Theoretical Computer Science 498 (2013) 46–57

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Theoretical Computer Science

www.elsevier.com/locate/tcs

Bounded-degree minimum-radius spanning trees in wirelesssensor networks

Min Kyung An, Nhat X. Lam, Dung T. Huynh ∗, Trac N. Nguyen

Department of Computer Science, University of Texas at Dallas, Richardson, TX 75080, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 August 2012Received in revised form 30 March 2013Accepted 28 May 2013Communicated by P. Widmayer

Keywords:Bounded-degreeBounded-diameterBounded-radiusSpanning treeDisk graphBicriteria approximation

In this paper, we study the problem of computing a spanning tree of a given undirecteddisk graph such that the radius of the tree is minimized subject to a given degreeconstraint �∗. We first introduce an (8,4)-bicriteria approximation algorithm for unit diskgraphs (which is a special case of disk graphs) that computes a spanning tree such thatthe degree of any nodes in the tree is at most �∗ + 8 and its radius is at most 4 · OPT,where OPT is the minimum possible radius of any spanning tree with degree bound �∗. Wealso introduce an (α,2)-bicriteria approximation algorithm for disk graphs that computesa spanning tree whose maximum node degree is at most �∗ + α and whose radius isbounded by 2 · OPT, where α is a non-constant value that depends on M and k with Mbeing the number of distinct disk radii and k being the ratio of the largest and the smallestdisk radius.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Given an undirected graph, the problem of computing a spanning tree which satisfies certain constraints such as nodedegree, diameter (or radius) or total cost has been the focus of many researchers as the tree can serve as the pre-definednetwork infrastructure in wireless sensor networks (WSNs) and other applications. Such problems are known as constrainedspanning tree problems in the literature.

The Minimum-Degree Spanning Tree (MDeST) problem is one of the constrained spanning tree problems, and its objectiveis to compute a spanning tree such that the maximum node degree of the tree is minimized. The NP-completeness ofthe problem has been proved in [1] for general graphs whereas [2] showed an iterative polynomial time approximationalgorithm, which is currently the best, that computes a spanning tree whose maximum node degree is at most �∗ + 1,where �∗ is the degree of the optimal tree.

A problem related to the MDeST problem is the Degree-Bounded Minimum-Cost Spanning Tree (DBMCST) problem. Givena nonnegative cost function on the edges of a graph and a degree constraint �∗ , its objective is to compute a minimumcost spanning tree with the maximum node degree �∗ . [3] proposed a bicriteria approximation algorithm that computes aspanning tree with maximum node degree O (�∗ + log |V |) and cost O (optc), where V is the set of nodes of the given graphand optc is the minimum cost of any spanning tree whose maximum node degree is at most �∗ . Later, [4] introduced animproved bicriteria approximation algorithm that computes a spanning tree with maximum node degree �∗ + 2 and costoptc . It has again been improved by a bicriteria approximation algorithm in [5] that yields a spanning tree with maximumnode degree �∗ + 1 and cost optc .

* Corresponding author.E-mail addresses: mka081000@utdallas.edu (M.K. An), lxnhat@utdallas.edu (N.X. Lam), huynh@utdallas.edu (D.T. Huynh), nguyentn@utdallas.edu

(T.N. Nguyen).

0304-3975/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.tcs.2013.05.033

M.K. An et al. / Theoretical Computer Science 498 (2013) 46–57 47

Besides bounding the node degree, the problem of computing a spanning tree with bounded-diameter has also beeninvestigated by several researchers. The objective of the Minimum-Diameter Spanning Tree (MDiST) problem is to compute aspanning tree such that the diameter of the tree is minimized. [6] described an O (|V |3) algorithm that solves the MDiSTproblem in the polynomial time. Later, [7] showed that the MDiST problem on general graphs is reducible to the absolute1-center problem [8] which can be solved in O (|V ||E| + |V |2 log |V |) time [9], where E is the set of edges of the givengraph.

While these works concerned only the node degree constraint or the diameter constraint, recently [10] has focused onbounding both the node degree and radius. The Bounded-Degree Minimum-Radius Spanning Tree (BDMRST) problem is to con-struct a spanning tree such that the radius of the tree is minimized subject to a given degree constraint �∗ . [10] introduceda (10,7)-bicriteria approximation algorithm for the BDMRST problem for UDGs that computes a spanning tree whose max-imum node degree is bounded by �∗ + 10 and whose diameter is at most 7 · OPT , where OPT is the minimum possibleradius for the given degree bound �∗ .

A problem closely related to BDMRST is the problem of computing a spanning tree of a given undirected edge-weightedunit disk graph such that the radius of the tree is minimized subject to a given degree constraint �∗ . The edges of the givengraph are endowed with a metric length function. [11] introduced an (O (log2 |V |), O (log |V |))-bicriteria approximationalgorithm that computes a spanning tree whose maximum node degree is bounded by O (�∗ · log2 |V |) and whose diameteris bounded by O (log |V | · OPT ′), where OPT ′ is the minimum possible diameter for the given degree bound �∗ . Later, [12]proposed a (1,

√log�∗ |V | )-bicriteria approximation algorithm that computes a spanning tree whose maximum node degree

is bounded by �∗ and whose diameter is bounded by O (√

log�∗ |V | · OPT ′). The algorithm in [12], however, works only forundirected complete unit disk graphs, and �∗ � 3.

In this paper, we continue the study of the Bounded-Degree Minimum-Radius Spanning Tree (BDMRST) problem on diskgraphs (DGs) as well as unit disk graphs (UDGs). Note that the sole existing work [10] studied the problem for unit diskgraphs (UDGs) only. We first introduce an (8,4)-bicriteria approximation algorithm that constructs a spanning tree for aUDG whose maximum node degree is at most �∗ + 8 and whose radius is bounded by 4 · OPT . This is an improvementover the bound (10,7) obtained in [10]. We also propose a simple (α,2)-bicriteria approximation algorithm that computesa spanning tree for a DG where the maximum node degree is at most �∗ + α and whose radius is bounded by 2 · OPT ,where α is a non-constant value that depends on M and k with M being the number of distinct disk radii and k be-ing the ratio of the largest and the smallest disk radius. To the best of our knowledge, the latter is the first result forDGs.

This paper is organized as follows. Section 2 describes our network model and defines the Bounded-Degree Minimum-Radius Spanning Tree (BDMRST) problem. Section 3 introduces a bicriteria approximation algorithm for the BDMRST problemfor unit disk graphs (UDGs), whereas Section 4 deals with a bicriteria approximation algorithm for disk graphs (DGs).

2. Preliminaries

2.1. Network model

A wireless sensor network (WSN) consists of a set V of sensor nodes, and each node u ∈ V is assigned a transmissionpower level p(u). The signal sent by a node u can be received by another node v if the distance between u and v , denotedby d(u, v), is � p(u). We consider the bidirectional case where two nodes u and v can communicate via a communicationedge which exists between nodes u and v , if and only if d(u, v) � p(u) and d(v, u)� p(v). In this paper, the communicationgraph is modeled as an undirected graph G(V , E), where E = {(u, v) | u, v ∈ V , d(u, v) � p(u) and d(v, u) � p(v)}. In theliterature, a communication graph is called a disk graph (DG) if all nodes are assigned various power levels, and if all nodesare assigned the same power level, it is called a unit disk graph (UDG).

We now introduce some definitions and notations that will be used subsequently:

• Graph Center: Given a communication graph G = (V , E), we call a node c a center node if the distance from c to thefarthest node from c is minimum.

• Maximal Independent Set (MIS): A subset V ′ ⊆ V of the graph G is said to be independent if for any vertices u, v ∈ V ′ ,(u, v) /∈ E . An independent set is said to be maximal if it is not a proper subset of another independent set.

• Connected Dominating Set (CDS): A dominating set (DS) is a subset V ′ ⊆ V such that every vertex v is either in V ′ oradjacent to a vertex in V ′ . A DS is said to be connected if it induces a connected subgraph.

2.2. Problem definition

Given a disk graph G = (V , E) and an integer constant �∗ � 2, the objective of the Bounded-Degree Minimum-RadiusSpanning Tree (BDMRST) problem is to construct a bounded-degree minimum-radius spanning tree T = (V , E T ⊆ E) of Gsuch that the radius of T is minimized while the degree of any node in T is at most �∗ . Formally, BDMRST is defined asfollows:

48 M.K. An et al. / Theoretical Computer Science 498 (2013) 46–57

Fig. 1. Example of an MIS that consists of black nodes. c is the center node of G .

Input. A disk graph G = (V , E), where V is a set of nodes and E is a set of communication edges, and a degree constraint�∗ � 2.Output. A minimum-radius spanning tree T = (V , E T ⊆ E) whose degree is bounded by �∗ .

In this paper, we focus on designing an (α,β)-bicriteria approximation for the BDMRST problem. An (α,β)-bicriteriaapproximation algorithm provides a spanning tree for any instance of BDMRST satisfying the following two conditions as in[10]:

1. The degree of any nodes in the spanning tree is at most α + �∗ .2. The radius of the spanning tree is at most β · OPT , where OPT is the minimum possible radius of any spanning tree

whose degree is bounded by �∗ .

2.3. NP-completeness of the BDMRST problem

First note that [13] showed that the Hamilton Path problem on grid graphs is NP-complete. Now, observe that a specialcase of the BDMRST problem is to compute Hamiltonian path on UDGs, which is clearly NP-complete [13] since grid graphsare a special case of UDGs. Thus, we have

Theorem 1. The Bounded-Degree Minimum-Radius Spanning Tree (BDMRST) problem is NP-complete.

3. Bounded-degree minimum-radius spanning tree on unit disk graphs

In this section, we introduce an (8,4)-bicriteria approximation algorithm for the Bounded-Degree Minimum-Radius Span-ning Tree (BDMRST) problem for unit disk graphs. Given a UDG G = (V , E) where all nodes are assigned the same powerlevel r, we assume without loss of generality that G is connected.

3.1. Algorithm

Our algorithm, BDMRST-UDG (Algorithm 1), starts with an initial tree which is the data aggregation tree T I , that is aspanning tree of G , computed in [14] as follows. A center node c of G is picked as the root node, and a breadth-first-search(BFS) tree T B F S (cf. [15]) on G is constructed. We then [14] find a Maximal Independent Set (MIS) of G using an algorithmin [16] starting with T B F S . The nodes in the MIS are called dominators, and the others are called dominatees. Here, the MISconstructed by [16] guarantees that the shortest hop-distance between any two complementary pairs, A and M I S \ A, whereA ⊆ M I S , is exactly two hops. For example, in Fig. 1, consider one pair A = {v1, v2} and B = M I S \ A = {v3, v4, v5, c}. Theshortest hop-distance between the sets A and B on G is exactly two hops.

Next, based on the MIS, we obtain a Connected Dominating Set (CDS) of G by connecting the dominators using someconnectors that were originally dominatees. If there exist dominatees that are not connected to the CDS, then each of suchdominatees is connected to its neighboring dominator that has the smallest hop-distance to c. We use the newly formedtree denoted by T I = (V , ET ⊆ E) as an initial tree. Fig. 2(a) shows an example for the construction of the CDS, and Fig. 2(b)shows an example for the construction of an initial tree T I . Once T I is obtained we compute a backbone tree which is asubtree T B ⊆ T I that consists of the dominators and their connectors, and the tree edges of T I connecting them.

Our algorithm is divided into three phases:

• Phase 1 (Removing Redundant Connectors): We remove redundant connectors on the initial backbone tree T B , andobtain the modified backbone tree T ′ .

B

M.K. An et al. / Theoretical Computer Science 498 (2013) 46–57 49

Algorithm 1 Bounded-Degree Minimum-Radius Spanning Tree of a UDG (BDMRST-UDG)Input: A UDG G = (V , E), and a degree constraint �∗ � 2Output: Minimum-radius tree T F whose node degree is bounded by �∗ + 8

1: T I ← data aggregation tree of G as constructed in [14]2: T B = (V B , E B ) ← the set V B ⊆ V of dominators and their connectors, and the tree edges E B of T I connecting them3: T ′

B ← RRC(T B ) // Phase 14: for each dominator u ∈ V do5: Partition u’s broadcasting disk into 8 sectors each of which has an angle of 2π

8 radian.6: end for7: T ′′

B ← BTC(T ′B ) // Phase 2

8: T L = (V L , EL) ← the set V L ⊆ V of dominators and their dominatees, and the tree edges EL of T I connecting them9: T F ← LCT-UDG((T ′′

B ∪ T L),�∗) // Phase 3

10: return Bounded-degree minimum-radius spanning tree T F

Fig. 2. (a) Example of the construction of a CDS which consists of the set of nodes that are connected by bold lines. Black nodes represent dominators andgray nodes represent dominatees. (b) Example of the construction of an initial tree T I whose edges are represented by bold lines. Black nodes representdominators, gray nodes represent connectors, and white nodes represent dominatees.

Fig. 3. Example of RRC (Phase 1). Black nodes represent dominators, gray nodes represent connectors, and white nodes represent dominatees.

• Phase 2 (Modifying Backbone Tree): We modify the backbone tree T ′B by reconnecting the connectors, and obtain the

modified backbone tree T ′′B .

• Phase 3 (Constructing Local Trees): For the dominatees, we build several local trees by reconnecting the dominatees,and obtain the final bounded-degree minimum-radius spanning tree T F .

Phase 1. Once T I is obtained, we remove redundant connectors so that the number connectors of a dominator is minimized.Denoting the level (depth) of a node u in a tree T by �T (u), let us consider a dominator u with �T B (u) = � and its2-hop-away dominators v and w with �T B (v) = �T B (w) = � + 2. Let v ′ and w ′ with �T B (v ′) = �T B (w ′) = � + 1 be thecorresponding connectors in T B for v and w , respectively. There may exist the case that v and w can be shared by one ofthe connectors, v ′ and w ′ (see Lemma 2). If that happens, we remove one of the connectors, and therefore three dominators,u, v and w , are connected via only one connector. This phase is denoted by RRC and its details are contained in Algorithm 2.Fig. 3 shows an example of RRC.

Phase 2. For each dominator u, we partition its broadcasting disk into 8 sectors each of which has an angle of 2π8 radian

(see Fig. 4). The ith sector of u is denoted by seci , 1 � i � 8. We note that the connectors of u in each seci form a completegraph. Now, for each seci , letting Cu be the set of u’s connectors in T ′

B , with �T ′B

= �T ′B(u) + 1, we choose a node v ′ ∈ Cu

as a local sink such that d(v ′, u) is smallest. As the next step, we build a local tree T v ′ rooted at v ′ with the remainingconnectors. In order to construct the local trees, we consider the following two cases (see Lemma 3):

50 M.K. An et al. / Theoretical Computer Science 498 (2013) 46–57

Algorithm 2 Removing Redundant Connectors (RRC)Input: Backbone tree T B = (V B , E B )

Output: The modified backbone tree T ′B

1: for each dominator u ∈ V B do2: E ′

B ← E B

3: X ← u’s 2-hop-away dominators in T B with �T B = �T B (u) + 24: Y ← u’s connectors in T B with �T B = �T B (u) + 15: for each connector v ′ ∈ Y do6: Z ← {w | (v ′, w) /∈ E ′

B , w ∈ X}7: for each dominator w ∈ Z do8: Pick w ′ which is the connector that connects u and w .9: if d(w, v ′) � r then

10: E ′B ← E ′

B − {(w, w ′)}, E ′B ← E ′

B ∪ {(w, v ′)}11: Z ← Z − {w}12: if ∃ no more dominators in Z connected to u via w ′ then13: Mark w ′ as a dominatee.14: Y ← Y − {w ′}15: end if16: end if17: end for18: end for19: end for20: return T ′

B = (V B , E ′B )

Fig. 4. Partitioning u’s broadcasting disk into 8 sectors.

• Case 1: 1 < |Cu | � 4.• Case 2: 4 < |Cu | � 8.

In Case 1, all remaining connectors are connected to the local sink v ′ , but v ′ is still connected to its dominator u. Then,v ′ is connected to at most 3 connectors as children in T ′′

B . In Case 2, we divide Cu into two subsets, C1u and C2

u , suchthat |C1

u | and |C2u | are almost equal. (If |Cu | is even, we divide it into the two equal-sized subsets, otherwise one sub-

set’s size is one greater than the other subset’s size.) Without loss of generality, assume that the local sink v ′ of seci isin C1

u . Then, we choose a node w ′ ∈ C2u arbitrarily as a temporary local sink of the set C2

u . Next, for each C1u and C2

u , theremaining connectors in each set are connected to their local sink. Then, we connect w ′ to v ′ , but v ′ is still connectedto its dominator u. Here, v ′ is connected to at most 4 connectors as children, and w ′ is connected to at most 3 connec-tors as children in T ′′

B . The details are contained in Algorithm 3, and Fig. 5 shows an example of Phase 2 for the case|Cu | = 8.

Phase 3. In this phase, we consider only dominatees. For each dominator u, the dominatees of u in each sector seci ,1 � i � 8, form a complete graph. Letting Du be the set of dominatees of u in each seci , we choose one node x ∈ Dusuch that d(x, u) is smallest. We then construct a local tree Tx rooted at x spanning all nodes in Du as follows. LetY ⊆ Du be the set of nodes that are not connected to Tx yet, and X ⊆ Du be the set of nodes that are connected toTx but their degree is 1, where the local sink x is initially an element of X . We select up to �∗ − 1 nodes closestto x from Y , if they exist, and connect them to x. Then, x is removed from X , and the (up to) �∗ − 1 newly se-lected nodes are removed from Y and added to X . Next, we pick one node x′ , whose level (that is its depth in thetree Tx) is smallest, from X , and we select (up to) �∗ − 1 nodes closest to x′ from Y , if they exist, and connect themto x′ . Then, x′ is removed from X , and the newly connected �∗ − 1 nodes are also removed from Y and become newelements of X . We repeat this process until all nodes in Du are connected in Tx . The details are contained in Algo-rithm 4.

M.K. An et al. / Theoretical Computer Science 498 (2013) 46–57 51

Algorithm 3 Backbone Tree Construction (BTC)Input: Backbone tree T ′

B = (V B , E ′B )

Output: Modified backbone tree T ′′B whose maximum node degree is bounded by 10

1: E ′′B ← E ′

B2: for each dominator u ∈ V B do3: for each sector seci of u do4: Cu ← all connectors of u in seci with �T ′

B= �T ′

B(u) + 1

5: Choose a local sink v ′ ∈ Cu such that d(v ′, u) is smallest.6: if 1 < |Cu | � 4 then7: for each connector w ′ ∈ (Cu − {v ′}) do8: E ′′

B ← E ′′B − {(u, w ′)}, E ′′

B ← E ′′B ∪ {(w ′, v ′)}

9: end for10: else if 4 < |Cu | � 8 then11: if |Cu | mod 2 = 0 then12: Divide Cu into two subsets, C1

u and C2u , such that |C1

u | = |C2u | = |Cu |

2 .13: else14: Divide Cu into two subsets, C1

u and C2u , such that |C1

u | = |C2u | + 1 = |Cu |+1

2 .15: end if16: for each C i

u ∈ {C1u , C2

u} do17: if v ′ ∈ C i

u then18: for each x′ ∈ (C i

u − {v ′}) do19: E ′′

B ← E ′′B − {(x′, u)}, E ′′

B ← E ′′B ∪ {(x′, v ′)}

20: end for21: else if v ′ /∈ C i

u then22: Choose a node w ′ ∈ C i

u arbitrarily.23: for each y′ ∈ (C i

u − {w ′}) do24: E ′′

B ← E ′′B − {(y′, u)}, E ′′

B ← E ′′B ∪ {(y′, w ′)}

25: end for26: end if27: end for28: E ′′

B ← E ′′B − {(w ′, u)}, E ′′

B ← E ′′B ∪ {(w ′, v ′)}

29: end if30: end for31: end for32: return T ′′

B = (V B , E ′′B )

Fig. 5. Example of BTC (Phase 2). Black nodes represent dominators, and gray nodes represent connectors.

Next, we consider the following two cases:

• Case 1: |Cu | � 1.• Case 2: |Cu | = 0.

In Case 1, there already exists the local sink v ′ , which is a connector, for seci which was chosen in Phase 2. We connect xto v ′ . In Case 2, there exist only dominatees in seci , and therefore we choose x as the local sink for seci . In this case, x isstill connected to its dominator u. The details are contained in Algorithm 4, and Fig. 6 shows an example of Phase 3 in caseof |Cu | � 1 with �∗ = 4.

3.2. Analysis

In this section, we analyze the BDMRST-UDG algorithm (Algorithm 1), and show that it gives an (8,4)-bicriteria approx-imation.

52 M.K. An et al. / Theoretical Computer Science 498 (2013) 46–57

Algorithm 4 Local Tree Construction (LTC-UDG)

Input: Tree T̂ = (T ′′B ∪ T L) and degree constraint �∗ � 2

Output: Tree T F

1: ET ← Tree edges of T̂2: Let Q be a set of dominatees in V .3: for each dominator u ∈ V do4: ET ← ET − {(x, y) | d(x, y)� r and x, y ∈ (Q ∪ {v})}5: for each sector seci of u do6: Du ← all dominatees of u in seci

7: Cu ← all connectors of u in seci

8: if |Du |� 1 then9: Choose one node x ∈ Du such that d(x, u) is smallest.

10: /* Steps 10–18: Construct a spanning tree Tx rooted at x spanning all nodes in Du */11: X ← {x}, Y ← Du − {x}, V x ← {x}, Ex ← ∅, Tx ← (V x, Ex)

12: while X = ∅ or Y = ∅ do13: Pick a node x′ ∈ X whose level (the depth on Tx) is smallest.14: Select �∗ − 1 nodes which are closest to x′ from Y . If |Y | < �∗ − 1, then pick all nodes in Y . Let Y ′ ⊆ Y be the set of such selected nodes.15: Connect each node y′ ∈ Y ′ to x′ .16: ET ← ET ∪ {(y′, x′)} Ex ← Ex ∪ {(y′, x′)}17: Y ← Y − Y ′ , X ← X − {x′}, X ← X ∪ Y ′18: end while19: if |Cu | � 1 then20: Let v ′ ∈ Y be the local sink in seci .21: ET ← ET ∪ {(x, v ′)}22: else if |Cu | = 0 then23: Let x be the local sink in seci .24: ET ← ET ∪ {(x, u)}25: end if26: end if27: end for28: end for29: return T F = (V , ET )

Fig. 6. Example of Phase 3 for the case |Cu | � 1 with �∗ = 4. Black nodes represent dominators, gray nodes represent connectors, and white nodes representdominatees.

Lemma 2. (See [14,17].) Suppose that dominators v and w are within 2 hops from dominator u, and v ′ and w ′ are the correspondingconnectors for v and w, respectively. If vuw � 2 arcsin 1

4 , then either w v ′ � r or v w ′ � r.

Lemma 3. The number of connectors in one sector is bounded by 8.

Proof. First, let sec(u, θ, r) denote a sector with an angle of θ radian of a circle (broadcasting disk) centered at a node uwith the radius of r (the gray sector in Fig. 7). Consider a dominator u and let us first bound the number of connectors ina sector seci(u, 2π

8 , r), 1 � i � 8, (one sector in Fig. 4) before removing redundant connectors.

Let sec(u, θ ′′,2r) be the union of the three sectors, sec(u, π2 ,2r), seci(u, 2π

8 ,2r) and sec(u, π2 ,2r), where θ ′′ = π

2 + 2π8 +

π2 = 5π

4 (see Fig. 7). Since a connector connects a dominator u to other dominators which are 2 hops away from u inthe CDS, the connectors in seci(u, 2π

8 , r) can connect u to only dominators which reside in the area of sec(u, θ ′′,2r) −sec(u, θ ′′, r). Therefore, the number of connectors in seci(u, 2π

8 , r) cannot exceed the number of dominators in the areaof sec(u, θ ′′,2r) − sec(u, θ ′′, r), and it is sufficient to bound the number of such dominators. Observing that θ ′′ < 4, let usbound the number of dominators in the area of sec(u,4,2r) − sec(u,4, r). We partition the area into 16 cells as in Fig. 8.In Fig. 8, p1 p0 p3 = p3 p0 p5 = p5 p0 p7 = 1 radian, where pk , 0 � k � 9, represents a point, and p1 p0 p2 = p3 p0 p4 = p5 p0 p6 = p7 p0 p8 = 0.5 radian. Here, the largest distance in each cell is � r [18], and therefore there exists at most 1dominator in each cell. This implies that there exist at most 16 connectors in seci(u, 2π , r).

8

M.K. An et al. / Theoretical Computer Science 498 (2013) 46–57 53

Fig. 7. Example of sec(u, θ, r) and sec(u, θ ′′,2r), where θ = 2π8 radian, θ ′ = π

2 radian, and θ ′′ = θ ′ + θ + θ ′ = 5π4 radian.

Fig. 8. Bounding the number of connectors in a sector. Radius of innermost circle is r, radius of dotted circle is 1.5r, and radius of outermost circle is 2r.

Next, let us bound the number of connectors after removing redundant connectors. Consider the case that we have themaximum number of connectors, namely 16. Without loss of generality, let us assume that a connector v j connects u toa dominator w j in the cell c j , where 1 � j � 16. Considering two dominators wq and wq+8, 1 � q � 8, and their corre-sponding connectors vq and vq+8, as wquwq+8 � 0.5 radian � (2 arcsin 1

4 ) radian, by Lemma 2, one of the two connectorsvq and vq+8 is removed during the removal procedure. Therefore, after removing all redundant connectors, we can have atmost 8 connectors remaining in seci(u, 2π

8 , r). �Lemma 4. The node degree of a dominator is bounded by 8.

Proof. Consider a dominator u. It is connected to nodes in its 8 sectors each of which has at most one local sink which isa connector or a dominatee. Therefore, the node degree of a dominator in T F is bounded by 8. �Lemma 5. The node degree of a connector is bounded by 10.

Proof. Consider a dominator u with �T B (u) = �. After Phase 1 (RRC), u is connected to at most 8 connectors with �T ′B(u) =

� + 1 in each sector seci , 1 � i � 8 (Lemma 3). Let us denote the set of such connectors in seci by C . After Phase 1, eachv ′ ∈ C is connected at most 5 connectors [19] in T ′

B (1 dominator with �T B = �T ′B

= �, and the other 4 dominators with�T B = �T ′

B= � + 2). In Phase 2, at most 4 connectors with �T B = �T ′

B= � + 1 become children of v ′ in T ′′

B (see Fig. 5). InPhase 3, at most 1 dominatee with �T B = � + 1 becomes a child of v ′ in T F (see Fig. 6). For each sector of every dominatorin the final tree T F , we can observe that a connector is connected to at most 10 nodes, and thus the lemma holds. �Corollary 6. The node degree of any nodes in the backbone tree T B of T F is at most 10.

Lemma 7. The radius R(T F ) of T F is at most 4 · OPT.

54 M.K. An et al. / Theoretical Computer Science 498 (2013) 46–57

Proof. Let us first bound the radius R(T B) of the backbone tree T B in T F . T B is constructed based on the initial tree T I

whose radius is R . After Phase 1, T B is modified to T ′B and its radius is still the same as the radius of T I , namely R . It is

because, during Phase 1, a dominator, denoted by v , with �T B (v) = � + 2 is reconnected to a dominator, denoted by u, with�T B (u) = � via the new connector, denoted by w ′ , with �T B (w ′) = � + 1 which is the same level as its original connector’s,denoted by v ′ (see Fig. 3). After Phase 2, T ′

B is transformed into T ′′B . Observe that during Phase 2, at each level of the tree,

a connector’s level is increased by at most 2. Therefore, after Phase 2, the radius of T ′B is increased to at most 3R which

implies that the radius of T ′′B is at most 3R . Next, let us bound the radius of the local trees. In Phase 3, each of the local

trees were constructed from a complete graph, and therefore it is an optimal tree in which every node has the node degree�∗ (except the leaves and the last parent). Thus the radius of each local tree is OPT . Furthermore, noting that R is also theradius of a graph G where the maximum node degree is at most �, having such a degree constraint with 2 � �∗ � � canonly increase the radius of a tree, and hence R � OPT . Therefore R(T F )� 3 · R + OPT = 4 · OPT . �Theorem 8. Algorithm BDMRST-UDG is an (8,4)-approximation algorithm.

Proof. The degree of any nodes in T F is at most max{�∗,10} � �∗ + 8 for any �∗ � 2. The radius of T F is at most 4 · OPTby Lemma 7. Therefore, BDMRST-UDG is an (8,4)-approximation algorithm. �

In BDMRST-UDG (Algorithm 1), Step 1 takes O (n2) time [14,16]. During Step 3 (the sub-procedure of removing redun-dant connectors, contained in Algorithm 2), the outermost for-loop in Algorithm 2 takes O (n) time, but all inner for-loopsin Algorithm 2 take constant time. Therefore, Step 3 takes O (n) time. Steps 4–6 simply take O (n) time. In Step 7 (the sub-procedure of constructing the backbone tree, contained in Algorithm 3), the outermost for-loop in Algorithm 3 takes O (n)

time, but all inner for-loops in Algorithm 3 take constant times, and therefore the Step 7 takes O (n) time. The last step(Step 9) for constructing the local spanning trees, contained in Algorithm 4, takes O (n3) time since the outermost for-loopin Algorithm 3 takes O (n) time and the inner while-loop takes O (n2) time. Thus, BDMRST-UDG has O (n3) time complexity.

4. Bounded-degree minimum-radius spanning tree on disk graphs

In this section, we introduce a simple (α,β)-bicriteria approximation algorithms for the Bounded-Degree Minimum-Radius Spanning Tree (BDMRST) problem on disk graphs. Given a disk graph G = (V , E), let P be the set of power levels{r1, r2, . . . , rM}, i.e., each node u ∈ V is assigned a power level p(u) ∈ P by a power assignment function p : V → P . Weassume that G is connected, and r1 � r2 � · · · � rM .

4.1. Algorithm

Like the algorithm for unit disk graph in Section 3, namely BDMRST-UDG, our new algorithm (Algorithm 5) also startswith an initial tree T I = (V , ET ⊆ E) which is a data aggregation tree in [14]. Once T I is obtained, we also compute abackbone tree which is a subtree T B ⊆ T I that consists of the dominators and their connectors, and the tree edges of T I

connecting them.

Algorithm 5 Bounded-Degree Minimum-Radius Spanning Tree of a Disk Graph (BDMRST-DG)Input: A disk graph G = (V , E) and a degree constraint �∗ � 2Output: Minimum-radius tree T F whose node degree is bounded by �∗

1: T I ← data aggregation tree of G as constructed in [14]2: for each dominator u ∈ V do3: Partition its broadcasting disk into several pieces as described in Section 4.1.4: end for5: T F ← LTC-DG(T I ,�

∗)6: return Bounded-degree minimum-radius spanning tree T F

In this algorithm, we focus on constructing local trees by replacing tree edges of T I . Let us denote a node u whose powerlevel p(u) = rm , 1 � m � M , by um . For each dominator um , we partition its broadcasting disk into 8 sectors each of whichhas an angle of 2π

8 radian (see Fig. 4). We again partition the broadcasting disk into m concentric circles (disks) centeredat um . Each of the concentric circles has radius r1, . . . , rm . Fig. 9 shows an example of partitioning the broadcasting disk ofu4 into 4 concentric disks with radius r1, r2, r3 and r4. Denoting the ith sector of um by seci , 1 � i � 8, and a concentricdisk with radius r j by d j , 1 � j � m, let ai1 represent the area of seci ∩ d1, ai2 represent the area of seci ∩ (d2 − d1), andin general, aij represent the are of seci ∩ (d j − d j−1). Then, the dominatees of um in each area aij , 1 � i � 8, 1 � j � m,form a complete graph. In each aij , we build a local tree rooted at one of the dominatees as follows. Let Q ij be the set ofdominatees located in aij that are connected to um . We pick a node w ∈ Q ij as a local sink such that d(w, um) is smallest.We then construct a local tree T w rooted at w spanning all nodes in Q ij as done in Phase 3 in BDMRST-UDG (Algorithm 1).The details are contained in Algorithm 6, and Fig. 10 shows an example of the construction of local trees.

M.K. An et al. / Theoretical Computer Science 498 (2013) 46–57 55

Fig. 9. Partitioning um ’s broadcasting disk into m concentric circles (disks) centered at um .

Fig. 10. Example of Local Tree Construction (LTC-DG) with �∗ = 4. Black nodes represent dominators, and white nodes represent dominatees. Nodes filledwith upward diagonals are dominatees that are selected as local sinks.

Algorithm 6 Local Tree Construction (LTC-DG)Input: Initial tree T = (V , ET ) and degree constraint �∗ � 2Output: Modified tree T F

1: Let Q be a set of dominatees in V .2: for each dominator um ∈ V do3: for i = 1 to 8 do4: Q i ← all dominatees of um in seci

5: if |Q i | � 2 then6: for j = 1 to M do7: Q ij ← {v | v ∈ Q i and v is located in aij}8: if |Q ij | � 2 then9: Choose one node w ∈ Q ij such that d(w, um) is smallest.

10: for each node w ′ ∈ (Q ij − {w}) do11: ET ← ET − {(w ′, um)}12: end for13: /* Steps 15–22: Construct a spanning tree T w rooted at w spanning all nodes in Q ij */14: X ← {w}, Y ← Q ij − {w}, V w ← {w}, E w ← ∅, T w ← (V w , E w )

15: while X = ∅ or Y = ∅ do16: Pick a node x′ ∈ X whose level (the depth on T w ) is smallest.17: Select �∗ − 1 nodes (which are closest to x′) from Y . If |Y | < �∗ − 1, then pick all nodes in Y . Let Y ′ ⊆ Y be the set of such selected

nodes.18: Connect each node y′ ∈ Y ′ to x′ .19: ET ← ET ∪ {(y′, x′)}, E w ← E w ∪ {(y′, x′)}20: Y ← Y − Y ′ , X ← X − {x′}, X ← X ∪ Y ′21: end while22: end if23: end for24: end if25: end for26: end for27: return T F = (V , ET )

56 M.K. An et al. / Theoretical Computer Science 498 (2013) 46–57

Fig. 11. Bounding the number of connectors.

4.2. Analysis

In this section, we analyze the BDMRST-DG algorithm (Algorithm 5), and show that it gives an (α,β)-bicriteria approxi-mation.

Lemma 9. (See [19].) The degree of each connector in T F is bounded by K = 6(3 log2 k� + 2), where k = rMr1

.

Lemma 10. Each dominator has at most 4π M(k + 1) connectors.

Proof. Consider a dominator um whose power level is rm , 1 � m � M . um is connected to 2-hop-away dominators via itsconnectors, and therefore it is sufficient to bound the number of such dominators to bound the number of connectors ofum . Let us bound the number of 2-hop-away dominators whose power level is ri , 1 � i � M . Let sec(v, θ, r) denote a sectorwith an angle of θ radian of a circle (broadcasting disk) centered at a node v with the radius of r, and consider a sectorsec(um, θ, rm + ri), where θ = ri

rm+ri. We partition the area of sec(um, θ, rm + ri) − sec(um, θ, rm) into 2 cells as in Fig. 11.

Here, the largest distance in each cell is � ri [18], and hence there exists at most 1 dominator whose level is ri in eachcell. This implies that there exist at most 2 connectors for the dominators in sec(um, θ, rm). As um has 2π(rm+ri)

risectors, the

number of connectors for the dominators whose power level is ri is Di = 2 · 2π(rm+ri)ri

= 4π( rmri

+ 1). Note that Di is largestwhen rm = rM , and the 2-hop-away dominators of um can be assigned different power levels in P . Therefore, the numberof connectors of a dominator is bounded by

∑Mi=1(4π( rM

ri+ 1)) = 4π · rM

∑Mi=1

1ri

+ ∑Mi=1 4π < 4π · rM · M

r1+ 4π · M =

4π M(k + 1). �Lemma 11. A dominator is connected to at most 8M dominatees in T F .

Proof. Consider a dominator um , where 1 � m � M . For each aij , 1 � i � 8, 1 � j � m, of um , at most one dominateewhich is a local sink is connected to um , and hence um is connected to at most 8m dominatees in T . Thus, the number ofdominatees that are connected to a dominator in T is bounded by 8M . �Corollary 12. The degree of each dominator in T F is at most L = 4π M(k + 1) + 8M.

Corollary 13. The degree of any node in T B is at most max{K , L}.

Lemma 14. The radius R(T F ) of T F is at most 2 · OPT.

Proof. It is obvious that the radius R(T B) of T B is R . Next, let us bound the radius of each local tree. For each dominatorum , 1 � m � M , a local tree in an area aij , 1 � i � 8, 1 � j � m, is constructed on a complete graph. Therefore, it is possibleto construct an optimal spanning tree whose maximum node degree is �∗ and whose radius is minimized subject to thedegree constraint �∗ in each area. Thus, the radius R(T F ) of T F is R + OPT . Since 2 � �∗ � �, R � OPT , it follows thatR(T F ) = R + OPT � OPT + OPT � 2 · OPT . �Theorem 15. It is an (α,β)-approximation algorithm, where α = K + L − 2 and β = 2.

Proof. The degree of any nodes in T F is at most max{�∗,max{K , L}} � �∗ + K + L − 2 for any �∗ � 2. The radius of thetree is at most 2 · OPT by Lemma 14. Therefore, it is an (α,β)-approximation algorithm, where α = K + L − 2 and β = 2. �

M.K. An et al. / Theoretical Computer Science 498 (2013) 46–57 57

In BDMRST-DG (Algorithm 5), Step 1 takes O (n2) time [14,16], and Steps 2–4 simply take O (n) time. In Step 5 forconstructing the local spanning trees in Algorithm 6, the outermost for-loop (Steps 2–26) takes O (n) time, and the secondfor-loop (Steps 3–25) takes constant time. The other for-loop (Steps 6–23) takes O (n) time, but Steps 10–21 including theinnermost while-loop take O (n2) time. Thus, BDMRST-DG has O (n4) time complexity.

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