Dual Power Assignment for Network Connectivity
in Wireless Sensor Networks
Jian-Jia Chen, Hsueh-I Lu, Tei-Wei Kuo, Chuan-Yue Yang, and Ai-Chun Pang Department of Computer Science and Information Engineering,
Graduate Institute of Networking and Multimedia, National Taiwan University, Taipei, Taiwan, ROC. Email:{r90079, hil, ktw, r92032, acpang}@csie.ntu.edu.tw Abstract— Strong connectivity has been an important feature explored
in many network applications, such as sensor networks. This research focuses on a dual power assignment problem, where each sensor node has two transmission power levels. The objective is to minimize the number of wireless sensor nodes assigned to transmit messages at the high transmission power level, while the resulting sensor network is strongly connected. We propose an efficient1.75-approximation algorithm for this challenging problem. We not only show that the approximation ratio of the proposed algorithm is tight but also demonstrate the capability of the proposed algorithm in terms of simulation experiments.
Keywords: Dual power assignment, Power level assignment, Strong connectivity, Wireless sensor network.
1. Introduction
With the advance of technology for wireless sensor nodes, sensor networks are now widely tested and deployed for different application domains. One major challenge on the deployment of a wireless sensor network is on the power consumption minimization issues. Such an observation triggered a number of studies and implementations in energy-efficient research topics, such as those for packet routing, node placement, power-level assignments, etc.
In sensor networks, connectivity is usually required to collect information sensed by nodes. In many applications, sensor nodes might even need to operate autonomously and form an ad hoc network. Sometimes nodes might even need to work together to detect some complicated events, such as those for user behaviors. Similar requirements for connectivity are often seen in the research areas for mesh networks and ad hoc networks. As a result, strong connectivity is identified in the past decade as an important topic in the literature for sensor networks, mesh networks, and ad hoc networks, e.g., [1, 3, 5, 8, 9, 11, 12]. When the available transmission power levels for each wireless sensor node are continuous in a range of reals, many researchers have proposed results for the strong connectivity of wireless sensor nodes in [1, 3, 5, 8, 9]. In particular, 2-approximation algorithms based on minimum spanning trees were proposed in [1, 9]. When wireless sensor nodes are deployed in the 2-dimensional or the 3-dimensional space, the power-level assignment problem was proved beingN P-hard [5, 7, 8]. Researchers also considered the power-level assignment problem for different topological constraints in [2–4, 6]. Minimization of the maximum power consumption of a node to establish a connected wireless network was explored in [9, 10].
This research is motivated by the reality in which wireless sensor nodes might only have a set of discrete power levels available for assignment. In particular, we are interested in a dual power assignment problem, in which there are two available power levels for each wireless sensor node. The objective of the dual power assignment problem is to minimize the number of nodes assigned to transmit messages at the high transmission power level, while the resulting network is strongly connected. Sholander, Frank, and Yankopolus [12] proposed a heuristic algorithm by grouping nodes This work is supported in part by grants from the NSC program 93-2752-E-002-008-PAE.
into two categories. The dual power assignment problem was proved beingN P-hard by Rong, Choi, and Choi [11], and a 2-approximation algorithm was given. In this paper, we propose an efficient ap-proximation algorithm for the dual power assignment problem. The approximation ratio is proved being 1.75, and the approximation ratio is shown to be tight. The proposed algorithm is highly efficient with O(n2) time complexity, where n is the number of wireless sensor
nodes in a network. The strength of the proposed algorithm algorithm is demonstrated by a series of experiments, for which significant improvement is observed, compared to the previous result in [11].
The rest of this paper is organized as follows: In Section 2, we formally define the system models under considerations and the dual power assignment problem for network connectivity. An approximation algorithm is presented in Section 3 with analysis of its properties. Section 4 reports the experimental results. Section 5 is the conclusion.
2. System Models and Problem Definition
We study a power-efficient networking problem to construct a strongly connected network for wireless sensor nodes with two transmission power levels. We are concerned with a set of n im-migrated wireless sensor nodes placed in a field, where each node is specified by its location. In such a network, a packet from a sensor node may need to be delivered through several hops before reaching its final destination. Two transmission power levels are given for the sensor nodes, where the higher/lower one is denoted as
high/lowtransmission power. Each transmission power level specifies
its transmission range. Let rH and rL with rH > rL denote the transmission ranges for the high and low transmission powers, respectively. A node is set to be high (respectively, low), if it is assigned with the high (respectively, low) transmission power level. There is a direct link from one wireless sensor nodeu to another wireless sensor node v if v could receive and decode any signal
fromu. That is, if the distance between u and v is no more than the
transmission range of the transmission power level assigned for u, v can receive and decode any signals from u. Under an assignment of transmission power levels for the sensor nodes, one routing path from a nodeu to another node v exists if the message delivered from u can reach v through a series of multi-hop transition. A network is said to be strongly connected if there exists at least one routing path fromu to v for any two different sensor nodes u and v when the sensor nodes transmit messages at their assigned transmission power levels. The objective of the dual power assignment problem for network connectivity is to minimize the power consumption for n wireless sensor nodes, which is defined as the total power of the assigned transmission power levels of these nodes, while the resulting network is strongly connected. Since assigning more nodes to be high results in larger power consumption, the objective is equivalent to the minimization of the number of nodes set to be high. In this paper, we consider non-trivial cases in which the network is strongly
connected if all nodes are set to be high (and the network is not strongly connected if some nodes are set to be low).
We formulate the dual power assignment problem for network connectivity by a graph-theoretic approach as follows. Specifically, we are given two directed graphsGH= (V, EH) and GL= (V, EL) on the same setV of n wireless sensor nodes, where a directed edge (u, v) from node u to node v in GH(respectively,GL) signifies that
node v can decode the signals from node u when node u is high
(respectively, low). The following properties hold forGHandGL: • GHis symmetric and strongly connected;
• GLis symmetric but not strongly connected; and • GLis a proper subgraph ofGH.
A power assignment is represented by a subset U of V whose members are set to be high. Let EH(U) consist of the outgoing edges ofGHfrom the nodes inU. Let G(U) = (V, EH(U) ∪ EL). For any setS, let |S| denote the cardinality of S. It is clear that setting all the nodes inU as high and the others as low results in a strongly connected network if and only if there exists a path fromu
tov in G(U) for any two nodes u and v. Therefore, the objective of
the dual power assignment problem is to derive a power assignment U ⊆ V with the minimum |U| such that G(U) is strongly connected. The dual power assignment problem for network connectivity can be defined as a graph-theoretical problem as follows:
Dual Power Assignment Problem
Input instance: Two directed graphs GH = (V, EH) and GL =
(V, EL) on the same vertex set V , where GL is a proper subgraph ofGH. BothGH andGLare symmetric.GHis strongly connected, whereasGLis not.
Objective: A subsetU of V such that G(U) is strongly connected
and|U| is minimized.
In [11], Rong, et al. has shown that the Dual Power Assignment problem isN P-hard and proposed a 2-approximation algorithm.
3. A1.75-Approximation Algorithm
In this section, we present our algorithm with the approximation ratio 1.75 for the Dual Power Assignment problem. Before we proceed with further discussion, some terminologies are defined as follows. For any directed graph G, let B(G) be the set of the strongly connected components in the directed graphG. Let C(S, u) denote the strongly connected component of G(S) that contains
node u for a power assignment S. Let Γ(S, u) consist of the
nodes v in V with (u, v) ∈ GH and C(S, u) = C(S, v). That is, Γ(S, u) contains the neighbors of u in GH that are not in the strongly connected componentC(S, u). A member of B(G(S)) is a
neighboring componentofu in G(S) if it contains at least one node
of Γ(S, u). Let G(S) be the undirected graph on B(G(S)), where each strongly connected component inB(G(S)) is a vertex in G(S), and two vertices inG(S) are adjacent if each of them contains a node
inV − S such that the two nodes are adjacent in GH. The strongly
connected component ofG(S) represented by a vertex w in G(S) is denoted byG−1(S, w).
For example, suppose that we are given 10 sensor nodes, where the corresponding directed graphs GH and GL are illustrated in Figures 1(a) and (b). Suppose thatS is {v1, v7}. We know EH(S) =
{(v1, v7), (v7, v1), (v1, v2), (v7, v8)} and G(S) is (V, EL∪EH(S)).
Letw1,w2,w3, andw4represent the strongly connected components
of G(S) containing v1, v3, v5, and v9, respectively. B(G(S)) is
{w1, w2, w3, w4}. Γ(S, v2) is {v3, v9}. w2 andw4are neighboring
components ofv2, whereasw3is not. Moreover, Γ(S, v8) is {v9}, w4
is a neighboring component ofv8, whereasw2andw3are not.G(S)
is an undirected graph onw1,w2,w3, andw4, shown in Figure 1(c).
Since v2 and v3 are adjacent inGH, there is an undirected edge
(w1, w2). The other edges are because v4 andv5 are adjacent (the
edge (w2, w3)), v6andv10are adjacent (the edge (w3, w4)), and v2
andv9are adjacent (the edge (w1, w4)). Similarly, if S is an empty
set,G(∅) is an undirected graph on w
1,w2,w3,w4, andw5, shown
in Figure 1(d), wherew
1,w2,w3,w4, andw5represent the strongly
connected components of G(∅) containing v1, v3, v5, v9, and v7,
respectively.
With the initialization of S as an empty set, our proposed
three-phase assignmentalgorithm, referred as Algorithm TPAand
summa-rized in Algorithm 1, inserts nodes intoS incrementally in a greedy manner. Algorithm TPAhas the following three phases:
1) First, every strongly connected component of B(G(S)) is unmarked. As long as there is still a node u in V that has more than one neighboring component inG(S), the first phase calls select(u, S), which is defined as follows, with the setting ofS as∅.
The subroutine select(u, S) first marks C(S, u) and in-sertsu into S. Then, for each neighborv of u in Γ(S, u),
ifC(S, v) is unmarked, the subroutine select(u, S)
recur-sively calls select(v, S).
After the recursive call returns, we insert all of the nodes in
S into S and unmark the strongly connected component of
B(G(S)) which contains u.
2) The second phase repeats the following procedure untilG(S) does not contain any cycle.
Let ¯C be a cycle of at least three vertices in G(S). Suppose that ¯C = (c0, c1, c2, . . . , c| ¯C| = c0). Let vi be a node
within the strongly connected component, represented by ci (i.e., G−1(S, ci)), such that the strongly connected component, represented byci+1 (i.e.,G−1(S, ci+1)), is a neighboring component ofviinG(S). The second phase then inserts these| ¯C| nodes v1, v2, . . . , v| ¯C|intoS. 3) After the above procedures, the resultingG(S) is a tree. Then,
the third phase repeats the following procedure until there is no edge exists inG(S).
Let u1 be a vertex inG(S) and u2 be a neighbor ofu1
in G(S). For notational brevity, let u3 = u1. For i ≤
2, let vi be a node of V within the strongly connected component represented byui(i.e.,G−1(S, ui)) such that the strongly connected component represented by ui+1 (i.e.,G−1(S, ui+1)) is a neighboring component of viin
G(S). The third phase then inserts nodes v1 andv2 into
S.
It is clear thatG(S) has only one vertex after executing Algorithm TPA. Therefore,G(S) is a strongly connected graph. Consider the input instance described byGH andGL in Figures 1(a) and 1(b) as an example for illustrating Algorithm TPA. Initially, S = ∅. Since v2 has 2 neighboring components, Algorithm TPA calls select(v2,
S) in the first phase, where S = ∅ initially. v
2, v3, v9, and
v8 are inserted into S during the recursive call of select(v2, S).
Then, v2, v3, v9, andv8 are inserted into S. The first phase then
terminates since no node has more than one neighboring component. After that,v1, v2, v3, v4, v7, v8, v9, andv10form a strongly connected
component, and so dov5andv6. Since no cycle exists forG(S), the
second phase inserts no node intoS. In the third phase, v4andv6are
inserted intoS. Therefore, the number of nodes in S is 6. The optimal solution for such an input instance is 5 by assigningv1, v3, v5, v8,
andv10as high.
A straightforward implementation of Algorithm TPA requires
O(|V |(|V |+|EH|)) time complexity. We could apply the disjoint-set
data structures in our implementations so that the time complexity is
v1 v2 v3 v4 v5 v6 v7 v 8 v9 v10 (a)GH v1 v2 v3 v4 v5 v6 v7 v 8 v9 v10 (b)GL w1 w2 w3 w4 (c)G({v1, v7}) w 1 w 5 w 2 w 3 w 4 (d)G(φ)
Figure. 1. An example for notation explanations: (a)GH and (b) GL, where each dot represents a sensor node and an arrow between two nodesvi
andvj indicates that there are two directed edges(vi, vj) and (vj, vi) in the graph. (c) G({v1, v7}), where w1,w2,w3, andw4represent the strongly
connected components ofG({v1, v7}) containing v1,v3, v5, andv9, respectively. (d)G(∅), where w1,w2,w3,w4, andw5represent the strongly connected
components ofG(∅) containing v1,v3, v5,v9, andv7, respectively.
Algorithm 1: Three Phase Assignment (TPA)
Input: (GH, GL);
Output: A subsetSofV such thatG(S)is strongly connected;
1: S ← ∅;
{first phase}
2: unmark every strongly connected component inB(G(S));
3: while there exists a node u ∈ V whose number of neighboring
components inB(G(S))is more than1do 4: call select(u, S← ∅);
5: insert all of the nodes inSintoS;
6: unmark the strongly connected componentC(S, u);
{second phase}
7: whilethere exists a cycleC = (c¯ 0, c1, c2, . . . , c| ¯C|= c0)inG(S)
do
8: letvibe a vertex inV, whereC(S, vi)is equal toG−1(S, ci)and
∃ u ∈ Γ(S, vi) ∩ u ∈ G−1(S, ci+1); 9: insertv1, v2, . . . , v| ¯C|intoS;
{third phase}
10: whilethere exists an edge(u1, u2) ∈ G(S)do
11: let vi be a vertex inV, where C(S, vi)is equal toG−1(S, ui)
and∃ w ∈ Γ(S, vi) ∩ w ∈ G−1(S, u(i+1)), fori ≤ 2; (remark:
u3= u1)
12: insertv1andv2 intoS;
13: returnS;
Procedure: select(u,S)
1: insertuintoSand mark the strongly connected componentC(S, u);
2: for allv ∈ Γ(S, u) do
3: ifC(S, v)is unmarked then
4: call select(v,S);
nodes under considerations.
Now we show that the approximation ratio of Algorithm TPAis 1.75. For notational brevity, let S2denote the setS after the second
phase of Algorithm TPA. For the rest of the section, letU be a power assignment ofV such that G(U) is strongly connected. We can derive two different lower bounds of|U|.
Lemma 1: |U| ≥ |B(GL)| ≥ 2.
Proof: For each connected component in B(GL), there must
be at least one node included inU. Otherwise, G(U) is not strongly connected.
Lemma 2: Let U be a subset ofV . If no cycle exists in G(U)
and each vertex in V has at most one neighboring component in G(U), then |U| ≥ 2(|B(G(U))| − 1).
Proof: We prove this lemma by contradiction with the
assump-tion of|U| < 2(|B(G(U))|−1). Let ¯G(U, ∅) be a directed graph on
B(G(U)). For notational brevity, vertices are labeled with the same
labels in ¯G(U, ∅) and G(U). Note that no edge exists in ¯G(U, ∅). ¯
G(U, U) is constructed as follows: For a node u in U, let v be the
vertex inG(U), where G−1(U, v) is equal to C(U, u). Since each vertex inV has at most one neighboring component in G(U), there is at most one vertexs in ¯G(U, ∅) which represents the neighboring
component of u in G(U). If such a vertex s exists, the directed edge (v, s) is added into ¯G(U, ∅). Let ¯G(U, U) be the resulted directed graph by considering all of the nodes inU and eliminating the duplicated directed edges in the above step.
SinceG(U) is strongly connected, ¯G(U, U) is strongly connected. However, at most one directed edge will be added into ¯G(U, U) for each element inU during the process in constructing ¯G(U, U). Because no cycle exists in G(U), and GH is strongly connected, G(U) is a tree. Besides, if a directed edge (u, v) is in ¯G(U, U), then there exists a corresponding undirected edge (u, v) in G(U). There are (|B(G(U))|−1) edges in G(U). ¯G(U, U) is strongly connected only if there are at least 2(|B(G(U))|−1) directed edges are added during the construction of ¯G(U, U). Thus, |U| < 2(|B(G(U))|−1) implies that ¯G(U, U) is not strongly connected. Therefore, G(U) is not connected. A contradiction is reached.
S2is a subset ofV which satisfies the properties of U stated in
Lemma 2. Combining with Lemma 1, we have
|U| ≥ max{|B(GL)|, 2(|B(G(S2))| − 1)}. (1)
In the following, we prove an upper bound on the cardinality of the power assignmentS derived from Algorithm TPA.
Lemma 3: |S| ≤ 1.5|B(GL)| + 0.5|B(G(S2))| − 2.
Proof: For Step 4 in each iteration of the while loop in
Algorithm 1, i.e., the recursive call of select(u, ∅), let k be the number of strongly connect components marked for the recursive call. That is, k nodes in V are inserted into S after the recursive call. Thesek marked strongly connected components become a connected component inG(S) after inserting these k nodes into S. Therefore, thesek − 1 strongly connected components are merged into one new strongly connected component. The number of nodes inserted into S for each of the k − 1 contracted strongly connected components is amortized to be k
k−1. Similarly, when a cycle ¯C is considered in the second phase, | ¯C| connected components become a connected component after inserting| ¯C| nodes into S. The number of nodes inserted intoS for each of the | ¯C| − 1 contracted connected compo-nents in this iteration is amortized to be | ¯C|−1| ¯C| . Letk1, k2, . . . , km denote the number of nodes inserted into S for Step 4 in each iteration of the while loop in the first phase of Algorithm 1. Let
¯
C1, ¯C2, . . . , ¯Cq denote the cycles considered in the second phase.
There are |B(GL)| − |B(G(S2))| strongly connected components
contractedafter the second phase of Algorithm TPA. Therefore, we
have |S2| = Pmi=1kik−1i (ki− 1) + Pq i=1| ¯C| ¯Ci|−1i| (| ¯Ci| − 1) ≤ 1.5(Pm i=1(ki− 1) + Pq i=1(| ¯Ci| − 1)) = 1.5(|B(GL)| − |B(G(S2))|),
becauseki≥ 3 and | ¯Ci| ≥ 3 for every i (kik−1i ≤ 1.5 and | ¯C| ¯Ci|−1i| ≤ 1.5 ). Since there are only (|B(G(S2))|−1) edges before we proceed
nodes intoS. Therefore, we have
|S| ≤ 1.5(|B(GL)| − |B(G(S2))|) + 2(|B(G(S2))| − 1)
= 1.5|B(GL)| + 0.5|B(G(S2))| − 2.
Theorem 1: Algorithm TPA is a polynomial-time
1.75-approximation algorithm for the Dual Power Assignment problem.
Proof: We prove this theorem by showing that
|S| |U| ≤ 1.5|B(GL)| + 0.5|B(G(S 2))| − 2 max{|B(GL)|, 2(|B(G(S2))| − 1)} < 1.75. If|B(GL)| ≥ 2(|B(G(S2))| − 1), then |S| |U| ≤ 1.5|B(GL)| + 0.25|B(GL)| − 1.5 |B(GL)| < 1.75. If|B(GL)| < 2(|B(G(S2))| − 1), then |S| |U|≤ 3(|B(G(S22(|B(G(S))| − 1) + 0.5|B(G(S2))| − 1) 2))| − 2 < 1.75.
After the approximation ratio of our algorithm is shown, we show the tightness of the approximation bound by presenting a set of input instances. Consider the input instance shown in Figure 2, where each dot represents a sensor node. For any two nodes u and v inside a circle in Figure 2, there is a directed edge (u, v) from u to v in both GH and GL; that is, the distance between u and v is no longer thanrL. In Figure 2(a), an arrow between two nodesu and v indicates that there are two directed edges (u, v) and (v, u) in GH; that is, the distance between u and v is longer than rL and no longer thanrH. As shown in Figure 2, the pattern ofA1, A2,
A3, and A4 repeatsk times. Totally, there are 4k + 2 circles. The
optimal power assignment assigns only 4k + 2 nodes to be high as shown in Figure 2(c), where the larger solid nodes are set to be high. Algorithm TPAreturns a solution with 7k + 3 nodes assigned to be high as shown in Figure 2(d), where the hollow nodes are included in the first phase and the larger solid nodes are included in the third phase, whereas no nodes are inserted in the second phase. Therefore, the approximation ratio of Algorithm TPA is tight for sufficiently largek.
4. Experimental Results
A. Experimental Setups and Performance Metric
Algorithm TPA is simulated extensively with comparison to the
algorithm proposed in [11] (denoted as AlgorithmSP), which is a
2-approximation algorithm for the Dual Power Assignment problem. We consider sensor nodes in theR2 space. Three types of
distribu-tions on sensor nodes are considered, in which the parameter setup is similar to that in [11]. For the first type of distributions of sensor nodes, both of thex-ordinate and y-ordinate of a sensor node are uniform random variables between 0 and 1000. For the second type of distributions, nodes are deployed according to a Poisson distribution by setting the mean value as 500. In addition, for the first and second types of distributions, simulations are conducted for different network sizes (n = 30, 50, 100). For the third type of distributions, we consider a specific application scenario, in which 300 sensor nodes are deployed homogeneously. Specifically, a 1000× 1000 R2 plane is divided into rectangular regions with equal size. Each
region is associated with two nodes. If (x1, y1) and (x2, y2) are
the ordinates of the left-bottom and the right-top points of a region, two nodes are deployed in this region in which the x-ordinates (respectively,y-ordinates) are uniform random variables between x1
and x2 (respectively, y1 and y2). After that, the other 100 sensor
nodes are deployed by setting both x-ordinate and y-ordinate as uniform random variables between 0 and 1000.
Given a deployment of sensor nodes, we have to determine the transmission ranges rH and rL. In our experiments, rH is set as the shortest transmission rage associated for each node such that assigning all of the sensor nodes to high makes the wireless sensor network strongly connected. In our experiments, we simulate the two algorithms by varying the ratio ofrL torH from 5% to 80%.
We compare the performance of the two algorithms with an estimated lower bound, derived from Equation (1). The ratio of
relative high power nodes of an algorithm for an input instance is
defined as the ratio of the number of nodes assigned to be high in the power assignment derived from the algorithm to the estimated lower bound of the input instance. The average and maximum ratios of relative high power nodes are measured and conducted from 500 independent experiments for each parameter configuration.
B. Simulation Results
Figures 3 (a)-(d) show the performance result when nodes are deployed by uniform distributions, and the ratio ofrL torH varies from 0.05 to 0.8 stepped by 0.05. The average ratios of relative high power nodes when n = 50 and n = 100 are reported in Figures 3 (a) and (b), respectively. The maximum ratios of relative high power nodes when n = 50 and n = 100 are reported in Figures 3 (c) and (d), respectively. Similar results were observed when n = 30. As shown in Figures 3(a)-(d), for the metric of the ratios of relative high power nodes, Algorithm TPAoutperforms AlgorithmSP
in either worst cases or average cases. Besides, the maximum ratio of relative high power nodes of Algorithm TPA is no more than 1.75,
which is as the same as the analysis in Section 3. The maximum ratio of relative high power nodes of Algorithm TPAis at most 1.7,
whereas that of AlgorithmSPis at most 1.9. When the ratio of rLto rHis close to 1, both of the ratios of relative high power nodes of the two algorithms tend to approach their theoretical bounds, i.e., 1.75 for Algorithm TPA and 2 for Algorithm SP. In the worst cases, as
the ratio ofrL torHincreases, the number of nodes in a connected component ofGLbecomes larger. Thus, the number of candidates of nodes to be set being high also increases. In both algorithms, once a node is set as high, one node of its neighboring components should be high. Therefore, assigning nodes in an incorrect sequence may be a poor choice compared to the estimated lower bound. As for average cases, there is a peak in Figures 3(a) and (b), i.e., when the ratio ofrL torHis 0.65. This is because that when the ratio of rL to rH is low, to make the network strongly connected, most nodes have to be assigned to be high either for power assignments derived from the two algorithms or for the corresponding estimated lower bound. Besides, when the ratio ofrL torHis large enough, in most cases, assigning most nodes to transmit message at the low power level could make the network strongly connected either in power assignments derived from the two algorithms or in the corresponding estimated lower bound.
Similarly, Figures 3(e)-(h) report the simulation results when nodes are deployed by Poisson distributions. The average ratios of relative high power nodes when n = 50 and n = 100 are reported in Figures 3 (e) and (f), respectively. The maximum ratios of relative high power nodes whenn = 50 and n = 100 are reported in Figures 3 (g) and (h), respectively. The results in Figures 3(e)-(h) are similar to those in Figures 3(a)-(d). However, the peaks in Figure 3(g)-(h) shift to left, where the ratio ofrL torH is about 0.5 or 0.55. This comes from the characteristics of Poisson distributions since most nodes are not very far away.
Figure 4 shows the results for the third type of deployment distributions by varying rL/rH from 0.05 to 0.8 stepped by 0.05.
1 2 3 · · · · k − 1 k A2 A3 A4 A1 · · · · · · · · · · · · (a)GH 1 2 3 · · · · k − 1 k A2 A3 A4 A1 · · · · · · · · · · · · (b)GL 1 2 3 · · · · k − 1 k A2 A3 A4 A1 · · · · · · · · · · · · (c) Optimal assignment 1 2 3 · · · · k − 1 k A2 A3 A4 A1 · · · · · · · · · · · · · · · · · · · · · · · · (d) Assignment of Algorithm TPA
Figure. 2. An input instance for a tight example: (a)GH, (b)GL, (c) an optimal assignment by assigning the larger solid nodes as high, and (d) an
assignment derived from Algorithm TPA, where the hollow nodes are included in the first phase and the larger solid nodes are included in the third phase.
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ratio of relative high power nodes
rL / rH TPA SP (a)n = 50 1 1.1 1.2 1.3 1.4 1.5 1.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ratio of relative high power nodes
rL / rH TPA SP (b)n = 100 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ratio of relative high power nodes
rL / rH TPA SP (c)n = 50 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ratio of relative high power nodes
rL / rH TPA SP (d)n = 100 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ratio of relative high power nodes
rL / rH TPA SP (e)n = 50 1 1.1 1.2 1.3 1.4 1.5 1.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ratio of relative high power nodes
rL / rH TPA SP (f)n = 100 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ratio of relative high power nodes
rL / rH TPA SP (g)n = 50 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ratio of relative high power nodes
rL / rH
TPA SP
(h)n = 100
Figure. 3. (a)-(b): average ratios of relative high power nodes forn = 50, 100 when nodes are deployed by a uniform distribution, respectively. (c)-(d):
maximum ratios of relative high power nodes forn = 50, 100 when nodes are deployed by a uniform distribution, respectively. (e)-(f): average ratios of
relative high power nodes forn = 50, 100 when nodes are deployed by a Poisson distribution, respectively. (g)-(h): maximum ratios of relative high power
nodes forn = 50, 100 when nodes are deployed by a Poisson distribution, respectively.
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ratio of relative high power nodes
rL / rH TPA
SP
(a) Average ratio
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Ratio of relative high power nodes
rL / rH TPA
SP
(b) Maximum ratio Figure. 4. (a) average ratios of relative high power nodes and (b) maximum
ratios of relative high power nodes forn = 300 when the plane is divided
into 100 regions and each region has at least 2 nodes.
The average and maximum ratios of relative high power nodes are reported in Figures 4 (a) and (b), respectively. The results in Figures 4 are similar to those in Figures 3.
5. Conclusion
In this paper, we explore a power-efficient networking problem to maintain the property of strong connectivity for wireless sensor networks so that the power consumption is minimized. We consider a dual power assignment problem for a set of wireless sensor nodes, where each node has two transmission power levels. The objective is to minimize the number of nodes assigned to transmit messages in the high power level, while the resulting network is strongly connected. Given the NP-hardness of the problem, we present an efficient approximation algorithm with the approximation ratio 1.75. We not only show the tightness of the ratio but also demonstrate the capability of the proposed algorithm in terms of simulation experiments, for which very encouraging results are shown. We must point out that even though the proposed algorithm is presented for power-level assignment of sensor networks, it could also be applied to the power-level assignment for other kinds of networks.
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