應用於無線傳感器網路之二步著色

國

立

交

通

大

學

應用數學系

碩

士

論

文

應用於無線傳感器網路之二步著色

A distance-two coloring with applications to wireless

sensor and actor networks

研 究 生：趙彥丞

指導教授：陳秋媛 教授

應用於無線傳感器網路之二步著色

A distance-two coloring with applications to wireless

sensor and actor networks

研 究 生：趙彥丞 Student：Yen-Cheng Chao

指導教授：陳秋媛 Advisor：Chiuyuan Chen

國 立 交 通 大 學

應 用 數 學 系

碩 士 論 文

A Thesis

Submitted to Department of Applied Mathematics College of Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

in

Applied Mathematics June 2013

Hsinchu, Taiwan, Republic of China

應用於無線傳感器網路之二步著色

研究生：趙彥丞 指導老師：陳秋媛 教授

國 立 交 通 大 學

應 用 數 學 系

摘 要

無線感測網路可廣泛的應用在環境監控。一個可以允許感測器與外界溝通的 有效方法，是利用一個或多個反應節點作為從無線感測網路中所取得資料的 接收者。一個無線傳感器網路是由多個隨機佈署的感測器以及少量的反應節 點所組成，而反應節點會組織感測器進而形成一個以其為中心的同心圓形狀 網路。定位、路由以及防止碰撞為三個主要的無線傳感器網路問題。本篇論 文的主要貢獻為：提出一個新的虛擬結構來做定位以解決防止碰撞問題，同 時對於我們所提出的虛擬結構的連接圖給出最佳的(在某些情況下是接近最 佳的)二步著色。 關鍵詞：無線傳感器網路，粗質定位，二步著色，防止碰撞。

A distance-two coloring with application to wireless

sensor and actor networks

Student: Yen-Cheng Chao

Advisor: Chiuyuan Chen

Department of Applied Mathematics National Chiao Tung University

Abstract

Wireless sensor networks (WSNs) have a wide array of applications in envi-ronment and infrastructure monitoring. An efficient solution to allow sensors to communicate with the outside world is making use of one or several actors as the receiver of the data harvested by the WSNs. A wireless sensor and actor network (WSAN) consists of many randomly deployed sensors and a few actors that orga-nize the sensors in their vicinity into an actor-centric network. Localization, routing, and collision avoidance are three fundamental problems in WSANs. The main con-tribution of this thesis is to solve the collision avoidance problem by proposing a new virtual infrastructure for the localization, and give optimal (in some cases, near-optimal) distance-two colorings for the adjacency graph of our virtual infras-tructure.

Keywords: Wireless sensor and actor network, Coarse-grain localization, Distance-two coloring, Collision avoidance

誌

謝

兩年的碩士生活結束了，由衷感謝陳秋媛老師在這段期間裡，課業上能給予 及時的幫助，使得此篇論文能順利完成，平時也會關心我的生活狀況，在我遇到 困難時，總能給予包容與傾聽，非常感謝老師這段時間的照顧。 另外，要感謝在求學期間學長們的幫忙，讓我在學習上更加順利，也能適當 地給我好建議。研究所同儕們的鼓勵，不論是學習上與生活上，讓我的碩士生活 更加精采。這短暫期間的系網頁的人員們、系辦小姐們，因為有你們才能讓我們 校園生活順利完整。還有交大裡許許多多的人，不論是打掃阿姨、餐廳的服務人 員…等等，真的非常謝謝你們，提供了這麼好的校園環境。 最後，我想感謝我的家人們、親人、情人，在我的求學過程給我不論是精神 抑或物質上的幫助，讓我能順利完成學業，我必會在社會上更努力，不會辜負你 們的期望。 iii

Contents

Abstract (in Chinese) i

Abstract (in English) ii

Acknowledgement iii Contents iv List of Figures v List of Tables v 1 Introduction 1 1.1 Localization . . . 1 1.2 Routing . . . 3 1.3 Collision avoidance . . . 3 1.4 Our contribution . . . 4

2 Our virtual infrastructure, basic definitions, and lower bounds 5 2.1 Our virtual infrastructure . . . 6

2.2 Basic definitions and lower bounds for distance-two coloring . . . 7

3 Our distance-two coloring algorithms 10 3.1 Optimal coloring for H` with ` = 3 · i . . . 10

3.2 Optimal coloring for H` with ` = 8 · i . . . 12

3.3 Optimal coloring for H4 . . . 16

3.4 Optimal coloring for H5 . . . 18

List of Figures

1 (a) The virtual infrastructure with 7 coronas and 8 sectors proposed in

[2, 3, 4, 14, 17, 18, 19]; the number of sectors in each corona will be the same. (b) The virtual infrastructure with 7 coronas and 4 sectors in corona

1 proposed in [7, 12, 13]; the number of sectors is doubled at coronas 2 and 4. 3

2 (a) The virtual infrastructure that starts with ` = 4 sectors. (b) Its

cor-responding adjacency graph H4; the six black, the five green, and the five

red vertices denote S(3,1), T(2,4), and T(4,13), respectively. . . 7

3 (a) S(c,s); (b) T(c,s) for even s; (c) T(c,s) for odd s. . . 9

4 The optimal distance-two 6-coloring for H3 produced by OP T 3. . . 11

5 (a) The 8 blocks of H8. (b) The distance-two 6-coloring for H8 produced

by OP T 8. . . 13

6 The distance-two 7-coloring for H4 produced by OP T 4; all the vertices

are colored by using A (along with permutations p0, p1, . . . , p6) except that

those highlighted are colored by using M4. . . 17

7 The distance-two 7-coloring for H5 produced by OP T 5; all the vertices

are colored by using A (along with permutations p0, p1, . . . , p6) except that

those highlighted are colored by using M5. . . 19

List of Tables

1 The best previous distance-two colorings for G`. . . 4

1

Introduction

A wireless sensor and actor network (WSAN) [1] consists of massively and randomly deployed tiny sensors and a few actors that organize the sensors in their vicinity into a short-lived actor-centric network to support a specific mission. These tiny and low-cost sensors have small (nonrenewable) energy supply and limited communication range, and, after deployed, are unaware of their location and are unattended. Actors are mobile along the area of deployed sensors to collect the sensed data from sensors within its transmission range and to aggregate and transmit to the outside world. Each actor is equipped with better processing capabilities, higher transmission power to send broadcasts for a distance, and a longer battery life than the sensors. Actor-centric sensor networks have many application in environment and infrastructure monitoring, and can detect emergent, unexpected and coherent behaviors and trends, and find immediate applications in environmental monitoring and homeland security.

In the study of WSANs, there are three fundamental problems: (i) localization, (ii) routing, and (iii) collision avoidance.

1.1

Localization

Due to the sensors constraints on the cost, size, energy consumption, and implemen-tation environment, most sensor nodes do not know their locations. The localization problem is to determine, for individual sensor nodes, as closely as possible their geo-graphic coordinates in the area of deployment. The sensed data could be meaningless if it is not related to the exact position or at least a sufficiently small region of the monitored area, and position information guiding sensors to transmit data has been studied on many geographic routing protocols. An immediate approach to provide the exact position of each sensor is based on localization systems (e.g., globally positioning system (GPS)), but this approach takes expensive cost and is not suitable for plenty of randomly deployed,

tiny and low-cost sensors in many applications. Hence, a coarse-grain location aware-ness is sufficient for WSANs with a trade-off: that an coarse-grain location awareaware-ness is lightweight, but the resulting positioning accuracy is only a rough approximation of the exact geographic location.

Training is referred to the task of allowing each sensor to acquire a coarse-grain lo-cation. Wadaa et al. [19] first proposed a training protocol in which each actor trains sensors in its vicinity, namely, the actor-region, to associate these sensors with coarse-grain coordinates related to the actor. More precisely, after training, each sensor in the actor-region will acquire two coordinates: the corona and the sector to which it belongs. A training protocol provides for free a clustering of the sensors and a virtual infrastructure, where a cluster consists of all sensors having the same coordinates.

The resulted virtual infrastructure of training protocols proposed in [2, 3, 4, 14, 17, 18] are identical (see Figure 1(a)); one consequence of these training protocols is that: the number of sectors in each corona are the same. By contrast, Navarra and Pinotti [12] presented a new virtual infrastructure in which the number of sectors is doubled at each corona i, for i is a power of 2; see Figure 1(b). The papers [7, 13] also used the same virtual infrastructure as [12]. One interesting result of [12] is that the ratio given by the area spanned by two clusters is at most 2. Notice that in [4, 12, 13, 17, 18, 19] the terminology sink-centric network was used instead of actor-centric network.

In [2], Bertossi et al. proposed two scalable energy-efficient training protocols for sensor networks. Navarra, et al. [13] proposed the protocol, called Cooperative. This protocol is the fastest training algorithm for asynchronous sensors, and it matches the running time of the fastest known training algorithm for synchronous sensors. Other training protocols for WSNs have been proposed in the literature [3, 4, 19].

actor 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 (a) actor 0 1 2 3 4 5 6 0 1 2 3 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 (b)

Figure 1: (a) The virtual infrastructure with 7 coronas and 8 sectors proposed in [2, 3, 4, 14, 17, 18, 19]; the number of sectors in each corona will be the same. (b) The virtual infrastructure with 7 coronas and 4 sectors in corona 1 proposed in [7, 12, 13]; the number of sectors is doubled at coronas 2 and 4.

1.2

Routing

In a trained actor-centric network, the routing can be easily performed as followed: the message can be trivially routed inward within a single sector to the actor or routed following several paths consisting of subpaths within a sector or within a corona (clockwise or counterclockwise, depending on which is the shortest path) and a subpath toward the actor within a sector. In addition, to help the actor to locate an event that has occurred in the network, each sensor can add on its coordinates to the sensed data before delivering the messages to the actor.

1.3

Collision avoidance

A wireless sensor network can be modeled as a graph with sensor nodes as vertices and the communication link, if it exists, between any two nodes as an edge. An ordinary coloring assigns each vertex a color such that two adjacent vertices receive distinct colors. Our graph-theoretic terminologies are standard; see [5, 21]. During data transmission, packet collisions (i.e., radio interference) may occur and lead to packet losses and

retrans-missions, which result in an overhead on energy consumption and transmission latency, and therefore shorten the network lifetime. There are two major types of collisions: the direct and the hidden collisions [6, 20]. The former occurs when a node simultaneously delivers and receives packets, and the latter occurs when a node simultaneously receives packets from more than one node. An ordinary coloring could solve the direct collision by scheduling two sensor nodes with a link to transmit in distinct time-slots (or chan-nels or frequencies). However, an ordinary coloring could not solve the hidden collision. Therefore, instead of using an original coloring, a distance-two coloring is needed, which assigns each vertex a color in such a way that two vertices receive distinct colors if they are of distance at most 2.

1.4

Our contribution

Throughout this thesis, we will follow the convention (see [12]) that ` is an integer

and ` ≥ 3. In [12], Navarra and Pinotti defined the adjacency graph G` for their virtual

infrastructure, which is: each vertex corresponds to a cluster and two vertices are adjacent if their corresponding clusters share the boundary of a corona or a sector, where ` is the number of sectors imposed in corona 1. They gave an optimal distance-two coloring for

G3 and a quasioptimal one for G4. Then, Das et al. [7] gave a distance-two coloring of

G` with 2` colors, and Navarra et al. [13] improved the distance-two coloring algorithms

of G`. We now list the best previous known results in Table 1.

G` # of colors lower bound optimal coloring

` = 3 · 2i, i ≥ 0 6 6 Yes [13]

` = 4 7 7 Yes [13]

` = 5 7 7 Yes [13]

` = 4 · i, i ≥ 2 8 6 No [13]

` ≥ 7 9 6 No [13]

Table 1: The best previous distance-two colorings for G`.

remainder of the integer division of i by j.

In this article, we propose a new virtual infrastructure and distance-two colorings for

the adjacency graph H` of our virtual infrastructure. We now list our results in Table 2.

H` # of colors lower bound optimal coloring

` = 3 · i, i ≥ 1 6 6 Yes ` = 4 7 7 Yes ` = 5 7 7 Yes ` = 8 · i, i ≥ 1 6 6 Yes ` = 10 or ` = 20 7 6 No ` ∈ {m, 2m, 4m}, odd m ≥ 7 and 3 - m 8 6 No [9]

Table 2: The performance of our distance-two colorings for H`.

The remaining part of this thesis is organized as follows. Section 2 gives our virtual infrastructure, basic definitions, and lower bounds for distance-two coloring. Section 3 proposes our distance-two colorings. Section 4 discusses the leader election problem for our virtual infrastructure. Concluding remarks are given in the final section.

2

Our virtual infrastructure, basic definitions, and

lower bounds

We first describe the WSAN model. In a WSAN, all sensors possess three basic

capabilities: sensory, computation, and wireless communication; and operate subject to the following constraints:

1. Each sensor is asynchronous — it wakes up for the first time according to its internal clock and it is not engaged in an explicit synchronization protocol, neither with the actor nor with the other sensors;

2. Individual sensors are unattended — once deployed, it is neither feasible nor practical to devote attention to individual sensors;

3. No sensor has global information about the network topology, but each sensor can receive transmissions from the sink;

4. The sensors are anonymous — they are not associated with unique IDs;

5. Each sensor has a modest non-renewable energy budget and a limited transmission range;

6. Sensors can transmit and receive on multiple frequency channels. Moreover, the number of channels and frequencies are the same for all the sensors.

A training protocol imposes a virtual coordinate system onto the sensor networks by establishing:

1. Coronas : The actor-region is divided into k coronas C0, C1, . . . , Ck−1 determined

by k concentric circles of radii r1 < r2 < · · · < rk centered at the actor.

2. Sectors : The actor-region is divided into h equiangular sectors S0, S1, . . . , Sh−1,

originated at the actor, each having a width of 2π_h radians.

For convenience, the coronas and sectors are referred by specifying their numbers;

thus, corona Cc and sector Ss will be referred to as corona c and sector s, respectively.

In a built virtual coordinate system, a cluster is the intersection between a corona c and a sector s. All sensors in a cluster acquire the same coordinates, denoted by (c, s). For

convenience, the radii ri’s are considered as ri = i for i = 1, 2, . . . , k.

2.1

Our virtual infrastructure

We now propose a new virtual infrastructure with ` sectors imposed in corona 1 and

the number of sectors is doubled at each corona c, where c = 2p, for p = 1, 2, . . . , bk−1₂ c.

Set

hc= ` · 2b

c 2c

for easy writing, which is the number of sectors in corona c. The formulated definition of the adjacency graph of our virtual infrastructure is given in the following definition.

Definition 1. The adjacency graph H` has one vertex (c, s), where 1 ≤ c ≤ k − 1 and

0 ≤ s < hc, for each cluster in the virtual infrastructure. Two vertices (c, s) and (c0, s0),

with c ≥ c0, are adjacent if

1. c = c0 and |s − s0| ≡ 1 (mod hc), or

2. c = c0+ 1 is odd and s = s0, or

3. c = c0+ 1 is even and s0 = bs₂c. (See Figure 2 for an illustration.)

(1 , 0) (2,0) (2, 1) ₍₃ ,0) (3, 1) (4 ,₀₎ (4 , 1) (4 , 2) (4, 3) (5 ,₀₎ (5 , 1) (5 , 2) (5, 3) (1, 1) (2,3) (2,2) (3,3) (3,2) (4, 7) (4, 6) (4, 5) (4, 4) (5, 7) (5, 6) (5, 5) (5, 4) (1 , 2) (2 ,4) (2, 5) (3 ,4) (3, 5) (4 ,₈₎ (4 , 9) (4 , 10) (4, 11) (5 ,₈₎ (5 , 9) (5 , 10) (5, 11) (1, 3) (2,7) (2,6) (3 ,7) (3,6) (4, 15) (4, 14) (4, 13) (4, 12) (5, 15) (5, 14) (5, 13) (5, 12) (a) y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy yy y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y i i i i i i i i i i i i i i i i i i i i i y y i i i i i i i i i i i i i ii ii i i y y y i i i i i i i i i i i i i i i i i i i i i y y i y y y i i i i i i i i i i i i y y y y y y (b)

Figure 2: (a) The virtual infrastructure that starts with ` = 4 sectors. (b) Its

correspond-ing adjacency graph H4; the six black, the five green, and the five red vertices denote

S(3,1), T(2,4), and T(4,13), respectively.

2.2

Basic definitions and lower bounds for distance-two coloring

It is easy to see that in adjacency graphs G` and H`, a vertex corresponds to a cluster

and two vertices are adjacent if their corresponding clusters share the boundary of a corona or a sector. For the rest of our discussion, we will not consider corona 0 since the sensors in it can retrieve the information with the actor by itself, so the scheduling of communication

in it is not necessary. Equivalently, it could be assumed that the transmission reaches the actor when it reaches corona 0 (see also [7, 12, 13]). We now give the definition of distance-two coloring.

Definition 2. A distance-two coloring of a graph G is an assignment of a color to each of the vertices of G in such a way that two vertices are assigned different colors whenever they are at distance one or two (i.e., they are adjacent or have a common neighbor). If the colors are chosen from a set of d colors, then the coloring is called a distance-two d-coloring.

Before going further, we introduce two notations that will be used in later discussion.

For odd c ≥ 3, define S(c,s) be a 6-element subset of the vertex set of H` such that

S(c,s)= {(c − 1, s), (c, |s − 1|hc), (c, s), (c, |s + 1|hc), (c + 1, |2s|hc+1), (c + 1, |2s + 1|hc+1)}.

For even c ≥ 2, define T(c,s) be a 5-element subset of the vertex set of H` such that

T(c,s)= {(c − 1, |bs−1₂ c|hc−1), (c − 1, |bs+1₂ c|hc−1), (c, |s − 1|hc), (c, s), (c, |s + 1|hc)}.

For S(c,s)or T(c,s), the vertex (c, s) will be called its center. See Figure 3 for an illustration.

Also, the six black, the five green, and the five red vertices shown in Figure 2(b) denote S(3,1), T(2,4), and T(4,13), respectively.

Immediately, we observe the following fact.

Lemma 2.1. All the vertices in S(c,s) have a pairwise distance of at most two. This is

also true for T(c,s).

This lemma is obvious and its proof is omitted. Do notice that it is impractical to consider a virtual infrastructure with three (or fewer) coronas. Therefore, all the adjacency

graphs G`’s and H`’s are assumed to have at least 4 coronas. This assumption is crucial to

A A A


y y y y y y (c − 1, s) (c, |s − 1|hc) (c, s) (c, |s + 1|hc) (c + 1, |2s|hc+1) (c + 1, |2s + 1|hc+1) (a)


A A A y y y y y (c, |s − 1|hc) (c, s) (c, |s + 1|hc) (c − 1, |bs−1₂ c|_hc−1) (c − 1, |bs+1₂ c|_hc−1) (b)


A A A y y y y y (c, |s − 1|hc) (c, s) (c, |s + 1|hc) (c − 1, |bs−1₂ c|_hc−1) (c − 1, |bs+1₂ c|_hc−1) (c)

Figure 3: (a) S(c,s); (b) T(c,s) for even s; (c) T(c,s) for odd s.

Navarra et al. [13] proved that any distance-two coloring of G` requires at least 6 colors;

and we prove the following lemma

Lemma 2.2. Any distance-two coloring of H` requires at least 6 colors.

Proof. By definition, S(3,1) = {(2, 1), (3, 0), (3, 1), (3, 2), (4, 2), (4, 3)}. Since S(3,1) has 6

vertices, it follows from Lemma 2.1 that any distance-two coloring of H` requires at least

6 colors.

The above lower bound can be sharpened for H4 and H5; see the following theorem.

Theorem 2.3. [9] Any distance-two coloring of H4 or H5 requires at least 7 colors.

In [13], Navarra et al. also proved that any distance-two coloring of G4 or G5 requires

at least 7 colors. Theorem 2.3 provides a much simpler proof for such a result since

the subgraph of H` induced by vertices in coronas 1 to 4 is isomorphic to the subgraph

of G` induced by vertices in the same coronas. Before ending this section, we give two

G is a connected graph other than a complete graph or an odd cycle, then the chromatic number of G is at most the maximum degree of G.

Lemma 2.4. The minimum number of colors required by a distance-two coloring of H`

is between 6 and 16.

Proof. A distance-two coloring of a graph G can be obtained from a coloring of the square

of G (i.e., G2). Since the maximum degree of H_`2 is at most 16, by Lemma 2.2 and Brooks’

Theorem, we have this lemma.

Lemma 2.5. If H` has a distance-two d-coloring, then so does H2`.

Proof. This lemma follows from the fact that H2` is an induced subgraph of H` (H2` can

be obtained form H` by removing vertices in coronas 1 and 2).

3

Our distance-two coloring algorithms

In this section, we propose algorithms OP T 3, OP T 8, OP T 4, OP T 5, and COL to

color H` for ` = 3 · i, ` = 8 · i, ` = 4, ` = 5, and ` ≥ 3 (the general case), respectively.

We will prove that the first four algorithms (i.e., OP T 3, OP T 8, OP T 4, and OP T 5) give optimal distance-two colorings, and the last algorithm COL gives a near-optimal one.

3.1

Optimal coloring for H

`

with ` = 3 · i

Let M (c, s) denote the value of the (c, s) entry in a matrix M . The idea of our coloring

algorithm is to design a 4-by-3 matrix with the following three properties (Ψ1, Ψ2, and

Ψ3) and to use this matrix to perform coloring.

Ψ1: For c, s, and s0, we always have M (|c|4, |s0|3) 6= M (|c|4, |s|3) if |s0|3 6= |s|3.

Ψ3: For c and s, we always have M (|c + 2|4, |s|3) = M (|c|4, |2s + 2|3). We now design a 4-by-3 matrix

A =          0 1 2 0 6 5 4 1 1 2 3 2 4 5 6 3 3 2 1          .

Then A(0, 0) = 6, A(0, 1) = 5, A(0, 2) = 4, etc. It is easy to verify that matrix A is

designed with properties Ψ1, Ψ2, and Ψ3. We now give a coloring algorithm for H` with

` = 3 · i, i ≥ 1; see Figure 4 for an illustration of this algorithm. Algorithm 1 OP T 3 (As Executed At Every Vertex)

1: vertex (c, s) gets the color A(|c|4, |s|3);

y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 1 2 3 4 5 6 4 5 6 3 2 1 3 2 1 6 5 4 6 5 4 6 5 4 6 5 4 1 2 3 1 2 3 1 2 3 1 2 3 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 _{6 4} 5 6 4 5 6

Figure 4: The optimal distance-two 6-coloring for H3 produced by OP T 3.

Theorem 3.1. Algorithm OP T 3 is distributed, takes constant time, and produces an

optimal distance-two 6-coloring for H` with ` = 3 · i, i ≥ 1.

Proof. It is obvious that OP T 3 is distributed (a vertex could look up matrix A and obtain its own color independently) and takes constant time. Since ` = 3 · i and i ≥ 1 and

hc = ` · 2b c

2c, we have 3 | h_c. Let f be the coloring produced by OP T 3. We now verify

that f is a distance-two coloring. Suppose (c, s) and (c0, s0) are two distinct vertices that

are of distance at most 2 and c ≤ c0. Then c0− c ≤ 2 and there are three cases.

Case 1: c0 = c. Then since 3 | hc, we have 1 ≤ |s0− s|hc ≤ 2 and hence 1 ≤ |s0− s|3 ≤ 2,

which implies |s0|3 6= |s|3. By Ψ1, f (c0, s0) − f (c, s) = A(|c|4, |s0|3) − A(|c|4, |s|3) 6= 0.

Case 2: c0 = c + 1. By Ψ2, f (c0, s0) − f (c, s) = A(|c + 1|4, |s|3) − A(|c|4, |s0|3) 6= 0.

Case 3: c0 = c + 2. Then either s0 = 2s or 2s + 1 occurs. In the former case, by Ψ3 and

then Ψ1, f (c0, s0) − f (c, s) = A(|c + 2|4, |2s|3) − A(|c|4, |s|3) = A(|c|4, |2(2s) + 2|3) −

A(|c|4, |s|3) = A(|c|4, |s + 2|3) − A(|c|4, |s|3) 6= 0. In the latter case, again by Ψ3 and

then Ψ1, f (c0, s0) − f (c, s) = A(|c + 2|4, |2s + 1|3) − A(|c|4, |s|3) = A(|c|4, |2(2s + 1) +

2|3) − A(|c|4, |s|3) = A(|c|4, |s + 1|3) − A(|c|4, |s|3) 6= 0.

Therefore, f is a distance-two coloring. It is obvious that f uses 6 colors. Thus by Lemma 2.2, OP T 3 is optimal and we have this theorem.

3.2

Optimal coloring for H

`

with ` = 8 · i

First we define seven permutations on colors 1, 2, . . . , 6: p0 = (3, 5), p1 = (1, 3),

p2 = (2, 6), p3 = (2, 5), p4 = (3, 4), p5 = (3, 6), and p6 = (1, 2), where a permutation (x, y)

exchanges colors x and y in a coloring (i.e., replaces x with y, and y with x), and for a color c we denote the operator ◦ by

c ◦ (x, y) =    y if c = x; x if c = y; c otherwise,

and c ◦ (x, y)(x0, y0) = (c ◦ (x, y)) ◦ (x0, y0).

We now give a coloring algorithm for H` with ` = 8 · i, i ≥ 1. Imagine that we

partition the vertices of H` into eight subsets (we also call them blocks) B0, B1, . . . , B7,

where Bb = {(c, s) :

j 8s `·2b c2c

k

H` by using OPT3; when 3 | `, we are done, and when 3 - `, we change the colors of

vertices in Bb by using the permutation Q

j<bpj = p0p1. . . pb−1. See Figure 5(b) for an illustration of this algorithm.

Algorithm 2 OP T 8 (As Executed At Every Vertex)

1: if |`|3 = 0 then // 3 | `.

2: vertex (c, s) gets the color A(|c|4, |s|3);

3: else 4: let b =j 8·s `·2b c2c k ; 5: if |`|3 = 1 then

6: vertex (c, s) gets the color A(|c + 2|4, |s|3) ◦Q_j<bpj;

7: else // |`|3 = 2.

8: vertex (c, s) gets the color A(|c|4, |s|3) ◦

Q j<bpj; 9: end if 10: end if y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy yy y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy y y y y y y y y y y y y y y y y y y y i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ii ii i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ii i i i i i i i i i i i i i i i i i i i B0 B1 B2 B3 B4 B5 B6 B7 (a) y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy yy y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy y y y y y y y y y y y y y y y y y y y i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ii ii i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ii i i i i i i i i i i i i i i i i i i i 1 2 4 5 6 4 3 2 1 5 6 5 4 6 3 4 6 3 3 5 1 4 6 1 6 5 3 2 2 4 1 2 4 1 6 4 6 2 5 4 1 5 3 2 6 4 1 4 5 1 3 5 1 3 3 4 6 5 1 6 1 4 3 2 2 5 6 2 5 6 1 5 1 2 3 1 2 5 1 2 3 5 6 3 5 2 3 5 6 2 3 6 2 4 6 ₂ 3 4 1 3 4 2 3 4 B0 (b)

Figure 5: (a) The 8 blocks of H8. (b) The distance-two 6-coloring for H8 produced by

OP T 8.

Theorem 3.2. Algorithm OP T 8 is distributed, takes constant time, and produces an

optimal distance-two 6-coloring for H` with ` = 8 · i, i ≥ 1.

OP T 8 performs in the same way as OP T 3; by Theorem 3.1, OP T 8 produces an optimal

distance-two 6-coloring. In the remaining proof, we consider |`|3 6= 0.

For convenience, let p7 = (2, 4), A0 = A, and A1, A2, . . . , A7 be matrices such that

Ab(c, s) = A(c, s) ◦Q_j<bpj. Then: A0 =          0 1 2 0 6 5 4 1 1 2 3 2 4 5 6 3 3 2 1          , A1 =          0 1 2 0 6 3 4 1 1 2 5 2 4 3 6 3 5 2 1          , A2 =          0 1 2 0 6 1 4 1 3 2 5 2 4 1 6 3 5 2 3          , A3 =          0 1 2 0 2 1 4 1 3 6 5 2 4 1 2 3 5 6 3          , A4 =          0 1 2 0 5 1 4 1 3 6 2 2 4 1 5 3 2 6 3          , A5 =          0 1 2 0 5 1 3 1 4 6 2 2 3 1 5 3 2 6 4          , A6 =          0 1 2 0 5 1 6 1 4 3 2 2 6 1 5 3 2 3 4          , A7 =          0 1 2 0 5 2 6 1 4 3 1 2 6 2 5 3 1 3 4          .

Since A0 is exactly A, it clearly has the properties Ψ1, Ψ2, and Ψ3. For b = 1, 2, . . . , 7,

Ab is obtained by renaming the colors in A. Thus A1, A2, . . . , A7 also have the properties

Ψ1, Ψ2, and Ψ3.

Let f be the coloring produced by OP T 8. Then

f (c, s) = Ab(|c + 2|4, |s|3) if |`|3 = 1;

Ab(|c|4, |s|3) if |`|3 = 2,

f or(c, s) ∈ Bb.

We now verify that f is a distance-two coloring. Suppose (c, s) ∈ Bb and (c0, s0) ∈ Bb0

are two distinct vertices that are of distance at most 2. Then |b0− b|8 ≤ 2 and there are

three cases.

Case 1: b0 = b. Then (c, s) and (c0, s0) belong to the same block and therefore get their

an argument similar to the one used in Theorem 3.1, we have

f (c0, s0) − f (c, s) = Ab(|c

0_{+ 2|4, |s}0_{|3) − Ab(|c + 2|4, |s|3) 6= 0 if |`|3 = 1;}

Ab(|c0|4, |s0|3) − Ab(|c|4, |s|3) 6= 0 if |`|3 = 2.

Case 2: b0 = |b + 2|8. This case occurs only when ` = 8 and c0 = c = 1 and |s0− s|8 = 2.

By checking the coloring in corona 1 of Figure 5(b), f (c0, s0) − f (c, s) 6= 0 holds.

Case 3: b0 = |b + 1|8. Then (c, s) and (c0, s0) belong to two adjacent blocks and there are

two subcases.

Subcase 3-1: c0 = c. In this subcase, 1 ≤ |s0− s|hc ≤ 2. When b = 0, 1, . . . , 6, we

have |s0|3 6= |s|3, and we observe that if the color Ab+1(|c|4, |s0|3) is not indicated

in pb, then Ab+1(|c|4, |s0|3) = Ab(|c|4, |s0|3) 6= Ab(|c|4, |s|3) by Ψ1; otherwise, for

some s00, Ab+1(|c|4, |s0|3) = Ab(|c + 1|4, |s00|3) 6= Ab(|c|4, |s|3) by Ψ2. When b = 7,

we have |s0 + hc|3 6= |s|3, and we observe that if the color A0(|c|4, |s0_{|3) is not}

indicated in p7, then A0(|c|4, |s0|3) = A7(|c|4, |s0 + hc|3) 6= A7(|c|4, |s|3) by Ψ1;

otherwise, for some s00, A0(|c|4, |s0|3) = A7(|c + 1|4, |s00|3) 6= A7(|c|4, |s|3) by Ψ2.

Thus, we have

f (c0, s0)−f (c, s) = A|b+1|8(|c

0_{+ 2|}

4, |s0|3) − Ab(|c + 2|4, |s|3) 6= 0 if |`|3 = 1;

A|b+1|8(|c0|4, |s0|3) − Ab(|c|4, |s|3) 6= 0 if |`|3 = 2.

Subcase 3-2: c0 6= c. Since (c, s) and (c0_{, s}0_{) belong to two adjacent blocks, in this}

subcase, (c0, s0) and (c, s) are of distance exactly two and |c0−c| = |s0_−s|

hc = 1. So we only need to check the colors used on the boundary of two adjacent blocks

(i.e., the boundary of B0 and B1, the boundary of B1 and B2, . . ., the boundary

and |`|3 = 2: |`|3 = 1 B0 B1 B2 B3 B4 B5 B6 B7 B0 c = 1 3 · · · 2 1 · · · 5 2 · · · 3 5 · · · 6 3 · · · 2 6 · · · 4 2 · · · 3 4 · · · 1 3 · · · c = 2 6 · · · 6 3 · · · 3 4 · · · 4 2 · · · 2 1 · · · 1 3 · · · 3 5 · · · 5 2 · · · 2 6 · · · c = 3 1 · · · 1 2 · · · 2 5 · · · 5 3 · · · 3 6 · · · 6 2 · · · 2 4 · · · 4 3 · · · 3 1 · · · c = 2 4 · · · 5 6 · · · 4 1 · · · 6 4 · · · 1 5 · · · 4 1 · · · 5 6 · · · 1 5 · · · 6 4 · · · c = 1 3 · · · 2 1 · · · 5 2 · · · 3 5 · · · 6 3 · · · 2 6 · · · 4 2 · · · 3 4 · · · 1 3 · · · .. . ... ... ... ... ... ... ... ... ... |`|3 = 2 B0 B1 B2 B3 B4 B5 B6 B7 B0 c = 1 1 · · · 1 2 · · · 2 5 · · · 5 3 · · · 3 6 · · · 6 2 · · · 2 4 · · · 4 3 · · · 3 1 · · · c = 2 4 · · · 5 6 · · · 4 1 · · · 6 4 · · · 1 5 · · · 4 1 · · · 5 6 · · · 1 5 · · · 6 4 · · · c = 3 3 · · · 2 1 · · · 5 2 · · · 3 5 · · · 6 3 · · · 2 6 · · · 4 2 · · · 3 4 · · · 1 3 · · · c = 4 6 · · · 6 3 · · · 3 4 · · · 4 2 · · · 2 1 · · · 1 3 · · · 3 5 · · · 5 2 · · · 2 6 · · · c = 5 1 · · · 1 2 · · · 2 5 · · · 5 3 · · · 3 6 · · · 6 2 · · · 2 4 · · · 4 3 · · · 3 1 · · · .. . ... ... ... ... ... ... ... ... ...

From the above lists, two vertices get different colors if they are on the bound-ary of two adjacent blocks and of distance exactly two.

From the above, f is a distance-two coloring. It is obvious that f uses 6 colors. Thus by Lemma 2.2, OP T 8 is optimal and we have this theorem.

3.3

Optimal coloring for H

4

We define a matrix M4 =    0 1 2 3 4 5 6 7 1 6 4 1 5 − − − − 2 1 3 7 2 6 3 7 2   

where “−” means the corresponding item is not used. By Lemma 2.5, H8 is a subgraph

of H4. Thus one way to color H4 is to extend a coloring of H8 and this leads to Algorithm

Algorithm 3 OP T 4 (As Executed At Every Vertex)

1: if c ≤ 2 then

2: vertices (c, s) get the color M4(c, s);

3: else // c ≥ 3.

4: if (c, s) = (3, 0) || (c, s) = (3, 4) then

5: vertex (c, s) gets the color 7;

6: else

7: let b = j 2·s

2b c2c k

;

8: vertex (c, s) gets the color A(|c + 2|4, |s|3) ◦Q

j<bpj; // use the |`|3 = 1 case in OP T 8. 9: end if 10: end if y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy yy y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ii ii i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 6 1 3 7 2 4 5 6 4 3 2 1 5 6 5 4 6 3 4 6 3 4 2 7 3 5 1 4 6 1 6 5 3 2 2 4 1 2 4 1 6 4 1 6 3 7 2 5 4 1 5 3 2 6 4 1 4 5 1 3 5 1 3 5 2 7 3 4 6 5 1 6 1 4 3 2 2 5 6 2 5 6 1 5

Figure 6: The distance-two 7-coloring for H4 produced by OP T 4; all the vertices are

colored by using A (along with permutations p0, p1, . . . , p6) except that those highlighted

are colored by using M4.

Theorem 3.3. Algorithm OP T 4 is distributed, takes constant number of steps, and

pro-duces an optimal distance-two 7-coloring for H4.

Proof. It is obvious that OP T 8 is distributed and takes constant time. Let f be the coloring produced by OP T 4. We now verify that f is a distance-two coloring. Suppose

(c, s) and (c0, s0) are two distinct vertices that are of distance at most 2. If at least one of

(c, s) and (c0, s0) is highlighted (see Figure 6), then f (c, s) 6= f (c0, s0) can be verified by a

in the same way as the |`|3 = 1 case of OP T 8; hence f (c, s) 6= f (c0, s0) by Theorem 3.2. From the above, f is a distance-two coloring. It is obvious that f uses 7 colors. Thus by Theorem 2.3, OP T 4 is optimal and we have this theorem.

3.4

Optimal coloring for H

5

We first define a matrix M5 =

            0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 1 2 5 4 3 − − − − − − − − − − − − − − − 2 4 5 6 4 1 7 6 1 5 7 − − − − − − − − − − 3 3 2 7 5 6 3 2 7 4 1 − − − − − − − − − − 4 6 5 4 6 3 4 2 1 4 2 1 4 5 1 3 5 2 6 5 2 5 1 2 7 1 2 5 3 7 5 3 6 2 7 6 2 4 3 7 4 3            

where “−” means the corresponding item is not used. By Lemma 2.5, H40 is a subgraph

of H5. Thus one way to color H5 is to extend a coloring of H40and this leads to Algorithm

OP T 5. See Figure 7 for an illustration of this algorithm. Algorithm 4 OP T 5 (As Executed At Every Vertex)

1: if c ≤ 5 then

2: vertices (c, s) get the color M5(c, s);

3: else // c ≥ 6.

4: let b = b 8·s

5·2b c2cc;

5: vertex (c, s) gets the color A(|c|4, |s|3)◦Q

j<bpj; // use the |`|3 = 2 case in OP T 8.

6: end if

Theorem 3.4. Algorithm OP T 5 is distributed, takes constant number of steps, and

pro-duces an optimal distance-two 7-coloring for H5.

Proof. It is obvious that OP T 8 is distributed and takes constant time. Let f be the coloring produced by OP T 5. We now verify that f is a distance-two coloring. Suppose

y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy_{y y y y y}yy yy yy y y y y y y y y y y y y y y y y y y y y y y y y y y y y y_{y y y y y}yy yy y y y i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ii_{i i i i i}ii ii ii i i i i i i i i i i i i i i i i i i i i i i i i i i i i i_{i i i i i}ii ii i i i 1 2 5 4 3 4 5 6 4 1 7 6 1 5 7 3 2 7 5 6 3 2 7 4 1 6 5 4 6 3 4 2 1 4 2 1 4 5 1 3 5 2 6 5 2 1 2 7 1 2 5 3 7 5 3 6 2 7 6 2 4 3 7 4 3 4 5 6 4 5 6 4 3 6 4 1 6 4 1 6 4 1 2 4 1 5 4 1 5 4 1 5 3 1 5 6 1 5 6 15 62 5 6 3 2 1 3 2 1 5 2 1 5 2 3 5 2 3 5 6 3 5 6 3 2 6 3 2 6 4 2 6 4 2 3 4 2 3 4 1 3 4 1

Figure 7: The distance-two 7-coloring for H5 produced by OP T 5; all the vertices are

colored by using A (along with permutations p0, p1, . . . , p6) except that those highlighted

are colored by using M5.

(c, s) and (c0, s0) are two distinct vertices that are of distance at most 2. If at least one of

(c, s) and (c0, s0) is highlighted (see Figure 7), then f (c, s) 6= f (c0, s0) can be verified by a

brute-force checking. If both of (c, s) and (c0, s0) are not highlighted, then OP T 5 performs

in the same way as the |`|3 = 2 case of OP T 8; hence f (c, s) 6= f (c0, s0) by Theorem 3.2.

From the above, f is a distance-two coloring. It is obvious that f uses 7 colors. Thus by Theorem 2.3, OP T 5 is optimal and we have this theorem.

4

The leader election problem

The leader election problem is to select a leader (from the sensors in a cluster) to perform certain tasks on each cluster. Because sensor networks contain many sensed data of the local environment, leader election can be used to combine or aggregate the data into meaningful information. More precisely, leader election has applications to coordination and data fusion, the latter is also called data aggregation and can be used to reduce the number of data to be communicated between the sensor node and the actor so that to

avoid information overload. Leaders paly the most important role of each cluster. Thus an efficient process for the election of a cluster leader (or data aggregator node) is essential. In [13], the authors mentioned that they use the uniform leader election for radio networks protocol in [15] (abbreviated as ULERNP) to select a leader for each cluster. Unfortunately, we find that this is incorrect. In ULERNP, the network has to be a single-hop network (i.e., every two nodes can communicate directly). Therefore to use ULERNP

to select a leader for each cluster in the virtual infrastructure G`, the nodes in each cluster

have to form a complete graph; however, it is usually impossible that every two nodes in a cluster can communicate directly. Furthermore, when the nodes are very dense, ULERNP usually produces dramatic communication overhead.

In [8], a hybrid approach that combines the energy conservation with the simplicity was introduced. This approach is based on four selection parameters: (1) the available energy, (2) the number of neighbouring sensor nodes, (3) the distance from the current group leader, and (4) the level of trust; for details, please refer to [8]. This approach

can be used in leader election for G` and H`. However, nodes may produce a lot of

communication overhead since G` and H` are usually multi-hop networks. For other

leader election protocols, please see [10, 16].

Before closing this section, we propose an idea of how to perform leader election in

a multi-hop network like G` and H`. We will only consider the parameter (1) and the

distance from the candidate node to the other nodes in the cluster (the leader should be easy accessed from the other nodes). If more than one node can be selected, we randomly select one of them as the leader.

5

The concluding remarks

In this thesis, we propose a virtual infrastructure called H` and an distance-two

to the sensors in a network and allows geographic routing. Our distance-two coloring algorithm can be used to assign the frequency channels (or colors) in a fully distributed manner and our algorithm uses fewer channels than the previous work [13]. In the fu-ture, we intend to determine an appropriate way for the leader election problem, because choosing the right leader can help enhancing the network lifetime and can make routing more easier. In real world applications, the environment may have obstruction in it. Thus it is also challenging to find a virtual infrastructure for such an environment.

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