國
立
交
通
大
學
應用數學系
碩
士
論
文
用於無線感測網路中達到較高覆蓋率的新適應定位演算法
New Adaptive Localization Algorithms That Achieve
Better Coverage for Wireless Sensor Networks
研 究 生：林紹鈞
指導教授：陳秋媛 教授
用於無線感測網路中達到較高覆蓋率的新適應定位演算法
New Adaptive Localization Algorithms That Achieve
Better Coverage for Wireless Sensor Networks
研 究 生：林紹鈞 Student：Shao-Chun Lin
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 ThesisSubmitted 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
用於無線感測網路中達到較高覆蓋率的新適應定位演算法
研究生：林紹鈞 指導老師：陳秋媛 教授
國 立 交 通 大 學
應 用 數 學 系
摘 要
許多無線感測網路的應用(例如:森林火災的偵測、環境監控、軍事應用、野 生動物的追蹤)都需要位置資訊來偵測並且記錄事件發生的地點。如何獲取 感測器的位置資訊變成一個重要的問題，並且通常被稱為定位問題。利用全 球定位系統(GPS)是一個可能的解決方案，但是並不實際。其原因在於一個 無線感測網路經常包含數千個感測器，以致於為每個感測器裝上 GPS 的價 格太過昂貴而難以實現這個解決方案。在文獻[3]中，黃等人研究了一個新的 最佳化問題，最小花費定位問題(minimum cost localization problem)，這個問 題透過利用最小數量的錨點來定位出所有感測器的位置。本篇論文的目的是 對於無線感測網路提出可調式演算法的概念。我們的演算法比文獻[3]的演算 法簡易，並且考慮了所有文獻[3]的演算法所考慮的情形；實驗數據也顯示我 們的演算法有比較好的覆蓋率。New Adaptive Localization Algorithms That Achieve
Better Coverage for Wireless Sensor Networks
Student: Shao-Chun Lin
Advisor: Chiuyuan Chen
Department of Applied Mathematics National Chiao Tung University
Abstract
Many applications in wireless sensor networks (such as forest fires detection, en-vironment monitoring, military, and wildlife tracking) require position information to detect and record events. How to obtain the position information of the sen-sors become an important problem and is usually called the localization problem. Using Global Positioning System (GPS) is a possible solution for localization but is impractical. The reason is that a wireless sensor network usually has thousands of sensors and it is too expensive to equip every sensor a GPS. In [3], Huang et al. studied a new optimization problem, minimum cost localization problem, which aims to localize all sensors by using the minimum number of anchors. The pur-pose of this thesis is to propur-pose adaptive algorithms for the localization problem. Our algorithms are simpler than the algorithms in [3] and cover all the cases in the algorithms in [3]; simulation results also show that our algorithms have better coverage.
Keywords: wireless sensor network, localization, algorithm, rigidity, globally rigid, trilat-eration, triangulation.
誌
謝
首先，碩士能夠畢業，我要感謝我的父母，若不是他們不停的支援我，做我 的後盾，我也不可能這麼快樂的念大學、研究所。接著非常感謝我的指導老師陳 秋媛老師，很感恩她這兩年來對我耐心的指導以及付出，才能夠讓我知道自己有 哪些地方需要改進，並且在這兩年不斷的訓練與成長。 另外，在碩士的兩年內，非常感謝修課的老師在我學習的期間教導我莫大幫 助的知識，包括了陳秋媛教授、翁志文教授、傅恆霖教授、符麥克教授、滕元翔 老師，除此之外，系辦的張慧珊姊姊、張麗君姊姊、陳盈吟姊姊也都在我需要幫 忙時，給了我必要的協助。 在研究的路上，常常感到許多挫折，除了老師們的支持之外，同學們也給了 不少的幫助，尤其是邱鈺傑學長、林武雄學長、袁智龍學長、王奕倫學長、呂竺 晏、趙彥丞、蕭禕廷、林立庭、徐志杰、王稟鈞、陳珈惠、吳惠敏、姚欣豪等同 學，都在我報告以及課堂學習中，給我非常有用的建議，並提醒我的缺失，若不 是他們的幫助，我也不可能順利的完成我的碩士論文；在他們的身上，也看到了 許多值得學習的優點，能夠和這些同學在同一個團體奮鬥，是我引以為榮的。
Contents
Abstract (in Chinese) i
Abstract (in English) ii
Contents v
Contents v
List of Figures vii
1 Introduction 1
2 Related works 4
2.1 Rigidity theory . . . 4
2.2 Coarse-grain localization . . . 7
2.3 Fine-grain localization . . . 8
3 The main result 13 3.1 Localization-Phase . . . 14 3.1.1 Trilateration . . . 15 3.1.2 Sweep2 . . . 15 3.1.3 Rigidity . . . 16 3.2 AnchorChoose-Phase . . . 17 4 Simulation results 20 4.1 The three parameters considered in our simulations . . . 21
4.2 Simulation results for the graph in [8] . . . 22
4.4 Simulation results for sparse graphs . . . 29 4.5 Simulation results for dense graphs . . . 33
List of Figures
1 (a) A initial-anchor set S for trilateration, which is not a 3-dominating set.
(b) A 3-dominating set for the graph in (a) requires at least four nodes. . . 3
2 Examples of a (a) non-rigid, (b) rigid, (c) redundantly rigid (also globally rigid) graph. (d) A rigid graph may have two localization solutions. . . 5
3 (a) A wireless sensor and actor network with two actors A and B. (b) A trained subnetwork. The source of this figure is [3]. . . 8
4 An illustration of the Euclidian method; this figure is from [11]. . . 11
5 A illustration of the Min-max method; this figure is from [11]. . . 13
6 Example of Rigid Algorithm. . . 18
7 (a) A subgraph of the original network. (b) A strategy without adaptivity: a node with the maximum degree is chosen as the next initial-anchor; here u is chosen. (c) The strategy with adaptivity: a node with the maximum (v.degree − v.ann) is chosen as the next initial-anchor; here v is chosen. (d) The contribution of v; those nodes colored black are localized. . . 19
8 The graph in [8]. . . 22
9 (a) COVERAGE and (b) CP of LocalTri + AdpativeChoose and Sweep2 + AdpativeChoose. . . 24
10 IAF for very sparse graphs. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase. . . 26
11 COVERAGE for very sparse graphs when the cardinality of an initial-anchor set is at most 100. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase. . . 27
12 COVERAGE for very sparse graphs when the cardinality of an initial-anchor set is at most 120. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase. . . 28
13 IAF for sparse graphs. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase. . . 30 14 COVERAGE for very sparse graphs when the cardinality of an
initial-anchor set is at most 30. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase. . . 31 15 COVERAGE for very sparse graphs when the cardinality of an
initial-anchor set is at most 50. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase. . . 32 16 IAF for dense graphs. (a) LocalTri is used in the Localization-Phase. (b)
Sweep2 is used in the Localization-Phase. . . 34 17 COVERAGE for very sparse graphs when the cardinality of an
initial-anchor set is at most 5. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase. . . 35 18 COVERAGE for very sparse graphs when the cardinality of an
initial-anchor set is at most 10. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase. . . 36 19 (a) LocalTri, (b) Sweep2 is used in the Localization-Phase. In this figure,
“≈” means the outcomes of the two methods differ in 2% and the better one will be shown first. . . 37
1
Introduction
Wireless sensor networks (WSNs) have many applications such as forest fires detection, environment monitoring, military, and wildlife tracking. In many applications, the sensors (nodes) need to know their positions in order to detect and record events. In this thesis, we use the terms sensor and node interchangeably. The process of computing the positions of the nodes is called localization. This thesis focus on the localization problem. Here we require a fine-grain location awareness although some researchers considered a coarse-grain location awareness; see [3].
It is not difficult to see that one way to solve the localization problem is to equip each sensor a GPS. Since a wireless sensor network usually has thousands of sensors, this approach is too expensive and therefore impractical. Novel approaches have been proposed to choose some sensors and equip them with GPS and the other sensors determine their positions by means of the Euclidean distances to their neighbors using different distance ranging methods (such as radio signal strength or time difference of arrival).
A wireless sensor network is usually modeled as a graph G = (V, E) and many often, a unit disk graph is used, meaning that each node has the same transmission range. Before going further, we introduce some terminologies. A graph G is generic if it has no three nodes collinear. A graph G is grounded if uv ∈ E implies that the distance between u, v can be measured or estimated via wireless communication. In this thesis, we assume the wireless sensor network forms a generic, grounded, unit disk graph; see also [8].
Two nodes are neighbors if they are adjacent. For convenience, let N (v) denote the set of neighbors of a node v and call N (v) the neighborhood of v. Many researchers call a node which is equipped with GPS an anchor or anchor node or beacon. In this thesis, however, for clarity, a node which is equipped with GPS is called an initial-anchor; a node which is not an initial-anchor and has determined its location is called an obtained-anchor. Both initial-anchors and obtained-anchors are called anchors. For each node v, only the
distances between u, v where u ∈ N (v) and the positions of anchors in N (v) can be used to determine the location of v. For convenience, let S denote the set of all initial-anchors and call S the initial-anchor set. The cardinality of S is denoted as |S|.
Trilateration is the most basic approach for localization and it has been used for thousands of years. This approach uses the known positions of anchors and the measured distance to the anchors. To determine the location of a node using trilateration alone, this node needs to hear from at least three anchors. That is, a sensor v computes the number of anchors in its neighborhood N (v) and if v has three anchors in N (v), then the location of v can be determined. In many localization algorithms, iterative trilateration (or multilateration) is used to localize nodes. In iterative trilateration, nodes measure distances to their neighbors and share their position information with their neighbors to collaboratively compute their positions. If a sensor whose position has been determined, it can act as a new anchor to localize other nodes by sharing its position with its neighbors. This iterative process continues until no nodes can be further localized.
How to find a good initial-anchor set S for a wireless sensor network is obviously an interesting problem. Do notice that finding an initial-anchor set for trilateration is not equivalent to finding a 3-dominating set. A 3-dominating set can serve as an initial-anchor set for trilateration but an initial-initial-anchor set for trilateration needs not to be a 3-dominating set. The following figure shows an example S with |S| = 3 and a 3-3-dominating set requires four nodes.
A graph G is localizable if the position of each node can be uniquely determined. A formulation of the localization problem is: given a graph G and an initial-anchor set S, determine if G is localizable by using S. We say S is feasible if the position of each node in G can be uniquely determined with S being the initial-anchor set. Some researchers do not ask for 100% of sensors being localized. In this situation, coverage is considered and is defined to be the percentage of sensors that can be localized.
(a) (b)
Figure 1: (a) A initial-anchor set S for trilateration, which is not a 3-dominating set. (b) A 3-dominating set for the graph in (a) requires at least four nodes.
Most localization algorithms focus on determining whether a wireless sensor network is localizable and/or how to localize as many nodes as possible with a “static” initial-anchor set; see [1, 5, 7, 13, 11, 12, 14, 15, 17]. Recently, in [8], Huang et al. studied a new optimization problem, minimum cost localization problem, which aims to localize all sensors using the minimum number of anchors given the distance measurements. The algorithm proposed in [8] is called sweep and it first applies trilateration; when no nodes can be further localized, it switches to consider pairs of sensors u, v such that u, v are neighbors and they can cooperate to obtain their positions. After no such u, v pairs can be found, the algorithm switches back to trilateration again, and so on. We will describe the algorithm sweep in detail in Sections 2 and 3.
This thesis considers the same problem as in [8], i.e., the minimum cost localization problem. The main contribution of this thesis is to propose an adaptive algorithm to choose an initial-anchor set S. For convenience, let S and S0 denote the initial-anchor sets generated by our algorithm and by the algorithm in [8], respectively. Simulation results show that:
1. Our algorithm always has a better coverage than the algorithm in [8].
2. For dense graphs, S always has a smaller cardinality than S0.
This thesis is organized as follows. In Section 2, we give an overview of related works. The main result is presented in Section 3. Simulations and comparisons are given in Section 4. Concluding remarks are given in the final section.
2
Related works
In this section, we review related works with respect to localization and this section consists of three subsections. In Subsection 2.1, we give results of “rigidity theory”. In Subsection 2.2, we review the results of coarse-grain localization. In Subsection 2.3, we briefly describe previous results of fine-grain localization.
2.1
Rigidity theory
In [6], Hendrickson proved that if G has a unique realization in Rd, then G is (d +
1)-connected and redundantly rigid (defined later). He also conjectured that every realization of a (d + 1)-connected and redundantly rigid graph in Rdis unique. Hendrickson’s conjec-ture is true for d = 1 but was disproved by Connelly for d ≥ 3 in [4]. In [9], Jackson and Jordsan resolved the remaining open case by showing that Hendrickson’s conjecture is true for d = 2. Jackson and Jordsan also deduced that every realization of a 6-connected graph as a two-dimensional generic framework is a unique realization.
A graph G is redundantly rigid in Rdif deleting any edge of G results in a graph which
is rigid in Rd. It is a long story to give a formal definition for “rigid”; therefore we will
adopt the following famous theorem, which was proposed by Laman.
Theorem 2.1. [2, 10] Laman’s Condition
A graph G = (V, L) with n vertices is rigid in R2 if and only if L contains a subset E consisting of 2n − 3 edges with the property that, for any nonempty subset E0 ⊂ E, the number of edges in E0 cannot exceed 2n0− 3, where n0 is the number of vertices of G which
Notice that in many literatures (for example [2, 9]), if a grounded graph has a unique localization solution then it will is called globally rigid. In [9], Jackson and Jordsan presented a method to determine if a graph has a unique localization solution.
Theorem 2.2. [2, 9] A grounded graph G is globally rigid in R2 if and only if either G
is a complete graph on at most three vertices or G is 3-connected and redundantly rigid.
The following theorem gives a sufficient condition for a graph to be globally rigid. Note that in [1], Ferruccio et al. tried to characterize a series of graphs which can be easily localized and they proved that a 6-connected graph can be localized by trilateration with three initial-anchors.
Theorem 2.3. [9] If a graph G is 6-connected, then G is globally rigid in R2.
Intuitively, if a graph is non-rigid, then it may have an infinite number of localization solutions. When a graph is rigid, it has a finite number of localization solutions. When a graph is globally rigid, it has a unique localization solution. We use Figure 2 to illustrate the definition of rigidity. The graph in Figure 2(a) is non-rigid, in Figure 2(b) is rigid since it satisfies Laman’s Condition, in Figure 2(c) is redundantly rigid since if we deleted any edge then the resultant graph is still rigid. Note that the graph in Figure 2(c) is also globally rigid since it is 3-connected and redundantly rigid.
(a) (b) (c) (d)
Figure 2: Examples of a (a) non-rigid, (b) rigid, (c) redundantly rigid (also globally rigid) graph. (d) A rigid graph may have two localization solutions.
From the above, the localization problem can be transformed into the problem of de-termining the globally rigidity of a graph, i.e., dede-termining whether a graph is 3-connected and redundantly rigid. It is not difficult to check if a graph is 3-connected. As for the redundantly rigidity, in [7], Hendrickson proposed a polynomial-time algorithm (for conve-nience, call this algorithm H-Algorithm) to determine the redundantly rigidity of a graph. Note that when H-Algorithm was proposed, it is not known that this algorithm could be used to solve the localization problem. Until Jackson and Jordsan proved Theorem 2.2 in [9], Connelly [5] combined the results of [7] and [9] (i.e., H-Algorithm and Theorem 2.2) and proposed an algorithm to determine if a graph is localizable. Connelly even re-proved the correctness of H-Algorithm. For convenience, we call Connelly’s method C-Algorithm. C-Algorithm is centralized and is a divide-and-conquer algorithm.
The rigidity theorem has two major drawbacks for the localization problem: (i) It characterizes the graphs which can be localized by exactly three anchors, but it does not characterize graphs which can be localized by four or more anchors. (ii) It provides a theoretical method (the answer is only “Yes” or “No”), not an algorithmic method, for the localization problem. Note that a wireless sensor network may not have a certain structure and its corresponding graph may not globally rigid. Thus it is very natural to ask: “How to localize a graph which is not globally rigid?” In [17], Yang et al. tried to use the wheel graph structure to solve the localization problem for a graph which is not necessarily globally rigid. They pointed out that if a wheel structure (in the given graph) contains three or more anchors, then this wheel structure is localizable since a wheel is 3-connected and redundantly rigid (hence is globally rigid). It is also natural to ask: “If there exist more than three anchors, then what kind of graphs can be localized?”. Several literatures have ever discussed the question.
2.2
Coarse-grain localization
In some applications, a wireless sensor and actor network (WSAN) is considered, which 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. After deployed, these tiny and low-cost sensors are unaware of their location and are unattended. Actors are mobile to collect the sensed data from sensors. 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.
In a WSAN, a coarse-grain location awareness is sufficient with a trade-off: that an coarse-grain location awareness 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 location. Wadaa et al. [16] 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. Recently, in [3], Barsi et al. proposed an energy-efficient training protocol. After training, each sensor in the actor-region will acquire two coordinates: the corona and the sector to which it belongs. See Figure 3 for an illustration. 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.
More precisely, a training protocol imposes a virtual coordinate system onto the WSAN 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,
(a) (b)
Figure 3: (a) A wireless sensor and actor network with two actors A and B. (b) A trained subnetwork. The source of this figure is [3].
2.3
Fine-grain localization
Different from the coarse-grain localization, a fine-grain localization determines the position of each sensors. This position may be related to the earth or a virtual coordinate system on the given network. In a localization algorithm, a sensor can acquire its position by two methods: (i) using GPS and (ii) by calculation (for example, using trilateration). We now give the details of trilateration. In this localization algorithm, a sensor (say, v) computes the number of anchors in N (v). If this number is ≥ 3, then v uses three of the anchors (say, a1, a2, a3) to compute its own position. First use the distances a1v
and a2v to construct two possible positions for v. Then use the distance a3v to determine
which of the two possible positions of v is correct. Notice that trilateration will fail if the three anchors a1, a2, a3 are collinear. Thus to use trilateration, the given graph has to be
a generic graph.
As mentioned in the previous section, in [8], Huang et al. considered the minimum cost localization problem. They proposed an algorithm called sweep. This algorithm first applies trilateration; when no nodes can be further localized, it switches to consider pairs
of sensors u, v such that u, v are neighbors and they can cooperate to obtain their positions. After no such u, v pairs can be found, the algorithm switches back to trilateration again, and so on. More precisely, when u, v satisfy the following three conditions, they can cooperate to fulfill localization: (i) u and v are neighbors (hence the distance between them is known), (ii) N (u) contains two anchors (say, a1 and a2) and N (v) also contains
two anchors (say, a3 and a4), and (iii) |{a1, a2} ∩ {a3, a4}| ≤ 1 (meaning that the anchors
used by u cannot be identical to those used by v). It is not difficult to see that if u, v satisfy (ii), then both u and v have two possible positions. If u, v do not satisfy (iii), then there are at least two possible pairs of positions and hence cannot be uniquely localized. If u, v satisfy (iii), then it is possible that only one pair of positions match the distance between u and v, which can be obtained if u, v satisfy (i).
Some of the algorithms for the localization problem are centralized, while the others are distributed. In centralized algorithms, some nodes have a more powerful computa-tion ability; hence these nodes can gather more informacomputa-tion of the given graph such as the maximum and the average degree of nodes. While in distributed algorithms, all sen-sors have a limited power and computation ability; hence sensen-sors can only gather the information within the vicinity.
In [11], Langendoen and Reijers compared three distributed algorithms that handle the localization problem when there are noises. One of the algorithms uses the bounding box approach to approximate the position of sensors. Another one uses the average distance approach to compute the distance between pairs of sensors, and then use this distance information to compute the position of sensors. The remaining one calculates the distance between anchors and sensors and uses the standard least square method to compute approximate positions; due to the the existence of noise, solutions may not exist. We briefly describe the difference between algorithms that assume noise-free and al-gorithms that allow the existence of noise. The first difference is that: for the latter
algorithms, the estimated distances may not be accurate due to the existence of noise. These algorithms have to use the distance information (which may not be accurate) to obtain positions which are closest to the actual positions. The second difference is that: for the former algorithms, the positions of anchors are only shared with one-hop neigh-bors, but for the latter algorithms, the positions of anchors are shared within two-hop or three-hops or n-hop neighbors for some n. In [13], an n-hop multilateration algorithm al-lowing the existence of noise was proposed and the positions of anchors are shared within n-hop neighbors. Note that there are various methods for estimating the distance between sensors and anchors within n-hop neighbors.
There are three representative algorithms that can localize a graph when there exist noises; see [14, 15, 12]. In [11], Langendoen and Reijers provided a comparison between these three algorithms. As was mentioned before, there are various methods for estimating the distance between sensors and anchors within n-hop neighbors. Each of the algorithms in [14, 15, 12] provides an interesting method to estimate the distance between sensors and anchors within n-hop neighbors. Before ending this subsection, we briefly describe the methods used in [14, 15, 12].
In [14], Savarese et al. proposed a method called DV-hop. In DV-hop, a sensor (say, v) computes the number k of hops between an anchor (say, a) and itself. Then, the distance between v and a is estimated by the average distance for one hop times k.
In [15], Savvides et al. proposed a method and it was called Sum-Dist in [11]. In Sum-Dist, anchors initiate a flood. More precisely, Sum-dist starts its process from the anchors, which send messages including the anchor id, anchor position, and a path length 0. Each sensor that receives the message adds its measured range to the path length. To avoid the flooding of messages, Savvides et al. limited the flooding range. At the end of the above process, most sensors will get several anchors’ positions and several estimated distances.
One drawback of DV-hop is that it fails for highly irregular network topologies, where the variance in actual hop distances is very large. In [12], Niculescu and Nath have proposed another method, called Euclidean, which is based on the local geometry of the nodes around an anchor. Again, anchors initiate a flood, but forwarding the distance is more complicated than [14, 15]. In particular, when a node has received messages from two neighbors that know their distance to the anchor, and to each other, it can calculate the distance to the anchor. We use Figure 4 to describe the method Euclidean. This figure shows a node (i.e., Self) that has two neighbors n1 and n2 with distance estimates (a and b) to an anchor (i.e., Anchor). Note that Self is not a neighbor of Anchor and currently does not have a distance estimate to Anchor. Self and Self’ are actually the two possible positions for the same node. The nodes n1, n2, and Self (or n1, n2, and Self’) form a K3 and therefore the distances c, d, and e are known. In this figure, r1 and r2
denote the distance estimates of Self to Anchor and Self’ to Anchor, respectively. The method Euclidean uses neighbor vote or common neighbor to determine which of r1 and r2 is correct.
Figure 4: An illustration of the Euclidian method; this figure is from [11].
position of each node. We now introduce two algorithms to calculate positions. The two algorithms are called Lateration and Min-max. In Lateration, each distance estimate contributes a constraint; using all the constraints, one can use the standard least square method to obtain an approximate position of a node.
In Min-max, a node (say, v) that needs to be localized will compute a region that it belongs to; this region is actually a rectangle and is called a bounding box. Figure 5 shows that v is localized by using (i) the positions of three anchors (called Anchor1, Anchor2, and Anchor3 in this figure) and (ii) the three estimated distances between Anchor1 and v, Anchor2 and v, and Anchor3 and v. For convenience, denote the three estimated distances as d1, d2, d3, respectively. Let the positions of Anchor1, Anchor2, and Anchor3
be (a1, b1), (a2, b2), and (a3, b3), respectively. Let (x, y) denote the position of v. Then x
lies in the intersection of the following three intervals:
[a1 − d1, a1+ d1], [a2 − d2, a2+ d2], [a3− d3, a3+ d3].
The values
minx = max{a1− d1, a2− d2, a3 − d3} and maxx = min{a1+ d1, a2+ d2, a3+ d3}
are calculated to bound the x-coordinate of v. Similarly, y lies in the intersection of the following three intervals:
[b1− d1, b1+ d1], [b2 − d2, b2+ d2], [b3− d3, b3 + d3].
The values
miny = max{b1− d1, b2− d2, b3 − d3} and maxy = min{b1+ d1, b2+ d2, b3+ d3}
are calculated to bound the y-coordinate of v. The above process also explains why this algorithm called Min-max. After calculating minx, maxx, miny, and maxy, the algorithm
chooses (minx+maxx
2 ,
miny+maxy
2 ) as the estimated position of v; this is the node labeled “est.
Figure 5: A illustration of the Min-max method; this figure is from [11].
3
The main result
This section gives the main result of this thesis. In particular, an algorithm called Main is proposed; see below. Throughout this section, v.ann denotes the number of anchors in N (v).
Algorithm Main uses the Boolean variable v.localized to indicate whether a node v has been localized. Initially, all nodes v have v.localized = false. Once v is localized, v.localized is set to true. Algorithm Main uses the following theorem to choose its initial-anchor set S; see lines 5-10 in Main.
Theorem 3.1. Let G = (V, E) be a graph that models a given wireless sensor network and S be any feasible initial-anchor set of G. For all v ∈ V with degree deg(v) ≤ 2, we have v ∈ S. In other words, any feasible initial-anchor set must contain all nodes with degree ≤ 2.
Proof. Suppose to the contrary that S is a feasible initial-anchor set and there is a node v 6∈ S such that deg(v) ≤ 2. Since S is feasible, v must be localized. However, if deg(v) = 1, then v has an infinite number of possible positions, which is a contradiction;
Algorithm 1 Main
Require: A wireless sensor network G = (V, E).
Ensure: A localization for G and the initial-anchor set S that is used in the localization.
1: S = ∅; 2: for each v ∈ V do 3: v.localized = false; 4: end for 5: for each v ∈ V do 6: if v.degree ≤ 2 then 7: S = S ∪ {v}; 8: v.localized = true; 9: end if 10: end for 11: run Localization-Phase;
12: while there exists any u such that (u.localized == false) do
13: v = the result of AnchorChoose-Phase;
14: S = S ∪ {v};
15: run Localization-Phase;
16: end while
if deg(v) = 2, then v has at least two possible positions, which is also a contradiction.
Then, Algorithm Main runs the Localization-Phase; see line 11 and Subsection 3.1. After the Localization-Phase, if there exits any node which is not localized, then Algorithm Main runs AnchorChoose-Phase to choose a node v as a new initial-anchor and add v to the initial-anchor set S; see lines 13-14 in Main. After v is added to S, Algorithm Main runs Localization-Phase again. Notice that we have proposed an adaptive algorithm for AnchorChoose-Phase and we will give the details in Subsection 3.2. The while-loop in lines 12-16 repeats until all nodes have been localized. When Algorithm Main stops, the initial-anchor set S is feasible.
3.1
Localization-Phase
Notice that the Localization-Phase can be implemented in many ways. In this thesis, we give three localization algorithms (called Trilateration, Sweep2, and Rigid) and each of them can be used in the Localization-Phase.
3.1.1 Trilateration
The pseudocode of Trilateration is as follows. Algorithm 2 Trilateration
Require: A wireless sensor network G = (V, E).
Ensure: G with those nodes of it that can be localized by trilateration.
1: for each v ∈ V such that (v.localized == false) do
2: compute v.ann;
3: end for
4: for each v ∈ V such that (v.localized == false) do
5: if v.ann ≥ 3 then
6: use three anchors in N (v) to determine the position of v; 7: v.localized = true;
8: run Trilateration;
9: end if 10: end for
3.1.2 Sweep2
We have mentioned the algorithm sweep that was proposed in [8] in Subsection2.3. In [8], algorithm sweep is implemented by either Greedy-Sweep-1 or Greedy-Sweep-2; thus there are two cases. Notice that Greedy-Sweep-2 actually contains all the statements in Greedy-Sweep-1. We find that it is unnecessary to have both Greedy-Sweep-1 and Greedy-Sweep-2. Therefore, in this thesis, we modify algorithm sweep and propose a new algorithm called Sweep2. Our algorithm Sweep2 is more general and it does not need to distinguish between the two cases in sweep. Similar to Trilateration, algorithm Sweep2 will compute the number of anchors in a node’s neighborhood. If there are two neighboring nodes u, v such that both u.annandv.ann equal to 2, then it is possible to localize u, v at once. Let the two possible positions for u be u1, u2 and the two possible positions for v
be v1, v2. Then algorithm Sweep2 can check if only one of the four distances u1v1, u1v2,
u2v1, and u2v2 equals to the distance uv. If this is true, then u, v can be localized at once.
Algorithm 3 Sweep2
1: for each v ∈ V such that (v.localized == false) do
2: Compute v.ann;
3: end for
4: run Trilateration;
5: for each v ∈ V such that (v.localized == false) do
6: if v.ann == 2 then
7: for each u ∈ N (v) do
8: if (u.ann == 2) and (u.localized == false) then
9: compute the two possible positions for u and denote them as u1, u2; 10: compute the two possible positions for v and denote them as v1, v2; 11: compute the distances u1v1, u1v2, u2v1, u2v2, and uv;
12: if ∃! distance (say, uivj) in {u1v1, u1v2, u2v1, u2v2} equals to uv then 13: use the distances uivj and uv to determine the positions of u, v; 14: u.localized = v.localized = true;
15: run Sweep2; 16: end if 17: end if 18: end for 19: end if 20: end for
In Sweep2, two nodes u, v each has only two anchors in their neighborhood but u, v try to cooperate to obtain their positions. The idea of Sweep2 can be generalized to obtain Sweep3, Sweep4, . . ., Sweepk. In general, Sweepk means k nodes try to cooperate to obtain their positions.
3.1.3 Rigidity
In this subsubsection, we propose a localization algorithm based on the rigidity theory; call this algorithm Rigidity. When algorithm Main runs, all nodes with degree ≤ 2 will be added into the initial-anchor set. Suppose Main uses Rigidity in its Localization-Phase. Then, each initial-anchor broadcasts a “Hello” message to all of its neighbors. Each node v (including the anchors) that receives a “Hello” message run the algorithm Trilateration at first and then uses C-Algorithm to find a rigid sub-graph H in N [v] if there exists a such graph. Algorithm Rigidity computes the position of a node by a method similar to
the one used in Sweepk. This solution will be unique since the subgraph H is globally rigid. The pseudocode of algorithm Rigidity is as follows.
Algorithm 4 Rigid
1: for each v ∈ V do
2: if v.localized == true then
3: v broadcasts a “Hello” message to all of its neighbors;
4: end if
5: end for
6: for each v ∈ V which receives the “Hello” message do
7: run Trilateration;
8: use C-Algorithm to find a globally rigid graph in N [v], containing at least non-anchor node, say this graph as H;
9: if there exists three anchors in H then
10: use the Sweepk to localize each u ∈ H;
11: for each u ∈ H and u.localized == false do
12: u.localized = true;
13: broadcast a “Hello” message to all of its neighbors;
14: end for
15: end if
16: end for
Figure 6 shows how Rigid algorithm works. First, u finds there are three anchors in N (u) and uses Trilateration to localize itself (see Figure 6(a) and (b)). Then, u will determine whether N [u] contains a globally rigid graph with at least one non-anchor node. Rigid finds there exists a wheel graph which is globally rigid in N [v] and computes the position of each node. Finally, all nodes of the wheel graph are localized (see Figure 6(c)).
3.2
AnchorChoose-Phase
In [8], Huang et al. also proposed a method to choose initial-anchors; for convenience, we call it HuangChoose. In this subsection, we will propose an adaptive algorithm for choosing anchors. We first give our observations and motivations. After the execution of line 11 in algorithm Main, there may exist nodes that have not been not localized yet. In order to localize all of the nodes, more initial-anchors are needed. How to choose a node which is not localized as an initial-anchor is obviously an important issue.
(a) (b)
(c)
Figure 6: Example of Rigid Algorithm.
Intuitively, among the non-localized nodes, a node v with the maximum degree should be chosen since v makes the largest number of nodes have one more anchor in their neighborhoods. However, simulation results show that nodes that have large degrees usually locate in some regions of the given graph. Therefore, it is very likely that the chosen anchors are from the same region of the graph and could not help with localizing nodes in the other regions. To overcome this difficulty, we will use an adaptive algorithm to choose initial-anchors. Let v.degree denote the degree of a node v. More precisely, instead of using v.degree, algorithm AdpativeChoose will use (v.degree − v.ann). For instance, if v.degree is 8 and v.ann is 3, then algorithm AdpativeChoose will treat v.degree as 5. A node v with the maximum (v.degree − v.ann) will be chosen as an initial-anchor. The pseudocode is as follows.
Figure 7 shows the performance of algorithm AdpativeChoose. In Figure 7(a), nodes colored with black are anchors and the others are non-anchors and therefore non-localized. As shown in Figure 7(b), a tradition algorithm will choose u as the next initial-anchor since u has the maximum degree. However, as shown in Figure 7(c), algorithm AdpativeChoose
Algorithm 5 AdpativeChoose
1: for each v ∈ V such that (v.localized == false) do
2: compute the degree v.degree;
3: v.degree = v.degree − v.ann;
4: end for
5: find v with the maximum v.degree which is not localized;
6: return v;
will choose v as next initial-anchor since v has the maximum (v.degree − v.ann). It is not difficult to check that v is a better choice than u; see Figure 7(d).
Algorithm AdpativeChoose is a greedy strategy. It is obvious that there are other greedy strategies; for example, HuangChoose is another greedy strategy. How to choose the most effective initial-anchor is still an open problem and is our future work. We will compare AdpativeChoose with HuangChoose later; see Section 4.
(a) (b)
(c) (d)
Figure 7: (a) A subgraph of the original network. (b) A strategy without adaptivity: a node with the maximum degree is chosen as the next initial-anchor; here u is chosen. (c) The strategy with adaptivity: a node with the maximum (v.degree − v.ann) is chosen as the next initial-anchor; here v is chosen. (d) The contribution of v; those nodes colored black are localized.
4
Simulation results
In this section, we provide our simulation results. For convenience, in the remaining figures, LocalTri means Trilateration. According to our simulations, the performance of Sweep3 is almost the same as Sweep2. However, the computation of Sweep3 is much more complicated than that of Sweep2. Therefore, we will not provide the results of Sweep3, . . ., Sweepk. In our simulations, we will compare the following six combinations of Localization-Phase + AnchorChoose-Phase:
LocalTri + AdpativeChoose, LocalTri + HuangChoose, LocalTri + MaxDegree,
Sweep2 + AdpativeChoose, Sweep2 + HuangChoose, and Sweep2 + MaxDegree.
Notice that MaxDegree will choose a node v such that v.localized = false and deg(v) is the maximum among all such nodes.
This section is divided into six subsections and is organized as follows. In Subsec-tion 4.1, we introduce three parameters (IAF, COVERAGE, and CP ) for evaluating the performance of the four combinations of Localization-Phase + AnchorChoose-Phase; our comparisons are based on the values of the three parameters (IAF, COVERAGE, and CP ). In Subsection 4.2, we compare the four combinations with respect to the graph in [8]. In Subsection 4.3, we compare the six combinations with respect to very sparse graphs. In Subsection 4.4, we compare the six combinations with respect to sparse graphs. In Subsection 4.5, we compare the six combinations with respect to dense graphs.
Before going further, we describe the four simulation environments that we will use: the graph in [8], very sparse graphs, sparse graphs, and dense graphs. Note that all the graphs used in our simulations are unit disk graphs. The first simulation environment is the graph in [8], which is a graph with 199 nodes randomly generated in a 1200m × 1000m rectangle region and the transmitting range of each node is set to 80m. The second simulation environment contains 200 very sparse graphs: each graph is with 200 nodes
randomly generated in a 1200m × 1000m rectangle region and the transmitting range of each node is set to 70m. The third simulation environment contains 200 sparse graphs: each graph is with 200 nodes randomly generated in a 800m × 600m rectangle region and the transmitting range of each node is set to 70m. The fourth simulation environment contains 200 dense graphs: each graph is with 200 nodes randomly generated in a 800m × 600m rectangle region and the transmitting range of each node is set to 100m.
4.1
The three parameters considered in our simulations
In this subsection, we introduce three parameters (IAF, COVERAGE, and CP) for evaluating the performance. In the remaining part of this thesis, we will use Anchor to denote the set of nodes that know their positions, that is, Anchor contains initial-anchors and those nodes that have been localized. Let G be the given graph, n be the number of nodes in G, and A be a combination of Localization-Phase + AnchorChoose-Phase.
The first parameter IAF is the cardinality of an initial-anchor set, i.e.,
IAFA(G) = |S|
where S is an initial-anchor set. The second parameter COVERAGE is the percentage of nodes that know their positions, i.e.,
COVERAGEA(G) =
|Anchor | n .
It is obvious that if an initial-anchor set is feasible, then COVERAGE will achieve 100%. When a localization algorithm is running, nodes can be added into the initial-anchor set; therefore we use Sc to denote the current initial-anchor set. The third parameter CP is
used to show the contribution of the current initial-anchor set and we define
CPA(G) =
|Anchor | |Sc|
.
4.2
Simulation results for the graph in [8]
The graph in [8] is shown in Figure 8.
Figure 8: The graph in [8].
By [8],
IAFLocalTri +HuangChoose(G) = 42 and IAFSweep2 +HuangChoose(G) = 33.
If our adaptive algorithm AdpativeChoose is used to choose anchors, then
IAFLocalT ri+AdaptiveChoose(G) = 40 and IAFSweep2+AdaptiveChoose(G) = 31.
To sum up,
40 = IAFLocalT ri+AdaptiveChoose(G) < IAFLocalTri +HuangChoose(G) = 42
and
31 = IAFSweep2+AdaptiveChoose(G) < IAFSweep2 +HuangChoose(G) = 33.
Thus AdaptiveChoose has a better (i.e., smaller) IAF than HuangChoose.
Figure 9(a) shows COVERAGE of LocalTri + AdpativeChoose and Sweep2 + Adpative-Choose. It can be seen that when the cardinality of an initial-anchor set is 15, the curve of
Sweep2 + AdpativeChoose suddenly rises up and Sweep2 + AdpativeChoose outperforms LocalTri + AdpativeChoose.
Figure 9(b) shows the CP of LocalTri + AdpativeChoose and Sweep2 + AdpativeChoose. Notice that a local maximum CP occurs when the curves in Figure 9(a) has a local maximum. CP decreases after some initial-anchors are used, indicating that the remaining un-localized nodes may have certain structures which are difficult to be localized.
(a)
(b)
Figure 9: (a) COVERAGE and (b) CP of LocalTri + AdpativeChoose and Sweep2 + AdpativeChoose.
4.3
Simulation results for very sparse graphs
We first compare IAF. Figure 10 shows that when IAF is considered, AdaptiveChoose performs almost equal to HuangChoose and in only very few cases, AdaptiveChoose per-forms worse than HuangChoose. In fact, the average IAF of LocalTri + AdpativeChoose is 134.63, LocalTri + MaxDegree is 156 and that of LocalTri + HuangChoose is 131.12. Moreover, the average IAF of Sweep2 + AdpativeChoose is 122.47, LocalTri + MaxDegree is 147 and that of Sweep2 + HuangChoose is 119.56.
To consider COVERAGE, we restrict the cardinality of an initial-anchor set to be at most 100 or at most 120. Figure 11 shows the COVERAGE value when the cardinal-ity of an initial-anchor set is restricted to at most 100. If shows that when LocalTri is used, AdaptiveChoose performs almost equal to HuangChoose and in only very few cases, AdaptiveChoose performs better than HuangChoose; however, when Sweep2 is used, Adap-tiveChoose performs almost equal to HuangChoose and in only very few cases, Adaptive-Choose performs worse than HuangAdaptive-Choose. In fact, the average COVERAGE of LocalTri + AdpativeChoose is 57.80%, LocalTri + MaxDegree is 57.80% and that of LocalTri + HuangChoose is 57.73%. Moreover, the average COVERAGE of Sweep2 + Adpative-Choose is 68.06%, Sweep2 + MaxDegree is 68.10% and that of Sweep2 + HuangAdpative-Choose is 65.64%.
Figure 12 shows the COVERAGE value when the cardinality of an initial-anchor set is restricted to at most 120. It shows that when LocalTri is used, AdaptiveChoose performs almost equal to HuangChoose and in many cases, AdaptiveChoose performs better than HuangChoose; however, when Sweep2 is used, AdaptiveChoose performs almost equal to HuangChoose and in only very few cases, AdaptiveChoose performs worse than Huang-Choose. In fact, the average COVERAGE of LocalTri + AdpativeChoose is 57.87%, Lo-calTri + MaxDegree is 57.87% and that of LoLo-calTri + HuangChoose is 57.92%. Moreover, the average COVERAGE of Sweep2 + AdpativeChoose is 68.06%, Sweep2 + MaxDegree
is 68.10% and that of Sweep2 + HuangChoose is 65.64%.
To sum up, for very sparse graphs, we find that AdaptiveChoose performs almost equal to or performs better than HuangChoose when LocalTri is used in the Localization-Phase and AdaptiveChoose performs almost equal to HuangChoose when Sweep2 is used in the Localization-Phase.
(a)
(b)
Figure 10: IAF for very sparse graphs. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase.
(a)
(b)
Figure 11: COVERAGE for very sparse graphs when the cardinality of an initial-anchor set is at most 100. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase.
(a)
(b)
Figure 12: COVERAGE for very sparse graphs when the cardinality of an initial-anchor set is at most 120. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase.
4.4
Simulation results for sparse graphs
We first compare IAF. Figure 13 shows that when IAF is considered, AdaptiveChoose performs almost equal to HuangChoose. In fact, the average IAF of LocalTri + Adpative-Choose is 61.585, LocalTri + MaxDegree is 61.625 and that of LocalTri + HuangAdpative-Choose is 62.26. Moreover, the average IAF of Sweep2 + AdpativeChoose is 50.36, Sweep2 + MaxDegree is 51.005 and that of Sweep2 + HuangChoose is 49.855.
To consider COVERAGE, we restrict the cardinality of an initial-anchor set to be at most 30 or at most 50. Figure 14 shows the COVERAGE value when the cardinality of an initial-anchor set is restricted to at most 30. If shows that no matter which one of LocalTri and Sweep2 is used, AdaptiveChoose performs better than HuangChoose in most cases. In fact, the average COVERAGE of LocalTri + AdpativeChoose is 48.66%, LocalTri + MaxDegree is 48.75% and that of LocalTri + HuangChoose is 24.26%. Moreover, the average COVERAGE of Sweep2 + AdpativeChoose is 60.23%, Sweep2 + MaxDegree is 60.47% and that of Sweep2 + HuangChoose is 49.69%.
Figure 15 shows the COVERAGE value when the cardinality of an initial-anchor set is restricted to at most 50. It shows that when LocalTri is used, AdaptiveChoose performs better than HuangChoose in most cases; however, when Sweep2 is used, AdaptiveChoose performs almost equal to HuangChoose. In fact, the average COVERAGE of LocalTri + AdpativeChoose is 88.58%, LocalTri + MaxDegree is 88.70% and that of LocalTri + HuangChoose is 76.64%. Moreover, the average COVERAGE of Sweep2 + AdpativeChoose is 97.78%, Sweep2 + MaxDegree is 97.58% and that of Sweep2 + HuangChoose is 97.08%. To sum up, for sparse graphs and when IAF is considered, AdaptiveChoose performs better than HuangChoose when LocalTri is used in the Localization-Phase and Adaptive-Choose performs worse than HuangAdaptive-Choose when Sweep2 is used in the Localization-Phase. For sparse graphs and when COVERAGE is considered, no matter which one of LocalTri and Sweep2 is used, AdaptiveChoose performs better than HuangChoose in most cases.
(a)
(b)
Figure 13: IAF for sparse graphs. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase.
(a)
(b)
Figure 14: COVERAGE for very sparse graphs when the cardinality of an initial-anchor set is at most 30. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase.
(a)
(b)
Figure 15: COVERAGE for very sparse graphs when the cardinality of an initial-anchor set is at most 50. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase.
4.5
Simulation results for dense graphs
We first compare IAF. Figure 16 shows that when IAF is considered, AdaptiveChoose performs better than HuangChoose. In fact, the average IAF of LocalTri + AdpativeChoose is 6.915, LocalTri + MaxDegree is 6.875 and that of LocalTri + HuangChoose is 11.72. Moreover, the average IAF of Sweep2 + AdpativeChoose is 5.81, Sweep2 + MaxDegree is 5.795 and that of Sweep2 + HuangChoose is 6.79.
To consider COVERAGE, we restrict the cardinality of an initial-anchor set to be at most 5 or at most 10. Figure 17 shows the COVERAGE value when the cardinality of an initial-anchor set is restricted to at most 5. If shows that no matter which one of LocalTri and Sweep2 is used, AdaptiveChoose performs better than HuangChoose in most cases. In fact, the average COVERAGE of LocalTri + AdpativeChoose is 80.49%, LocalTri + MaxDegree is 82.09% and that of LocalTri + HuangChoose is 12.19%. Moreover, the average COVERAGE of Sweep2 + AdpativeChoose is 85.50%, Sweep2 + MaxDegree is 87.31% and that of Sweep2 + HuangChoose is 68.59%.
Figure 18 shows the COVERAGE value when the cardinality of an initial-anchor set is restricted to at most 10. It shows that no matter which one of LocalTri and Sweep2 is used, AdaptiveChoose performs better than HuangChoose in most cases. In fact, the average COVERAGE of LocalTri + AdpativeChoose is 99.92%, LocalTri + MaxDegree is 99.88% and that of LocalTri + HuangChoose is 70.36%. Moreover, the average COVERAGE of Sweep2 + AdpativeChoose is 99.99%, Sweep2 + MaxDegree is 99.97% and that of Sweep2 + HuangChoose is 99.69%.
To sum up, for dense graphs and when IAF is considered, no matter which one of LocalTri and Sweep2 is used, AdaptiveChoose performs better than HuangChoose. For dense graphs and when COVERAGE is considered, we also have the same result.
(a)
(b)
Figure 16: IAF for dense graphs. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase.
(a)
(b)
Figure 17: COVERAGE for very sparse graphs when the cardinality of an initial-anchor set is at most 5. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase.
(a)
(b)
Figure 18: COVERAGE for very sparse graphs when the cardinality of an initial-anchor set is at most 10. (a) LocalTri is used in the Localization-Phase. (b) Sweep2 is used in the Localization-Phase.
5
Concluding remarks
Many applications in wireless sensor networks require position information to detect and record events. In [8], Huang et al. studied the minimum cost localization problem, which aims to localize all sensors by using the minimum number of anchors. The purpose of this thesis is consider the same problem but using an adaptive algorithm. In particular, we propose an adaptive algorithm called AdaptiveChoose to choose initial-anchors. Our algorithms are simpler than the algorithms in [3] and cover all the cases in the algorithms in [8]; Simulation results show that AdaptiveChoose usually outperforms HuangChoose for dense graphs and AdaptiveChoose performs almost the same as HuangChoose for sparse or very sparse graphs. The overview of the simulation is shown in Figure 19. The first column is the simulation environment. The first row shows the two parameters that we consider. The algorithm with better result will be shown in the corresponding fields.
(a) (b)
Figure 19: (a) LocalTri, (b) Sweep2 is used in the Localization-Phase. In this figure, “≈” means the outcomes of the two methods differ in 2% and the better one will be shown first.
In the coming future, we would like to consider the followings problems. How to combine AdpativeChoose and HuangChoose to obtain better results? Given a certain initial-anchor set, determine what kind of graphs are localizable. It is known that a globally rigid graph can be localized by using three anchors; what will happen if more anchors are used? Design a distributed version of AdaptiveChoose. We have tried to
generalize algorithm Sweep2 to be Sweepk. In Sweep2, two nodes try to cooperate to obtain their positions; in general, in Sweepk, k nodes try to cooperate to obtain their positions. Although we found that Sweep3 does not improve Sweep2, we conjecture that Sweepk does improve Sweep2 whenever k ≥ 4. Finally, it is interesting to generalize the algorithm Rigid to Rigidk, meaning that instead of using one-hop neighborhoods, k-hop neighborhoods are used.
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