Definitions and Preliminaries

This chapter targets to design a distributed algorithm which constructs a CDS in O(∆) time and has O(n) message complexity.

4.1 Network Model

We assume that all the nodes (i.e., devices) are deployed in a rectangular region in the plane and they have an equal transmission range of one unit. We also assume (i) each node has unique ID and an absolute location information in the plane, (ii) all the nodes in the network are stationary, (iii) the communication between two nodes is achieved through a single radio transmission if the nodes are close enough to receive each other’s transmission, and (iv) a node knows the IDs and location information of its 1-hop neigh- bors and the corners of the rectangular region. The model of distributed computation that we adopt is the standard fully synchronous model and we do not consider collisions or other transmission failures in the network. The aforesaid assumptions in our network model are feasible and are widely used in the literature [62].

Observation 4.2.1. All nodes in a cell are within the communication (transmission) range of any other node in that cell (see Figure 4.1(d)).

(a) (b)

1

√3/2

(c)

pi 1

(d)

Figure 4.1: (a) A set of points (nodes) in the plane, (b) the UDG corresponding to the points, (c) hexagonal partition of the rectangular region containing the points, and (d) a unit disk centered at a point pi circumscribes the cell in which pi lies.

The proposed distributed algorithm to obtain a CDS is divided into two phases: (i) construction of a dominating set (DS): one node is selected from every non-empty cell Hi to cover the rest of the nodes in that cell. We refer to this node as the dominator of that cell and is denoted by di, wheredi ∈Hi. Therefore, the set of all such dominators is a dominating set D of G, and (ii) connect the nodes obtained in (i) by a subset of nodes in V \ D. In the rest of the chapter, we refer to the DS obtained in (i) as D, and the connector nodes chosen in (ii) as C. The dominating set D, together with the connector set C forms a CDS, say M. Therefore, M=D ∪ C and |M| =|D|+|C|.

The following definitions, Lemma 4.2.11 and Lemma 4.2.12 play crucial role to find the set C of connector nodes during the second phase.

Definition 4.2.2. A pair of dominators (d_i, d_j) such that d_i ∈ H_i and d_j ∈ H_j is said to be a pair of k-hop dominators if the number of edges on a shortest path between di

and d_j inG is at most k.

Definition 4.2.3. A pair of 2-hop dominators (di, dj) such that di ∈ Hi and dj ∈ Hj

is said to be a pair of 2-hop special dominators if the node connecting them, say u, is either in Hi or Hj. The path di ∼ u∼dj is said to be a 2-hop special path (see Figure 4.2(a)).

Definition 4.2.4. A pair of 3-hop dominators (di, dj) such that di ∈Hi and dj ∈Hj is said to be a pair of 3-hop special dominators if there exist two nodesu and v such that u ∈Hi, v ∈Hj, and u, v are adjacent. The path di ∼u ∼v ∼ dj is said to be a 3-hop special path (see Figure 4.2(b)) and the nodes u and v are called connector nodes.

H_i Hj

d_i d_j u

(a)

H_i Hj

d_j v di

u

(b)

Hi H_j H_k H_i⁰ H_i⁰⁰

H_k⁰ H_k⁰⁰

(c)

Figure 4.2: (a) di, dj are a pair of 2-hop special dominators and u is a connector node, and (b) d_i, d_j are a pair of 3-hop special dominators and u, v are connector nodes.

We use (1, 2, 3) - special dominators to represent 1-hop, 2-hop special, and 3-hop special dominators, and SD(d_i) to represent the set of all (1, 2, 3) - special dominators of the dominator di. In Figure 4.3, the set of all special dominators ofdi are d_j, d_k, d_l, d_m, d_n, d_o, and d_p, hence, SD(d_i) ={d_j, d_k, d_l, d_m, d_n, d_o, d_p}. Note thatd_i and

d_q are 2-hop dominators but not 2-hop special dominators. Similarly, d_i and d_r are not 3-hop special dominators as the node adjacent todr does not lie in the cell wheredrlies.

Definition 4.2.5. A special path between two nodes in SD(di) is a path having the connector nodes in the same cells as that of the dominators (see Figure 4.3).

d_i d_j d_l

d_m d_n d_o

d_r d_k

d_q d_p

Figure 4.3: Special paths betweendi and other dominators (exceptdr) using connectors.

Definition 4.2.6. A triplet of hexagons Hi, Hj, and Hk in the partition are said to be a linear successive cell triplet if the centers of the hexagons are collinear and Hj is adjacent to Hi and Hk (see Figure 4.2 (c)). The cellHj is referred as the middle cell of the linear successive cell triplet. Hi andHk are referred to as the end cells of the linear successive cell triplet.

Definition 4.2.7. The distance between setsD⁰, D⁰⁰ ⊆ D is the minimum hop distance between nodes di and dj, where di ∈D⁰ and dj ∈D⁰⁰.

Lemma 4.2.8. Any pair of non-empty complementary subsets of D are at most 3-hop away from each other. In other words, the distance between any pair of non-empty complementary subsets of D is at most 3.

di dj

u

v

w

Figure 4.4: Demonstration of Lemma 4.2.8.

Proof. Let D and D⁰⁰ be two subset of D such that D⁰ ∩D⁰⁰ = ∅ and D⁰ ∪D⁰⁰ = D. The objective is to show that D⁰ and D⁰⁰ are at most 3-hop away. On contrary, assume that D⁰ and D⁰⁰ are at least 4-hop away. Let u, v, and w are the nodes connecting the dominators di and dj, where di ∈ D⁰ and dj ∈ D⁰⁰ (see Figure 4.4). Since v is a non-dominator node, it must be dominated by a node in D, sayd_k. AsD⁰∩D⁰⁰ =∅ and D⁰∪D⁰⁰=D, dk must be either in D⁰ orD⁰⁰. If dk ∈D⁰, thendk and dj are 3-hop away from each other viav andw. If dk ∈D⁰⁰, then di and dk are 3-hop away from each other via u and v. In either case we arrived at contradiction. Thus the lemma follows.

Lemma 4.2.9. For any dominator d_i ∈ D (|D| > 1), there exists at least 1 special dominator of di in D, i.e., |SD(di)| ≥1.

Proof. Consider the dominator d_i in H_i and an edge e(u, v) ∈ G such that u is in H_i and v is in Hj,i6=j. Note that such an edge exists in the connected graph G for some j, except for the trivial case where G is entirely within a single cell and |D| = 1. Let dj be the dominator in Hj. We consider all possible cases as follows: (i) if u = di and v =d_j, then d_i, d_j is a pair of 1-hop dominators, (ii) if u=d_i, v 6=d_j oru 6=d_i, v =d_j, then di, dj is a pair of 2-hop special dominators, and (iii) ifu6=di, v 6=dj, thendi, dj is a pair of 3-hop special dominators. In all the cases there is always a special dominator of di. Thus the lemma follows.

Corollary 4.2.10. If (D⁰, D⁰⁰) is a pair of non-empty complementary subsets ofD, then there exists at least one pair of dominators (di, dj)∈ (D⁰, D⁰⁰) such that (di, dj) form a pair of (1, 2, 3) - special dominators, i.e., d_j ∈SD(d_i) and also d_i ∈SD(d_j).

Proof. Follows from Lemma 4.2.8 and Lemma 4.2.9.

Alzoubi et al. [4] first obtained a maximal independent set and considered pairs of nodes that are at most 3-hop away in order to obtain the connector set C. The authors showed that the number of nodes in the maximal independent set that are at most three hops away from an arbitrary node in the maximal independent set is at most 47. In our

algorithm, we consider only pairs of 2-hop special and 3-hop special dominators inD to obtain C and get a bound of 18 on the total number of (1, 2, 3) - special dominators from any dominator in D.

Lemma 4.2.11. |SD(di)| ≤18 for any di ∈ D, if |D|>1.

17 7 16

18 1 2 8

14 0 3 15

11 5 6 9

13 10 12

4

i

Figure 4.5: The cells within unit distance from Hi.

Proof. Consider a non-empty cell Hi. Letdi be its dominator. The objective is to show that the cardinality of the set of (1, 2, 3) - special dominators ofdi is at most 18. Recall that (1, 2, 3) - special dominators of di are 1-hop, 2-hop special, and 3-hop special dominators. Therefore, the (1, 2, 3) - special dominators of di are the dominators lying in the cells that are within a unit distance apart fromHi and there are 18 such cells (see Figure 4.5). Thus the lemma follows.

To connect di to all dominators in SD(di), we might have to chose a path between at most 18 dominator pairs (di, dj), where dj ∈ SD(di). However, using the Lemma 4.2.12, we improve the bound on the number of such dominator pairs that have to be considered to establish connection between di to all nodes inSD(di).

Lemma 4.2.12. Letdi ∈ D. If there is a special path between a pair of nodes inSD(di) that are in the end cells of a linear successive cell triplet, then the connector node(s) on

the special path also connect(s) the dominator (if any) in the middle cell of the linear successive cell triplet.

Proof. Consider a linear successive cell triplet consisting of cells H_a,H_b, and H_c, where Hb is the middle cell andHa and Hc are the end cells. AssumingHa, Hb, and Hc are to be non-empty, letd_a,d_b, and d_cbe their respective dominators. If there is a special path betweendaand dc(i.e.,dc∈SD(da)), then there can be at most two connector nodes on the special path. Letuandv be the connector nodes on the special path, whereu∈H_a, v ∈ Hc, ande(u, v) ∈ E. If there is only one connector node on the special path, then either u =d_a or v = d_c. We first prove that any node in H_b is adjacent to at least one of the nodes u orv.

Hc

Ha

k l

m n o p Hb

δ(m) δ(p)

Figure 4.6: Demonstration of Lemma 4.2.12.

The shaded regions shown in Figure 4.6 inH_a (resp. H_c) represents the region that is within unit distance from Hc (resp. Ha) i.e., the region where u (resp. v) lies. Let k, l, m, n, o,and, p be the set of corners of the middle cell in the triplet. Also, let δ(u), δ(v), δ(p), and δ(m) be the unit disks centered at u, v, p, and m, respectively. Observe that uand v lie in the intersection region of the disks δ(p) andδ(m). This implies that, p and m lie inside the disksδ(u) and δ(v), respectively. Hence, k and l lie in δ(u) and o, n lie in δ(v) and the regular hexagon H_b entirely lies inside δ(u)∪δ(v). Hence, any

node in H_b is adjacent to either u or v. From the above argument, d_b is adjacent to at least one of u orv. Therefore, the connector nodes on the special path between da and d_c also connect d_b.

Corollary 4.2.13. Letdi ∈ D. Special paths between at most 12 dominator pairs(di, dj) such that d_j ∈ SD(d_i) is sufficient to ensure paths between d_i and each dominator in SD(di).

Proof. Let us consider a cell H_i and its surrounding 18 cells (see Figure 4.5). Let d_i be the dominator in Hi. By Lemma 4.2.11, |SD(di)| ≤ 18. That is, there can be at most 18 (d_i, d_j) (1, 2, 3) - special dominator pairs. In Lemma 4.2.12, we proved that if we can connect di with the dominator in the farthest hexagon to Hi in the triplet (where H_i is an end cell), then the connector node(s) also connect(s) the dominator in the adjacent cell ofHi in the triplet. There are six triplets correspond to six sides ofHi. Thus, the connectors on the special paths betweend_i and the dominators in the farthest hexagons to Hi in the triplets also connect the dominators in the six adjacent cells of H_i. Therefore, the maximum number of dominator pairs (d_i, d_j) needed to connect all 18 dominators in SD(di) is 6 + (18−6×2) = 12.