.

Step 2: Connecting to the Nearest Neighbor

With respond messages, every member selects the closest neighbor. Once a neighbor is chosen, the connect message is forwarded via the minimum-hop shortest path. The connect message is used to establish a connection between nodes in the multicast group. At the same time, all relay nodes on the minimum-hop shortest path record this pair of members, previous hop and the next hop on the path. When all receivers send the connect message, a “temporary tree” among all mutlicast group members including the source is constructed.

C. Phase 3: Eliminating Cycles

In Phase 2, we construct a “temporary tree” made up of multicast group members. However, when other nodes are

Algorithm 2 Request Forwarding

1: for all node u receiving request message do

2: dist = klocation of u - sender locationk

3: if dist < rc then

4: add u to node sequence

5: H ← H + 1

6: newDist = klocation of u - location of previous hopk

7: p ← p + newDist

9: if u is in the multicast group then

10: nodeSourceDist = klocation of u - source locationk

11: senderSourceDist = ksender location - source locationk

12: if nodeSourceDist < senderSourceDist then

13: send respond message back to the sender

14: end if

15: end if

16: end if

17: end for

added to it as relays, cycles might be formed. In particular, when paths connecting different pairs of multicast members share the same relay nodes, such node may receive redundant information, which indicates that cycles come into being.

Therefore, we check the existence of cycles in this phase and eliminate them if any.

Suppose a node u acts as a relay for k (k > 1) pairs of nodes in the multicast group, which are directly connected in the temporary tree, denoted as (R11, R12), (R21, R22),..., (Rk1, Rk2).

Let us assume that in each pair, Ri1 is closer to source than Ri2 (1 ≤ i ≤ k). A relay stores its previous and the next hop of the path from Ri1 to Ri2, and they are denoted as P Hi

and N Hirespectively. Then it chooses one pair randomly, say, (Rj1, Rj2) and keeps the information: (Rj1, Rj2, P Hj, N Hj).

For other pairs (Ri1, R_i2) where Ri16= Rj1, the relay modifies their information as (Rj1, Ri2, P Hj, N Hi). Define a set Q, where Q = {q | q =< Ri1, R_i2, P H_i >, ∀ R_i16= Rj1}. Last, it sends “Eliminate message Q” and its previous hops delete unnecessary edges accordingly. In Algorithm 3, we show how to wipe out the cycles.

D. Proof of Tree Topology

The previous subsections describe how we can connect mutlicast group members using our TST algorithm. Now let us prove that the topology constructed by TST algorithm is exactly a tree. We first show that temporary tree formed in the second phase has a tree topology in Lemma 1. But relays are added into the temporary tree to connect receivers, which might result in the existence of cycles. In Lemma 2 we show that cycle elimination can in fact guarantee the tree topology.

Lemma 1: The temporary tree connecting m receivers has a tree topology.

Proof: Assume that each wireless node is identified with a unique label ni. We order these nodes based on their distance from the source node, and we have an ordered set:

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{n₁, n₂, ..., n_m} such that n_j is closer to the source than ni

for 1 ≤ j < i. It is easy to show that n_i can only connect to node in the set {n₁, ..., n_i−1} according to our construction algorithm. Such ordering guarantees that the generated topology is acyclic. Besides, every multicast member tries to connect to another member that is closer to source, so every member can find a path to the source. Naturally the generated topology is also connected. A connected and acyclic graph is a tree.

Algorithm 3 Cycle Elimination

1: for all node u engaged in paths between k pairs of members do

2: choose an integer j such that 1 ≤ j ≤ k

3: for all i such that 1 ≤ i ≤ k and i 6= j do

4: forward Eliminate message Qi=< R_i1, R_i2, P H_i>

5: end for

6: end for

7: for all node w receiving “Eliminate message Qi” do

8: if w is exactly P Hi then

9: if w is not in the multicast group then

10: P Hi← previous hop of w on path (Ri1, Ri2)

11: forward the modified Qi to the previous hop

12: eliminate information: (Ri1, Ri2, P Hi, N Hi)

13: end if

14: end if

15: end for

Based on Lemma 1, we have the following lemma:

Lemma 2:The topology connecting nodes generated by TST algorithm is a tree that spans all multicast group members.

Proof: The existence of cycles means that some nodes in the multicast tree may receive redundant messages, i.e., some nodes have more than one previous hop. For these nodes, they send “Eliminate messages” and ensure that they have only one previous hop. When all nodes in the multicast tree have only one previous hop, no cycle exists.

Multiple previous hops also indicate that multiple paths may exist between two nodes. Once some previous hops are unnecessary, the paths involving these hops can also be eliminated. Thus Algorithm 3 can eliminate these unnecessary paths, and this completes our proof.

E. Illustration

We use an example to illustrate our TST algorithm in Figure 4.1. Nodes are distributed in the unit square as shown in Figure 1(a). Solid nodes represent source nodes labeled by “S”, or multicast members labeled by “R”. The hollow nodes can be chosen as relays. The first step is to build a temporary tree spanning all multicast members. The dashed lines denote virtual connections between two members. Then nodes on the minimal-hop shortest path are engaged as relays between two neighboring members. They form the topology as shown in Figure 1(b). Note that there exists a cycle marked with dotted rectangular box. The last step is to eliminate unnecessary edges as is done in Figure 1(c). Finally we obtain the multicast tree as is shown in Figure 1(d).

R R

R

R R

R R R

R

S

(a) Building a temporary tree spanning multicast group members

(b) Adding relay nodes

(c) Eliminating redundant edges and maintaining the topology of tree

(d) Constructing the multicast tree with relays added

Fig. 4.1: Steps of the TST algorithm

V. LENGTHANALYSIS

The previous section described our Toward Source Tree algorithm. In the next three sections, we will discuss its performance in terms of tree length, time complexity, and energy efficiency. In this section, we discuss the length of TST.

We first obtain the length of temporary tree, first assuming uniform distribution nodes and then extending to a general setting. Next we explore the length of minimal-hop path that connects two receivers. Combining the length of temporary tree and the path, we can derive the upper bound for the multicast tree length.

A. Temporary Tree in Uniform Distribution

We start by discussing the tree length of the temporary tree.

Lemma 3:Assume nodes are uniformly distributed in a unit square. The expected length of the temporary tree spanning m receivers is upper bounded by c√

m, where c = 5.622.

Proof: See Appendix A.

B. Temporary Tree in General Distribution

Based on the conclusions of tree length in uniform distribution, we further study the case that nodes are non-uniformly distributed. We partition the unit square into k small squares, where m = k^1+γ and 0 < γ < 1. We construct trees among nodes in each square, and then connect nodes in different cells so that all nodes in the network are connected. For each square, the source is outside the square and we still apply the TST algorithm for the tree construction. Lemma 4 can estimate the intra-square edge length, and we study the inter-square edge

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length in Lemma 5. With both inter- and intra- square edge estimation, we derive upper bound for temporary tree length in general distribution.

S

S’

R

^x

y

Fig. 5.1: Approximate neighbor region when the source is located outside the square

Lemma 4: (Intra-square edges) Let m nodes be independently distributed in a unit square with density function f (x).

The source S is located outside the square. Let each node connect to the closest neighbor that has shorter Euclidean distance to S than the node it self. If no such receiver exists, it does not connect to other nodes. A tree can be constructed among m nodes, and the expected length of such a tree is upper bounded by c√

m, where c = 5.622.

Proof: For those nodes that not closest to S, they can always find another node to connect to. For the node that is closest to S, it will be connected to by other nodes. We can prove that the topology formed by m nodes is exactly a tree with Lemma 1. We denote this tree as T .

There are two differences of this lemma from Lemma 3.

One is that the source is located outside the square, and the other is that a node won’t connect to others when it can’t find another one that has shorter Euclidean distance to the source.

Now we find a point S⁰ that is closest to S in the boundary of square region, as is shown in Figure 5.1. With S⁰ as the source, a temporary tree as mentioned in TST algorithm can be established spanning all nodes in the network. We denote the temporary tree as T⁰. In the following we demonstrate that the tree length of T⁰ can be used to estimate the upper bound of length of T .

For a node R, we use NRto denote the regions where nodes might be selected by R as a its neighbor. We use a rectangular region as approximate neighbor region N_R⁰, and N_R⁰ ⊆ NR. The approximate neighbor region is the region marked with parallel lines in Figure 5.1. We use the method adopted in the proof of Lemma 3 to estimate the length of T .

It can be observed that the approximate neighbor regions are the same in both cases that we take S as the source and that we take S⁰ as the source. There are some details that need to be clarified. Firstly, when we only consider the nodes in approximate neighbor region, the estimated tree length is larger than actual length, because we ignore the nodes that are closer to node R. Secondly, if neighbors exist in the

approximate region, the estimations of edge length are the same for both T and T⁰. Thirdly, if no neighbor is found in approximate region for a node, we assume that it does not connect to others in T but it connects to the source in T⁰ in our calculation. From the analysis above, we can conclude that length of T is upper bounded by the length of T⁰.

Also recall that in our proof of Lemma 3, and 5.622√ m is the upper bound for temporary tree length wherever S⁰ is. In summary, we can directly use estimated tree length in Lemma 3 as the upper bound of the tree length of T . This completes our proof.

Lemma 5: (Inter-square edges) Let m nodes be independently distributed in a unit square with density function f (x).

The unit square [0, 1] × [0, 1] can be partitioned into k square cells with edge length of^√¹

k, where m = k^1+γand 0 < γ < 1.

The length of inter-square edges connecting k cells in the unit square is o(√

m).

Proof: We know that the expected number of nodes in each square cell is greater than ^m_k1 = k^γ1. To compute the minimal distance between two nodes in adjacent squares, we partition the cell with edge length of ^√¹

k into smaller grids with edge length of _k¹α, where α > ¹₂.

We claim that if α−γ < ¹₂, the minimal length between two adjacent cells is in an order of oÄ ₁

√ k

ä. This comes from the observation that we can connect adjacent cells by connecting nodes in adjacent grids whose edge length is _k¹α, as is shown in Figure 5.2. In this figure, the yellow and the black squares are two adjacent cells with edge length of ^√¹

k. The blue grids contained in them are the smaller squares with edge length of

1

k^α. Green lines are used to show that nodes in the adjacent grids are connected.

As we can see from Figure 5.2, for two adjacent cells with edge length of ^√¹

k, k^α−1/2 pairs of nodes in adjacent grids might exist. Denote P1 as the probability that a node exists in a grid with edge length of 1/k^α. Since the area of each square is very small, we can regard nodes in the same square uniformly distributed. We have

P1= 1 − (1 − 1

k^2α−1)^m1^k .

Denote P₂ as the probability that nodes exist in both of the adjacent grids.

P2= 1 − P₁².

There are k^α−1/2 pairs of nodes in adjacent grids, and we denote P as the probability that at least one pair exist. We have

P = 1 − P₂^k^α−1/2. Hence we have

P = 1 − (1 − (1 − (1 − 1

k^2α−1)^m1^k )²)^k^α−1/2 (5-1) In order to let k squares connected by inter-square edges, it should hold that P^k → 1. Therefore, we need the following condition

1 − k⁻^kα−1/2¹ (1 − (1 − 1

k^2α−1)^m1^k )². (5-2)

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The expression that r1(k) r₂(k) means that r2(k)/r₁(k) → 0 as k → ∞. Condition (5-2) is equivalent to condition (5-3).

log k k^−1/2+γ+α1

1

k^2α−1, (5-3)

Condition (5-3) can be satisfied when α < γ + ¹₂. With

1

2 < α < γ + ¹₂, we can evaluate P . By (5-1) it can be verified that

P ∼ 1 − exp(−2₁k^1/2−α+γ− ₁

log 2k^1/2−α). (5-4) It is easy to show that P^k → 1 with the expression (5-4), which means such pairs of nodes exist for all adjacent cells with high probability. Since (1 − P ) is exponentially decaying to zero, the expectation of total path length needed to connect k cells is

k^1−αP^k+ k^1/2k(1 − P ) ∼ k^1−α= o(k^1/2) (5-5) Due to the fact that k = o(m), the expected path length for inter-square connection is in the order of o(√

m).

1/kα

1/k

^1/2

1/k

^1/2

Fig. 5.2: Inter-square edges between nodes in adjacent square cells

Lemma 6: Let m nodes be independently distributed with density function f (x). The expectation for the total length of temporary tree E[LV] is smaller than c√

m. We have E[LV] ≤ c√

mR

x∈[0,1]²pf(x)dx, where c ≈ 5.622.

Proof: See Appendix B.

C. Path With Minimal Hops

Receivers are connected by the minimal-hop path. In this part, we study the relationship between the path length and Euclidean distance between two nodes.

Lemma 7: Let n nodes be independently and identically distributed over [0, 1] × [0, 1] with distribution function f (x).

Suppose that the Euclidean distance between two nodes u and v is x. The following properties hold:

(a) The expectation of fewest relays that are needed to connect u and v converges to ^x_r as n approaches ∞;

(b) The length expectation of the path connecting uv and involving the fewest relays converges to Euclidean distance x.

Proof: See Appendix C.

D. Multicast Tree

We divide the [0, 1]×[0, 1] network region into k squares. In each square, we construct a tree and connect nodes with intra- square edges. Adjacent squares are connected by the inter- square edges. All nodes are connected by intra- and inter- square edges, and they can be used to estimate the tree length.

Theorem 1: Let n nodes be independently and identically distributed in a unit square and their distribution satisfies the density function f (x). We construct a multicast tree spanning m receivers as well as the source with TST algorithm. When m and n are both very large, the expected length of the tree is upper bounded by c√

mR

x∈[0,1]²pf(x)dx, where c = 5.622.

Proof: Denote ei,j as the edge connecting Receivers i and j in the temporary tree TV, li,j as the length of the minimal-hop path between the two receivers. Since redundant edges will be eliminated, E(L_M) ≤ P

ei,j∈TV

E(l_i,j). And the path length converges to Euclidean distance as network size goes to ∞ according to Lemma 7. So we have:

E(L_M) ≤ c√ m

Z

x∈[0,1]²

»f (x)dx. (5-6)

Remark: We derive an upper bound for the Toward Source Tree, but it is not a tight bound. In Section VIII, we will show that TST algorithm has even better empirical performance than our theoretical bound.

Lemma 8:Suppose Xi, 1 ≤ i < ∞, are independent random variables with distribution µ having compact support in R^d, d ≥ 2. If the monotone function ψ satisfies ψ(x) ∼ x^α as x → 0 for some 0 < α < d, then with probability 1

lim

n→∞n^{−(d−α)/d}M (X₁X₂, ..., X_n) = c(α, d) Z

Rd

f (x)^(d−α)/ddx (5-7) Here f denotes the density of the absolutely continuous part of µ and c(α, d) denotes a strictly positive constant which depends only on the power α and the dimension d [26].

Given a graph with some nodes and edges, building a minimal length tree spanning a subset of nodes with relays appropriately added is formulated as Steiner Tree Problem. If no relay nodes are allowed, then the tree with minimal length is called minimal spanning tree. However, Steiner tree can only optimize tree length by a constant ratio compared with the minimal spanning tree.

Lemma 9:Let P be a set of n points on the Euclidean plane.

Let ls(P ) and lm(P ) denote the lengths of the Steiner minimum tree and the minimum spanning tree on P respectively.

The inequality holds: [28]

l_s(P ) ≥

√3

2 l_m(P ) (5-8)

Combining the two lemmas above, we can conclude that the length of Steiner tree spanning m receivers is:

LST ≥

√3 2 c1

√m Z

[0,1]²

»f (x)dx (5-9)

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Here c1 is the constant equal to c(1, 2) mentioned in Lemma 8. Roberts estimated that c₁= 0.656 [27].

From (5-6) and (5-9), we prove that the length of Toward Source Tree is in the same order as that of Steiner tree, and the difference between them is only a constant ratio no larger than 10.

Remark 1: Many works have shown the lengths of any well-designed spanning tree consisting of m nodes can reach the order of O(√

m), but the order-optimality of our Toward Source Tree in tree length is not a trivial result. TST is not a simple spanning tree of m multicast group members, since other relay nodes must be selected carefully and be involved in multicasting due to the limited transmission range of wireless sensors. Hence minimum multicast tree is formulated as a Steiner tree problem instead of minimum spanning tree problem. Constructing a minimum spanning tree takes polynomial time, while constructing a Steiner tree is proved to be NP- hard. Asymptotic tree length of TST is a result based on quanlitative analysis of this hard problem in graph theory. As far as we know, few works have given asymptotic length of their multicast trees.

One may also doubt that when the network is dense, we can choose infinitely many relay nodes such that the path length between two multicast members approaches their Eu- clidean distance. In this way, the tree can also reach the order-optimality in length. However, we should notice that it would bring terribly long delay and large energy consumption since too many extra nodes are involved in multicasting. The technique to balance the tree length and delay is that we choose minimum-hop shortest path between multicast members, as is described in Section IV.

The minimum length multicast tree has potential benefits of energy efficiency and delay reduction. The asymptotic length of TST not only indicates its order-optimality, but also shows that TST is quite a good approximation of Steiner tree since the approximation ratio is no larger than 10.

VI. RUNNINGTIMEANALYSIS

Time efficiency is another important aspect to evaluate the quality of multicast routing algorithms. In practice, it is expected that the multicast tree can be constructed with small time costs. Now let us derive the time complexity of TST.

Theorem 2: Let n nodes be independently and identically distributed in unit square. The running time of TST algorithm is O(√

n log n).

Proof: There are three serial phases in TST algorithm, so we discuss the time cost of each phase one by one.

In Phase 1, the messages containing location information of the source are broadcast in the network. The furthest distance between the source and another node is O(1), so at most O ¹_r

relays are needed for a message to reach one node.

In expectation, there are πr²2n = O nr²

nodes within transmission range of a node and hence a node has to wait for O nr² time slots to transmit a message. The time needed for Phase 1 is:

OÇ… 1 r

å

≤ E(t1) ≤ O (nr) . (6-1)

In Phase 2, the dominant time cost is searching for neighboring receivers. In the k^thsearch session, the coverage range is 2^kr. We need O 2^k relays to forward request messages from one receiver to any other nodes within its search coverage range. Since the coverage range does not exceed √

2, the number of search sessions cannot be more than†

log₂

√2 r

£.

E(t2) ≤ O Ölog₂

√2 r

X

i=0

2ⁱnr² è

≤ O(nr). (6-2)

In Phase 3, the worst case is that relays on the path whose length is O(1) form cycles. Time for cycle elimination is

E(t₃) = OÅ 1 rnr²

ã

= O(nr). (6-3)

The total running time is E(t) =

i=3

P

i=1

E(ti), so we have

O

Å… n

log n ã

≤ E(t) ≤ O(p

n log n). (6-4) which completes our proof.

O(√n/log n ) O(√n log n) O(c ) Complexity Approximation ratio

of tree length

1 10

{

Infeasible region

n

Fig. 6.1: Relationship between tree length and time complexity Remark: For any algorithm to construct a multicast tree among a group of nodes, broadcast in Phase 1 is necessary.

Since no node has a knowledge of the multicast group except the source, such information has to be forwarded to every node in the network so that they can know whether they should participate in multicasting. The lower bound of time for multicast tree construction is O» _n

log n

. Since TST achieves the time complexity upper bounded by O(√

n log n), the minimal time cost to construct a multicast tree is also upper bounded by O(√

n log n). Hence the time complexity of TST algorithm shares the same upper and lower bounds as the minimal time cost, and the ratio between these two bounds is only O(log n).

The length of multicast trees have a great influence on communication quality in terms of transmission delay and wireless interference. Construction of minimum-length trees is an NP-hard problem, and takes exponential time. Approximate algorithms achieve larger tree length with lower time complexity. Now we explore the relationship between tree length and time complexity in Figure 6.1. Since the lower bound of

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time needed for multicast tree construction is O

» _n

log n

, the region with time complexity smaller than O» _n

log n

is infeasible. Accurate solution to Steiner tree problem achieves the approximation ratio of 1 at the cost of exponential time, and our algorithm achieves the ratio of 10. The approximation ratio of other algorithms like those in Table 2.1 approaches 1 but they have larger time costs.

VII. ENERGYEFFICIENCY

Energy is a primary consideration in wireless sensor networks since sensors are battery-powered and their energy is limited. We consider the following factors: 1) the energy consumed to construct such multicast trees; and 2) the energy needed to send messages along the tree constructed by this algorithm. The former one is usually measured by the amount of exchanged messages to run distributed routing algorithms;

and the latter directly depends on the number of nodes participating in the transmission. We focus on both aspects.

A. Message Complexity

The following theorem quantifies the message complexity in TST.

Theorem 3: Let n nodes be independently and identically distributed in the unit square. The message complexity of TST algorithm is O(n).

Proof: See Appendix D.

Remark: Since each node needs a message telling them whether they are chosen as receivers, the lower bound of message complexity is O(n). Hence TST algorithm is an order-optimal solution in terms of message complexity.

B. Number of Forwarding Nodes

Since the transmission range is fixed, the number of trans- mitters in the tree determines the energy consumption for information propagation. We evaluate the number of forwarding nodes in this subsection.

Theorem 4: Let n nodes be independently and identically distributed in the unit square. The number of forwarding nodes in the multicast tree is

NT ST =







Θ»_mn

log n

, m = OÄ _n

log n

ä;

Θ(m), m = ωÄ _n

log n

ä.

(7-1)

When m = O (n/ log n), the number of forwarding nodes is order-optimal.

Proof: Let T_V be the virtual tree, e_i,j be an edge in the virtual tree connecting two receivers i and j, and di,j be the Euclidean distance between them. When m is small, relay nodes form the dominant part of the forwarding nodes in our multicast tree. The total number of transmitting nodes, NT ST

in the Toward Source Tree is: NT ST = Θ Ç

P

ei,j∈TV

dij

r

å

= ΘÄ^√_m

r

ä. As m grows larger, receivers are close to each other and thus fewer relay nodes are added. Therefore, receivers are dominant in the multicast tree, NT ST = Θ(m).

We should discuss the number of forwarding nodes in two cases, and there exists a critical value for m that determines in

which case it should be discussed. The critical value satisfies:

ΘÄ^√_m_c

r

ä= Θ(m_c), so mc= Θ _r¹₂ .

Denote Nmin as the minimal number of relay nodes that are engaged in propagating the messages from one source to m receivers. [3] gives the lower bound of Nmin under the assumption that all nodes are uniformly distributed. Now we use its method and explore Nmin in the case of general distribution. When m is small, the distance between two receivers is large compared with the transmission range.

Nmin = ΩÄ^√_m

r

ä. This lower bound is achievable with our algorithm, so Nmin= ΘÄ^√_m

r

ä. When m is very large, there exist many receivers within the transmission range of one node, so that one transmission can deliver messages to a large number of receivers. In this case, we only need to choose a connected dominating set from m receivers, and N_min is exactly the size of minimum connected dominating set. We will give the definitions of both connected dominating set and minimum connected dominating set.

Definition 1 (Connected dominating set): D is the connected dominating set of a graph G if and only it satisfies two properties:

(a) Any node in D can reach any other node in D by a path that stays entirely within D.

(b) Every vertex in G either belongs to D or it is adjacent to a vertex in D.

Definition 2 (Minimum connected dominating set): MD is the minimum connected dominating set of graph G if MD is the connected dominating set containing the smallest number of nodes.

We still need to discuss Nmin in two cases. There also exists a critical value md, and Θ(md) = ΘÄ^√_m_d

r

ä, so md= ΘÄ _n

log n

ä.

Nmin=







Θ»_mn

log n

, m = OÄ _n

log n

ä; ΩÄ _n

log n

ä, m = ωÄ _n

log n

ä. (7-2) From (7-1) and (7-2), we can find when m = OÄ _n

log n

ä, the number of forwarding nodes in the multicast tree is optimal in order sense.

Remark:When m = ωÄ _n

log n

ä, the number of forwarding nodes in TST tree may not be order-optimal. However, in graph theory, finding the minimum connected dominating set of a given graph is proved to be NP-complete [29]. And it also requires global information of network topology. So we consider it an acceptable sacrifice of energy to achieve the feasibility and time-efficiency in practice.

VIII. SIMULATIONS ANDAPPLICATIONS

We first perform extensive simulations to evaluate the empirical performance of Toward Source Tree algorithm, in terms of the length of the multicast tree, message complexity and the number of forwarding nodes engaged in the tree, and then present concrete examples of how Toward Source Tree algorithm can be applied to realistic scenarios. In the simulations, we mainly consider two common distribution patterns: uniform distribution and normal distribution.

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200 400 600 800 1000 6

9 12 15 18 21

TreeLength

m (multicast group size) Steiner Tree

Toward Source Tree

(a) Uniform distribution

200 400 600 800 1000

10 15 20

TreeLength

m (multicast group size) Steiner Tree

Toward Source Tree

(b) Normal distribution

Fig. 8.1: Tree length comparison

0 50 100 150 200

0 2 4 6 8 10

Messages(x1000)

m (multicast group size) n=200

n=600 n=1000

(a) Uniform distribution

0 50 100 150 200

0 2 4 6 8 10

Messages(x1000)

m (multicast group size) n=200

n=600 n=1000

(b) Normal distribution

Fig. 8.2: Message complexity

0 2000 4000 6000 8000

0 2000 4000 6000 8000 10000

Nodenumber

m (multicast group size) (a) Uniform distribution

0 2000 4000 6000 8000

0 2000 4000 6000 8000 10000

Nodenumber

m (multicast group size) (b) Normal distribution

Fig. 8.3: Number of forwarding nodes

A. Performance Evaluation

1) Uniform Distribution: We first consider nodes are uniformly distributed in a unit square and transmission range is set to be r =»_{log n}

n . We explore the effect of multicast group size m on the tree length. Assuming that the network size is fixed as 1000, we obtain the lengths of the Stenier tree and TST tree when the value of m varies. The length of the Steiner tree can be obtained via NewBossa in [30]. Two curves in Figure 8.1(a) describe the relationship between m and the length of the Toward Source Tree as well as the Steiner Tree. It is shown that the length of TST tree is larger than that of the Steiner Tree but quite close to it. According to simulation statistics, the ratio of the tree length achieved by the two algorithms is 1.114 on average. When nodes are uniformly distributed,

Toward Source Tree is a good approximation of the Steiner Tree.

Then we evaluate the message complexity in the construction of TST tree, and explore the relationship among the network size n, the multicast group size m and message complexity. We set the network size n = 200, 600, 1000 respectively, and record the quantity of exchanged messages when multicast group size m varies. Different curves cor- respond to different network sizes in Figure 8.2(a). As can be seen in the figure, the quantity of exchanged messages increases with the multicast group size as well as the network size. It is quite intuitive that the larger network size can result in more exchanged messages. Since the transmission power necessary to maintain the connectivity is less in dense networks than in sparse networks, more relays are engaged in

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multicasting as the network size increases. Hence messages used to contact nodes and inquire routing information become more.

We find that more messages are exchanged when we fix the network size and add more multicast group members. This result is not so intuitive. On the one hand, multicast group members become closer to each other when the multicast group size increases, so they need to search for the appropriate neighbors in a smaller coverage range. Fewer nodes are in- quired within the coverage range, and fewer request messages are sent. On the other hand, when a multicast group member looks for neighbors, more other members might find they are closer to the source. Hence more response messages might be sent back. Total messages increase as more nodes join the multicast group.

Finally, we consider the number of forwarding nodes engaged in multicasting. To derive the statistical properties of TST, we set the network size as 100, 000. In Figure 8.3(a), when the multicast group is small, the first part of the curve indicates that the number of forwarding nodes is O(√

m).

As there are more multicast group members, the number of forwarding nodes grows linearly with the group size.

2) Non-uniform Distribution: We randomly choose the location of source, xs, within the unit square first. For the case of non-uniform distribution, we consider that nodes satisfy the normal distribution: f (x) = ^√¹_2πe⁻^kx−xsk

2

2 , where kx − xsk is the Euclidean distance between the node and the source.

It is possible that the nodes are scattered outside unit square during simulations. If this happens, we relocate these nodes until they are within this unit square.

We evaluate the TST algorithm still in terms of tree length, message complexity and the number of forwarding nodes.

We find the results are quite similar to those in the uniform distribution. In Figure 8.1(b), the length of TST tree is only a little larger than the optimal length in our simulations. The statistics show that the ratio between them is 1.110 on average.

As for the message complexity, Figure 8.2(b) shows that more messages are exchanged among more multicast group members or in denser networks, and the quantity of messages is still O(n). As is shown in Figure 8.3(b), the number of transmitting nodes in the multicast tree is linear with √

m, and becomes linear with m when more nodes participate in the multicast group assuming that the network size keeps unchanged.

B. Practical Applications

TST algorithm can be implemented in practical systems as well as be integrated into sensor network architecture and protocol design to improve the network performance. Many sensor network systems need multicast transmission as they involve different kinds of sensors and interactions between various modules such as the LED lighting system in [31], the localization system in [32] and the building monitoring system in [33]. These make multicast groups naturally form within the systems and multicasting through a minimum multicast tree is a desirable way for these energy-constrained sensor applications. Therefore, we can apply TST algorithm for multicast routing within the aforementioned systems. In

addition, during the tree construction process, TST algorithm only requires single-hop transmission, which conforms to the communication protocols (e.g. zigbee) used in these systems.

Besides, currently, a notable trend in sensor network study is to adopt the IPv6-based architecture [34], in which multicasting is frequently used for scope addressing, discovery and configuration [34]. In those IPv6-based architectures, TST algorithm exhibits an apparent advantage for multicast routing over other algorithms in the sense that the sender does not need to have any prior knowledge of geographical locations of intended destinations before the tree construction. Theses locations can be acquired in the first phase of TST algorithm.

Such property caters to the most common situation of multicast in IPv6 [35].

IX. CONCLUSION ANDFUTUREWORKS

In this paper, we propose a novel algorithm, which we call Toward Source Tree, to generate approximate Steiner Trees in wireless sensor networks. The TST algorithm is a simple and distributed scheme for constructing low-cost and energy-efficient multicast trees in the wireless sensor network setting. We prove its performance measures in terms of tree length, time complexity, and energy efficiency. We show that the tree length is in the same order as, and is in practice very close to, the Steiner tree. We prove its running time is the shortest among all existing solutions. We prove that its message complexity and the number of nodes that participate in forwarding are both order-optimal, yielding high energy efficiency for applications.

For future research, we present several directions to extend the current study of minimum multicast tree construction as follows.

• Our present work mainly focuses on the optimization of path length, energy and computation costs in multicast of wireless sensor networks. It is also interesting to develop an algorithm which optimizes other metrics jointly such as the throughput [39], load balancing [37] and congestion control [37]. The idea that utilizes multiple multicast trees to provide backup routing paths for load balancing [37] is particularly enlightening.

• Our algorithm is currently designed for static networks. It is of great interest to consider multicast tree construction in time-varying networks [38], [39] or mobile networks, where the major technique of tree construction may differ greatly from that in static networks.

• The proposed TST algorithm in our work is a distributed algorithm, and the analysis is performed in the setting of stochastic networks. It suggests an inspiring direction to combine combinatorial optimization methods with stochastic optimization methods [38], [39] to solve multicast routing problems in sensor networks.

ACKNOWLEDGMENT

This work was supported by NSF China (No. 61532012, 61325012, 61271219, 61521062, 61428205, 61602303 and 91438115) and China Postdoctoral Science Foundation.

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APPENDIXA

TREELENGTH IN THEUNIFORMCASE

When we analyze the temporary tree made up of the source and m receivers, transmission range can be ignored since it does nothing with the temporary tree construction.

We establish a two-dimensional coordinate system shown in Figure 1.1. Let the source be the origin, and X-axis as well as Y-axis parallel to the square edge. The whole network is divided into m square cells, and the edge length of each cell is

√1

m. In the coordinate system, the edge length is normalized to be 1, so the intersections in the network have integer coordinates. We use the coordinate of the vertex that is farthest