The Orphan Problem in
ZigBee Wireless Networks
Meng-Shiuan Pan, Student Member, IEEE, Chia-Hung Tsai, and
Yu-Chee Tseng, Senior Member, IEEE
Abstract—ZigBee is a communication standard which is considered to be suitable for wireless sensor networks. In ZigBee, a device (with a permanent 64-bit MAC address) is said to join a network if it can successfully obtain a 16-bit network address from a parent device. Parent devices calculate addresses for their child devices by a distributed address assignment scheme. This assignment is easy to implement, but it restricts the number of children of a device and the depth of the network. We observe that the ZigBee address assignment policy is too conservative, thus usually making the utilization of the address pool poor. Those devices that cannot receive network addresses will be isolated from the network and become orphan nodes. In this paper, we show that the orphan problem can be divided into two subproblems: the bounded-degree-and-depth tree formation (BDDTF) problem and the end-device maximum matching (EDMM) problem. We then propose algorithms to relieve the orphan problem. Our simulation results show that the proposed schemes can effectively reduce the number of orphan devices compared to the ZigBee strategy.
Index Terms—Graph theory, IEEE 802.15.4, network formation, orphan problem, wireless sensor network, ZigBee.
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I
NTRODUCTIONT
HE recent progress of wireless communication and embedded microsensing MEMS technologies has made wireless sensor networks (WSNs) feasible. A lot of research works have been dedicated to this area, including energy-efficient MAC protocols [11], [27], routing and transport protocols [8], [13], self-organizing schemes [16], [24], sensor deployment and coverage issues [14], [22], and localization schemes [6], [23]. Applications of WSNs include habitat monitoring [2], wildfire monitoring [1], mobile object tracking [21], [25], and navigation [20], [26].Recently, several WSN platforms have been developed, such as MICA, MICAz, Imote2, TelosB [4], TI CC2431 [5], and Jennic JN5121 [3]. For interoperability purpose, most platforms have adopted ZigBee [29] as their communication protocols. ZigBee adopts IEEE 802.15.4 standard [15] as its physical and MAC protocols and solves the interoperability issues from the physical layer to the application layer.
ZigBee supports three kinds of network topologies, namely star, tree, and mesh networks. A ZigBee coordinator is responsible for initializing, maintaining, and controlling the network. In a star network, all devices have to directly connect to the coordinator. For tree and mesh networks, devices can communicate with each other in a multihop fashion. The network is formed by one ZigBee coordinator and multiple ZigBee routers. A device can join a network as an end device by associating with the coordinator or a router. Fig. 1 shows a ZigBee tree network.
In ZigBee, each node has a permanent 64-bit MAC address. A device is said to successfully join a network if it can obtain a 16-bit network address from the coordinator or a router. Using a short network address is for simplicity and for saving communication bandwidths. Before forming a network, the coordinator needs to decide three important system parameters: the maximum number of children of a router (Cm), the maximum number of child routers of a router (Rm), and the depth of the network (Lm). Note that a child of a router can be a router or an end device, so Cm Rm. Given Cm, Rm, and Lm, ZigBee has suggested a distributed address assignment scheme. While simple, the scheme may prohibit a node from accepting a child router/device as constrained by these parameters. We say that a node becomes an orphan node when it cannot associate with any parent router but there are still unused address spaces remaining. We call this the orphan problem. For example, in Fig. 1, the router-capable device A has two potential parents B and C. Router B cannot accept A as its child because it has reached its maximum capacity of Cm¼ 5 children. Router C cannot accept A either because it has reached the maximum depth of Lm ¼ 2. So, A will become an orphan node. The orphan problem will worsen as the network scares up. We will further support this claim in Section 2.2 through simulations and real experi-ments. The orphan problem can be relieved if proper actions are taken. For example, in Fig. 1, if router E is connected to router D, router B will have capacity to accept A.
Given Cm, Rm, and Lm, we show that the orphan problem can be divided into two subproblems: 1) connect-ing as many routers as possible to form a tree and 2) connecting as many end devices as possible to the above tree. The first subproblem involves the router-capable devices only and can be modeled as a bounded-degree-and-depth tree formation (BDDTF) problem. We prove that this
. The authors are with the Department of Computer Science, National Chiao-Tung University, 1001 Ta Hsueh Road, Hsinchu 30010, Taiwan, ROC. E-mail: {mspan, chiahung, yctseng}@cs.nctu.edu.tw.
Manuscript received 21 Sept. 2007; revised 7 Nov. 2008; accepted 3 Mar. 2009; published online 12 Mar. 2009.
For information on obtaining reprints of this article, please send e-mail to: tmc@computer.org, and reference IEEECS Log Number TMC-2007-09-0291. Digital Object Identifier no. 10.1109/TMC.2009.60.
subproblem is in fact NP-complete. The second subproblem needs to connect as many end devices to the above tree as possible constrained by router’s capacities and can be modeled as an end-device maximum matching (EDMM) problem. We prove that the EDMM problem is computa-tionally feasible and then exist an optimal algorithm to solve it. To summarize, our approach involves two stages. The first stage will try to relieve the BDDTF problem by connecting more routers. Based on the result, the second stage will be able to connect the largest number of end devices to the tree.
Several works have investigated the bounded-degree spanning tree problem. Czumaj and Strothmann [10] propose polynomial-time graph algorithms when additional con-nectivity and maximum degree of a graph are given. However, the depth constraint is not considered. Konemann and Ravi [18] introduce an approximation algorithm, which can find a spanning tree with a maximum degree of OðK þ logjV jÞ, where K is the degree constraint and V is the set of nodes in the graph. The result is not applicable to our case because it does not consider the depth constraint and the number of children of a node is not bounded. In [17], a polynomial-time algorithm is proposed to construct a spanning tree with a bounded degree and a bounded diameter. However, this algorithm is designed for complete graphs, which is not the case in a ZigBee network. Also, these works are not tailored to ZigBee specifications. Some works have focused on address configuration. Bhatti and Yue [7] propose a network address assignment scheme based on the address assignment rule for an n-dimensional hypercube. Interestingly, when the ZigBee network struc-ture is close to an n-cube, this scheme can indeed reduce the waste of address space. However, in practice, a WSN is typically randomly deployed on a 2D plane, which is unlikely to be similar to a high-dimensional n-cube. In fact, the scheme still suffers from the compatibility issue when
the n-cube is incomplete and the orphan problem may still exist. Besides, additional overhead will be incurred to ensure that no duplicate addresses are assigned to nodes. Kulkarni et al. [19] organize a network into concentric tiers around the sink and do not employ unique per-node addressing. When transmitting, a node will randomly choose an identifier for one-hop routing. This scheme is address light, but it is only suitable for reporting scenarios and cannot support point-to-point routing. In [28], an adaptive block addressing scheme is introduced for net-work auto-configuration purpose. It takes into account the actual network topology and thus is fully topology adaptive. However, because the size of the address pool allocated by the coordinator is depended on the topology, addition of new nodes can cause the whole network to conduct address update. Moreover, this scheme needs two phases to initialize its adaptive tree, which is different from the ZigBee association procedure and is thus not compatible with ZigBee.
The main contributions of this paper are threefold. First, this is the first work that points out the orphan problem in ZigBee wireless networks. Second, we show that the existence orphan is an inherent concern no matter how one sets the Cm, Rm, and Lm constraints. We verify this claim through different configurations and parameter settings. A larger Cm or Rm will impose more memory requirement on routers and packets, while a larger Lm will also induce longer network delays. Third, we connect the orphan problem to NP-complete and classical algorithms and then propose network formation heuristics that can effectively reduce the number of orphans with given Cm, Rm, and Lm.
The rest of this paper is organized as follows: Prelimin-aries are given in Section 2. Sections 3 and 4 present our algorithms. Simulation results are given in Section 5. Finally, Section 6 concludes this paper.
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RELIMINARIES2.1 Overview of IEEE 802.15.4 and ZigBee Standards
IEEE 802.15.4 [15] specifies the physical and data link protocols for low-rate wireless personal area networks (LR-WPAN). In the physical layer, there are three frequency bands with 27 radio channels. Channel 0 ranges from 868.0 to 868.6 MHz, which provides a data rate of 20 kbps. Channels 1-10 work from 902.0 to 928.0 MHz and each channel provides a data rate of 40 kbps. Channels 11-26 are located from 2.4 to 2.4835 GHz, each with a data rate of 250 kbps.
IEEE 802.15.4 devices are expected to have limited power, but need to operate for a longer period of time. Therefore, energy conservation is a critical issue. Devices are classified as full function devices (FFDs) and reduced function devices (RFDs). IEEE 802.15.4 supports star and peer-to-peer topologies. In each PAN, one device is designated as the coordinator, which is responsible for maintaining the net-work. An FFD has the capability of serving as a coordinator or associating with an existing coordinator/router and becoming a router. An RFD can only associate with a coordinator/router and cannot have children.
According to ZigBee standard [29], a ZigBee network is formed by the following procedures. Devices that are coordinator capable and do not currently join a network can be a candidate of a ZigBee coordinator. A device that desires to be a coordinator will scan all channels to find a suitable one. After selecting a channel, this device broadcasts a beacon containing a PAN identifier to initialize a PAN. A device that hears a beacon of an existing network can join this network by performing the association procedures and specifying its role, as a ZigBee router or an end device. If the device hears multiple beacons, it chooses the beacon sender with the smallest hop count to the coordinator. The beacon sender will determine whether to accept this device or not by considering its current capacity and its permitted association duration. If the device is successfully associated, the association response will contain a short 16-bit address for the request sender. This short address will be the network address for that device.
In ZigBee, network addresses are assigned to devices by a distributed address assignment scheme. The coordinator determines Cm, Rm, and Lm. The coordinator and each router can have at most Rm child routers and at least Cm Rmchild end devices. Devices’ addresses are assigned in a top-down manner. For the coordinator, the whole address space is logically partitioned into Rm þ 1 blocks. The first Rm blocks are to be assigned to the coordinator’s child routers and the last block is reserved for the coordinator’s own child end devices. From Cm, Rm, and Lm, each node computes a parameter called Cskip to derive the starting addresses of its children’s address pools. The Cskip for the coordinator or a router in depth d is defined as
CskipðdÞ ¼ 11þ Cm ðLm d 1Þ;þ Cm Rm CmRmLmd1 if Rm¼ 1;
1 Rm ; otherwise: (
ð1Þ
The coordinator is said to be at depth 0; a node which is a child of another node at depth d is said to be at depth dþ 1. Address assignment begins from the ZigBee co-ordinator by assigning address 0 to itself. If a parent node at depth d has an address Aparent, the nth child router is
assigned to address Aparentþ ðn 1Þ CskipðdÞ þ 1 and
nth child end device is assigned to address Aparentþ Rm
CskipðdÞ þ n. An example of the address assignment is shown in Fig. 1. The Cskip of the coordinator is obtained from (1) by setting d ¼ 0; Cm ¼ 5; Rm ¼ 3, and Lm ¼ 2. Then, the child routers of the coordinator will be assigned to addresses 0 þ ð1 1Þ 6 þ 1 ¼ 1, 0 þ ð2 1Þ 6 þ 1 ¼ 7, 0þ ð3 1Þ 6 þ 1 ¼ 13, etc. The address of the only child end device of coordinator is 0 þ 3 6 þ 1 ¼ 19. Note that, in ZigBee, the maximum network address capacity is 216¼ 65;536. This restricts that the coordinator cannot
decide the Cm, Rm, and Lm arbitrarily. 2.2 The Orphan Problem
By the above rules, the coordinator and routers can accept more routers and devices if they still have capacities. However, when a node cannot join the network because all its neighbors have run out of their address capacities, we say the node has become an orphan. This situation may be relieved if there are remaining address spaces in other places of the network. Fig. 1 is a small-scale orphan problem. Here, we present some real implementation results of the ZigBee network formation procedure based on Jennic JN5121 [3]. Fig. 2 shows a deployment of 49 routers on a 360 cm 360 cm grid area. The grid size is 60 cm 60 cm. Nodes’ transmission power is set to 150 mW, which can reach a transmission range around 100-200 cm. For each combination of (Rm; Lm), we conduct five experiments and observe the average number of orphans and the average end-to-end delay from the deepest node to the coordinator. Table 1 shows our experimental results. We can see that regardless of different (Rm; Lm)
combinations, there always exist 30-70 percent orphans. Although a smaller Rm can lead to fewer orphans, it also results in longer end-to-end delay.
Since it is infeasible to conduct large-scale real tests, we also use simulations to make more observations. In Fig. 3, 800 nodes are randomly deployed on a circular field with a radius of 230 m. Nodes’ transmission range is 25 m. To
reduce orphans, given an Rm, we will set Lm to the maximum possible value. So, we set ðRm; LmÞ ¼ ð4; 7Þ, (3, 9), and (2, 15) (these Lm values are the maximum possible ones for the given Rm) and Cm ¼ Rm (which means no end devices). In Fig. 3a, since Lm ¼ 7, the network cannot grow too deep, so a lot of nodes are left as orphans. In Fig. 3b, since a larger Lm ¼ 9 is used, there are much fewer orphans. However, there are still a lot of nodes at the edge unable to connect to the network. In Fig. 3c, with a larger Lm ¼ 15, orphans are significantly reduced. However, with the same setting, Fig. 3d shows a more extreme case where all neighboring nodes of one of the coordinator’s children have been associated with other routers, making it a leaf node. This actually wastes a lot of address spaces. A smaller Rm may result in a nonshortest path from a router to the coordinator, thus causing a longer transmission delay and even more orphans if their routing path lengths exceed the constraint of Lm. In fact, assuming Cm ¼ Rm, a router at depth d serving as a leaf implies a loss of1Rm1RmLmdþ1 address spaces. This is why a larger part of the network at the lower right side is unable to join the network. Note that this could happen
TABLE 1
The Percentages of Orphans and End-to-End Delays under Different Combinations of (Rm; Lm) in the Test Scenario of Fig. 2
Fig. 3. ZigBee network formation examples with (Rm; Lm) equal to (a) (4, 7), (b) (3, 9), and (c-d) (2, 15). There are 461, 341, 120, and 351 orphan nodes, respectively.
because the ZigBee tree formation is asynchronous and nodes will compete to connect to nearby routers. These observations motivate us to design our schemes by trying to maintain sufficient children for nodes nearby the coordinator.
While both routers and end devices may become orphans, there capabilities are different. A router may accept more routers/devices, while an end device cannot. Further, their address calculation rules are also different as reviewed in Section 2.1. For these reasons, we divide the orphan problem into two subproblems: BDDTF and EDMM problems. In the first BDDTF problem, we consider only router-capable devices and model the network by a graph Gr ¼ ðVr; ErÞ, where Vrconsists of all router-capable devices
and the coordinator t and Er contains all symmetric
communication links between nodes in Vr. Given
para-meters Cm; Rm, and Lm such that Cm Rm, the goal is to assign parent-child relationships to nodes such that as many vertices in Vrcan join the network as possible. Below,
we formulate this problem to a tree formation problem. Definition 1. Given Gr¼ ðVr; ErÞ, Rm, Lm, and an integer
N jVrj, the BDDTF problem is to construct a tree T rooted
at t from Grsuch that T satisfies the ZigBee tree definition and
T contains at least N nodes.
In [12], it is shown that the Degree-Constrained Spanning Tree (DCST) as defined below is NP-complete.
Definition 2. Given Gc¼ ðVc; EcÞ and a positive integer
Kc jVcj, the DCST problem is to find a spanning tree Tc
from Gcsuch that no vertex in Tchas a degree larger than Kc.
Theorem 1.The BDDTF problem is NP-complete.
Proof.To prove that the BDDTF problem is NP-complete, we first show that the problem belongs to NP. Given a tree T in Gr, it is easy to check whether T satisfies the
constraints of Rm and Lm and contains more than N nodes in polynomial time. Next, to prove that the BDDTF problem is NP-complete, we reduce the DCST problem to it. Let Gc¼ ðVc; EcÞ and integer Kc represent
an arbitrary instance of the DCST problem. We can transform Gc to an instance of the BDDTF problem Gr
by setting Vr¼ Vc, Er¼ Ec, N ¼ jVcj, Rm ¼ Kc, and
Lm! 1 in polynomial time. We now claim that we can find a Tcfor the DCST problem if and only if we can find
a ZigBee-conformed tree T containing N nodes. To prove the if part, if there is a ZigBee-conformed tree T in Gr to connect N ¼ jVcj ¼ jVrj nodes with parameters
Rm¼ Kc and Lm ! 1, we can find a tree Tc in Gc to
connect N ¼ jVcj nodes as a spanning tree in Gc such
that no vertex in Tc has a degree larger than Kc.
Conversely, to prove the only if part, suppose that there is a spanning tree Tc to connect the nodes in Gc. Since
Rm¼ Kc and Lm ! 1, there must exist a
ZigBee-conformed tree T ¼ Tc in Gr containing N ¼ jVcj jVrj
nodes. So, the theorem is proved. tu By Theorem 1, we can see that the first subproblem is intractable. Definition 1 and Theorem 1 imply that the orphan problem is inevitable with any Rm and Lm. This also implies that there is no optimal decision for choosing Cm, Rm, and Lm to avoid the orphan problem.
After solving the BDDTF problem, we already have a tree T containing the coordinator and some routers. In the second EDMM subproblem, we will connect nonrouter-capable devices to the tree T constructed earlier following the ZigBee definition such that as many end devices are connected to T as possible. Toward this goal, we model the network by a bipartite graph Gd¼ ðf ^Vr[ Veg; EdÞ, where ^Vr
consists of the coordinator and all routers in T , excluding those at depth Lm (note that those at depth Lm are unable to accept more children), Veconsists of all end devices, and
Edcontains all symmetric communication links between ^Vr
and Ve. Each vertex v 2 ^Vr can accept at most Cv
ðCm RmÞ end devices. From Gd, we construct another
bipartite graph ~Gd¼ ðf ~Vr[ ~Veg; ~EdÞ as follows:
1. From each vertex v 2 ^Vr, generate Cv vertices
v1; v2; . . . ; vCv in ~Vr.
2. From each vertex u 2 Ve, generate a vertex u in ~Ve.
3. From each edge (v; u) in Ed, where v 2 ^Vrand u 2 Ve,
connect each of the Cv vertices v1; v2; . . . ; vCv
gener-ated in rule 1 with the vertex u genergener-ated in rule 2. These edges form the set ~Ed.
Intuitively, we duplicate each v 2 ^Vr into Cv vertices,
and each edge ðv; uÞ 2 Ed into Cv edges. These Cv vertices
and Cv edges reflect the capability of router v to accept
end devices. It is clear that ~Gd is a bipartite graph with
edges connecting vertices in ~Vr and vertices in ~Ve only.
Since each vertex in ~Vr is connected to at most one vertex
in ~Ve, this translates the problem to a maximum matching
problem as follows:
Definition 3.Given a graph ~Gd¼ ðf ~Vr[ ~Veg; ~EdÞ, the EDMM
problem is to find a maximum matching of ~Gd.
Given router tree T , the maximum matching problem in Definition 3 can be solvable in polynomial time [9]. Note that even with maximum matching, it does not guarantee that all end devices will be connected, so orphan end devices may still exist after solving the second subpro-blem. Below, we will propose several schemes for these two subproblems.
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LGORITHMS FOR THEBDDTF P
ROBLEMWe propose two algorithms for the BDDTF problem. In our algorithms, we will repeatedly generate several BFS trees from Gr. For each tree being generated, we may decide to
truncate some nodes if the tree is not conformed to the ZigBee definition. The truncation is done based on nodes’ association priorities in the tree. Below, we show how such priorities are defined, given a BFS tree T in Gr:
. A node x has a higher priority than another node y if the subtree rooted at x in T has more nodes than the subtree rooted at y.
. If the subtrees rooted at nodes x and y have the same number of nodes, the one with less potential parents has a higher priority. A node regards a neighbor as a potential parent if this neighbor has a smaller hop count distance to the root in T than itself.
The above definitions are based on the considerations of address space utilization. The first rule is so defined
because node x may have a better utilization. The second rule is so defined because a node with less potential parents is more likely to encounter difficulty when trying to attach to the network. For example, in Fig. 4, if Rm ¼ 3, the coordinator will choose nodes A; B, and C as its child routers since they have larger subtrees. Similarly, B will choose D, E, and F as its child routers. However, if Rm ¼ 2, the coordinator will choose A and B as its child routers. Further, B will choose D and E as its child routers. Node F is not selected because it has more (two) potential parents and thus has a higher probability to be connected in later stages of the formation.
3.1 Centralized Span-and-Prune Algorithm
Given a graph Gr¼ ðVr; ErÞ, our goal is to find a tree T ¼
ðVT; ETÞ from Gr conforming to the ZigBee tree definition.
The algorithm consists of a sequence of iterations. Initially, T contains only the coordinator t. Then, in each iteration, there are two phases: Span and Prune. In the Span phase, we will pick a node in T , say x, and span from x a subtree T0to include as many nodes not yet in T as possible. Then, we attach T0 to T to form a larger tree. However, the new
tree may not satisfy the ZigBee definition. So, in the Prune phase, some of the newly added nodes in T0 may be trimmed. The resulting tree is then passed to the next iteration for another Span-and-Prune phases. This is repeated until no more nodes can be added. Each node in the network will be spanned at most once. To keep track of the nodes yet to be spanned, a queue Q will be maintained. The algorithm is presented below.
1. Initially, let queue Q contains only one node t. Let the depth of t to zero. Also, let the initial tree T ¼ ðftg; ;Þ.
2. Span phase: Check if Q is empty. If so, the algorithm is terminated and T is the final ZigBee tree. Otherwise, let x ¼ dequeueðQÞ and construct a spanning tree T0 from x as follows: Assuming the depth of x in T to be depthðxÞ, we try to span a
subtree from x with height not exceeding Lm depthðxÞ in Grin a breadth-first manner by including
as many nodes in Vr VT[ fxg as possible. Let the
resulting tree be T0.
3. Prune phase: Attach T0 to T by joining node x. Still,
name the new tree T . Since some of the nodes in T0
may violate the Rm parameter, we traverse nodes in T0from x in a breadth-first manner to trim T . a. When visiting a node, say y, set y as “traversed”
and check the number of children of y. If y has more than Rm children, we will compute their priorities based on T0(refer to the definitions of
nodes’ priorities in a tree given in the beginning of this section). Only the Rm highest prioritized children will remain in T , and the other children will be pruned from T .
b. When each node, say y0, that is pruned in step 3a
or 3b, let treeðy0Þ be the pruned subtree rooted at
y0. Since treeðy0Þ is pruned, we will try to attach y0
to another node n in T0if n satisfies the following conditions: 1) n is neighboring to y0 but not a
descendant of y0, 2) n is not traversed yet, and
3) depthðnÞ þ 1 þ heightðtreeðy0ÞÞ Lm. If so, we
will connect the subtree treeðy0Þ to node n. If
there are multiple such candidates, the one with a lower depth is connected first. If no such node n can be found, y prunes all its children. Then, for each pruned child, we recursively perform this step 3b to try to reconnect it to T0. This is repeated
until no further reconnection is possible. 4. After the above pruning, call the resulting tree T . For
nodes that are newly added into T in step 3, insert them into queue Q in such a way that nodes with lower depth values are inserted first (these nodes will go through Span-and-Prune phases again). Then, go back to step 2.
To summarize, step 3a is to prune those nodes violating the Rm constraint. In order to allow more vertices to join the network, step 3b tries to recursively reconnect those pruned subtrees to T0. Step 4 prepares newly joining nodes in Q for
possible spanning in step 2.
Fig. 5 illustrates an example. When being traversed, y decides to prune y0 and keep A; B, and C as children.
Step 3b will try to reconnect y0 to C or D, which are the
neighbors of y0in T0and are not traversed. In this example, only C can be considered because connecting to D violates the depth constraint Lm.
The computational complexity of this algorithm is analyzed as follows: The iteration from step 2 to step 4 will be executed at most jVrj times. In each iteration, the
complexity of constructing the tree T0 in step 2 is OðN2Þ,
where N ¼ jVrj jVTj is the number of nodes still not
connected to T . Step 3 checks all nodes in T0 and will be executed at most OðNÞ times. For a run in step 3 (assume visiting node y), the cost contains: 1) In step 3a, y can use a linear search method to find Rm highest prioritized children and the computational cost is OðDÞ, where D is the degree of Gr. 2) Since the subtree size of y is at most OðNÞ and a pruned
node checks at most OðDÞ neighbors to find its new parent,
Fig. 4. Examples of priority assignment in our algorithm. (The numbers in triangles indicate sizes of the corresponding subtrees.)
the cost of step 3b in a run is OðNDÞ. So, in one iteration, the time complexity of step 3 will be OðNðD þ NDÞÞ ¼ OðN2DÞ.
Step 4 sorts new nodes of T according to their depth values, so the time complexity is OðN2Þ. The complexity in
each iteration is OðN2þ N2Dþ N2Þ ¼ OðN2DÞ ¼ OðjV rj2DÞ.
Since there are at most jVrj iterations, the overall time
complexity of this algorithm is jVrj OðjVrj2DÞ ¼ OðjVrj3DÞ.
Although this complexity looks somewhat too high, we believe that using jVrj to bound N is too strong. Our
experimental experience reveals that the value of N will degrade quickly because most nodes will be connected to the tree T after several iterations. So, the time complexity of an iteration is quite small in practice.1
3.2 Distributed Depth-then-Breadth-Search Algorithm
The above Span-and-Prune algorithm is a centralized one. In this section, we present a distributed algorithm, which does a depth-first search followed by a breadth-first-like search. The depth-first search tries to form some long, thin back-bones, which are likely to pass through high-node-density areas. Then, from these backbones, we span the tree in a breadth-first-like manner. The algorithm is presented below. 1. Depth probing: Given a graph Gr¼ ðVr; ErÞ, the
coordinator t needs to probe the depth of the tree first. A Probeðsender addr; current depth; LmÞ pack-et is used for this purpose. The Probe packpack-ets are flooded in a BFS-like manner, until a depth Lm is reached. Note that following the definition of ZigBee, before the final tree is determined, nodes will use their 64-bit MAC addresses to communicate with each other in this stage.
This algorithm begins by the coordinator t flood-ing a ProbeðAddrðtÞ; 0; LmÞ packet in the network, where AddrðtÞ is t’s address. When a node v receives a Probeðsender addr; current depth; LmÞ packet, it does the following:
a. If this is the first time v receiving a Probe() packet, v sets its parent parðvÞ ¼ sender_addr and its depth depthðvÞ ¼ current depth þ 1. If depthðvÞ < Lm; v rebroadcasts a ProbeðAddrðvÞ; depthðvÞ; LmÞ packet.
b. If this is not the first time v is receiving a Probe() packet, it checks if depthðvÞ > current depth þ 1 is true. If so, a shorter path leading to the coordinator is found. So, v sets its parent parðvÞ ¼ sender addr and its depth depthðvÞ ¼ current depthþ 1. If depthðvÞ < Lm, v rebroad-casts a ProbeðAddrðvÞ; depthðvÞ; LmÞ packet. Note that to ensure reliability, a node may periodi-cally rebroadcast its Probe() packet. And each node can know the number of its potential parents by the Probe() packet.
2. Probe response: After the above probing, a BFS-like tree is formed. Each node then reports to its parent a Report() packet containing 1) the size of the subtree rooted by itself and 2) the height of the subtree rooted by itself. In addition, each node v will compute a tallest childðvÞ, which records the child of v whose subtree is the tallest among all child subtrees.
3. Backbone formation: After the coordinator t receives all its children’s reports, it will choose at most Rm children with the larger subtree sizes as backbone nodes. This is done by sending a Backbone() message to each of the selected children. When a node v is receiving a Backbone() message, it further invites its child with the tallest subtree, i.e., node tallest childðvÞ, into the backbone by sending a Backbone() packet to tallest childðvÞ. After this phase, t has constructed a backbone with up to Rm subtrees, each as a long, thin linear path.
4. BFS-like spanning: After the above backbone forma-tion, the coordinator can broadcast beacons to start the network. A node can broadcast beacons only if it has successfully joined the network as a router (according to ZigBee, this is achieved by exchanging Association_Request and Association_Response with its parent). In our rule, a backbone node must associate to its parent on the backbone, and its parent must accept the request. For each nonback-bone node, it will compete with each other in a distributed manner by its association priority, where the association priority is defined by the size of the subtree rooted by this node in the BFS-like tree formed in step 1. A nonbackbone node sends its association requests by specifying its priority. A beacon sender should wait for association requests for a period of time and sorts the received requests by their priorities. Then, the beacon sender can accept the higher priority ones until its capacity (Rm) is full.
1. By our simulation, in average, almost 75 percent of nodes can be connected to the tree T in first iteration. After second iteration, almost 88 percent of nodes can be connected to the tree T .
Compared to the ZigBee protocol, this algorithm requires nodes to broadcast three extra packets (Probe(), Report(), and Backbone()). A Probe() packet needs to flood to the whole network and thus needs an efficient broadcast scheme (this is beyond the scope of this paper). Let n be the total number of nodes in the network. Below, we will show that the additional time and message complexity against ZigBee are OðLmÞ and OðnÞ, respectively.
To see the additional time complexity, observe that the coordinator t will issue Probe() to check the depth of the tree and a node v will rebroadcast it only when depthðvÞ < Lmor it can find a shorter path to t. So, the additional time complexity will be bounded by OðLmÞ. In the process of finding the tallest child, each node will report to its parent. Because the reporting is started from leaf nodes, the additional time complexity is also bounded by OðLmÞ. Finally, the backbone formation will be triggered by t to construct a long, thin linear path. Again, the additional cost is OðLmÞ. Overall, the additional time complexity of our algorithm against ZigBee is bounded by OðLmÞ.
To see the additional message complexity of our algo-rithm, observe that the probing step is similar to a BFS tree construction, so the message complexity is OðnÞ. In the step of finding tallest child, because each node will only report to its parent once, the message cost is also OðnÞ. Finally, t will send Backbone() packets to its Rm selected children, who will further invite their children with the tallest subtrees. The cost is at most OðRm þ Rm ðLm 1ÞÞ. Overall, the additional message complexity against ZigBee is OðnÞ.
4
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LGORITHMS FOR THEEDMM P
ROBLEMIn Section 2.2, we have formulated the EDMM problem as a maximum matching problem in a bipartite graph. It is already known that there exists optimal polynomial-time algorithms to solve this problem. Below, we show how to use the maximum matching algorithm in [9] to solve our problem. Recall that after connecting routers in the BDDTF problem, we will obtain a bipartite graph ~Gd¼ ðf ~Vr[ ~Ve; ~EdgÞ. From
~
Gd, we can find a maximum matching as follows:
1. Try a greedy approach by first matching those vertices with small degrees. We denote this matching edge set as M. Then, we transform the undirected graph ~Gd to a directed graph ~Gd
0
by directing the edges in M to point from ~Veto ~Vr and directing the
edges not in M to point from ~Vrto ~Ve.
2. Apply a DFS search on ~Gd 0
starting from any node of ~
Vr. If ~Gd 0
has any alternating path [9] P staring from ~Ve
and ending at ~Vr, we mark all edges of P belonging
to M as not belonging to M, and vice versa. (It is easy to see that P must be of an odd length.) Then, we reconstruct ~Gd
0
based on the new M.
3. Repeat step 2 until each node in ~Vr has been
searched once. Then, the final M is a maximum matching of ~Gd.
As shown in [9], the complexity of the above procedure is Oððj ~VrjÞðj ~Vrj þ j ~Vej þ j ~EdjÞÞ.
The above algorithm is a centralized one. In practice, we need a distributed algorithm to allow routers to connect end
devices in a decentralized way. Below, we present a distributed algorithm, which has a greedy phase followed by a probing phase. In the greedy phase, the routers will accept end devices, which have less potential associable routers. Then, each orphan router will try to probe a three-hop alternating path P as discussed above to relieve its orphan situation. The probing process can be executed before a timer Tprobe expires. After Tprobe expires, an end device cannot
change its parent.
1. Greedy phase: Each router will periodically broadcast beacon packets with a reserve bit to indicate whether it still has capacity to accept more end devices. Each end device e will overhear beacons from routers and, based on these beacons, compute the number Neof
its neighbor routers with their reserve bits on. In the case of Ne¼ 0, e is a potential orphan. If Ne> 0, e
will try to perform the association procedure by providing its Ne value to routers. Routers simply
accept as many end devices as possible with smaller Nefirst (intuitively, a smaller Nemeans less potential
parents).
2. Probing phase: After the greedy phase, each asso-ciated end device will broadcast its new Ne value
(note that this value counts its parent as well as those neighboring routers which still have remaining capacities). For an orphan end device e (with Ne¼ 0), it can try to resolve its situation as follows:
a. A Probe() packet2 can be sent by e to any neighboring router r.
b. When r receives the Probe() from e; r can check if it has a child end device e0such that Ne0 2. If
so, r will send a Probe() packet to e0to ask e0to
switch to another router.
c. On reception of r’s Probe(), e0 will try to
associate with another router other than its current one. If it succeeds, a Probe_Ack() will be returned to r; otherwise, a Probe_Nack() will be returned.
d. When r receives the result from e0, a Probe_Ack()
or a Probe_Nack() will be returned to e accord-ingly. In the former case, e will associate with r. In the latter case, e will try another router by going back to step a, until timer Tprobe expires.
The above protocol allows an orphan to probe three-hop paths. It is not hard to extend this protocol to allow probing longer paths at higher costs (we leave it to the audience). Next, we analyze the additional time and message complex-ity required for this protocol against the original ZigBee. The additional time complexity will be bounded by
OðTprobeÞ. The additional message complexity is incurred
by the probing phase. Our protocol has a progressive property because each probe may reduce one orphan end device. So, the extra cost will be bounded by a polynomial function of the number of end devices. If longer alternating paths are explored, the cost will be higher. However, one may use the timer Tprobe to bound the cost.
5
S
IMULATIONR
ESULTSA simulator has been implemented based on Java. First, we compare our Span-and-Prune algorithm (SP) and Depth-then-Breadth-Search algorithm (DBS) against the ZigBee algorithm (ZB) in their capabilities to relieve the BDDTF problem under random and regular node deployment. Next, through varying the combinations of Cm, Rm, and Lm, we further show the superiority of SP and DBS even under different-node-density environment. We also investigate in more details the advantage of the backbone probing in our DBS scheme. Finally, we will show the performance of our distributed EDMM scheme to connect end devices.
5.1 Random versus Regular Networks
In Fig. 6, we test a 90-degree-sector area with a radius of 200 m and with 400 randomly deployed router-capable nodes each with a transmission range of 32 m. We set Cm ¼ Rm¼ 2 and Lm ¼ 8. ZB, SP, and DBS algorithms incur
110.2, 13.7, and 37.9 orphan routers, respectively, in average. DBS only incurs slightly more orphans than the centralized SP does. In particular, we see that both SP and DBS may leave some nodes nearby the coordinator unconnected due to the Rm constraint but can reach farther nodes. Fig. 7 considers a 25 25 regular grid with a grid distance of 10 m. Nodes’ transmission distances are 23 m. We set Cm ¼ Rm¼ 4 and Lm ¼ 7. In this case, ZB, SP, and DBS incur 70.2, 37.2, and 40.4 orphan routers, respectively. ZB per-forms the worst. DBS perper-forms closely to the centralized SP. 5.2 Impact of Link Density on the BDDTF Problem We simulate 800 randomly distributed router-capable devices in a circular region with a radius of 200 m with the coordinator at the center. We restrict Cm ¼ Rm and vary Rm and Lm to observe the number of orphan routers. Table 2 shows the address spaces of different combinations of Cm, Rm, and Lm, which can clearly accommodate much
Fig. 6. Network formation results in a 90-degree-sector area by (a) ZB, (b) SP, and (c) DBS.
Fig. 7. Network formation results in a 25 25 grid area by (a) ZB, (b) SP, and (c) DBS. TABLE 2
more than 800 nodes ideally. We set nodes’ transmission ranges to 35 m and 60 m. Since the network area and the number of nodes are fixed, a larger transmission range actually means denser links among neighboring nodes. As Fig. 8 shows, denser links do lead to much less orphans. However, transmission range depends on hardware fea-tures as well as deployment needs, which are sometimes uncontrollable. In addition, we see that in all cases, SP performs the best, followed by DBS and then ZB.
5.3 Impact of Rm and Lm on the BDDTF Problem Fig. 8 indicates that increasing Lm can more effectively reduce orphan routers as opposed to increasing Rm. In Fig. 9, we further fix Lm and vary Rm to conduct our tests. We see that the orphan situation can benefit less by enlarging Rm under low link density. However, as the link density is higher, enlarging Rm is still quite effective. This is because a higher link density will allow a node to have more potential children. Our scheme can save space for Rm and thus allow a larger space for Lm. For example, in Fig. 8a, SP incurs nearly the same number of orphan routers in the (3, 7) case (respectively, the (3, 8) case) as ZB does in the (3, 8) case (respectively, the (3, 9) case). In Fig. 9b, SP incurs nearly the same number of orphan routers when Rm ¼ 6 as ZB does when Rm ¼ 11. Saving the space for Rm can allow a larger Lm, which can in turn relieve the orphan problem. This shows the benefit of our SP scheme.
5.4 Impact of the Backbone Formation in DBS In DBS, there is a backbone formation to choose subtrees of larger sizes. We modify DBS to a DBS-NB (NB ¼ nonbackbone) scheme, which works similar to DBS but does not form backbones as in DBS (i.e., all nodes in step 4 are considered as nonbackbone ones). The results are in Fig. 10, which clearly shows the importance of the formation process.
Fig. 9. Comparison of the number of orphan routers by fixing Lm and varying Rm when the transmission range is (a) 35 m and (b) 60 m. Fig. 8. Comparison of the number of orphan routers when the transmission range is (a) 35 m and (b) 60 m.
Fig. 10. Comparison of DBS and DBS-NB schemes (800 routers; a circular sensing field of a radius of 200 m; nodes’ transmission range 35 m).
5.5 The EDMM Problem
In these experiments, we simulate the networks with both routers and end devices. We randomly place 800 routers and 8,000 end devices in a circular area of radius 200 m with the coordinator at the center. Routers’ transmission ranges are 35 m, and end devices’ are 15-30 m. An end device can only associate to a router located within its transmission range. We set Cm ¼ 15, Rm ¼ 3, and Lm¼ 8. We use SP to connect routers and then apply the centralized maximum matching scheme (Max-Match), our distributed matching scheme (Dis-Match), or ZigBee (ZB) to connect end devices. In all cases of end devices’ transmission ranges, Fig. 11 shows that Dis-Match can significantly reduce orphan end devices as opposed to ZB, and perform quite close to Max-Match.
6
C
ONCLUSIONSIn this paper, we have identified a new orphan problem in ZigBee-based wireless sensor networks. We show that the problem is nontrivial because a device is not guaranteed to join a network even if there are remaining address spaces in other places of the network. We model this orphan problem as two subproblems, namely the BDDTF problem and the EDMM problem. We prove that the BDDTF problem is NP-complete and propose a two-stage network formation policy, which can effectively relieve the orphan problem. Compared to the network formation scheme defined in ZigBee, our algorithms can significantly reduce the number of orphan routers. Contrarily and interestingly, we show that the EDMM problem is solvable in an optimal way in polynomial time by a centralized algorithm and propose a distributed matching algorithm. Our simulations also show that our distributed algorithm performs quiet closely to the max-imum matching algorithm. These results are expected to significantly enhance the connectivity of ZigBee networks.
A
CKNOWLEDGMENTSY.-C. Tseng’s research is cosponsored by MoE ATU Plan, by NSC grants 95-2221-E-009-058-MY3, 96-2218-E-009-004, 97-3114-E-009-001, 97-2221-E-009-142-MY3, and 97-2218-E-009-026, by MOEA under grant 94-EC-17-A-04-S1-044, by ITRI, Taiwan, and by III, Taiwan.
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[29] ZigBee Specification Version 2006, ZigBee Document 064112, 2006. Meng-Shiuan Pan received the BS and MS degrees from the National Chung Cheng University and the National Tsing Hua Uni-versity, Taiwan, in 2001 and 2003, respec-tively. He received the PhD degree from the Department of Computer Science, National Chiao Tung University, Taiwan, in 2008. His research interests include mobile computing and wireless communication. He is a student member of the IEEE.
Chia-Hung Tsai received the BS and MS degrees in computer science and information engineering from the National Chiao-Tung Uni-versity, Taiwan, in 2004 and 2006, respectively. He is currently pursuing the PhD degree in the Department of Computer Science, National Chiao-Tung University, Taiwan. His research interests include wireless communication and sensor networks.
Yu-Chee Tseng obtained the PhD degree in computer and information science from the Ohio State University in January 1994. He is a professor (2000-present), chairman (2005-pre-sent), and associate dean (2007-present) at the Department of Computer Science, National Chiao-Tung University, Taiwan. He is also an adjunct chair professor at the Chung Yuan Christian University (2006-present). He received the Outstanding Research Award from the National Science Council, ROC, in both 2001-2002 and 2003-2005, the Best Paper Award from the International Conference on Parallel Processing in 2003, the Elite I.T. Award in 2004, and the Distinguished Alumnus Award from the Ohio State University in 2005. His research interests include mobile computing, wireless communication, and parallel and distributed computing. He serves on the editorial boards for Telecommunication Systems (2005-present), the IEEE Transactions on Vehicular Technology (2005-2009), the IEEE Transactions on Mobile Computing (2006-present), and the IEEE Transactions on Parallel and Distributed Systems (2008-present). He is a senior member of the IEEE.
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