- Email: [email protected]
A note on the complexity of minimum latency data aggregation scheduling with uniform power in physical interference model
Theoretical Computer Science 569 (2015) 70–73
Contents lists available at ScienceDirect
Theoretical Computer Science www.elsevier.com/locate/tcs
Note
A note on the complexity of minimum latency data aggregation scheduling with uniform power in physical interference model Nhat X. Lam a , Tien Tran a , Min Kyung An b , Dung T. Huynh a,∗ a b
Dept. of Computer Science, University of Texas at Dallas, Richardson, TX 75080, USA Dept. of Computer Science, Sam Houston State University, Huntsville, TX 77341, USA
a r t i c l e
i n f o
Article history: Received 2 September 2014 Received in revised form 20 November 2014 Accepted 29 November 2014 Available online 12 December 2014 Communicated by D.-Z. Du
a b s t r a c t In this paper we prove that the Minimum Latency Aggregation Scheduling (MLAS) problem in the Signal-to-Interference-Noise-Ratio (SINR) model is APX-hard in the uniform power model. © 2014 Elsevier B.V. All rights reserved.
Keywords: Wireless sensor network Algorithm Complexity APX-hardness
1. Introduction Data aggregation is one of main applications of Wireless Sensor Networks (WSNs), and its main purpose is to collect data periodically from the sensor nodes and forward it to a destination called the sink node. As these tiny devices have limited energy resources, researchers have focused on finding ways to avoid sensor nodes’ unnecessary retransmissions of their collected data in order to extend the network lifetime. One approach is to compute schedules with the minimum number of timeslots such that data can be aggregated without any collision or interference. This problem is known as the Minimum Latency Aggregation Scheduling (MLAS) problem. In the literature, wireless networks are commonly modeled as graphs where any two nodes are connected via a communication edge if they are covered by each other’s transmission range. When considering the MLAS problem on such networks, choosing the interference model is a crucial step. While a substantial amount of research results have been obtained for the graph-based interference model, recently, several researchers have started investigating the problems in the more realistic physical interference model, also known as Signal-to-Interference-Noise-Ratio (SINR), which, unlike the graph model, more adequately captures real world phenomena. As the SINR model has been introduced only recently, few works exist and algorithms that guarantee theoretical performances are scarce. Along with the interference models, researchers have adopted one of two power models, uniform power and non-uniform power models, concerning the MLAS problem. The uniform power model assumes no power control, i.e., a uniform power level is typically used, whereas in the non-uniform model, determining the right power levels to be assigned to sending nodes
*
Corresponding author. E-mail address: [email protected] (D.T. Huynh).
http://dx.doi.org/10.1016/j.tcs.2014.11.034 0304-3975/© 2014 Elsevier B.V. All rights reserved.
N.X. Lam et al. / Theoretical Computer Science 569 (2015) 70–73
71
could also be part of the problem (also known as power control). The non-uniform power model is divided into three cases: the bounded power, the unlimited power, or the discrete power model. In the bounded power model, each node u is assigned a transmission power level p u ∈ [ p min , p max ], and in the unlimited power model, u is assigned a transmission power level p u ∈ [ p min , ∞]. In the discrete power model, each node u is assigned a transmission power level p u ∈ { p 1 , p 2 , ..., pk }, where k is the number of power levels used in the network. The MLAS problem in the graph-based interference model has been investigated by many researchers over the last several years. Assuming the uniform power model, in the collision-free graph model, Chen et al. [3] proved the NP-hardness of the MLAS problem and proposed a (Δ − 1)-approximation algorithm, where Δ is the maximum node degree. Later, Huang et al. [6] proposed a nearly-constant factor approximation algorithm whose latency is bounded by 23R + Δ − 18, and Yu et al. [15] introduced a distributed algorithm whose latency is bounded by 24D + 6Δ + 16, where R is the radius and D is the diameter of the network. Subsequently, Xu et al. [13,14] introduced a better constant factor approximation algorithm whose latency is bounded by 16R + Δ − 14, and Wan et al. [12] proposed three algorithms whose latency is log R bounded by 15R + Δ − 4, 2R + O (log R ) + Δ, and (1 + O ( √3 )) R + Δ, respectively. While only collision was considered R
in these papers, some researchers have studied the problem taking into consideration interference as well. This is done in the collision-interference-free graph model. Wan et al. [12], An et al. [1] proposed constant factor approximation algorithms whose latency is bounded by O ( R + Δ). In the SINR model, Li et al. [10] introduced the first constant factor approximation algorithm whose latency is bounded by O ( R + Δ). Assuming the nonuniform power model, An et al. [1] proved an Ω(log n) approximation lower bound in the metric model, where n is the number of nodes. It was investigated without power control in the collision-interference-free graph model with discrete power levels. In the SINR model with bounded power, Lam et al. [8] studied the MLAS problem with power control, and showed the first constant factor approximation algorithm whose latency is bounded by O ( R + log n). Later, Du et al. [4] proposed another constant factor approximation algorithm whose latency is also bounded by O ( R + log n) in the same model. In the unbounded power model with power control, Li et al. [9] proposed a distributed algorithm that yields O (χ ) timeslots, where χ is the link length diversity, and a centralized algorithm whose latency is O (log3 n) which was improved by Halldórsson and Mitra [5] to O (log n). In the discrete power model without power control, Lam et al. [7] showed not only an Ω(log n) approximation lower bound in the metric SINR model, but also its NP-hardness in the geometric SINR model. Lam et al. [7] has been extended in An et al. [2] and introduced two constant factor approximation algorithms whose latencies are bounded by O ( R + Δ) assuming the dual power model, i.e., each node is assigned either the high power level or the low power level. In this paper, we continue the study of the Minimum Latency Aggregation Scheduling (MLAS) problem in the metric SINR model with discrete power levels, but without power control. Assuming the most restricted model of uniform power, we prove the APX-hardness of the problem in the metric SINR model. The rest of this paper is organized as follows. In Section 2, we describe our network model and introduce the definitions used in this paper. In Section 3, we prove the APX-hardness. Finally, Section 4 contains some concluding remarks. 2. Models and definitions In this section, we introduce two SINR models, the metric model and the geometric model. For the former, we model a WSN in a metric space as ( V , D ), where V is a set of sensor nodes, and D : V × V −→ R + is the distance function that satisfies the triangle inequality. Considering a communication link li (si , r i ), where si is a sender and r i is a receiver, let CL(li ) be the set of other links concurrently sending at the time as li . According to the physical interference model, we have
SINR(li ) =
N+
p si D (si , r i )−α
l j ∈C L (li )
p s j D (s j , r i )−α
where N is the ambient noise, α is the path loss, and p u is the power level assigned to node u. Then, the receiver r i can successfully receive the signal from the sender si if and only if its SINR value exceeds a given threshold β ≥ 1. So a node u p with power p u can send signals to only nodes in the distance d where dα ≤ N uβ . We call these nodes u’s neighbors. In this paper, we are specifically concerned with the uniform power model. We use a single transmission power level denoted by p where each node u in V uses the transmission power p to communicate. In the restricted geometric model, the set V of sensor nodes are deployed on the plane and the distance function D is defined as the Euclidean distance between two nodes. Regarding the definition of neighboring nodes, we assume that a p sender u can send data to only nodes in the distance d, where dα ≤ γ βuN , for some constant γ > 1. The data aggregation problem for either model is defined as follows. A schedule is defined to be a sequence of timeslots, at each of which, several nodes are scheduled to send its aggregated data to one of its neighbors, and every node can be scheduled as a sender only once. Formally, at each timeslot t, we have an assignment vector πt = (lt1 , lt2 , ..., lt w ), in which lt i is a directed link from st i to rt i satisfying the SINR threshold inequality. And a schedule, as a sequence of assignment vectors, is denoted as Π = (π1 , π2 , ..., π L ), where L is the length of schedule or the schedule latency. Given a set of source nodes and the sink node s, the objective of the data aggregation problem is to find the minimum latency schedule to aggregate data from all source nodes to the given sink.
72
N.X. Lam et al. / Theoretical Computer Science 569 (2015) 70–73
Fig. 1. The corresponding MDAS instance.
3. The APX-hardness In this section, we prove the APX-hardness for the MLAS problem in the metric model with uniform power. From the reduction in the proof, it follows that MLAS is NP-complete. Theorem 1. Finding the minimum latency aggregation schedule in the metric SINR model with uniform power is APX-hard. Proof. We construct a polynomial-time L-reduction from the MINIMUM B-K-SET COVER problem (MBKSC) that is known to be APX-complete [11]. This problem is a variation of the MINIMUM SET COVER problem, where the cardinalities of all sets are bounded from above by a fixed constant K and the number of occurrences of any element is bounded by a fixed constant B ≥ 2. Let I S be an instance of MBKSC consisting of a collection S of subsets of a finite set E of elements. Let n and m denote the cardinalities of E and S, respectively. A solution to I S is a subset S ⊆ S such that every element e ∈ E is in at least one set A ∈ S . Let OPT S denote an optimal solution to I S . Given the instance I S of the set cover problem, we construct in polynomial time an instance I M of the MLAS problem as follows. I M consists of n + m + 1 nodes. The set V of nodes in I M is partitioned into three subsets of nodes. The first subset contains only a single node s as the sink node. The second subset C S and the third subset C E consist of m and n nodes, respectively. The nodes in C E are source nodes (see Fig. 1). The m nodes in C S correspond to the m sets in the collection S. For each node a ∈ C S , let S (a) be the set in S corresponding to a. Similarly, the set C E of n nodes corresponds to the n elements in the set E of I S . And e (b) denotes the element in E that corresponds to node b in C E . Let p be a constant denoting the uniform power level of all sensor nodes. According to the definition of the SINR model, p a node can send data to another nodes at distance d if dα ≤ γ β N . Letting d0 be d0 = function D on V × V for the metric model of the MLAS problem as follows:
α
p
γ β N , we now define the distance
⎧ d0 ⎪ ⎪ ⎨√ α
u ∈ C S and v = s β d0 u , v ∈ C S D (u , v ) = d0 u ∈ C s , v ∈ C E , e ( v ) ∈ S (u ) ⎪ ⎪ ⎩√ α β d0 u , v ∈ C E
√
All other distances follow from symmetry or are induced by the shortest path metric. Since 0 ≤ α β d0 ≤ 2d0 , the triangle inequality holds for D.√ Next observe that α β d0 > d0 (β > 1). Hence, it follows that there does not exist any direct connection between nodes in C E . Similarly, there is no direct connection between any node a in C E and nodes in C S whose corresponding sets do not contain e (a). Let SOL M denote a solution to I M . We construct the corresponding solution SOL S to I S as follows: SOL S = { S (a)|a ∈ C S , a appears in SOL M }. Let us denote OPT M as an optimal solution to I M . The following two results will be shown in Lemma 1 and Lemma 2 below.
L (SOL M ) = |SOL S | + n
|OPT S | K K ( B − 1) + 1 ≥ n Thus,
L (OPT M ) = |OPT S | + n ≤ |OPT S | + |OPT S | K K ( B − 1) + 1
≤ K K ( B − 1) + 1 + 1 |OPT S | ≤ ε |OPT S |
In addition, we also have:
N.X. Lam et al. / Theoretical Computer Science 569 (2015) 70–73
73
|SOL S | − |OPT S | = L (SOL M ) − n − L (OPT M ) − n = L (SOL M ) − L (OPT M ) This implies that the MBKSC problem is L-reducible to the MLAS problem. As it has been shown in [11] that the MBKSC is APX-complete, the theorem follows. 2 In order to prove Lemma 1 and Lemma 2, we need the following fact. Fact 1. While a node in V is sending, no other nodes can send data without any interference. Proof. Consider nodes i , v ∈ C S and node u ∈ C E . Let i send data to s and u send data to v at the same time. Then
SINR(u , v ) =
α p .d− 0
N + p .(d0
√ α
β)−α
=
γ βN N +γN
=
γβ 1+γ
<β
Similarly, we can show that 2 nodes in C E or C S cannot send data at the same time.
2
Lemma 1. L (SOLM ) = |SOL S | + n. Proof. According to Fact 1, we can split the schedule into two independent phases. In the first phase, all nodes in C E send their data to selected nodes in C S . In the second phase, selected nodes in C S send received data to s. According to Fact 1, it is trivial that we need exactly n timeslots for the first phase. And also based on Fact 1, the number of timeslots required for the second phase is the number of selected nodes in C S . Hence the lemma follows. 2 Lemma 2. |OPT S | K ( K ( B − 1) + 1) ≥ n. Proof. Consider the sets in S which are sorted in non-increasing order based on their cardinalities. Pick the largest one and move this set to the selected group. Then remove all subsets that covered by the selected group. Repeat this procedure until all sets are removed. Since every element occurs at most B times (B ≥ 2), and the cardinality of each set is bounded by K , there are at most K ( B − 1) + 1 sets to be removed at every step. Since the minimum number of sets is Kn , there are at least K ( K ( Bn−1)+1) steps with
n K ( K ( B −1)+1)
selected sets. Hence,
n K ( K ( B −1)+1)
≤ OPT S . 2
4. Conclusion In this paper, we studied the Minimum Latency Aggregation Scheduling (MLAS) problem in the metric SINR model without power control assuming the uniform power model, i.e., each node is assigned a single power level. We have proposed a formal proof to show the APX-hardness of the problem. The precise complexity of MLAS in the geometric SINR model with uniform power remains an interesting open question. References [1] M.K. An, N.X. Lam, D.T. Huynh, T.N. Nguyen, Minimum data aggregation schedule in wireless sensor networks, Int. J. Comput. Appl. 18 (4) (2011) 254–262. [2] M.K. An, N.X. Lam, D.T. Huynh, T.N. Nguyen, Minimum latency data aggregation in the physical interference model, Comput. Commun. 35 (18) (2012) 2175–2186. [3] X. Chen, X. Hu, J. Zhu, Data gathering schedule for minimal aggregation time in wireless sensor networks, Int. J. Distrib. Sens. Netw. 5 (4) (2009) 321–337. [4] H. Du, Z. Zhang, W. Wu, L. Wu, K. Xing, Constant-approximation for optimal data aggregation with physical interference, J. Global Optim. 56 (4) (2013) 1653–1666. [5] M.M. Halldórsson, P. Mitra, Wireless connectivity and capacity, in: SODA, 2012, pp. 516–526. [6] S.C.H. Huang, P.-J. Wan, C.T. Vu, Y. Li, F. Yao, Nearly constant approximation for data aggregation scheduling, in: INFOCOM, 2007, pp. 6–12. [7] N.X. Lam, M.K. An, D.T. Huynh, T.N. Nguyen, Minimum latency data aggregation in the physical interference model, in: MSWiM, 2011, pp. 93–102. [8] N.X. Lam, M.K. An, D.T. Huynh, T.N. Nguyen, Scheduling problems in interference-aware wireless sensor networks, in: ICNC, 2013, pp. 783–789. [9] H. Li, Q.S. Hua, C. Wu, F.C.M. Lau, Minimum-latency aggregation scheduling in wireless sensor networks under physical interference model, in: MSWiM, 2010, pp. 360–367. [10] X.-Y. Li, X. Xu, S. Wang, S. Tang, G. Dai, J. Zhao, Y. Qi, Efficient data aggregation in multi-hop wireless sensor networks under physical interference model, in: MASS, 2009, pp. 353–362. [11] C.H. Papadimitriou, M. Yannakakis, Optimization, approximation, and complexity classes, J. Comput. System Sci. 43 (3) (1991) 425–440. [12] P.-J. Wan, S.C.-H. Huang, L. Wang, Z. Wan, X. Jia, Minimum-latency aggregation scheduling in multihop wireless networks, in: MOBIHOC, 2009, pp. 185–194. [13] X. Xu, X.Y. Li, X. Mao, S. Tang, S. Wang, A delay-efficient algorithm for data aggregation in multihop wireless sensor networks, IEEE Trans. Parallel Distrib. Syst. 22 (2011) 163–175. [14] X. Xu, S. Wang, X. Mao, S. Tang, P. Xu, X.-Y. Li, Efficient data aggregation in multi-hop WSNs, in: GLOBECOM, 2009, pp. 3916–3921. [15] B. Yu, J. Li, Y. Li, Distributed data aggregation scheduling in wireless sensor networks, in: INFOCOM, 2009, pp. 2159–2167.
Contents lists available at ScienceDirect
Theoretical Computer Science www.elsevier.com/locate/tcs
Note
A note on the complexity of minimum latency data aggregation scheduling with uniform power in physical interference model Nhat X. Lam a , Tien Tran a , Min Kyung An b , Dung T. Huynh a,∗ a b
Dept. of Computer Science, University of Texas at Dallas, Richardson, TX 75080, USA Dept. of Computer Science, Sam Houston State University, Huntsville, TX 77341, USA
a r t i c l e
i n f o
Article history: Received 2 September 2014 Received in revised form 20 November 2014 Accepted 29 November 2014 Available online 12 December 2014 Communicated by D.-Z. Du
a b s t r a c t In this paper we prove that the Minimum Latency Aggregation Scheduling (MLAS) problem in the Signal-to-Interference-Noise-Ratio (SINR) model is APX-hard in the uniform power model. © 2014 Elsevier B.V. All rights reserved.
Keywords: Wireless sensor network Algorithm Complexity APX-hardness
1. Introduction Data aggregation is one of main applications of Wireless Sensor Networks (WSNs), and its main purpose is to collect data periodically from the sensor nodes and forward it to a destination called the sink node. As these tiny devices have limited energy resources, researchers have focused on finding ways to avoid sensor nodes’ unnecessary retransmissions of their collected data in order to extend the network lifetime. One approach is to compute schedules with the minimum number of timeslots such that data can be aggregated without any collision or interference. This problem is known as the Minimum Latency Aggregation Scheduling (MLAS) problem. In the literature, wireless networks are commonly modeled as graphs where any two nodes are connected via a communication edge if they are covered by each other’s transmission range. When considering the MLAS problem on such networks, choosing the interference model is a crucial step. While a substantial amount of research results have been obtained for the graph-based interference model, recently, several researchers have started investigating the problems in the more realistic physical interference model, also known as Signal-to-Interference-Noise-Ratio (SINR), which, unlike the graph model, more adequately captures real world phenomena. As the SINR model has been introduced only recently, few works exist and algorithms that guarantee theoretical performances are scarce. Along with the interference models, researchers have adopted one of two power models, uniform power and non-uniform power models, concerning the MLAS problem. The uniform power model assumes no power control, i.e., a uniform power level is typically used, whereas in the non-uniform model, determining the right power levels to be assigned to sending nodes
*
Corresponding author. E-mail address: [email protected] (D.T. Huynh).
http://dx.doi.org/10.1016/j.tcs.2014.11.034 0304-3975/© 2014 Elsevier B.V. All rights reserved.
N.X. Lam et al. / Theoretical Computer Science 569 (2015) 70–73
71
could also be part of the problem (also known as power control). The non-uniform power model is divided into three cases: the bounded power, the unlimited power, or the discrete power model. In the bounded power model, each node u is assigned a transmission power level p u ∈ [ p min , p max ], and in the unlimited power model, u is assigned a transmission power level p u ∈ [ p min , ∞]. In the discrete power model, each node u is assigned a transmission power level p u ∈ { p 1 , p 2 , ..., pk }, where k is the number of power levels used in the network. The MLAS problem in the graph-based interference model has been investigated by many researchers over the last several years. Assuming the uniform power model, in the collision-free graph model, Chen et al. [3] proved the NP-hardness of the MLAS problem and proposed a (Δ − 1)-approximation algorithm, where Δ is the maximum node degree. Later, Huang et al. [6] proposed a nearly-constant factor approximation algorithm whose latency is bounded by 23R + Δ − 18, and Yu et al. [15] introduced a distributed algorithm whose latency is bounded by 24D + 6Δ + 16, where R is the radius and D is the diameter of the network. Subsequently, Xu et al. [13,14] introduced a better constant factor approximation algorithm whose latency is bounded by 16R + Δ − 14, and Wan et al. [12] proposed three algorithms whose latency is log R bounded by 15R + Δ − 4, 2R + O (log R ) + Δ, and (1 + O ( √3 )) R + Δ, respectively. While only collision was considered R
in these papers, some researchers have studied the problem taking into consideration interference as well. This is done in the collision-interference-free graph model. Wan et al. [12], An et al. [1] proposed constant factor approximation algorithms whose latency is bounded by O ( R + Δ). In the SINR model, Li et al. [10] introduced the first constant factor approximation algorithm whose latency is bounded by O ( R + Δ). Assuming the nonuniform power model, An et al. [1] proved an Ω(log n) approximation lower bound in the metric model, where n is the number of nodes. It was investigated without power control in the collision-interference-free graph model with discrete power levels. In the SINR model with bounded power, Lam et al. [8] studied the MLAS problem with power control, and showed the first constant factor approximation algorithm whose latency is bounded by O ( R + log n). Later, Du et al. [4] proposed another constant factor approximation algorithm whose latency is also bounded by O ( R + log n) in the same model. In the unbounded power model with power control, Li et al. [9] proposed a distributed algorithm that yields O (χ ) timeslots, where χ is the link length diversity, and a centralized algorithm whose latency is O (log3 n) which was improved by Halldórsson and Mitra [5] to O (log n). In the discrete power model without power control, Lam et al. [7] showed not only an Ω(log n) approximation lower bound in the metric SINR model, but also its NP-hardness in the geometric SINR model. Lam et al. [7] has been extended in An et al. [2] and introduced two constant factor approximation algorithms whose latencies are bounded by O ( R + Δ) assuming the dual power model, i.e., each node is assigned either the high power level or the low power level. In this paper, we continue the study of the Minimum Latency Aggregation Scheduling (MLAS) problem in the metric SINR model with discrete power levels, but without power control. Assuming the most restricted model of uniform power, we prove the APX-hardness of the problem in the metric SINR model. The rest of this paper is organized as follows. In Section 2, we describe our network model and introduce the definitions used in this paper. In Section 3, we prove the APX-hardness. Finally, Section 4 contains some concluding remarks. 2. Models and definitions In this section, we introduce two SINR models, the metric model and the geometric model. For the former, we model a WSN in a metric space as ( V , D ), where V is a set of sensor nodes, and D : V × V −→ R + is the distance function that satisfies the triangle inequality. Considering a communication link li (si , r i ), where si is a sender and r i is a receiver, let CL(li ) be the set of other links concurrently sending at the time as li . According to the physical interference model, we have
SINR(li ) =
N+
p si D (si , r i )−α
l j ∈C L (li )
p s j D (s j , r i )−α
where N is the ambient noise, α is the path loss, and p u is the power level assigned to node u. Then, the receiver r i can successfully receive the signal from the sender si if and only if its SINR value exceeds a given threshold β ≥ 1. So a node u p with power p u can send signals to only nodes in the distance d where dα ≤ N uβ . We call these nodes u’s neighbors. In this paper, we are specifically concerned with the uniform power model. We use a single transmission power level denoted by p where each node u in V uses the transmission power p to communicate. In the restricted geometric model, the set V of sensor nodes are deployed on the plane and the distance function D is defined as the Euclidean distance between two nodes. Regarding the definition of neighboring nodes, we assume that a p sender u can send data to only nodes in the distance d, where dα ≤ γ βuN , for some constant γ > 1. The data aggregation problem for either model is defined as follows. A schedule is defined to be a sequence of timeslots, at each of which, several nodes are scheduled to send its aggregated data to one of its neighbors, and every node can be scheduled as a sender only once. Formally, at each timeslot t, we have an assignment vector πt = (lt1 , lt2 , ..., lt w ), in which lt i is a directed link from st i to rt i satisfying the SINR threshold inequality. And a schedule, as a sequence of assignment vectors, is denoted as Π = (π1 , π2 , ..., π L ), where L is the length of schedule or the schedule latency. Given a set of source nodes and the sink node s, the objective of the data aggregation problem is to find the minimum latency schedule to aggregate data from all source nodes to the given sink.
72
N.X. Lam et al. / Theoretical Computer Science 569 (2015) 70–73
Fig. 1. The corresponding MDAS instance.
3. The APX-hardness In this section, we prove the APX-hardness for the MLAS problem in the metric model with uniform power. From the reduction in the proof, it follows that MLAS is NP-complete. Theorem 1. Finding the minimum latency aggregation schedule in the metric SINR model with uniform power is APX-hard. Proof. We construct a polynomial-time L-reduction from the MINIMUM B-K-SET COVER problem (MBKSC) that is known to be APX-complete [11]. This problem is a variation of the MINIMUM SET COVER problem, where the cardinalities of all sets are bounded from above by a fixed constant K and the number of occurrences of any element is bounded by a fixed constant B ≥ 2. Let I S be an instance of MBKSC consisting of a collection S of subsets of a finite set E of elements. Let n and m denote the cardinalities of E and S, respectively. A solution to I S is a subset S ⊆ S such that every element e ∈ E is in at least one set A ∈ S . Let OPT S denote an optimal solution to I S . Given the instance I S of the set cover problem, we construct in polynomial time an instance I M of the MLAS problem as follows. I M consists of n + m + 1 nodes. The set V of nodes in I M is partitioned into three subsets of nodes. The first subset contains only a single node s as the sink node. The second subset C S and the third subset C E consist of m and n nodes, respectively. The nodes in C E are source nodes (see Fig. 1). The m nodes in C S correspond to the m sets in the collection S. For each node a ∈ C S , let S (a) be the set in S corresponding to a. Similarly, the set C E of n nodes corresponds to the n elements in the set E of I S . And e (b) denotes the element in E that corresponds to node b in C E . Let p be a constant denoting the uniform power level of all sensor nodes. According to the definition of the SINR model, p a node can send data to another nodes at distance d if dα ≤ γ β N . Letting d0 be d0 = function D on V × V for the metric model of the MLAS problem as follows:
α
p
γ β N , we now define the distance
⎧ d0 ⎪ ⎪ ⎨√ α
u ∈ C S and v = s β d0 u , v ∈ C S D (u , v ) = d0 u ∈ C s , v ∈ C E , e ( v ) ∈ S (u ) ⎪ ⎪ ⎩√ α β d0 u , v ∈ C E
√
All other distances follow from symmetry or are induced by the shortest path metric. Since 0 ≤ α β d0 ≤ 2d0 , the triangle inequality holds for D.√ Next observe that α β d0 > d0 (β > 1). Hence, it follows that there does not exist any direct connection between nodes in C E . Similarly, there is no direct connection between any node a in C E and nodes in C S whose corresponding sets do not contain e (a). Let SOL M denote a solution to I M . We construct the corresponding solution SOL S to I S as follows: SOL S = { S (a)|a ∈ C S , a appears in SOL M }. Let us denote OPT M as an optimal solution to I M . The following two results will be shown in Lemma 1 and Lemma 2 below.
L (SOL M ) = |SOL S | + n
|OPT S | K K ( B − 1) + 1 ≥ n Thus,
L (OPT M ) = |OPT S | + n ≤ |OPT S | + |OPT S | K K ( B − 1) + 1
≤ K K ( B − 1) + 1 + 1 |OPT S | ≤ ε |OPT S |
In addition, we also have:
N.X. Lam et al. / Theoretical Computer Science 569 (2015) 70–73
73
|SOL S | − |OPT S | = L (SOL M ) − n − L (OPT M ) − n = L (SOL M ) − L (OPT M ) This implies that the MBKSC problem is L-reducible to the MLAS problem. As it has been shown in [11] that the MBKSC is APX-complete, the theorem follows. 2 In order to prove Lemma 1 and Lemma 2, we need the following fact. Fact 1. While a node in V is sending, no other nodes can send data without any interference. Proof. Consider nodes i , v ∈ C S and node u ∈ C E . Let i send data to s and u send data to v at the same time. Then
SINR(u , v ) =
α p .d− 0
N + p .(d0
√ α
β)−α
=
γ βN N +γN
=
γβ 1+γ
<β
Similarly, we can show that 2 nodes in C E or C S cannot send data at the same time.
2
Lemma 1. L (SOLM ) = |SOL S | + n. Proof. According to Fact 1, we can split the schedule into two independent phases. In the first phase, all nodes in C E send their data to selected nodes in C S . In the second phase, selected nodes in C S send received data to s. According to Fact 1, it is trivial that we need exactly n timeslots for the first phase. And also based on Fact 1, the number of timeslots required for the second phase is the number of selected nodes in C S . Hence the lemma follows. 2 Lemma 2. |OPT S | K ( K ( B − 1) + 1) ≥ n. Proof. Consider the sets in S which are sorted in non-increasing order based on their cardinalities. Pick the largest one and move this set to the selected group. Then remove all subsets that covered by the selected group. Repeat this procedure until all sets are removed. Since every element occurs at most B times (B ≥ 2), and the cardinality of each set is bounded by K , there are at most K ( B − 1) + 1 sets to be removed at every step. Since the minimum number of sets is Kn , there are at least K ( K ( Bn−1)+1) steps with
n K ( K ( B −1)+1)
selected sets. Hence,
n K ( K ( B −1)+1)
≤ OPT S . 2
4. Conclusion In this paper, we studied the Minimum Latency Aggregation Scheduling (MLAS) problem in the metric SINR model without power control assuming the uniform power model, i.e., each node is assigned a single power level. We have proposed a formal proof to show the APX-hardness of the problem. The precise complexity of MLAS in the geometric SINR model with uniform power remains an interesting open question. References [1] M.K. An, N.X. Lam, D.T. Huynh, T.N. Nguyen, Minimum data aggregation schedule in wireless sensor networks, Int. J. Comput. Appl. 18 (4) (2011) 254–262. [2] M.K. An, N.X. Lam, D.T. Huynh, T.N. Nguyen, Minimum latency data aggregation in the physical interference model, Comput. Commun. 35 (18) (2012) 2175–2186. [3] X. Chen, X. Hu, J. Zhu, Data gathering schedule for minimal aggregation time in wireless sensor networks, Int. J. Distrib. Sens. Netw. 5 (4) (2009) 321–337. [4] H. Du, Z. Zhang, W. Wu, L. Wu, K. Xing, Constant-approximation for optimal data aggregation with physical interference, J. Global Optim. 56 (4) (2013) 1653–1666. [5] M.M. Halldórsson, P. Mitra, Wireless connectivity and capacity, in: SODA, 2012, pp. 516–526. [6] S.C.H. Huang, P.-J. Wan, C.T. Vu, Y. Li, F. Yao, Nearly constant approximation for data aggregation scheduling, in: INFOCOM, 2007, pp. 6–12. [7] N.X. Lam, M.K. An, D.T. Huynh, T.N. Nguyen, Minimum latency data aggregation in the physical interference model, in: MSWiM, 2011, pp. 93–102. [8] N.X. Lam, M.K. An, D.T. Huynh, T.N. Nguyen, Scheduling problems in interference-aware wireless sensor networks, in: ICNC, 2013, pp. 783–789. [9] H. Li, Q.S. Hua, C. Wu, F.C.M. Lau, Minimum-latency aggregation scheduling in wireless sensor networks under physical interference model, in: MSWiM, 2010, pp. 360–367. [10] X.-Y. Li, X. Xu, S. Wang, S. Tang, G. Dai, J. Zhao, Y. Qi, Efficient data aggregation in multi-hop wireless sensor networks under physical interference model, in: MASS, 2009, pp. 353–362. [11] C.H. Papadimitriou, M. Yannakakis, Optimization, approximation, and complexity classes, J. Comput. System Sci. 43 (3) (1991) 425–440. [12] P.-J. Wan, S.C.-H. Huang, L. Wang, Z. Wan, X. Jia, Minimum-latency aggregation scheduling in multihop wireless networks, in: MOBIHOC, 2009, pp. 185–194. [13] X. Xu, X.Y. Li, X. Mao, S. Tang, S. Wang, A delay-efficient algorithm for data aggregation in multihop wireless sensor networks, IEEE Trans. Parallel Distrib. Syst. 22 (2011) 163–175. [14] X. Xu, S. Wang, X. Mao, S. Tang, P. Xu, X.-Y. Li, Efficient data aggregation in multi-hop WSNs, in: GLOBECOM, 2009, pp. 3916–3921. [15] B. Yu, J. Li, Y. Li, Distributed data aggregation scheduling in wireless sensor networks, in: INFOCOM, 2009, pp. 2159–2167.