國 立 交 通 大 學
應用數學系
碩 士 論 文
無線網路中單位圓盤圖的不完美比例
On the Imperfection Ratio of Unit Disc Graphs
研 究 生：何恭毅
指導教授：陳秋媛 教授
中 華 民 國 一 百 年 六 月
無線網路中單位圓盤圖的不完美比例
On the Imperfection Ratio of Unit Disc Graphs
研 究 生：何恭毅 Student：Kung-Yi Ho
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 2011
無線網路中單位圓盤圖的不完美比例
研究生：何恭毅
指導老師：陳秋媛 教授
國 立 交 通 大 學
應 用 數 學 系
摘 要
無線網路上的干擾問題是重要且困難的，該問題對應到單位圓盤圖的著色問題。 在文獻[9]中，Mani和Petr對隨機網路的單位圓盤圖做了大量的模擬，並且觀察到 單位圓盤圖的完全子圖數 ( ) G 和著色數 ( ) G 是非常接近的。為了估計 ( ) G 與 ( )G 的接近程度，Mani和Petr採用了「不完美比例」 ( ) sup ( ') ( ') R G imp G G 。這裡 的G'是由G轉換過來的圖，而這個sup是考慮所有權重向量R之後計算所得。已 有學者證明出imp G 的理論上界是( ) 2.155，imp G 若且唯若( ) 1 G是完美圖。基 於模擬所得之結果，Mani和Petr認為，對單位圓盤圖而言，一個切合實際的上界 是imp G ( ) 1.2079，這個上界值遠小於猜測中的上界值1.5，也遠小於理論上界 值2.155。本論文之目的在於證明確實存在單位圓盤圖其imp(G)1.2079，且猜 測中的上界值imp G ( ) 1.5是可達到的。特別地，我們證明了：若G是長度5的 奇圈或者是Harary圖H2 ,3m m2（其中 m 是奇數），則imp G ( ) 1.5；若G是輪圖W ，6 則imp G ( ) 4 / 3。 關鍵詞：無線干擾，完全子圖數，著色數，單位圓盤圖，不完美比例。 中 華 民 國 一 百 年 六 月On the Imperfection Ratio of Unit Disc Graphs
Student: Kung-Yi Ho
Advisor: Chiuyuan Chen
Department of Applied Mathematics National Chiao Tung University
Abstract
The interference problem between nodes in a wireless network is important and difficult and it corresponds to the coloring problem in Unit Disc Graphs (UDGs). In [9], Mani and Petr performed extensive simulations with UDGs of random networks and observed that in a UDG G, the clique number ω(G) and the chromatic number χ(G) were typically very close to one another. To evaluate the closeness of χ(G) and ω(G), Mani and Petr used the measure “imperfection ratio” imp(G) = supRχ(Gω(G00)).
Here G0 is a graph transformed from G and the supremum is computed over all
possible weight vectors R. It has been proven that the theoretical bound of imp(G) is 2.155 and imp(G) = 1 if and only if G is perfect. Based on the simulation results, Mani and Petr concluded that a practical bound for UDGs is imp(G) ≤ 1.2079, which is far less than the conjectured upper bound of 1.5 or the theoretic upper bound of 2.155. The purpose of this thesis is to show that there exist UDGs such that imp(G) > 1.2079 and the conjectured upper bound imp(G) = 1.5 is achievable. In particular, we show that: if G is an odd cycle of length ≥ 5 or is the Harary
graph H2m,3m+2where m is odd, then imp(G) = 1.5, and if G is the wheel W6, then
imp(G) = 4/3.
Keywords: Wireless interference, clique number, chromatic number, unit disk graph, imperfection ratio.
誌 謝
很快的兩年碩士生活要過去了，在這期間最需要感謝的
就是陳秋媛老師，除了對論文提出了許多適當的方向，使得
此篇論文能夠順利產生，平常更會關心我的生活狀況，在我
碰到困難的時候提供足夠的幫助和包容，非常感謝老師在這
段期間的照顧。
另外，我也要感謝我的父母在精神上以及物質上提供的
支持，讓我能夠沒有後顧之憂的專注在求學上，同時也希望
自己未來能夠有所成就，才不會辜負他們的期待。
最後要感謝邱鈺傑學長以及蔡詩妤提出的一些建議，讓
我獲益良多，也很感激其他同學和學弟妹們在這些日子裡提
供的幫助。
雖然還有許多想要感謝的人，但沒有辦法一一列舉，最
後讓我再次感謝所有幫助過我的人，謝謝！
Contents
Abstract (in Chinese) i
Abstract (in English) ii
Acknowledgement iii
Contents iv
List of Figures v
1 Introduction 1
2 UDGs with imp(G) > 1.2079 6
3 Our coloring algorithm and simulation results 11
3.1 Randomly generated unit disk graphs . . . 11 3.2 Randomly generated weighted unit disk graph . . . 13 3.3 Randomly generated unit disk graphs with different density of nodes . . . . 14 3.4 Randomly generated unit disk graphs that allow the addition of nodes . . . 16
List of Figures
1 This graph has chromatic number 3. If a balanced load scenario occurs,
then three timeslots are required. . . 3
2 Transforming a weighted graph G into an un-weighted graph G0; the weights on vertices a, b, c in G are 3, 1, 2, respectively. . . 4
3 W6, the wheel graph with 6 vertices. . . 5
4 The average P R(G). . . 12
5 The maximum P R(G). . . 13
6 An example of P R(G) = 1.211. . . 13
7 An example of different density of nodes. . . 15
8 The average P R(G) in UDGs with different density of nodes. . . 16
1
Introduction
A wireless ad hoc network (or simply a wireless network) consists of a set of nodes that communicate with each other without any physical infrastructure or centralized adminis-tration. The interference problem between nodes in a wireless network is important and difficult and it can be modeled using graph theoretic techniques, in particular, the theory associated with Unit Disc Graphs (UDGs). As we will see below, the chromatic number of a UDG model of a wireless network is directly related to the interference problem. The chromatic number is a graph invariant. The clique number is another graph invariant and is closely related to the chromatic number. In some special cases, the clique number is equal to the chromatic number.
Before going further, we give some definitions. Our graph terminology and notation are standard; see [2] and [11] except as indicated. All graphs in thesis are assumed simple. Let G = (V, E) be a graph. We say that G is k-colorable if the vertices of G can be colored by using at most k colors such that no two adjacent vertices receive the same color. The chromatic number of G, denoted by χ(G), is defined to be the smallest k such that G is k-colorable. A clique of G is a complete subgraph in G. A maximum clique of G is a clique of the largest possible size in G. The clique number of G, denoted by ω(G), is the number of vertices in a maximum clique in G.
It is well known that the graph coloring problem is NP-complete and that even the problem of approximating the chromatic number within any constant ratio is NP-hard [6]. In [3], Clark et al. proved that the coloring problem remains NP-complete for UDGs. In fact, Clark et al. proved that the problem of determining, given a UDG G, whether G is 3-colorable is NP-complete. Notice that in [1], Breu and Kirkpatrick have proved that the problem of determining, given a graph G, whether G is a UDG is NP-hard. In [5], Graf et al. improved the result of Clark et al. by showing that the problem of determining, given
a UDG G and a fixed integer k, whether G is k-colorable remains NP-complete for any fixed k ≥ 3; they also proposed a 3-approximation algorithm for the coloring problem.
It is clear that for any graph G, the chromatic number is always lower bounded by the clique number, i.e., χ(G) ≥ ω(G). For the special case of “perfect graphs”, the chromatic number and the clique number have equal values in every induced subgraph. While computing χ(G) is still NP-complete for UDGs, computing ω(G) can be done in polynomial time for UDGs [3].
We assume that the given wireless network has n nodes and their respective position coordinates is in 2D. The transmission range (TR) of a given node is defined as the maximum distance at which the nodes transmission can be successfully received, and all nodes that lie within the transmission range of a given node are called its communicating neighbors. The interference range (IR) is defined as the maximum distance at which a given node’s transmission can interfere with or corrupt a simultaneous transmission or reception attempt by another node, and all nodes that lie within interference range of a given node are called its interfering neighbors. Clearly, all communicating neighbors are interfering neighbors and vise versa.
Recently, in [9], Mani and Petr treated the case in which IR is the same for all nodes. The graph model is a UDG and is called an interference graph. More precisely, a UDG G is formed by taking the nodes in the wireless network as its vertices, and there is an edge between vertices u and v if and only if the Euclidean distance between u and v, denoted by d(u, v), is less than or equal to 1. Notice that we will use the terms node and vertex interchangeably. If two nodes share an edge, then it means that they are mutually interfering and hence cannot transmit simultaneously in the same timeslot. There are two possible scenarios: balanced load scenario and unbalanced load scenario. In the former case, each node require the identical number of transmission timeslots per second to suit their traffic requirements; as a result, the chromatic number gives the minimum number
of timeslots per second. See Figure 1. However, a balanced load scenario rarely occurs in the real world.
Figure 1: This graph has chromatic number 3. If a balanced load scenario occurs, then three timeslots are required.
In [9], Mani and Petr considered the unbalanced load scenario, wherein the traffic rates of each node need not be identical. In particular, for each node vi, let ri be the number
of timeslots required per second by vi to satisfy its traffic needs. The UDG now becomes
a weighted UDG such that each vertex vi has a weight ri associated with it. To find out
the optimal (i.e., minimum) number of timeslots required per second for a weighted UDG G = (V, E), Mani and Petr used weighted vertex coloring [4] algorithms, which is simply normal (un-weighted) coloring done on a transformed graph G0. The graph G0 = (V0, E0) is obtained from G by replacing each vertex vi in G by a clique of size ri and the edge set
E is augmented to obtain E0 in such a way that if two vertices u, v are neighbors in G, then in G0 every node in the clique corresponding to u is also a neighbor of every node in the clique corresponding to v. (See Figure 2.)
As was mentioned above, the chromatic number χ(G) of a UDG model of a wireless network is directly related to interference. Closely related to χ(G) is the clique number ω(G). For most classes of graphs, computing χ(G) and ω(G) are both NP-complete. But for UDGs, while computing χ(G) is still NP-complete, computing ω(G) can be done in polynomial time. This raises the question: How close is ω(G) to χ(G)? For general
a
b
c
G
G’
Figure 2: Transforming a weighted graph G into an un-weighted graph G0; the weights on vertices a, b, c in G are 3, 1, 2, respectively.
graphs, χ(G)/ω(G) can be very large. In [10], Peeters has observed that χ(G) ≤ 3ω(G) − 2 if G is a UDG.
See also [7]. In [9], Mani and Petr performed extensive simulations with UDGs of random networks and observed that in a UDG G, the clique number ω(G) and the chromatic number χ(G) were typically very close to one another. To evaluate the closeness of χ(G) and ω(G), Mani and Petr used the measure “imperfection ratio”
imp(G) = sup
R
χ(G0) ω(G0)
of a transformed weighted graph, defined as the supremum of the ratio of its chromatic number to its clique number. Here the supremum is computed over all possible weight vectors R.
It has been proven that the theoretical bound of imp(G) is 2.155 and imp(G) = 1 if and only if G is perfect [4]. Based on the simulation results, Mani and Petr concluded that a practical bound for UDGs is imp(G) ≤ 1.2079, which is far less than the conjectured upper bound of 1.5 or the theoretic upper bound of 2.155. The following is Mani and Petr’s simulation scenario: they assume the simulation area is a disk of radius 1 and place n nodes in randomly chosen locations within the disc. Node vi is assigned an integer
in 1, 2, . . . , K, where K corresponds to the maximum weight. They varied n as 10, 25, 50, 75 and 100 and independently varied K as 1, 5 10, 20, 30, 40, and 50. The smallest mean size of the UDG (in terms of number of vertices) is 10 and the largest is 2550. They observed that in a UDG G, ω(G) and χ(G) were typically very close to one another. Based on the simulation results, Mani and Petr concluded that a practical bound for UDGs is imp(G) ≤ 1.2079 and ω(G) can be used as a very good approximation to χ(G); in particular, they said that a practical bound of χ(G) ≤ 1.21ω(G) could be used if G is a UDG.
The purpose of this thesis is to show that there exist UDGs such that imp(G) > 1.2079, and moreover, the conjectured upper bound imp(G) = 1.5 is achievable. In particular, we show that: if G is an odd cycle of length ≥ 5 or is the Harary graph H2m,3m+2 where m
is odd, then imp(G) = 1.5, and if G is the wheel W6 (see Figure 3), then imp(G) = 4/3.
We also propose an algorithm to color the nodes of a UDG and perform simulations to compare the number of colors used by our algorithm and that used by the First-Fit coloring algorithm.
Figure 3: W6, the wheel graph with 6 vertices.
This thesis is organized as follows. In Section 2, we gives UDGs with imp(G) > 1.2079. In Section 3, we propose a coloring algorithm and compare the results of our algorithm with the classical First-Fit coloring algorithm. Concluding remarks are given in the final section.
2
UDGs with imp(G) > 1.2079
The fact that imp(G) ≥ χ(G)/ω(G) will be used throughout this section. Lemma 2.1. There exists a general graph G such that imp(G) → ∞.
Proof. This lemma follows from the fact that we can use Mycielski construction [11] to obtain a new triangle-free graph G∗ from a given triangle-free graph G such that χ(G∗) = χ(G) + 1 and ω(G∗) = ω(G) = 2.
Before going further, we introduce some notations. Let G be a weighted graph. We use wG(v) to denote the weight of a vertex v in G and use G0 to denote the un-weighted
transformed graph of G; see Section 1 and Figure 2 for an illustration of G0. For conve-nience, we use Sv to denote the set of vertices in G0 that correspond to a vertex v in G.
Let Cn denote a cycle of length n. It is not difficult to verify that Cn is a UDG. Cn is
called an odd cycle if it is of odd length. We have the following theorem. Theorem 2.2. If G is an odd cycle of length ≥ 5, then imp(G) = 1.5.
Proof. Since χ(G) = 3 and ω(G) = 2, we have imp(G) ≥ 1.5. On the other hand, let V (G) = {v1, v2, . . . , vn}, wG(vi) = ri for each i, and E(G) = {v1v2, v2v3, . . . , vn−1vn, vnv1}.
Without loss of generality, we may assume that r1+ r2 = ω(G0). Clearly,
(
r2 + r3 ≤ r1+ r2
rn+ r1 ≤ r1+ r2.
Thus r3+ rn ≤ r1+ r2 = ω(G0). By the pigeonhole principle, r3 ≤ 0.5ω(G0) or rn ≤
0.5ω(G0) must occur. Suppose rn ≤ 0.5ω(G0) occurs; the case that r3 ≤ 0.5ω(G0) occurs
can be proven in a similar way and we omit its proof. To prove this theorem, it suffices to prove that G0 is 1.5ω(G0)-colorable.
Since |Sv1∪ Sv2∪ Svn| ≤ 1.5ω(G
0), the subgraph induced by S
v1∪ Sv2∪ Svn is 1.5ω(G
0
)-colorable. Denote the set of colors on Sv1∪ Sv2 by C
1,2 and denote the set of colors on S vn
by Cn. Notice that C1,2∩ Cn= ∅. Then, color the vertices in S
vi according to the ordering
i = 3, 4, . . . , n − 1. For each i, before Svi is to be colored, the vertices in Svi−2∪ Svi−1 have
already been colored; denote the set of colors on Svi−1 by C
i−1. Since G is an odd cycle of
length ≥ 5, in G0 there is no edge joining a vertex in Svi and a vertex in Svi−2. Hence it is
possible to colors the vertices in Svi by using the colors in C
1,2\ Ci−1. The above process
proves that G0 is 1.5ω(G0)-colorable.
A wheel graph with n vertices, denoted by Wn, is the graph obtained by adding a new
vertex to the cycle Cn−1 and making this new vertex joining each vertex of Cn−1. It is
not difficult to see that W6 is a UDG. We have the following theorem for W6.
Theorem 2.3. If G = W6, then imp(G) = 4/3.
Proof. Clearly, imp(G) ≥ χ(G)ω(G) = 4/3. Suppose V (G) = {v1, v2, . . . , v6}, wG(vi) = ri for
each i, and E(G) = {v1v2, v2v3, . . . , v4v5, v5v1} ∪ {viv6|1 ≤ i ≤ 5}. Let H be the subgraph
of G induced by {v1, v2, . . . , v5}. Then χ(G 0) ω(G0) = χ(H0)+w G(v6) ω(H0)+w
G(v6) and if we want to make
χ(G0)
ω(G0)
as large as possible, then we must have wG(v6) = 1. Without loss of generality, we may
assume that r1 + r2 + 1 = ω(G0). Set ω = r1 + r2 + 1 for easy writing. There are two
cases.
Case 1: r3 ≤ ω/3 or r5 ≤ ω/3. Suppose r5 ≤ ω/3 occurs; the case that r3 ≤ ω/3 occurs
can be proven in a similar way and we omit its proof. To prove this theorem, it suffices to prove that G0 is 43ω-colorable. Since |Sv1 ∪ Sv2 ∪ Sv5| ≤
4
3ω − 1, the subgraph induced
by Sv1 ∪ Sv2 ∪ Sv5 is (
4
3ω − 1)-colorable. Denote the set of colors on Sv1 ∪ Sv2 by C
1,2 and
denote the set of colors on Sv5 by C
5. Notice that C1,2∩ C5 = ∅. Then, color the vertices in
Svi according to the ordering i = 3, 4. For each i, before Svi is to be colored, the vertices
in Svi−2∪ Svi−1 have already been colored; denote the set of colors on Svi−1 by C
i−1. Since
H is an odd cycle of length 5, in G0 there is no edge joining a vertex in Svi and a vertex
in Svi−2. Hence it is possible to colors the vertices in Svi by using the colors in C
Assign v6 any color that is not used on Sv1 ∪ Sv2 ∪ Sv3 ∪ Sv4 ∪ Sv5. The above process
proves that G0 is 43ω-colorable.
Case 2: r3 > ω/3 and r5 > ω/3. It suffices to prove that G0 is 43ω-colorable. Since
|Sv1 ∪ Sv2| = ω − 1, the subgraph induced by Sv1 ∪ Sv2 is (ω − 1)-colorable. Denote the
set of colors on Sv1 and Sv2 by C
1 and C2, respectively. Let C0 be a set of ω/3 colors such
that C0∩ (C1∪ C2) = ∅. Then, color the vertices in S
v3 by using the colors in C
0 as their
first choice and the colors in C1 as their secondary choice. Since |S
v3| = r3, the number
of colors used on Sv3 in the set C
1 equals r
3− ω/3. Let C3 denote the set of colors on Sv3.
Then, color the vertices in Sv5 by using the colors in C
0 as their first choice and the colors
in C2 as their secondary choice. Since |Sv5| = r5, the number of colors used on Sv5 in the
set C2 equals r
5− ω/3. Let C5 denote the set of colors on Sv5. Then, color the vertices in
Sv4 by using the colors in C
1∪ C2. Since r
1 ≥ r3 > ω/3 and r1 + r5 < ω together imply
that r5 < 2ω/3, we have
r1+ r2 ≥ r3+ r4 > r3+ r4+ r5− 2ω/3 = (r3− ω/3) + r4+ (r5− ω/3)
and therefore it is possible to color the vertices in Sv4 by using the colors in (C
1∪ C2) \
(C3∪ C5). Assign v
6 any color that is not used on Sv1 ∪ Sv2 ∪ Sv3 ∪ Sv4 ∪ Sv5. The above
process proves that G0 is 4
3ω-colorable.
Let H be an induced subgraph of G. We now show that it is possible that neither imp(G) > imp(H) nor imp(G) < imp(H) holds.
Observation 2.4. There exists a UDG G such that imp(G) > imp(H1) and imp(G) <
imp(H2) for some induced subgraph H1 and H2 of G.
Proof. Let G = W6, H1 = C3, and H2 = C5. Clearly, imp(C3) = 1. By Theorems 2.3
and 2.2, imp(W6) = 4/3 and imp(C5) = 1.5. Thus we have imp(W6) > imp(C3) and
We now define the Harary graph Hk,n; see also [11]. Given k < n, place n vertices
around a circle, equally spaces. Let the vertices be 0, 1, . . . , n − 1; the edges are added in the following ways.
Case 1: k = 2m. Add an edge between i and j whenever i − m ≤ j ≤ i + m (mod n). Case 2: k = 2m+1 and n is even. Construct Hk,n from Hk−1,n by adding an edge between
i and i + n/2 for each 1 ≤ i ≤ n/2.
Case 3: k = 2m + 1 and n is odd. Construct Hk,n from Hk−1,n by adding an edge between
0 and (n − 1)/2, 0 and (n + 1)/2, i and i + (n + 1)/2 for each 1 ≤ i ≤ (n − 1)/2.
Theorem 2.5. Consider the Harary graph G of the form H2m,3m+2. Then imp(G) = 1.5
if m is odd, 3m+22m+2 ≤ imp(G) ≤ 1.5 if m is even.
Proof. Let V (G) = {v1, v2, ..., v3m+2}. This theorem holds if we can prove that imp(G) =
d3m+2 2 e
m+1 . We first prove that imp(G) ≥
d3m+2 2 e
m+1 . An independent set of a graph is a subset
of its vertex set such that each pair of vertices in this subset are not adjacent. Since an independent set of a Harary graph is of size at most 2, each color can be used at most twice. Consequently, χ(G) ≥l|V (G)|2 m=3m+22 . Define a coloring
f : V (G) → {1, 2, . . . ,3m+2 2 }
as follows:
(i) If m is odd, let f (vi) = f (vi+3m+12 ) = i for each 1 ≤ i ≤ 3m+1
2 and let f (v3m+2) =
3m+2
2 .
(ii) If m is even, let f (vi) = f (vi+3m+22 ) = i for each 1 ≤ i ≤ 3m+22 .
Hence χ(G) ≤3m+22 . Therefore χ(G) = 3m+22 . Notice that ω(G) = m + 1. Thus we have imp(G) ≥ d
3m+2 2 e
m+1 .
easy writing. To prove this theorem, it suffices to prove that G0 is 1.5ω-colorable. Let k = min{0 ≤ k ≤ m|Pk+1 i=1 ri > 0.5ω}. Then ( rm+2+ rm+3 + · · · + rm+k+1 ≤ r1+ r2+ · · · + rk ≤ 0.5ω r2m+k+3+ r2m+k+4+ · · · + r3m+2 ≤ rk+2+ rk+3+ · · · + rm+1 < 0.5ω.
Since there is no edge joining a vertex in Svi and a vertex in Svj for all m+2 ≤ i ≤ m+k+1
and 2m + k + 3 ≤ j ≤ 3m + 2, {∪m+k+1i=1 Svi} ∪ {∪
3m+2
i=2m+k+3Svi} is 1.5ω-colorable. Denote
the set of colors on Svi by C
i for 1 ≤ i ≤ m + 1. Then, Pi j=1|C j| > Pm+i+1 j=m+k+2|Svj| for i = k + 1, k + 2, . . . , m, Pm+1 j=i |Cj| > P2m+k+2
j=2m+i+1|Svj| for i = k + 1, k, . . . , 1 and
Pm+1
j=1 |Cj| ≥
P2m+k+2
j=m+k+2|Svj|. This implies that we can color the vertices in Svi according
to the ordering i = m + k + 2, m + k + 3, . . . , 2m + 1, 2m + k + 2, 2m + k + 1, . . . , 2m + 2 by using the colors in ∪m+1i=1 Ci. Thus G0 is 1.5ω-colorable.
We now list some imperfection ratios of Harary graphs H2m,3m+2. It is not difficult to
see that imp(H2m,3m+2) → 1.5 when m is even.
m 3m + 2 χ(G) ω(G) lower bound of imp(G) upper bound of imp(G)
1 5 3 2 1.5 1.5 2 8 4 3 1.333 1.5 3 11 6 4 1.5 1.5 4 14 7 5 1.4 1.5 5 17 9 6 1.5 1.5 6 20 10 7 1.429 1.5 7 23 12 8 1.5 1.5
3
Our coloring algorithm and simulation results
For convenience, let c(G) by the number of colors used by a given algorithm. The performance of an algorithm is defined by P R(G) = c(G)/ω(G). Let First-Fit (also called a greedy coloring algorithm) denote the coloring algorithm that examines the vertices of a graph in an arbitrary order and assigns each vertex the smallest-indexed color not already used on its examined neighbors. To improve First-Fit, we examines the vertices of a graph in the order obtained by breadth-first search (BFS) form any node and we call our algorithm BFS-First-Fit. Since Mani and Petr [9] mentioned that χ(G)/ω(G) is at most 1.2079 in their simulation results, if BFS-First-Fit obtains P R(G) > 1.2, then we will run BFS-First-Fit again by choosing another vertex as the root of the BFS until P R(G) ≤ 1.2 or the above process has been repeated too many times (in this thesis, the threshold value of 10 is chosen). Notice that we can adjust the value 1.2 in P R(G) > 1.2 and the number of times that the root of BFS-First-Fit is changed to get a better performance.
This section is divided into four subsections. In Subsection 3.1, we consider randomly generated UDGs. In Subsection 3.2, we consider randomly generated weighted UDGs. In Subsection 3.3, we consider randomly generated UDGs in which the nodes are not evenly distributed. And in Subsection 3.4, we consider randomly generated UDGs that allow the addition of new nodes. For each subsection, simulations results for First-Fit and BFS-First-Fit are obtained.
3.1
Randomly generated unit disk graphs
To perform the simulations, we randomly construct 500 connected UDGs with n nodes in a 100m × 100m area, where n is ranged from 100 to 500, with an increment of 50. The interference range of each node is assumed to be 25m.
Figure 4 shows the average P R(G) obtained by First-Fit and BFS-First-Fit. Both of them increase as the number of nodes increases. The performance of BFS-First-Fit is better than that of First-Fit in all cases and the difference between them increases as the number of nodes increases.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 100 150 200 250 300 350 400 450 500 number of nodes av er ag e P R (G ) BFS-First-Fit First-Fit
Figure 4: The average P R(G).
Figure 5 shows the maximum P R(G) obtained by First-Fit and BFS-First-Fit. We observe that maximum P R(G) is about 1.2 if BFS-First-Fit is used and about 1.5 if First-Fit is used.
Among our 4500 simulations, P R(G) ≤ 1.2 occurs for almost all cases, and there are only 7 simulations with P R(G) > 1.2. Among the 7 simulations with P R(G) > 1.2, the maximum P R(G) of them is 1.2381, which is larger than 1.2079. There is an example of P R(G) = 1.211 in Figure 6; this UDG has 200 nodes and we color the edges of its maximum clique in color red. Based on our simulation results, we conclude that BFS-First-Fit has P R(G) ≤ 1.2 for most of the cases.
0 0.5 1 1.5 2 100 150 200 250 300 350 400 450 500 number of nodes m ax im um P R (G ) BFS-First-Fit First-Fit
Figure 5: The maximum P R(G).
Figure 6: An example of P R(G) = 1.211.
3.2
Randomly generated weighted unit disk graph
In this subsection, we consider weighted UDGs and we use the same parameters as in [9] to compare the P R(G) obtained by BFS-First-Fit with the simulation results in [9].
More precisely, we assume that n nodes are chosen randomly from a disk of radius 1 and each node has interference range 1. Node i has weight ri. The weights are chosen
randomly from 1, 2, . . . , K, where K corresponds to the maximum weight. We vary the number of nodes n as 10, 25, 50, 75 and 100. For each n, we also vary the maximum weight K as 1, 5, 10, 20, 30, 40 and 50. For each (n, K) pair, we perform 500 simulations. Table 2 shows the average value of P R(G), and Table 3 shows the maximum value of P R(G). From Table 2, we observe that P R(G) increases as the number of nodes increases, but there are no obvious relation between P R(G) and the maximum weight K. In Table 3, the maximum value of P R(G) is 1.222.
n K = 1 K = 5 K = 10 K = 20 K = 30 K = 40 K = 50 10 1.007 1.007 1.005 1.005 1.005 1.006 1.005 25 1.039 1.033 1.033 1.034 1.032 1.032 1.034 50 1.063 1.060 1.055 1.059 1.056 1.056 1.056 75 1.075 1.072 1.074 1.070 1.071 1.068 1.070 100 1.086 1.082 1.081 1.079 1.079 1.080 1.081
Table 2: The average value of P R(G).
n K = 1 K = 5 K = 10 K = 20 K = 30 K = 40 K = 50 10 1.167 1.111 1.138 1.145 1.135 1.213 1.153 25 1.222 1.175 1.138 1.146 1.149 1.126 1.166 50 1.176 1.169 1.181 1.136 1.172 1.153 1.158 75 1.160 1.167 1.188 1.195 1.159 1.185 1.187 100 1.182 1.208 1.186 1.179 1.186 1.166 1.182
Table 3: The maximum value of P R(G).
3.3
Randomly generated unit disk graphs with different density of nodes
In this subsection, we consider UDGs with different density of nodes. We randomly construct 500 connected UDGs with n nodes in a 100m × 100m area (for convenience, denote this area by A), where n is ranged from 100 to 500, with an increment of 50. The interference range of each node is assumed to be 25m. We consider four scenarios as follows.
(a) One of the four corner areas of A has more nodes. (See Figure 7(a).) (b) The center area of A has more nodes. (See Figure 7(b).)
(c) The area near one side of A has more nodes. (See Figure 7(c).)
(d) The area near the middle line of A has more nodes. (See Figure 7(d).)
Figure 7: An example of different density of nodes.
From our simulations, we observe that the selection of the root of the BFS will effect the performance of BFS-First-Fit. Choosing the root in the area with high density of nodes will make BFS-First-Fit have a better performance but the difference is not big; thus we omit the details of these simulation results.
Figure 8 shows the average P R(G) obtained by First-Fit and BFS-First-Fit. We can observe that the average P R(G) obtained by First-Fit and BFS-First-Fit are very close in (a) and (b). The difference between them decreases as the number of nodes increases
and this is because the chromatic number is very close to the clique number; so there is no big improvement. In (c) and (d), the improvement is larger.
點數 100 150 200 250 300 350 400 450 500 BFS-First-Fit1.00348 1.00448 1.0059 1.00806 1.00765 1.00658 1.00811 1.00578 1.00926 First-Fit 1.02221 1.01955 1.01636 1.01575 1.01341 1.01299 1.01369 1.01341 1.01149 點數 100 150 200 250 300 350 400 450 500 BFS-First-Fit1.02582 1.03034 1.03371 1.03558 1.03485 1.03858 1.03624 1.03573 1.03933 First-Fit 1.05676 1.05755 1.0567 1.05717 1.05428 1.05465 1.05569 1.05162 1.0529 點數 100 150 200 250 300 350 400 450 500 BFS-First-Fit1.04803 1.0639 1.06278 1.05541 1.06069 1.06125 1.06509 1.06256 1.06225 First-Fit 1.11726 1.13568 1.14825 1.15668 1.16768 1.16578 1.17231 1.1726 1.16571 點數 100 150 200 250 300 350 400 450 500 BFS-First-Fit1.07514 1.08232 1.07438 1.08505 1.09266 1.0907 1.08988 1.09582 1.09176 First-Fit 1.13881 1.16245 1.167390 1.18117 1.19182 1.1896 1.19282 1.19547 1.19387 200 400 600 1 2 3 4 5 6 7 8 9
number of nodes
點數 BFS-First-Fit First-Fit 0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 100 150 200 250 300 350 400 450 500(a)
av
er
ag
e
P
R
(G
)
BFS-First-Fit
First-Fit
1 1.01 1.02 1.03 1.04 1.05 1.06 100 150 200 250 300 350 400 450 500(b)
0.95 1 1.05 1.1 1.15 1.2 100 150 200 250 300 350 400 450 500(c)
1 1.05 1.1 1.15 1.2 1.25 100 150 200 250 300 350 400 450 500(d)
Figure 8: The average P R(G) in UDGs with different density of nodes.
3.4
Randomly generated unit disk graphs that allow the addition of
nodes
In real world, many networks have some nodes that do not exist initially but are added later. In this case, BFS-First-Fit can only be applied on initial nodes and when there are addition nodes, BFS-First-Fit must be restart for all nodes if we want to use it.
In this section, we simulate the UDGs which allows nodes to be added. We randomly construct 500 connected UDGs with initial n nodes in a 100m × 100m area, where n is 100 or 200. And each time we add n/4 nodes and totally we add the nodes for five times. The interference range of each node is assumed to be 25m. We use BFS-First-Fit to color
the initial n nodes and use First-Fit to color the added. We compare results of such a BFS-First-Fit plus First-Fit manner with results that only use First-Fit.
Figure 9 shows the average P R(G) of BFS-First-Fit and First-Fit. We observe that when the number of added nodes is more than 3n/4, the P R(G) of BFS-First-Fit and First-Fit become very close. Thus we suggest that when the wireless network allows the addition of nodes, BFS-First-Fit must be restarted if the number of added nodes is more than 3n/4. 點數 100 125 150 175 200 225 BFS-First-Fit 1.0806 1.13392 1.159 1.1672 1.1811 1.19357 First-Fit 1.16196 1.17217 1.1863 1.19048 1.20413 1.21309 點數 200 250 300 350 400 450 BFS-First-Fit1.11539 1.16879 1.18957 1.20362 1.21935 1.22946 First-Fit 1.22144 1.22943 1.23936 1.24641 1.25596 1.26421 1 1.05 1.1 1.15 1.2 1.25 100 125 150 175 200 225
number of nodes
av
er
ag
e
P
R
(G
)
BFS-First-Fit
First-Fit
1 1.05 1.1 1.15 1.2 1.25 1.3 200 250 300 350 400 450Figure 9: The average P R(G) in UDGs that allow the addition of nodes.
4
Concluding remarks
In this thesis, we propose UDGs with imp(G) > 1.2079 and we propose simulations to compare the difference between the clique number and the chromatic number obtained by our algorithm and by First-Fit; some different types of random UDGs have been considered. We find that in almost all cases, our algorithm can color the graph G with χ(G) < 1.2ω(G) colors. In the future, the theoretical bound of our algorithm will be considered.
References
[1] H. Breu and D. G. Kirkpatrick, “Unit disk graph recognition is NP-hard,” Technical Report 93-27, Department of Computer Science, University of British Columbia, 1993.
[2] G. Chartrand and L. Lensniak, Graph and Digraphs, Wadsworth, Monterey, CA, 1981.
[3] B. N. Clark, C. J. Colbourn, and D. S. Johnson, “Unit disk graphs,” Discrete Math-ematics, vol. 86, pp. 165-177, 1990.
[4] S. Gerke and C. McDiarmid, “Graph imperfection,” J. Combinatorial Theory, vol. 83, no. 1, pp. 58-78, Sept. 2001.
[5] A. Graf, M. Stumpf, and G. Weienfels, “On coloring unit disk graphs,” Algorithmica, vol. 20, no. 3, pp. 277-293, 1998.
[6] C. Lund and M. Yannakakis, “On the hardness of approximating minimization prob-lems,” in Proc. 25th Annual ACM Symposium on the Theory of Computing, pp. 286-293, New York, 1993.
[7] M. V. Marathe, H. Breu, H. B. Hunt III, S. S. Ravi, D. J. Rosenkrantz, “Simple heuristics for unit disk graphs,” Networks, vol. 25, pp. 59-68, 1995. (see also [8]). [8] M. V. Marathe, H. B. Hunt III, and S. S. Ravi, “Geometry based approximations
for intersection graphs,” in Proc. Fourth Canadian Conference on Computational Geometry, pp. 244-249, 1992.
[9] P. Mani and D. Petr, “Clique number vs. chromatic number in wireless interference graphs: simulation results,” IEEE Communications Letters, vol. 11, no. 7, pp. 592-594, 2007.
[10] R. Peeters, “On coloring j-unit sphere graphs,” FEW 512, Department of Economics, Tilburg University, 1991.
[11] D. B. West, Introduction to Graph Theory, 2nd ed. Prentice Hall, Upper Saddle River, NJ, 2001.
