Dual Power Assignment Optimization For k-Edge Connectivity in WSNs

2011 8th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks

Dual Power Assignment Optimization For k -Edge Connectivity in WSNs Nhat X. Lam, Trac N. Nguyen, Min K. An and D.T. Huynh Department of Computer Science University of Texas at Dallas Richardson, Texas 75083-0688 E-mail: {nxl081000,nguyentn,mka081000,huynh}@utdallas.edu

Abstract—Power consumption is one of the crucial issues in Wireless Sensor Networks (WSNs). It therefore has been the focus of many researchers. An important problem concerning power consumption is how to minimize the number of maximum power nodes while maintaining a desired network topology. As fault tolerance is vitally important in practice, it is desirable that the constructed network topology is k-edge-connected (or k-connected). In this paper, we study the dual power assignment problem for k-edge connectivity (kEDP ) in WSNs. While other studies consider only the special case k = 2, our goal is to address the general problem. We prove the NP-completeness of kEDP problem in the geometric case and provide a 2-approximation algorithm using linear programming techniques. To our knowledge, this approximation ratio is currently the best one. We also introduce a heuristic whose performance is better compared with an approximation algorithm in [1].

Keywords: dual power, connectivity, NP-complete, approximation, heuristic, wireless sensor network I. I NTRODUCTION Wireless sensor networks (WSNs) have been widely used in military and civilian applications in the last two decades. Due to the wide applications of WSNs and the emergence of new technology, WSNs are now a very active research area. One of the core issues concerning WSNs is energy consumption since each sensor is equipped with a small limited battery. A well known approach to achieve energy efficiency is to use topology control by assigning the power usage at each node thereby establishing an energy efficient network with desired features, such as connectivity, low interference and fault tolerance. In regard to connectivity, considerable attention has been given to the problems of minimizing the maximum power utilized at any node as well as the problem of minimizing the total power consumption of the network. While polynomial time algorithms are known for the former problem [2], [3], the latter one was proved to be NP-hard by Kirousis et al.[4]. Following this result, several approximation algorithms have been proposed in [4], [5], [2]. Another interesting problem is to minimize the number of maximum-power nodes. The importance of this problem has been analyzed in the studies of Poojary et al. [6] and Shah et al. [7]. In [8], Lloyd et al. constructed an approximation preserving reduction from minimizing the number of maximumpower nodes to minimizing the total power. This implies that

978-1-4577-0093-4/11/$26.00 ©2011 IEEE

any α-approximation of the former is also an α-approximation of the latter. In this paper, we consider the dual power model and address the issue of fault tolerance in the form of k-edge connectivity. We investigate the problem of minimizing the number of maximum-power nodes in such a way that results in a kedge-connected topology. Recall that in the dual power model, only two discrete power levels are available to assign to the nodes. It is easy to see that in this model, minimizing number of maximum-power nodes is equivalent to minimizing total power. The connectivity problem in this model was first studied by Rong et al. [9]. The authors provided an NP-completeness proof and proposed a 2-approximation algorithm. The approximation ratio was later improved to 1.75 by Chen et al. [10]. While these studies only focused on the asymmetric version, in [8] Lloyd and Liu proved the NP-hardness of the symmetric version and gave an approximation algorithm with the ratio of 1.67. Addressing fault tolerance, [11] was the first to study the 2-edge connectivity and bi-connectivity problems. The authors proposed a 6-approximation for the former and a 5approximation for the latter. Wang et al. proposed another algorithm to improve the ratio to 4 for both problems [1]. They later improved the ratio to 3.67 [1] by utilizing the algorithm of Lloyd et al. [8]. In this paper, we address a more general problem, namely the dual power assignment for k-edge connectivity. We first prove that this problem is NP-hard even in the (planar) geometric case. Then by formulating the problem as an integer linear programming problem, we design a 2-approximation algorithm using the iterated rounding method proposed by [12]. We also study the performance of the 3.67-approximation algorithm in [1] and our new heuristic through simulation. The rest of this paper is organized as follows. Section II provides the definitions and explanations used in this paper. Section III is devoted to the NP completeness result, and Section IV contains the 2-approximation algorithm. Section V discusses our heuristic and compares its performance against the approximation algorithm in [1] through simulation. Finally, Section VI contains some concluding remarks.

566

x

II. P RELIMINARIES

~x

p

p

p

~x

A. Network Model

r

p

Consider a set V of transceivers (nodes) in the plane. Each node u is assigned a power level denoted by p(u). The signal transmitted by node u can only received by a node v if the distance between u and v, denoted by d(u, v), is ≤ p(u). We only consider the bidirectional case [1] in which a communication edge (u, v) exists between two nodes u and v if and only if p(u) ≥ d(u, v) and p(v) ≥ d(u, v). Thus, the set V of nodes in the plane together with the power assignment p that assigns power levels to the nodes define a geometric (also known as intersection) graph Gp = (V, Ep ).

x

o r

p

o

x

r

r

o

p

r

o

x

o

r

r o

r

~x

p

~x

o r o

p x

r p p

p

Fig. 1.

p

o

o

r

o

~x

o r

~x

x

A single variable gadget

B. k-Edge Connectivity Dual Power Assignment Model In this paper we are specifically concerned with the dual power model as described in [1]. In this model, we use only two transmission power levels denoted by ph (high power) and pl (low power) where (ph > pl ). A node is called high (low) power node if it is assigned high (low) power level ph (pl ). Similarly, we define Eh = {e(u, v)|ph ≥ d(u, v) > pl } and El = {e(u, v)|d(u, v) ≤ pl } as sets of high power and low power edges, respectively. The objective of our problem is to construct a power assignment so that the induced graph is k-edge connected while minimizing the number of high power nodes, thereby minimizing the total power consumption in the network. (Recall that a k-edge connected graph is a graph in which removing any k−1 edges does not disconnect it.) The problem of minimizing the number of high-power nodes to obtain a k-edge-connected graph, denoted kEDP , can be formally defined as follows. Instance: Given a set of N nodes V = {v1 , v2 , ..., vN } on a plane, a Euclidean distance function d : V × V −→ R+ , a set of two power levels P = {pl , ph } at which a node can transmit, and a positive integer Q ≤ N . Question: Is there a power assignment to each node so that it induces a symmetric k-connected graph and the number of nodes assigned power level ph is bounded by Q?

C

x ~x

o

x o o ~x

~x

o

o

x o

o

o

o x

o

o

o

o

o

~z

o

o o

o o

~z

~y

o ~y

z

o o

o

z

y y

o

o

o

~y

x

z

o

o

y

z

~z y

o o

o x ~x

~x

o

~y o

o

~z

z ~y

y ~x

o o

y

~y

~z ~z

z

Fig. 2. The chain of variable gadgets contains a series of single and variable gadget

3 literals has a satisfying assignment such that exactly two literals per clause have the value TRUE. Note that if we negate the literals in an instance of Pl-1EX3SAT, which was proven to be NP-complete in [13], we have an instance of Pl-2EX3SAT. Hence, the Pl-2EX3SAT problem is also NPcomplete. From an instance of Pl-2EX3SAT we construct an instance of 2EDP as follow: d2

d2

~x t

C

u

~y

t

d1

III. NP C OMPLETENESS In this section, we prove the NP-completeness of the kEDP problem for geometric graphs. To obtain this result, it is sufficient to show the NP-completeness for a specific value of k. The following theorem is a result for the case k = 2. Theorem 1 2EDP is NP-complete. Proof: Given a power assignment we can verify in polynomial time that the resulting graph is 2-edge-connected by removing any edge and check for connectivity, and that the number of nodes that are assigned power level ph is ≤ Q. To show that 2EDP is NP-hard, we construct a polynomialtime reduction from the Planar 2 Exact 3-SAT (Pl-2EX3SAT) problem. The Pl-2EX3SAT problem is the problem of determining if a CNF formula in which each clause has exactly

~z

t d2

Fig. 3.

Placing bridge nodes from variable nodes to a clause node

Let F be an instance of Pl-2EX3SAT. Each clause in F is represented by a node called clause node in the instance of 2EDP . Let k be the maximum number of occurrences of a variable and its negation in F . For each variable x we create a variable gadget with 2 ∗ k literal nodes x and ˜ x (See Figure 1). Between every pair of a variable node and its negation in the gadget, we add one extra node (node p in Figure 1). For convenience, let us call a variable node and its negation variable nodes, and the new extra node variable-control node. The variable and variable-control nodes are connected to

567

~x d1

o

~x

d1

r

t

r Fig. 4. node

p

Placing bridge nodes from a variable node to a variable-connect d2

k

d1

x

d2

~x p

t

k

r

x

d1

Fig. 6. Placing node between a variable-control node and two neighboring variable nodes connected by perpendicular edges

r Fig. 5. Placing node between a variable-control node and two neighboring variable nodes connected by collinear edges

nodes (o’s and r’s) placed on an inner cycle inside the variable gadget. These nodes (o’s and r’s) are called variable-connect nodes. (In Figure 1, nodes x’s and ˜ x’s are variable nodes, nodes p’s are variable-control nodes, and nodes o’s and r’s are variable-connect nodes.) Each variable gadget uses 2 variable nodes to connect with other variable gadgets. Consecutive variable gadgets are connected by edges that link all variable gadgets together (See Fig. 2). To connect a variable node to a clause node, only variable nodes with degree 3 are used, and each such variable node is used to connect to at most one clause node if the variable or its negation appears in a clause in F . Thus, the degree of every node is bounded by 4. We now use Valiant’s result [14] to embed a planar graph with maximum degree 4 into the Euclidean plane: A planar graph with maximum degree 4 can be embedded in the plane using O(|V |) area in such a way that its vertices are at integer coordinates and its edges are drawn so that they are made up of line segments of form x = i or y = j, for integers i and j. After the embedding, we modify the edges of the graph already embedded in the plane to create an instance of 2EDP as follows. Let d be a unit distance in the plane. We define two distances d2 = d/10 and d1 = d2 /10. 1) On the edge connecting a variable node with a clause node, starting at distance d2 from the variable node, we add consecutive nodes at distance d1 , i.e., adjacent nodes are at distance d1 apart. We call these nodes auxiliary nodes. We now place additional nodes to the auxiliary nodes to form bridges to ensure 2-edge connectivity in certain parts of the graph. Consider the three edges connecting a clause node with variable nodes. Being embedded in the plane, two of these edges are collinear and the other

is perpendicular. On each of the collinear√edges, one set of new nodes are added at distance (d1 / 2) from each consecutive pair of auxiliary nodes to create isosceles right triangles (See Figure 3). On the perpendicular edge from the variable node to a clause node, we add a similar set of new nodes to the auxiliary nodes. However, the last new node is added at distance d1 from the auxiliary nodes (See node u in Figure 3). (This adds the additional property of planarity in the resulting graph.) Let us call all these newly added nodes bridge nodes, and the last auxiliary nodes at distance d2 from a variable node variable-interface nodes (see nodes t’s). 2) On the edge connecting a variable node to a variableconnect node (nodes o’s), we add auxiliary nodes such also that consecutive nodes are at distance d1 apart. We √ add bridge nodes as done above at distance (d1 / 2) from each pair of consecutive auxiliary nodes to create an isosceles right triangles forming bridges as before (See Figure 4). 3) On the two edges connecting a variable-control node (nodes p’s) to both of its neighboring variable nodes, starting from distance d2 from both variable nodes, we add auxiliary nodes at distance d1 from each other. The first added nodes are also called variable-interface nodes (nodes k’s and t’s in Figures 5 and 6).√As before, we also add bridge nodes at distance (d1 / 2) from each pair of consecutive auxiliary nodes to create isosceles right triangles forming bridges (Figure 5 and Figure 6). 4) On the remaining edges we place nodes at distance d1 from each other. We now have a set of nodes on the plane. To complete the construction of the instance of 2EDP problem, we define the power levels pl , ph , and the positive integer Q as follows:

568

pl = d 1 ,

ph = d 2 ,

Q = n ∗ (3 ∗ k ∗ u) + 2 ∗ m

where n and m are the numbers of variables and clauses in F , respectively. As this construction is clearly computable in polynomial time, we proceed to show the correctness of the reduction. For the only-if direction, suppose that the Pl-2EX3SAT instance F is satisfiable. Let φ be a Boolean assignment that satisfies F such that each clause in F has exactly 2 literals with the value TRUE. We assign the power level ph to the nodes as follows: 1) If a variable x is assigned value TRUE, then the variable node x in the 2EDP instance is assigned power level ph ; otherwise the variable node ˜ x is assigned the power level ph . 2) The variable-interface nodes which ”interface” with the variable nodes having power level ph are assigned power level ph . 3) All remaining nodes are assigned power level p1 . Thus, in every variable gadget for variable x either k variable nodes x or their negations ˜ x are assigned power level ph . The variable-interface nodes of each of these variable nodes within the gadget are also assigned ph . Hence, in each gadget there are 3∗k nodes with power level ph . Similarly, for the three variable nodes connected to a clause node, two of them have power level ph , and therefore two variable-interface nodes are assigned the power level ph . Thus, the total number of nodes assigned power level ph is n ∗ (3 ∗ k ∗ u) + 2 ∗ m which is ≤ Q. Now let us examine the graph resulting from this power assignment and show that it is 2-edge-connected. First consider the edges that link all variable gadgets together in a circular fashion. Obviously, at least two nodes/edges must be removed for the graph to be disconnected. Also, due to the bridges, each of the connections within a variable gadget requires at least 2 edges to be removed for the graph to be disconnected. Finally, for the connections from variable nodes to a clause nodes, at least 2 nodes have to be removed for the graph to be disconnected since there are exactly two literals with value TRUE in a clause. For the if direction suppose that for the 2EDP instance there is a power assignment that results in a 2-edge-connected geometric graph such that the total number of nodes assigned power level ph is ≤ Q. Without loss of generality assume that the number of nodes with power level ph is minimum. We show how to construct a Boolean assignment that satisfies F . We have these observations: 1) Each clause node must be connected to at least 2 different variable nodes for the graph to be 2-edge-connected. This requires that at least 2 variable-interface nodes of each clause have power level ph . While one variableinterface node with power level ph does not make the graph 2-edge-connected, three variable-interface nodes with power level ph would violate the fact that number of nodes with power level ph is mimimum since one of the three variable-interface nodes with ph can be assigned power level pl , and 2-edge-connectivity is

preserved. 2) For each variable gadget, at least 2 ∗ k nodes on the edges from the variable-control nodes to the variable nodes must have power level ph for the graph be 2edge-connected. 3) For the minimum number of 3∗k nodes in each variable gadget to have power level ph , exactly k variable nodes x or their negations ˜ x must have the power level ph to connect to the variable-control nodes. From the above observations, we can assign a variable in F the value TRUE if its variable node in the 2EDP instance has the power level ph . Observation 1 ensures that each clause will have exactly 2 variables with value TRUE, whereas Observations 2 and 3 guarantee that the Boolean assignment is consistent. This concludes the proof of Theorem 1. Remarks. The reader should note that the proof of Theorem 1 also holds for vertex connectivity. Thus, the dual power assignment optimization problem for k-vertex connectivity is also NP-complete for geometric graphs. Furthermore, the reduction in the proof yields a geometric graph that is planar, i.e., no edge crosses another. Hence the results apply to planar geometric graphs as well. IV. 2-A PPROXIMATION A LGORITHM In this section, we first construct a linear programming formulation for kEDP and then utilize the approach in [12] to design for it a 2-approximation algorithm. A. Formulation Let δE (S) := {e(u, v) ∈ E|u ∈ S ⊕ v ∈ S} where S ⊂ V (E). In other words, δE (S) is the set of edges in E with exactly one endpoint in S. In addition, we use E(v) to denote a subset of E containing all edges incident to v and V (E) to denote a set of all vertices incident with edges in E. Since the objective of the kEDP problem is to minimize the number of high power nodes, all low power edges can be used freely without increasing the value of the objective function. So the main issue is to efficiently pick appropriate high power edges to obtain a k-edge connected graph. For every high power edge e we use variable xe to express whether this edge is used xe = 1 or not xe = 0. We also need to formulate the requirement of k-edge connectivity. Recall that a k-edge connected graph is a graph in which every cut has at least k edges crossing it. So this requirement can be formulated as a restriction on every cut of the induced graph. Now the kEDP problem can be formulated as follows. IP1 min

 v∈V

s.t.

max xe

e∈Eh (v)



xe ≥ k − |δEl (S)|

for all S ⊆ V

e∈δEh (S)

xe ∈ {0, 1}

for all e ∈ E

The system of inequalities in the above formulation means that any feasible solution which is a selection x of xe , e ∈ Eh ,

569

satisfies that requirement that for every cut in the induced graph, there are at least k edges crossing this cut. The relaxation of IP1 is: LP2 min

 v∈V



max xe

e∈Eh (v)



s.t.

xe ≥ k − |δEl (S)|

xe

0 ≤ xe ≤ 1

for all e ∈ E

Although the objective function is not linear, LP2 is equivalent to the following linear programming formulation.  yv min v∈V

x e ≤ yv 

yv ≤

for all v ∈ V and e ∈ Eh (v) xe

for all v ∈ V

e∈Eh (v)



xe ≥ k − |δEl (S)|

for all S ⊆ V

e∈δEh (S)

0 ≤ yv ≤ 1 0 ≤ xe ≤ 1

for all v ∈ V for all e ∈ E

B. IR − AP P ROX Algorithm In this subsection we describe a 2-approximation algorithm in which we first solve LP2 fractionally and then use the iterated rounding technique first introduced in [12] to convert a fractional solution of LP2 into an integral solution which can be shown to be a 2-approximate solution of IP1. Let x∗ be an optimal basic solution of LP2 found by the given separation oracle. Denoting F = {e|x∗e ≥ 12 }, we round x∗e for all e in F to 1 and solve the residual linear programming problem which can be formulated as follows.

min

 v∈V \V (F )

s.t.

max xe

e∈Eh (v)



xe ≥ k − |δEl ∪F (S)|

for all S ⊆ V

e∈δEh \F (S)

0 ≤ xe ≤ 1

for all e ∈ Eh \ F

1 0

e∈F e ∈ Eh \ F

IR-APPROX(V, Eh , El ) Input: + set V of nodes in the plane + set Eh of all high power edges + set El of all low power edges Output: number of high power nodes 1 F =∅ 2 while F ∪ El is not k-edge-connected 3 Find an optimal basic feasible solution x to LP3 4 F = F ∪ {e|xe ≥ 12 } 5 return |V (F )| Fig. 7.

However, it is not straightforward that LP2 is solvable in polynomial time since the number of constraints is exponential. Recall that while running the Ellipsoid method, each iteration of this algorithm either checks if the center is feasible or returns a violated constraint. And it is known that finding a violated constraint for any infeasible solution can be done in polynomial time by utilizing the Max-flow Min-cut Theorem. The solution is not feasible if and only if there exists a pair of nodes whose min-cut corresponds to a violated inequality. So we assume that such a separation oracle for solving LP2 is given.

LP3

=

 We also have v∈V maxe∈Eh (v) xe = |V (F )|. So for convenience, we call F an integral solution of IP1. The pseudo-code of this algorithm can be found in Figure 8.

for all S ⊆ V

e∈δEh (S)

s.t.

We iteratively apply this rounding method for the residual problem to construct F until the graph induced by El ∪ F is k-edge connected. The corresponding integral solution to IP1 can be obtained as follows.

I TERATED ROUNDING

C. Performance Analysis In this subsection, we will show that IR − AP P ROX produces a 2-approximate solution in polynomial time. First, let us recall the fundamental result of [12] for the iterated rounding technique. Lemma 2 In any basic solution to LP2 and LP3, there is one component whose value is at least 12 . From Lemma 2, it follows that at every iteration at least one edge is rounded, and hence, the number of iterations is bounded by |Eh |. Thus, in order to prove that IR−AP P ROX is in P, it is sufficient to show how to find a basic solution to LP3 efficiently. It is clear that LP3 has a tight relation to LP2. Therefore, we can also construct a separation oracle for LP3 from the given one for LP2. The construction can be found in [12]. We now derive the approximation ratio of 2. Let the optimal ∗ , respectively. We have values of LP2 and LP3 be z ∗ and zres the following lemma and theorem which are similar to the ones in [12]. Lemma 3 If Q is an integral solution of LP3 with value at ∗ , then Q ∪ F is an integral solution to LP2 with most 2zres value at most 2z ∗ . Proof: Clearly, Q ∪ F is a feasible integral solution to LP2. Notice that the projection of x∗ on V \V (F ) is a feasible

570

solution to LP3, hence we have  ∗ zres ≤ z∗ − v∈V (F )

max xe

e∈Eh (v)

For any v ∈ V (F ), we know that there exists e ∈ Eh (v) such that xe ≥ 12 . Hence it follows that

B. Heuristic In this subsection, we provide a simple greedy heuristic. The basic idea of this heuristic comes from the following lemma, which follows from Proposition 3.1.3 in [15]. Recall that an H-path of a subgraph H of G is defined to be a simple path starting from a node s ∈ H, going through several nodes v ∈ G \ H and stopping at a node t ∈ H. Lemma 5 Given a 2-edge connected graph G. Starting from some 2-edge connected subgraph H of G, we can constructed a 2-edge connected spanning subgraph by successively adding H-paths to H.

∗ 2z ∗ ≥ 2zres + |V (F )| ∗ . Therefore By the assumption, we have |V (Q)| ≤ 2zres

2z ∗ ≥ |V (Q)| + |V (F )| ≥ |V (Q ∪ F )|

Theorem 4 IR-APPROX produces a 2-approximation. Proof: The optimal solution to IP1 is also a feasible solution to LP2. Hence, we have 2opt ≥ 2z ∗ . So, applying Lemma 3 inductively we obtain the approximation ratio. V. H EURISTIC AND P ERFORMANCE In this section we first review the approximation algorithm proposed by Wang et al. in [1], and then present a new heuristic whose performance is much better in experiments. The rest of this section is organized as follows. The first two subsections describe the approximation algorithm in [1] and our heuristic. The last subsection contains the experimental results that show the better performance of our heuristic. A. Wang et al. ’s 3.67-approximation algorithm In the following, we will briefly describe the Candidate Sets Filtering Algorithm proposed by Wang et al. [1] to solve the dual power assignment problem for 2-edge connectivity. This 3.67-approximation algorithm consists of two steps. The first step is to utilize the 1.67-approximation algorithm proposed by Lloyd and Liu [8] to construct a rooted spanning tree. In the second step, the algorithm modifies this tree to obtain a 2edge connected graph. This can be done by first using the low power edges to cover ”bridges”, which are defined to be edges whose removal disconnects the graph, and then trying with the maximum-power edges. It is easy to check if low power edges could help to cover bridges in the first step. The main issue is how to pick good high power edges in the second step so as to minimize the number of high power nodes. The authors define a candidate set of high power edges for each node. Their algorithm traverses the tree in a depth-first manner to construct candidate sets and also filter them. Finally, these sets contain only ”good” edges. It is sufficient to pick one in each set to obtain the desired graph. The reader is referred to [1] for further details.

Our heuristic proceeds in a greedy fashion. It starts with a trivial 2-edge connected component H and successively selects in each iteration a proper H-path to expand the current subgraph H until it becomes a 2-edge-connected spanning subgraph of the original graph G. The major issue is the greedy property used to determine the proper H-path at each step. Since our objective is to minimize the number of high power nodes, it is intuitively reasonable to consider the number of high power nodes required in an Hpath as its cost. Following this approach we define the cost of an H-path T as the amount of additional (based on the current status) high power nodes required in order to construct the connection through all nodes in T . So at each greedy iteration, we simply pick a minimum-cost H-path to expand the current subgraph H. The heuristic is described in the pseudo-code in Figure 8. H-PATH -C ONSTRUCTION(V, Eh , El ) Input: + set V of nodes in the plane + set Eh of all high power edges + set El of all low power edges Output: number of high power nodes 1 H = { any node u ∈ V } 2 while |V (H)| = |V | 3 Find a minimum cost H-path T 4 H = H ∪T 5 Eh = Eh \ set of edges in H 6 El = El \ set of edges in H 7 return number of high power nodes in H Fig. 8.

H-PATH -C ONSTRUCTION

The remainder of this subsection focuses on designing an algorithm to find a minimum-cost H-path. For a given node s ∈ H, ws (u, p) denotes the cost of a minimum-cost path from s to u, where u is assigned power level p ∈ {0, pl , ph }. Once all values ws (u, p) are computed, we simply pick the minimum-cost (s0 , u0 , p0 ) = argmin{ws (u, p)|s ∈ H, u ∈ H \ {s}, p ∈ {0, pl , ph }} to construct the desired H-path.

571

It is easy to compute ws (u, p) following the approach of Bellman-Ford algorithm [16]. We also use relaxation (Figure 9) to progressively decrease an estimate ws (u, p) on the value of a minimum-cost path from s to each vertex u ∈ V until it becomes the actual minimum one. Figure 10 is the pseudocode of the algorithm to find a minimum-cost H-path starting from a given source node s in H. The reader should note that an H-path may be a cycle, and we can compute minimum-cost H-cycles as follows. Given s ∈ H, for each edge (s, u), u ∈ / H, we find the minimumcost path from u to s. Together with (s, u), this will form a minimum-cost cycle. Finally, we pick the best one among all cycles found. To compute the minimum-cost paths, we use πs (u, p) to keep track the predecessor of (u, p) in the path from s. This algorithm will set πs so that we can traverse from any node back to s to obtain the corresponding optimum path.

M INIMUM -C OST-H-PATH(G, H, s, Ts ) Input: + graph G(V, E = El ∪ Eh ) + subgraph H + source node s ∈ H Output: minimum cost H-path Ts starting from s 1 Initialization 2 for i = 1 to 3|V | − 1 3 for each edge (u, v) ∈ E 4 RELAX(u, v, ws , πs ) 5 Pick the minimum ws (u0 , p0 ) 6 Construct Ts by traversing πs from (u0 , p0 ) 7 return Ts Fig. 10.

M INIMUM -C OST-H-PATH

C. Simulation In our experiments, we only consider the geometric model. We randomly generate 100 nodes on an area of size of 1000 x 1000. We then find the minimum power level pmin that induces a 2-edge-connected graph. Eventually, ph is picked randomly from the interval (pmin , dmax ), where dmax is the maximum distance between any pair of nodes, and pl is picked randomly from the interval (0, pmin ). We generate 100 cases in our simulation. For each case, we run the two algorithms and compare their performance. The results are shown in Table I. As seen in this table, in 78 out of 100 cases, the results of our heuristic are 19.50% better on average. In the 7 cases where our heuristic performs worse, its results are only 1.98% worse on average. While among the worse cases, the worst value is at most 6.25%, there is a better case where our heuristic performs 56.25% better. Thus, our heuristic performs much better than the approximation algorithm of Wang et al. overall.

RELAX(u, v, ws , πs ) 1 if (u, v) ∈ Eh 2 if ws (v, ph ) > ws (u, pl ) + 2 3 ws (v, ph ) = ws (u, pl ) + 2 4 πs (v, ph ) = (u, pl ) 5 if ws (v, ph ) > ws (u, ph ) + 1 6 ws (v, ph ) = ws (u, ph ) + 1 7 πs (v, ph ) = (u, ph ) 8 else // (u, v) ∈ El 9 if ws (v, pl ) > ws (u, pl ) 10 ws (v, pl ) = ws (u, pl ) 11 πs (v, pl ) = (u, pl ) 12 if ws (v, pl ) > ws (u, ph ) 13 ws (v, pl ) = ws (u, ph ) 14 πs (v, pl ) = (u, ph ) 15 if ws (v, ph ) > ws (u, pl ) + 1 16 ws (v, ph ) = ws (u, pl + 1) 17 πs (v, ph ) = (u, pl ) 18 if ws (v, ph ) > ws (u, ph ) + 1 19 ws (v, ph ) = ws (u, ph ) + 1 20 πs (v, ph ) = (u, ph ) Fig. 9.

(over 100 cases) Cases Maximum value Average value

Worse 7 6.25% 1.98%

Better 78 56.25% 19.50%

TABLE I E XPERIMENT R ESULTS

RELAXATION

It is straightforward to see that the M INIMUM -C OST-H-PATH algorithm runs in time O(N 3 ). The Initialization step which simply sets all values as infinity, requires O(N ) time while each of the O(V ) outer iterations checking over all edges takes O(N 2 ) steps. Thus, the time complexity of this procedure is O(N 3 ). This procedure is applied for each source node s in H. Therefore, each iteration in H-PATH -C ONSTRUCTION requires O(N 4 ) time. In addition, after every iteration, we expand H by at least one node. Hence, the number of iterations is bounded by the number of vertices. Thus, the heuristic has O(N 5 ) time complexity.

VI. C ONCLUSIONS In this paper, we have studied the kEDP problem on the geometric networks. We assumed that there are two power levels which can be used to assign for every node and have shown that this problem is NP-complete. We also introduced a 2-approximation algorithm using linear programming techniques. This approximation ratio beats the current record proposed by Wang et al. [1]. In addition, we proposed a new heuristic which also beats the algorithm of Wang et al. in experiments. However, this paper only focuses on k-edgeconnectivity. As to future work, we plan to study the dual

572

power assignment problem for k-vertex-connectivity which concerns node failure. R EFERENCES [1] C. Wang, J. Wilson, M. Park, A. Farago, and W. Wu, “On dual power assignment optimization for biconnectivity,” in Journal of Combinatorial Optimization, pp. 174–183, 2010. [2] E. L. Lloyd, R. Ramanathan, R. Liu, S. S. Ravi, and M. V. Marathe, “Algorithmic aspects of topology control problems for ad hoc networks,” in Proceedings of the Third ACM International Symposium on Mobile Ad Hoc Networking and Computing, pp. 123–134, 2002. [3] R. Ramanathan and R. Rosales-hain, “Topology control of multihop wireless networks using transmit power adjustment,” in Proceedings of the IEEE Infocom, pp. 404–413, 2000. [4] L. M. Kirousis, E. Kranakis, D. Krizanc, and A. Pelc, “Power consumption in packet radio networks,” Theoretical Computer Science, vol. 243, pp. 289–305, 1997. [5] G. Calinescu and P.-J. Wan, “Range assignment for biconnectivity and k-edge connectivity in wireless ad hoc networks,” in Mobile Networks and Applications, pp. 121–128, 2006. [6] N. Poojary, S. V. Krishnamurthy, and S. Dao, “Medium access control in a network of ad hoc mobile nodes with heterogeneous power capabilities,” in IEEE International Conference on Communications (ICC), pp. 872–877, 2001. [7] V. Shah, S. V. Krishnamurthy, and N. Poojary, “Improving the mac layer performance in ad hoc networks of nodes with heterogeneous transmit power capabilities,” in IEEE International Conference on Communications (ICC), 2004. [8] E. L. Lloyd, R. Liu, and S. S. Ravi, “Approximating the minimum number of maximum power users,” in Ad hoc Networks, Mobile Networks and Applications, pp. 129–142, 2006. [9] Y. Rong, H. Choi, and H. Choi, “Dual power management for network connectivity in wireless sensor networks,” in International Parallel and Distributed Processing Symposium, p. 255b, 2004. [10] J.-J. Chen, H.-I. Lu, C.-Y. Yang, and A.-C. Pang, “Dual power management for network connectivity in wireless sensor networks,” in IEEE GLOBECOM, 2005. [11] M. Park, C. Wang, W. I. JKV, W. Wu, and A. Farago, “Faulttolerant dual-power assignment in wireless sensor networks,” Tech. Rep. UTDCS–52–06, Dept. of Computer Science, University of Texas at Dallas, 2006. [12] K. Jain, “A factor 2 approximation algorithm for the generalized steiner network problem,” Mathematics Subject Classification, 2000. [13] M. Dyer and A. Frieze, “Planar 3dm is np-complete,” in Journal of Algorithms, pp. 174–184, 1986. [14] L. Valiant, “Universality considerations in vlsi circuits,” in IEEE Transactions on Computers, pp. 135–140, 1981. [15] R. Diestel, Graph Theory. New York Springer Science and Business Media, 2000. [16] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction To Algorithms, pp. 588–591. MIT Press, Cambridge, MA, 2 ed., 2001.

573