An SPT-based topology control algorithm for wireless ad hoc networks
Szu-Chi Wang
a, David S.L. Wei
b, Sy-Yen Kuo
a,*aDepartment of Electrical Engineering, National Taiwan University, Taipei, Taiwan bDepartment of Computer and Information Sciences, Fordham University, Bronx, NY, USA
Available online 17 January 2006
Abstract
In this paper, we present a localized Shortest-Path-Tree (SPT) based algorithm that copes with the topology control problem in wireless ad hoc networks. Each mobile node determines its own transmission power based only on its local information. The proposed algorithm first constructs local SPTs from the initial graph, after which the total power consumption is further reduced by allowing each mobile node to search the replaceable links individually. The constructed topology ensures network connectivity, and possesses the following desirable energy-efficient features: (i) the power stretch factor is bounded and can be predetermined, (ii) the power consumption is evenly distributed among the mobile nodes, and (iii) the total power consumption is lower than that obtained by the best known algorithms. The performance improvements of the proposed algorithm are demonstrated through exten-sive simulations. We conclude our work with a discussion of future research directions toward more integrated mobile network architectures.
2005 Elsevier B.V. All rights reserved.
Keywords: Ad hoc networks; Energy efficiency; Localized algorithm; Topology control
1. Introduction
Wireless ad hoc networks have been proposed as an alternative to cellular networks for use in areas where the existing communication infrastructure has been disrupted or destroyed (e.g., due to an earthquake or flood), or where construction of a fixed infrastructure would be inconve-nient or impossible (e.g., on a battlefield or in space). In such networks each node must serve as a user and as a router. A wireless link between any two nodes can be estab-lished if the radio transmission range of each node can cov-er the othcov-er. In the case of insufficient radio transmission range between the two nodes, multiple ‘‘hops’’ may be required, whereby intermediate nodes re-broadcast the messages until the destination node is reached. The reliance on wireless multi-hop communications to maintain connec-tivity among nodes adds new complexity to the design and operation of the network. Furthermore, the lack of a
phys-ical backbone poses a strong need for topology control of the network. It has been shown that the performance of a protocol for an ad hoc network can be enhanced if the pro-tocol is based on overlaying a virtual infrastructure on the ad hoc network. Also, due to the finite power supply of a mobile node, power conservation has been widely used as a primary control parameter in the design of protocols for wireless ad hoc networks. Therefore, the problem of power-efficient topology control has attracted increasing attention from researchers in the area of wireless networking.
In this paper, we propose an algorithm for constructing an energy-efficient topology for wireless ad hoc networks. Such networks can be modeled by a weight directed graph G = (V, E), where V represents the set of all mobile nodes and E represents the set of interconnections between the nodes. For each edge (u, v)2 E, node v must be in the trans-mission range of node u. We use iuvi to denote the Euclid-ean distance between nodes u and v. The weight of the edge (u, v), denoted by w (u, v), can be formulated as t Æ iuvia+ rp (u, v) in the most widely used power-attenuation model, where t is a threshold related to the signal-to-noise ratio 0140-3664/$ - see front matter 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.comcom.2005.11.006 *
Corresponding author.
E-mail address:[email protected](S.-Y. Kuo).
at node u, and a is a constant between 2 and 5 depending on the wireless transmission environment. The first part of the equation is typically called the transmitter power, which is the power consumed by transmitting a signal from node u to node v. The remaining part is the power con-sumed by the receiver and is called the receiver power. We assume that all receivers have the same power thresh-old for signal detection, and the value of t is thus an appro-priate constant. Hereafter, the sum of the transmitter power and the receiver power is called the transmission power. Also, throughout the paper, we use the terms link and edge interchangeably.
We assume that all mobile nodes are distributed on a two-dimensional plane and each mobile node has a GPS receiver that acquires information about the node’s loca-tion. Initially, all mobile nodes are operated at full trans-mission power and have the transtrans-mission radius equal to one unit by a proper scaling. Consequently, the resulting graph G can be modeled as a unit-disc graph (denoted as UDG (V)) and there is an edge between two nodes if and only if their Euclidean distance is at most one. We assume that UDG (V) is strongly connected. All the mobile nodes have unique identifiers (ID) numbered from 1 to N, where N = |V|. Each mobile node can individually adjust its own transmission power. We also assume that omni-directional antennas are used by all the mobile nodes to transmit and receive signals. Wireless ad hoc networks are power con-strained; thus, it is undesirable to require each mobile node to always transmit at maximum power. Otherwise, the total power consumption would often be unnecessarily high and transmission interference would occur more fre-quently. In fact, it has been shown that mobile nodes expend most of their power in communications [1]. As a result, each mobile node should adjust its transmission radius to reduce its power consumption, while still main-taining network connectivity. Due to the infrastructureless nature of ad hoc networks, it is preferable that the network topology should be constructed in a localized manner to avoid flooding the network. Stojmenovic and Lin gave the definition of a localized algorithm in [2]. A distributed power control algorithm is called localized if each node can decide its transmission power based only on the information of the nodes reachable in a small con-stant number of hops.
In this paper, we adopt several definitions given in [3]. Let f be a complete transmission power assignment on V, and Gf be the associated communication graph. Clearly,
Gf˝ G. The total power consumption of f (denoted by
tpc (f)) is defined asPu2Vf (u), where f (u) is the minimum
transmission power needed to reach all the neighbors of u in Gf. A given path, (u, v), from node u to node v in Gf
can be expressed as (u, v) = v0v1 vh1vh, where u = v0,
v = vh. The length of the path (u, v) (denoted by
| (u, v)|) is h, and its transmission power is defined as pYðu; vÞ¼X
h
i¼1
wðvi1; viÞ.
Given a communication graph H, the minimum-energy path between nodes u and v, denoted by QHminðu; vÞ, is a path whose total power consumption is the minimum among all the paths that connect these two nodes in H. Let pH(u, v) denote pðQHminðu; vÞÞ. The power stretch factor
of the graph Gfwith respect to G is then defined as
psfGfðGÞ ¼ max
u;v2V ;u6¼vðpGfðu; vÞ=pGðu; vÞÞ.
In the literature, the two most widely used energy conservation approaches are: (i) reducing the transmis-sion power of each node; and (ii) reducing the total power consumed by all nodes involved in one communi-cation session. The latter can be achieved by preserving the minimum-energy paths of the given UDG (V). How-ever, these two approaches may offset each other; a rel-evant discussion can be found in [4]. The major focus of our work is to develop a localized topology control algorithm in which each mobile node makes a decision about its transmission power based solely on its local information. As these locally made decisions collectively ensure global network connectivity, the network topology controlled by each node’s transmission power must be energy-efficient. More specifically, the proposed algo-rithm must achieve the following objectives: (i) it must be capable of adjusting the transmission radii of the nodes with low total power consumption; and (ii) it must have a constant bounded power stretch factor. Fig. 1
illustrates that the strongly connected property could be preserved under the reduction of transmission radii and the excision of links. Moreover, we consider several prac-tical issues, particularly the integration of ad hoc net-works into IP-enabled services. IPv6 is designed to satisfy various future networking requirements and a more advanced support of mobility can be achieved accordingly. Therefore, we examine relevant studies and identify some practical challenges that will be important in next-generation networks.
The remainder of this paper is organized as follows. Sec-tion2briefly introduces related works. Section3describes the basic concepts and properties of our algorithm. In Sec-tion4, the superiority of our algorithm is demonstrated via simulations. Also, using several important metrics, we compare the energy-efficiency of our topology with that of other algorithms. In Section 5, we propose an efficient distribution method to deal with node mobility. Finally, in Section6, we present our conclusions and indicate some future research directions.
2. Related works
In[5], Rodoplu and Meng described a distributed proto-col for constructing a topology that guarantees the preser-vation of the minimum-energy path between every pair of nodes connected on the original graph G. The concept of relay region was first introduced in their paper. Recently, based on the results in [5], Li and Halpern [6]proposed a
protocol that performs better and is computationally sim-pler. In [7], Li and Wan presented a distributed position-based protocol to construct an enclosure graph that con-serves power in one communication session. Their protocol is more efficient in time and space than that in[5]. All the above works focus on constructing a subgraph of G that includes the union of all of the minimum-energy paths. In addition, the problem of finding a complete transmis-sion power assignment using some optimization criteria has been studied in [8–10,17]. Although the objective of the approaches in[9,10,17]is to minimize total power con-sumption, they are all centralized and cannot be trans-formed into localized algorithms. The trade-off between sparseness and energy efficiency in a topology has been extensively investigated. In particular, Li et al.[3]discussed the energy efficiency property of several well-known prox-imity graphs, such as the constrained Gabriel graph (denot-ed by GG (G)), the constrain(denot-ed relative neighborhood graph (denoted by RNG (G)), and the constrained Yao graph (denoted by YGk(G)), over a (directed) graph
G. They showed that the total power consumption of these geometric structures can be arbitrarily larger than the min-imum total power needed to maintain the strong connectiv-ity of the network. In some of their works[3,11], however, they assume that the receiver power is zero, which is not practical according to[12].
The concept of a visible neighborhood, i.e., the topolog-ical view of each node based on its local information, was introduced in [13,14,16] for example. Li et al. [13] devel-oped a topology control algorithm in which each node builds its local minimum spanning tree using one-hop neighborhood information. The network topology derived by their algorithm has the following properties: (i) network connectivity is preserved, (ii) the node degree is bounded,
and (iii) all uni-directional links can be removed. It has been validated by simulations that their algorithm has advantages over [5,21] in terms of some metrics. In [16], Li proposed a localized algorithm to construct a subgraph of G whose total edge length is bounded by a constant mul-tiplied by that of the minimum spanning tree. Unfortunate-ly, it has also been shown that the total power consumption of the constructed subgraph can be O (|V|a) larger than the optimal. The algorithms in [13,21,16] cannot guarantee a constant bounded power stretch factor, which is desirable in applications sensitive to the worst-case behavior of topology control.
3. Our topology control algorithm 3.1. Observations
For simplicity, like other works, we initially assume that each node in a network is stationary. We then adapt our algorithm to the mobile environments described in Section
5. Considering a complete transmission power assignment f, there is a tradeoff between tpc (f) and psfGfðGÞ, i.e., over reducing the transmission power of each individual node may diminish minimum-energy paths or vice versa. The problem of finding a complete transmission power assign-ment f whose total power consumption is the minimum among all the complete transmission power assignments is usually called the min-total assignment problem. A similar problem is to find a solution that contains bi-directional links only. However, to the best of our knowledge, no localized algorithm for the above two problems has been
developed. Moreover, both problems are NP-hard
[8,9,17]and the best approximation ratio in either case is still an open question.
p1 p2 p3 p5 p6 p7 p4 p1 p2 p3 p4 p5 p6 p7
Therefore, our algorithm is designed heuristically as fol-lows. First, we construct an edge subgraph G0 from G;
where G0has a power stretch factor of 1. The logical link
set of node u in G0 is denoted as LL (u). We assume that
a tag is attached to each link to describe its attribute in LL (u). Initially, LL (u) contains all the edges of node u in UDG (V) and each link is tagged as regular. We then try to minimize the total power consumption as much as pos-sible. The basic idea is to let each node excise some logical links of the subgraph while keeping the power stretch fac-tor bounded by a constant cb. Unlike the algorithms pro-posed in [3], we do not try to construct a topology that guarantees a constant bounded node degree, though our algorithm may generate one. The reason is that, as shown in [3], a geometrical structure with a constant bounded node degree may contain very few minimum-energy paths between any two nodes, and our design goal is to preserve the maximum number of minimum-energy paths possible. Moreover, as mentioned in [4], further eliminating edges may result in more congestion and hence degrade network throughput and fault tolerance in the long run. For each node u, since no global knowledge of the network topology is available, each operation can only use the information from the nodes in its vicinity. Our solution is to let u construct a local topology view, LTV (u, k), based on the location and logical link information of the nodes within its k-hop neighborhood.
Information about one-hop neighbors can be obtained by using some form of beacon messages that are sent peri-odically and asynchronously by each node to announce its presence. To obtain information about two-hop neighbors, a common solution is to allow each node to attach its own one-hop neighborhood information while sending the bea-con messages. These two kinds of information are used extensively to facilitate message routing and broadcasting in wireless ad hoc networks; therefore, the cost of maintaining such information can be amortized. A similar concept can be generalized to gather k-hop (k > 2) neigh-borhood information. Note that, in practice, k should be small compared to the network diameter D. The definition of LTV (u, k) is as follows.
Definition 1 (Local topology view:). The local topology view of node u, denoted by LTV (u, k) = (V0, E0), is a subgraph of G such that: (i) a node vi2 V0 if the hop
distance between vi and u is no more than k; and (ii) an
edge (vi, vj)2 E0 if (vi, vj)2 LL and both viand vjbelong to
V0. The tag of each edge in E0is also recorded.
Suppose that a subgraph of G is associated with a trans-mission power assignment f. For each node u, if a logical link (u, v) satisfies the equation w (u, v) = f (u), then (u, v) is called a critical link of node u. Assume that all critical links of u are excised. If LL (u)„ B, we define ps (u) as (f (u) f0(u))/e (u), where f0(u) is the transmission power
needed to maintain the remaining logical link(s) of node u, and e (u) is the number of critical link(s) of node u; other-wise, ps (u) = 0. The priority of node u is a pair pri (u) =
Æps (u), ID (u)æ. Let pri (v1) = (ps (v1), ID1) and pri (v2) =
(ps (v2), ID2). Then, pri (v1) > pri (v2) if ps (v1) > ps (v2), or
ps (v1) = ps (v2) and ID1< ID2.
3.2. Two-phase localized algorithm
The proposed algorithm is comprised of two phases, namely, local shortest tree construction and path search replacement. These two phases are loosely coupled, i.e., the techniques introduced in each phase can be used indi-vidually in the design of any other topology control algo-rithm with proper modifications.
3.2.1. Phase (I) local shortest tree construction
In this phase, each node needs to know the IDs and the locations of its one-hop neighbors. This information can be gathered easily, since we assume that each node initially sends a beacon message with its maximum transmission power. More precisely, such information is included in LTV (u, 1), so the weight of each edge can thus be derived. Note that the edge weight includes both the transmitter power and the receiver power between the two end nodes of the edge. Each node u applies Dijkstra’s algorithm inde-pendently to get the shortest-paths from the source node u to the other nodes in LTV (u, 1). As a result, the local short-est path tree of node u, denoted by LSPT (u), can be obtained. The direct children of node u, DC (u), are defined as DC (u) = {v2 V0|h (LSPT (u), v) = 1}, where h (LSPT
(u), v) is the height of a child node v in LSPT (u). Node u then removes the logical links {(u, w)|w62 DC (u)} from LL (u). The topology generated under the above descrip-tions is denoted as GI.
Since for each node u, only one-hop neighborhood information is available for constructing LSPT (u), some links in GI may be uni-directional; however, such links are unfavorable in wireless ad hoc networks. In[13,15], it is shown that network topologies free of uni-directional links are much more beneficial for MAC layer control mechanisms, link-level acknowledgments, and package transmissions/re-transmissions. Our method of removing uni-directional links is simple: since at the end of Phase I, all nodes are aware of the logical links of their one-hop neighbors, each node deletes its uni-directional links. The resulting topology is denoted as GII. In Section3.4, we will prove that GIInot only preserves network connectivity, but also preserves all minimum-energy paths of G.
3.2.2. Phase (II) path search replacement
At the beginning of this phase, it is assumed that each node perceives the logical link(s) of its one-hop neighbors. As a result, for each node u, LTV (u, k) can be acquired by sending k 1 beacon messages. Node u can then reduce its transmission power further by eliminating the critical link(s) that are replaceable with alternative paths. That is, for each critical link (u, v), node u tries to find another path that reaches node v based on LTV (u, k). We call such a path the replacing path of (u, v). The entire replacing path(s)
of node u is denoted as RP (u). Whenever a node u finds that it has the highest priority within its k-hop neighbor-hood and ps (u) > 0, it starts a search for RP (u). The search process is also based on Dijkstra’s algorithm. After obtain-ing LTV (u, k), node u applies Dijkstra’s algorithm to it to search for the shortest path kmin(u, v). Note that all
the original critical links of each node should be excluded during the search process, which can be achieved by setting their edge weights to 1 temporally. Assume that the weights of all the shortest paths that originated in u are stored in an array dist[N] ordered by the IDs of the destina-tion nodes. Note that if a node v is unreachable from u in LTV (u, k), then dist[ID (v)] =1. Node u saves the result in an ID list, including all the intermediate nodes of kmin(u, v). Let cb be a predetermined constant. If no
replacing path exists, or p (kmin(u, v)) is more than cb
times larger than the weight of edge (u, v), the search pro-cess is terminated, RP (u) is set as an empty list, and ps (u) is set to 0. Otherwise, kmin(u, v) is the eligible
replacing path of (u, v), and (u, v) can be excised. An exam-ple is illustrated in Fig. 2. The gray line inFig. 2(a) is the transmission range of p1 with full transmission power.
After Phase I, the area covered by the dashed line in
Fig. 2(b) is the new transmission range of p1, which is
decided by LSPT (p1). After Phase II, if p1found an
alter-native path (p1, p6) = p1p4p5p7p6 in LTV (p1, 2) and
p ( (p1, p6)) < cb Æ w (p1, p6), then p1 can excise (p1, p6). In
this way, the transmission range (and thus the transmission power) of p1 can be substantially reduced, as shown in
Fig. 2(c).
If all critical links of u can be excised, ps (u) is set to 0 and f0(u) becomes the minimum operational power
need-ed to cover the remaining logical links. In other words, each node executes the search process at most once in Phase II. Node u then advises its k-hop neighbors about its replacing path(s) by sending a notification message. The message is a tuple nm =Æid, LL, RPæ that records the ID, logical links, and replacing path(s) of the sender node. Each constituent link in the replacing path(s) is
called a replacing link. If a node u is notified that its link (u, v) is a replacing link, the tag of (u, v) is switched to replacing. For each node w with ps (w) > 0, if w receives a notification message, it first checks if any of its critical links is tagged as replacing. If such links exist, w sets ps(w) to 0 and disseminates its new priority to its k-hop neighborhood. Otherwise, w updates LTV (w, k) according to nm. If w finds that its priority is the local maximum within its k-hop neighborhood, it starts to search the replacing path(s) of its critical link(s) follow-ing the above steps. Finally, after Phase II, the transmis-sion range of each node u can be taken as the power level it needs to reach its remaining logical links in the constructed topology. The constructed topology resulting from Phase II is denoted as GIII.
We believe that by considering a trade-off between local-ization and efficiency, choosing k = 2 should suffice. We show that it can be very difficult for each node u to inde-pendently excise some of its critical link(s) based only on LTV (u, 1) while keeping the power stretch factor bounded by a constant cb > 1. Consider the topology illustrated in
Fig. 3, where u1 unand v1 vnare equally scattered along
two rings, respectively. Assume that the receiver power is negligible and the maximum transmitter radius is R. For 1 6 i 6 n, node ui, and node vi are collinear, iuivii = R,
0 < iuiui+1i < R. Since for each node vi, 1 6 i 6 n, only
LTV (vi, 1) is available, so node vidoes not know whether
an alternative path (vi, ui) exists. Therefore, in case
ivi vi+1ia+ iui+1uiia> (cb 1) Æ Ra and some node vi
deci-des to excise link (vi, ui), the new power stretch factor will
exceed cb. Moreover, since the one-hop information of each vi is identical, the constructed graph can be
discon-nected if each viexcises its critical link. On the other hand,
it is obvious that if each node decides not to excise its critical link, the total power consumption will be much larger than the optimal if iuiui+1i/R and ivivi+1i/R are close
to 0 and ivi vi+1ia+ iui+1uiia>(cb 1) Æ Ra. As a result,
introducing LTV (u, k) with k > 1 is necessary to further reduce the total power consumption.
p1 p2 p3 p4 p5 p6 p7 p1 p2 p3 p4 p5 p6 p7 p1 p2 p3 p4 p5 p6 p7 a b c
3.3. Heuristics to improve the performance
Clearly, there has to be a tradeoff while searching for the relaying path(s), e.g., if some critical links of low-priority nodes are chosen to construct the replacing paths of the nodes with higher priorities, the transmission power of the low-priority nodes cannot be reduced. However, if node u excludes all the critical links in LTV (u, k) while applying Dijkstra’s algorithm, the possibility of failing to find an eligible replacing path will increase. We use a simple heuristic to further reduce the total power consumption: for each link (vi, vj) in LTV (u, k), if ps (vi) = 0 or (vi, vj) is
not a critical link of node vi, we modify the weight of (vi, vj)
by multiplying it by a dilation factor df (vi), defined as
1 + (ps (vi)/ps (u)), before applying Dijkstra’s algorithm. If
all the eligible replacing paths can not be found, Dijkstra’s algorithm is applied to the original LTV (u, k), and the above steps are repeated. We can also extend cb to be a real number, e.g., 1.5, instead of using an integer constant such that more flexibility can be introduced while Phase II is applied. In fact, the experiment results prove that using 1 < cb < 2 can further improve the power stretch factor at the cost of only a slight increase in total power consumption.
3.4. Properties of the constructed topology
We now prove some critical properties of the topology generated by our algorithm. For clarification, we use wG (u, v) to denote the weight of edge (u, v) in G.
Lemma 1. The minimum-energy path between any two nodes in G is preserved in GI.
Proof. For every node pair u and v, if (u, v)2 E and is deleted after the construction of LSPT (u), $ a path (u, v) in the initial LTV (u, 1) such that p ((u, v)) < wG(u, v). Assume a deleted link, (u, v), exists on the minimum-energy path m of G. Since LTV (u, 1) is a subset of G, (u, v)
should be included in G. If we replace (u, v) with in m, then the transmission power of the new path will be
smaller thanm, and a contradiction is derived. h
Lemma 2. The minimum-energy path between the two end nodes of each deleted link in GIis preserved in GII.
Proof. We sort all the links deleted in the uni-directional link removal procedure by the following rank formula: (u1, v1) > (u2, v2) if wG(u1, v1) > wG(u2, v2); or wG(u1, v1) = w-G
(u2, v2) and ID (u1) > ID(u2); or wG(u1, v1) = wG(u2, v2) and
ID (u1) = ID (u2) and ID (v1) > ID (v2). Since the order of
link deletion does not change the resulting subgraph, with-out loss of generality, we assume that the links are deleted according to their ranks in ascending order. We prove by induction that GII preserves the minimum-energy path of each deleted link.
Basis: k = 1, Consider the first deleted link (u1, v1). Clearly, for node v1, there is a path (v1, u1) in LSPT (v1) such that p ((v1
, u1)) < wG(v1, u1). According to our energy model, wG(v1, u1) = wG(u1, v1),and each link (uh, vh) that comprises p ((v1
, u1)) < wG(v1, u1) must satisfy wG(uh, vh) < wG(u1, v1). Therefore, each link that belongs to (v1
, u1) is bi-directional and is preserved in GII. Induction: Assume Lemma 2holds for all deleted links {(ui, vi)|i = 1, 2, . . . , k 1}. Now, we prove that Lemma 2
also holds for edge (uk, vk). Consider link (uk, vk). Clearly, for node vk, there is a path (vk
, uk) in LSPT (vk) such that p ( (vk
, uk)) < wG(uk, vk). Similarly, each edge (uh, vh) that comprises (vk
, uk) must satisfy wG(uh, vh) < wG(uk, vk). There are two cases:
Case 1. (uh, vh) is bi-directional; as a result, (uh, vh) is not deleted.
Case 2. (uh, vh) is uni-directional. Since wG(uh, vh) < wG(uk, vk), the minimum-energy path between (uh, vh) is reserved. In other words, there is a path, p ( (uh
, vh)), com-posed of bi-directional links that has the minimum total power consumption. This implies that the deletion of (uk, vk) does not affect the minimum-energy path between node ukand node vk. h
Lemma 3. GIIpreserves the network connectivity of G. Proof. Follows directly from Lemmas 1 and 2, since the minimum-energy path between any two nodes is preserved
in GII. h
Lemma 4. psfGIIIðGÞ is bounded by cb.
Proof. Clearly, the power stretch factor of GIIis one. Con-sider each constituent link (u, v) of the minimum-energy path, where (u, v) can only be substituted by a replacing path (u, v) such that p ( (u, v)) 6 cb Æ wG
(u, v). Since the
ui vi u2 v2 v3 u3 u1 v1
Fig. 3. This topology demonstrates that one-hop neighborhood informa-tion is insufficient for reducing the total power consumpinforma-tion while keeping a constrained power stretch factor.
end nodes of each replacing link are in the k-hop neighbor-hood of node u, they will not begin their search process until u finishes Phase II. Moreover, each replacing link of u is marked by a replacing tag and will not be excised in the future. As a result, the power stretch factor of GIII is at most cb times larger than GII. h
Theorem 1. GIIIpreserves the network connectivity of G and has a constant bounded power stretch factor cb.
Proof. Follows directly from Lemmas 1–4. h 4. Performance comparisons
Via simulations, we compared the performance of our algorithms with that of other approaches in terms of the total power consumption, the power stretch factor, and the node degree of the constructed topologies. The experi-ment results are summarized inTables 1–4. Our algorithm is denoted as ESPT. For study purposes, the first part of our algorithm is denoted as ESPT1, and the part consisting of Phases I and II is denoted as ESPT2. A unit disk graph (UDG) is used as the basis for comparison. We chose the SMECN algorithm proposed by Li and Halpern [6], since it performs significantly better than the algorithm proposed by Rodoplu and Meng [5] in terms of total power con-sumption, and it has a power stretch factor of 1.
Similarly, the constrained Gabriel graph (GG) outper-forms those described in [3] in terms of total power con-sumption and the power stretch factor. It is thus chosen for comparison here. The algorithm proposed by Li [16]
(AMST) is based on the constrained relative neighborhood graph. However, Li’s paper only offers a theoretical analy-sis without the experimental performance, and is of interest to us. The MST-based topology control algorithm (LMST) proposed in [13] is also used for comparison, because its total power consumption is lower than that of SMECN (though its power stretch factor is larger). Our algorithm operates with a = 4.0, k = 2, and cb = 1.5 and 2.0, respec-tively. In order to make a fair comparison, it is assumed that each node in SMECN broadcasts its neighbor discov-ery message (NDM) with the maximum operational power while gathering necessary local information. As for LMST, to balance the trade-off between tpc and psm, we do not incorporate the removal of uni-directional links.
We observe the following metrics of each constructed topology H: (i) the total power consumption (tpc) associat-ed with H; (ii) the power stretch mean (psm), definassociat-ed as psf ¼ ðPu;v2V pHðu;vÞ
pGðu;vÞÞ=N
2;1
(iii) the maximum power stretch factor (max psf) is observed throughout the experiment;2 (iv) the variance of transmission power (var tp);3 (v) the average node degree (avg nd); and (vi) the maximum node degree (max nd) is observed throughout the experiment. The last two metrics are chosen to facilitate a better under-standing of the topology’s characteristics. The values of tpc, psm, var tp, and avg nd are the average of 50 indepen-dent experimental results. Note that tpc is normalized to lie between 0.0 and 1.0 by dividing its values by the total power consumption of UDG. The transmitter range, R, is fixed at 500 m. The map sizes are equal to s· R by s· R for s = 1, 3, 5, 7, where s is directly related to the net-work’s density. The x and y coordinates of each node are selected at random in the interval [0, m], where m is the map size. The experiment was performed for N = 100.
From Tables 1–4, we observe that the topology con-structed by our algorithm (ESPT1/ESPT2) has a much low-er tpc than those of GG and SMECN in all cases, and the increase in the psm of ESPT2is almost negligible (less than 0.04). This means that the total power consumption of our algorithm can be reduced significantly, while preserving most of the minimum-energy paths. For comparison with LMST, we first consider denser networks (see Tables 1 and 2). The tpc values of our algorithm are slightly higher than those of LMST. However, since the differences are rel-atively small with respect to UDG, we believe that our algorithm has advantages over other energy-efficiency met-rics, such as psm and max psf. Note that since LMST does not take the power stretch factor into consideration, the subgraph generated by LMST may have a very large max psf (about 9 in this case), while our algorithm guarantees Table 1
The performance measurements with s = 1 s = 1
tpc psm max psf var tp avg nd max nd
UDG 1.0 1.0 1.0 0.0 96.5312 99 GG 0.000666 1.0 1.0 4.68E + 15 3.5992 8 SMECN 0.000328 1.0 1.0 1.20E + 15 2.6996 6 AMST 0.000253 1.01491 2.51846 8.76E + 14 2.3386 5 IJVBT 0.000099 1.1967 9.70118 9.30E + 13 1.98 4 ESPT1 0.000178 1.0 1.0 2.43E + 14 2.4252 5 ESPT2, cb = 2.0 0.000123 1.03482 1.99171 1.27E + 14 2.1356 5 ESPT2, cb = 1.5 0.000137 1.016 1.49991 1.52E + 14 2.2036 4
1 psm is a modification of the power stretch factor, since most of the above topologies clearly have a power stretch factor bounded by a very small constant. This metric is more suitable to further evaluate the energy conservation of a communication session.
2 max psf shapes the worst-case behavior of applications that operate over the constructed topology.
3 Due to the uniform power threshold in our model, for convenience, we set t = 1 when counting the transmission power.
a bounded max psf (1.5 and 2.0 in this case). For sparser networks (seeTables 3 and 4), ESPT2outperforms LMST in both tpc and psm. Although the max psf of LMST decreases as s is increases in our experiments, LMST does not provide a theoretical upper bound on the max psf. Both ESPT1 and ESPT2 have smaller tpc values compared to AMST in all cases. Note that although the topology con-structed by AMST tends to have a small psm (which is still larger than our algorithm), its power stretch factor can be as large as N 1 [16]. Clearly, if the receiver power is a non-zero constant, the power stretch factor of both LMST and AMST becomes even larger.
Both Kirousis et al.[9]and Cheng et al.[17]propose an MST-based centralized solution that approximates the optimal total power consumption with a factor of 2. Since there is no known localized topology control algorithm that guarantees a better approximation factor, we compare the ratio of the tpc of ESPT2(with cb = 1.5 and 2.0) to that
of their algorithm in Fig. 4. The ratio remains small and stable as s changes, which shows that our localized algo-rithm performs almost as well as the centralized algoalgo-rithm. Finally, we also observe that our algorithm performs well in var tp, especially in the case of ESPT2with s > 1. Since, overall, our algorithm performs better than the other approaches in both the summation and the variance of transmission power, it has a better chance of avoiding node power depletion. The above experimental results also indi-cate that, if the source and destination nodes are chosen randomly, our algorithm effectively curtails both the total power consumption and the variance of transmission power. In addition, with its small psm and constrained max psf, we believe that the topology constructed by our algorithm can provide a basis for power efficient communi-cations, and thus prolong lifetime of the nodes and the net-work. The topology constructed by each algorithm compared in this section is shown inFig. 5.
Table 3
The performance measurements with s = 5 s = 5
tpc psm max psf var tp avg nd max nd
UDG 1.0 1.0 1.0 0.0 10.4232 27 GG 0.245597 1.0 1.0 2.29E + 20 3.4496 8 SMECN 0.159539 1.0 1.0 1.47E + 20 2.6596 5 AMST 0.128633 1.01305 2.49099 1.15E + 20 2.3204 4 LMST 0.10403 1.03698 4.38695 9.96E + 19 2.135 4 ESPT1 0.122629 1.0 1.0 1.08E + 20 2.45 5 ESPT2, cb = 2.0 0.086227 1.03671 1.9971 6.51E + 19 2.1564 5 ESPT2, cb = 1.5 0.09287 1.01372 1.49999 6.87E + 19 2.2236 5 Table 4
The performance measurements with s = 7 s = 7
tpc psm max psf var tp avg nd max nd
UDG 1.0 1.0 1.0 0.0 5.6392 14 GG 0.380877 1.0 1.0 3.46E + 20 2.9548 8 SMECN 0.299838 1.0 1.0 3.01E + 20 2.44 6 AMST 0.261474 1.01752 2.8558 2.66E + 20 2.1856 4 LMST 0.254784 1.0228 3.14811 2.71E + 20 2.1376 4 ESPT1 0.27593 1.0 1.0 2.86E + 20 2.3468 5 ESPT2, cb = 2.0 0.228194 1.03382 1.99931 2.32E + 20 2.1096 5 ESPT2, cb = 1.5 0.237019 1.0135 1.4938 2.40E + 20 2.158 5 Table 2
The performance measurements with s = 3 s = 3
tpc psm max psf var tp avg nd max nd
UDG 1.0 1.0 1.0 0.0 25.5408 53 GG 0.05339 1.0 1.0 2.82E + 19 3.59 8 SMECN 0.026874 1.0 1.0 8.10E + 18 2.7016 6 AMST 0.020503 1.01413 2.46733 5.71E + 18 2.3388 4 LMST 0.010074 1.08511 6.31301 2.01E + 18 2.0258 4 ESPT1 0.01495 1.0 1.0 2.60E + 18 2.4284 5 ESPT2, cb = 2.0 0.010102 1.03485 1.99906 8.56E + 17 2.1372 5 ESPT2, cb = 1.5 0.011188 1.01516 1.49855 9.95E + 17 2.2044 4
We discuss the time complexity of each phase of our algorithm separately. We use Ni to denote the number of
nodes in the i-hop neighborhood, and Ei to denote the
number of edges in the i-hop neighborhood. First of all, it is well known that the cost of Dijkstra’s algorithm is O (|V|lg|V| + E), provided that the priority queue is imple-mented in a Fibonacci heap. Consequently, each node needs O (N1lgN1+ E1) time to complete Phase I. In the
worst case, the time complexity is O (|V|2). In Phase II, since each node performs the search process at most once, the computation cost of each node is O (NklgNk+ Ek). As in
Phase I, the worst-case time complexity of Phase II is also O (|V|2). Therefore, the time complexity for constructing both GIIand GIIIis O (|V|2).
5. Dealing with mobility
5.1. Limitations and potentialities
When considering node mobility, each node must be able to adjust its transmission power dynamically. An intu-itive idea to achieve network reconfiguration is that all mobile nodes send beacon messages with full transmission power and run the proposed algorithm periodically. In this chapter, we consider the case of modest movement of the nodes. With high mobility, basically a smaller k should be adopted in Phase II and the time period for sending bea-con messages should be very short. In fact, if the topology changes too fast, it is extremely difficult for a topology con-trol algorithm to effectively guarantee network connectivi-ty. On the other hand, the above idea may be too costly in the case of low node mobility. As mentioned in previous works (e.g., [7]), node movement can be viewed as two events, namely node addition and node deletion. Therefore, our task is to find an efficient way to add a new node to and remove a node from a network. We assume that a moving
node, say node v, broadcasts a node deletion notification before its movement and broadcasts its new position infor-mation to notify node addition at the end of each beacon interval, both with full transmission power. For node addi-tion, if node u received the new node information from node v, it recalculates LSPT (u) and checks if (u, v) is a log-ical link of this new LSPT (u). If it is, node u re-executes the search process and updates its logical link(s); node u informs node v to add link (v, u) if necessary. For node dele-tion, consideration should be taken only if (u, v) is a logical link of node u. In this case, node u recalculates LSPT (u) and re-executes the search process; node u also checks if (u, v) is a replacing link of some node. If node u becomes aware that the movement of node v will break the replacing path of some node w, u informs w to re-execute the search process and disseminates the new replacing path if there is one. It is not hard to see that node v only affects the logical link(s) of the nodes within its k-hop neighborhood. 5.2. Practical considerations
Through the simulations of ESPT2we observed that, in most cases, the average number of nodes affected by removal of a relaying link (denoted as anm) is close to one, with cb set to either 2.0 or 1.5 (seeFig. 6). In other words, the links that comprised the replacing paths were sporadic and evenly distributed. Thus, at the beginning of each beacon interval, node u checks if there is a change in the transmission radius after deciding the new logical links. If there is a change, node u updates its critical link(s). If node u finds that its ps value becomes larger than 0, it disseminates its new priority and re-executes the search process described in Section5.2. No recalculation is needed for node u if no nodes are added to its one-hop neighbor-hood and no nodes are deleted from its replacing path. Furthermore, the communications and computations of the reconfiguration process can be performed locally, which is preferred in the context of mobile ad hoc networks.
Another important issue is to provide address autocon-figuration for IP-enabled appliances for future ubiquitous computing, particularly in the case of IPv6 [18]. In fact, various related RFCs have been presented in recent years (e.g., [19,20]). IP-address autoconfiguration protocols should avoid central administration and be robust in order to cope with the dynamic environment. It would be appro-priate, therefore, to integrate them with the proposed local-ized topology control scheme.
6. Concluding remarks and future directions
In this paper, we have demonstrated how to construct and maintain an energy-efficient topology in wireless ad hoc networks in a distributed and localized manner. Our algorithm requires only local information to construct and maintain a topology on a given unit disk graph. The concept of k-redundant edges has been proposed by Li 0 1 2 3 4 5 0 2 4 6 8 10 tpc Ratio Network Density (s) cb = 2.0 cb = 1.5
Fig. 4. Comparison of the average total power consumption of ESPT2 with that of a well-known 2-approximation MST-based centralized algorithm.
0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 Topology by UDG 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 Topology by GG 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 Topology by SMECN 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 Topology by AMST 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 Topology by LMST 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 Topology by ESPT1 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 Topology by ESPT2, cb = 1.5 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 Topology by ESPT2, cb = 2.0
and Halpern[6]. However, the algorithm in[6]only deals with 2-redundant edges. The contributions of our algo-rithm are manifold. First, the approaches in Phase I deter-mine k-redundant edges for k P 2. In other words, our approach can produce a minimum-power topology with lower total power consumption that is totally different to the topologies in [5,6]. Our algorithm is also simpler in computation than those in[5,6]. Second, the topology after Phase II has several desirable features, such as a bounded power stretch factor, low total power consumption and small variance in transmission power. The simulation results show that our algorithm outperforms other approaches in terms of various important metrics. Third, our algorithm provides the flexibility of pre-determining the power stretch factor of the derived subgraph.
In addition, the topology constructed by our algorithm helps other programs save energy in the two ways suggest-ed in[6]: (i) the transmission power computed by our algo-rithm is sufficient for each node to send beacon messages; and (ii) the routing algorithm can be restricted to using the edges of the current subgraph to find an energy-efficient path for a communication session. In summary, the pro-posed algorithm effectively and efficiently constructs a
vir-tual backbone that can support energy-aware
communication applications over wireless ad hoc net-works. Nonetheless, an important question arises: If an algorithm can only guarantee better total power consump-tion, but has a poor power stretch factor (or vice versa), how can we judge if it guarantees a longer node and net-work lifetime? Furthermore, it has been pointed out that a suitable definition of network lifetime is application-de-pendent (e.g.,[21,22]). A main concern of our future work is to explore, with empirical evaluation, an efficient solu-tion of network reconfigurasolu-tion that can effectively handle various asynchronous communication settings in a mobile ad hoc environment.
Finally, IPv4 cannot satisfy the potential requirements of various emerging mobile networks, including wireless ad hoc networks [23]. Therefore, other solutions, such as Mobile IPv6 (MIPv6) are attracting more attention, because they can augment the Internet with mobility man-agement capacity [20,24]. Consequently, extending our work to provide viable next-generation network architec-tures that would merge ad hoc networks with the Internet and incorporate certain favorable features of IPv6 is of great interest. Unfortunately, integration of this kind poses many technical challenges, such as IP address handoff and seamless message routing (see, e.g.,[18,24,25]). Therefore, another future extension of our work will be to investigate techniques that could gracefully integrate the proposed topology control algorithm with IPv6-based networks.
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Sy-Yen Kuo is the Dean of the College of Elec-trical Engineering and Computer Science, National Taiwan Ocean University, Keelung, Taiwan. He is also a Professor at the Department of Electrical Engineering, National Taiwan Uni-versity where he is currently on leave and was the Chairman at the same department from 2001 to 2004. He received the BS (1979) in Electrical Engineering from National Taiwan University, the MS (1982) in Electrical & Computer Engi-neering from the University of California at Santa Barbara, and the Ph.D. (1987) in Computer Science from the University of Illinois at Urbana-Champaign. He spent his sabbatical years as a Visiting Professor at the Computer Science and Engineering Department, the Chinese University of Hong Kong from 2004 to 2005 and as a visiting researcher at AT&T Labs-Research, New Jersey from 1999 to 2000, respectively. He was the Chairman of the Department of Computer Sci-ence and Information Engineering, National Dong Hwa University, Tai-wan from 1995 to 1998, a faculty member in the Department of Electrical and Computer Engineering at the University of Arizona from 1988 to 1991, and an engineer at Fairchild Semiconductor and Silvar-Lisco, both in California, from 1982 to 1984. In 1989, he also worked as a summer faculty fellow at Jet Propulsion Laboratory of California Institute of Technology. His current research interests include dependable systems and networks, software reliability engineering, mobile computing, and reliable sensor networks.
Professor Kuo is an IEEE Fellow. He has published more than 240 papers in journals and conferences. He received the distinguished research award (1997-2005) from the National Science Council, Taiwan. He was also a recipient of the Best Paper Award in the 1996 International Sym-posium on Software Reliability Engineering, the Best Paper Award in the simulation and test category at the 1986 IEEE/ACM Design Automation Conference (DAC), the National Science Foundation’s Research Initia-tion Award in 1989, and the IEEE/ACM Design AutomaInitia-tion Scholarship in 1990 and 1991.
Szu-Chi Wang received the BS degree in Computer Science and Information Engineering and the Ph.D. degree in Electrical Engineering from National Taiwan University in 1995 and 2005, respectively. His research interests lie in the areas of designing, implementing and analyzing distrib-uted algorithms for wireless ad hoc networks.
David S.L. Wei received the Ph.D. degree in Computer and Information Science from the University of Pennsylvania, Philadelphia, in 1991. He is currently a Professor of Computer and Information Science Department at Fordham University. From May 1993 to August 1997 he was on the Faculty of Computer Science and Engineering at the University of Aizu, Japan (as an Associate Professor and then a Professor). Dr. Wei has authored and co-authored more than 70 technical papers in the areas of distributed and parallel processing, wireless networks and mobile computing, and optical networks in various archival journals and conference proceedings. He served on the program committee and was a session chair for several reputed international conferences. He served as a co-chair of Power Aware Communication and Software, Minitrack in the Software Track at the 34th Hawaii International Conference on Systems Sciences (HICSS-34). He is a lead guest editor of IEEE Journal on Selected Areas in Commu-nications for the special issue on Mobile Computing and Networking, and a guest editor of IEEE Journal on Selected Areas in Communications for the special issue on Peer-to-Peer Communications and Applications. Currently, Dr. Wei focuses his research effort on wireless networks, mobile computing, and peer-to-peer communications.
