New Adaptive Localization Algorithms That Achieve Better Coverage for Wireless Sensor Networks Advisor: Chiuyuan Chen Student: Shao-Chun Lin Department of Applied Mathematics National Chiao Tung University 2013/8/11

Introduction

Sensor能夠彼此間在傳送半徑內傳送訊息 藉由接力的方式將訊息傳到一個主控台 使用者能夠獲得這些資訊來判斷到底這個區域發生什麼事情 圖片來源:http://embedsoftdev.com/embedded/wireless-sensor-network-wsn/

transmission range (𝑅) Node : Sensor Disk radius: transmission range (𝑅) 𝑅

Unit Disk Graph Node : Sensor Disk radius: Transmission range (𝑅)

Applications of wireless sensor networks (WSNs) Wildlife tracking, military, forest fire detection, temperature detection, environment monitoring Why localization? To detect and record events. When tracking objects, the position information is important. …

Related works and Main Results

Definitions initial-anchor a node equipped with GPS initial-anchor set (𝑺) the set containing all initial-anchors anchor a node knows its position. feasible The initial-anchor set 𝑆 is called feasible if the position of each node in the given graph can be determined with 𝑆. 為了要解答這個問題，我們需要下面幾個定義 以同一個圖來說，不同的定位演算法會有不同的initial anchor set

Informations can be used to localize 𝑣 For each node 𝑣, the distances between 𝑢,𝑣 where 𝑢∈𝑁(𝑣) The positions of anchors in 𝑁 𝑣

*Fine-grain Localization Coarse-grain Localization Localization types *Fine-grain Localization Coarse-grain Localization Resitrict 𝑆 =3 [Rigid Theory] Consider noise [11~14, 2001~] *Find a feasible 𝑆 with |𝑆| as small as possible [8, 2011 Huang] Best Coverage [11~14, 2001~] 圖片來源:Efficient Location Training Protocols for Heterogeneous Sensor and Actor Networks

*Fine-grain Localization Coarse-grain Localization Localization types *Fine-grain Localization Coarse-grain Localization Resitrict 𝑆 =3 [Rigid Theory] Consider noise [11~14, 2001~] *Find a feasible 𝑆 with |𝑆| as small as possible [8, 2011 Huang] Best Coverage [11~14, 2001~] 圖片來源:Efficient Location Training Protocols for Heterogeneous Sensor and Actor Networks

Rigidity Theory non-rigid : localization solution is infinite. rigid : localization solution is finite. globally rigid : localization solution is unique.

Non-rigid Infinite Initial-anchor Unknown

Rigid graph Finite

Globally rigid graph Unique

Characterize globally rigid graph A graph which exists 3 anchors has unique localization solution if and only if the graph is globally rigid. redundantly rigid: After one edge is deleted, the remaining graph is a rigid graph. Laman’s Condition ([2], 1970 Laman) A graph 𝐺=(𝑉,𝐿) with 𝑛 vertices is rigid in 𝑅 2 if and only if 𝐿 contains a subset 𝐸 consisting of 𝟐𝒏 − 𝟑 edges with the property that, for any nonempty subset 𝐸′⊂ 𝐸, the number of edges in 𝐸′ cannot exceed 𝟐𝒏′ − 𝟑, where 𝑛’ is the number of vertices of 𝐺 which are endpoints of edges in 𝐸′.

Characterize globally rigid graph 1982, Lovasz and Yemini shows 6-connected graph is redundantly rigid. [7] 1992, Hendrickson proposed a polynomial-time algorithm to determine the redundantly rigidity of a graph. Hendrickson’s Conjecture A graph is called globally rigid if and only if the graph is 3-connected and redundantly rigid. [9] 2005, Jackson et al. proved that Hendrickson’s Conjecture is true. [5] 2005, Connelly mentioned that there is an algorithm to determine if a graph is globally rigid (i.e. localizable) in polynomial-time. C-algorithm.

Characterize globally rigid graph C-algorithm cannot compute position. 2006, Aspnes shows that to compute position in globally rigid with 3 anchors is NP-hard

*Fine-grain Localization Coarse-grain Localization Localization types *Fine-grain Localization Coarse-grain Localization Resitrict 𝑆 =3 [Rigid Theory] Consider noise [11~14, 2001~] *Find a feasible 𝑆 with |𝑆| as small as possible [8, 2011 Huang] Best Coverage [11~14, 2001~] 圖片來源:Efficient Location Training Protocols for Heterogeneous Sensor and Actor Networks

Choose node to become initial-anchor Grounded, generic, UDG A graph G Choose node to become initial-anchor Nodes with degree ≤2 Trilateration *Tri + Sweep2 Localization-Phase AnchorChoose-Phase *Tri + Rigid Check if all nodes are localized No Yes HuangChoose[2011] Output a feasible initial-anchor set 𝑆 *MaxDegree Choose *AdaptiveChoose

Choose node to become initial-anchor Grounded, generic, UDG A graph G Choose node to become initial-anchor Nodes with degree ≤2 Trilateration *Tri + Sweep2 Localization-Phase AnchorChoose-Phase *Tri + Rigid Check if all nodes are localized No Yes HuangChoose[2011] Output a feasible initial-anchor set 𝑆 *MaxDegree Choose *AdaptiveChoose

The graph we considered in this thesis Unit Disk Graph grounded ([2], 2005 Aspnes et al.) A graph 𝐺=(𝑉,𝐸) is grounded if 𝑢𝑣∈𝐸 implies that the distance 𝑢𝑣 can be measured or estimated via wireless communication. generic A graph is called generic if node coordinates are algebraically independent over rationals.

Choose node to become initial-anchor A graph G Choose node to become initial-anchor Nodes with degree ≤2 Localization-Phase AnchorChoose-Phase Check if all nodes are localized No Yes Output a feasible initial-anchor set 𝑆

Theorem: Let 𝑆 be any feasible initial-anchor set of 𝐺. For all 𝑣∈ 𝑉 with degree deg 𝑣 ≤2, we have 𝑣∈𝑆.

Choose node to become initial-anchor A graph G Choose node to become initial-anchor Trilateration *Tri + Sweep2 Localization-Phase AnchorChoose-Phase *Tri + Rigid Check if all nodes are localized No Yes Output a feasible initial-anchor set 𝑆

Localization-Phase Trilateration Sweep2+Tri Rigid+Tri anchor unknown initial-anchor

Trilateration anchor unknown initial-anchor

Trilateration anchor unknown initial-anchor

Sweep2(+Tri) 𝑢 𝑣

Sweep2(+Tri) 𝑢 𝑣

Sweep2 2006 Goldenberg first propose this idea, and called this as sweep. [8] 2011, Huang modified it to 2 neighbors version by two cases. In 2013, this thesis simplifies it and achieves the same performance, called this algorithm as Sweep2.

Rigidity Theory A point formation is a set of node positions. A point formation is called a localization solution of a grounded graph 𝐺=(𝑉,𝐸) if the following condition holds: for each 𝑢𝑣∈𝐸, distance between 𝑢, 𝑣 calculated from the their positions equals the distance 𝑢𝑣 in grounded graph 𝐺. A framework is that for each node v, v has its position. Framework:每個節點的位置 A framework is called as localization solution if the following condition holds;for each pair nodes u,v in G, if uv in G then uv distance satisfy the distance in grounded graph.

Rigid(+Tri) Localized subgraph Subgraph 𝑢

Rigid(+Tri) Localized subgraph Subgraph 𝑢

Rigid(Tri) Localized subgraph Subgraph 𝑢

Choose node to become initial-anchor A graph G Choose node to become initial-anchor Localization-Phase AnchorChoose-Phase Check if all nodes are localized No Yes HuangChoose[2011] Output a feasible initial-anchor set 𝑆 *MaxDegree Choose *AdaptiveChoose

AnchorChoose-Phase 𝒗.ann : # of anchors in 𝑁(𝑣) MaxDegreeChoose (a straightforward approach) HuangChoose ([8] 2011, Huang et al.) 𝒗.𝒓𝒂𝒏𝒌𝟎=# of 𝑢∈𝑁 𝑣 with 𝑢.ann =0 𝒗.𝒓𝒂𝒏𝒌𝟏=# of 𝑢∈𝑁 𝑣 with 𝑢.ann=1 𝒗.𝒓𝒂𝒏𝒌𝟐=# of 𝑢∈𝑁 𝑣 with 𝑢.ann=2 Choose 𝒗 with maximum 𝒗.𝒓𝒂𝒏𝒌𝟐 -> 𝒗.𝒓𝒂𝒏𝒌1 -> 𝒗.𝒓𝒂𝒏𝒌0 AdaptiveChoose (This thesis) Choose 𝒗 with maximum 𝒅𝒆𝒈 𝒗 −𝒗.ann

Choose node to become initial-anchor A graph G Choose node to become initial-anchor Localization-Phase AnchorChoose-Phase Check if all nodes are localized No Yes Output a feasible initial-anchor set 𝑆

𝑆 ≤𝑘 A graph G Choose node to become initial-anchor Localization-Phase AnchorChoose-Phase Check if all nodes are localized or 𝑆 ≥𝑘 No Yes 𝑨𝒏𝒄𝒉𝒐𝒓𝒔: The set of nodes that know their positions Output an initial-anchor set 𝑆 and |𝐴𝑛𝑐ℎ𝑜𝑟𝑠|

Simulation

Simulation Localization-Phase AnchorChoose-Phase Trilateration (LocalTri) Sweep2 AnchorChoose-Phase HuangChoose ([8] 2005, Huang et al.) AdaptiveChoose MaxDegreeChoose (MaxDegree)

Simulations Notation IAF: cardinality of an initial-anchor set 𝐴: Algorithm 𝐴𝑛𝑐ℎ𝑜𝑟𝑠: The set of nodes that know their positions 𝑆: initial-anchor set 𝑛: # of nodes IAF: cardinality of an initial-anchor set 𝐼𝐴 𝐹 𝐴 𝐺 =|𝑆| COVERAGE: the percentage of nodes that know their positions, 𝐶𝑂𝑉𝐸𝑅𝐴𝐺𝐸 𝐴 𝐺 = |𝐴𝑛𝑐ℎ𝑜𝑟𝑠| 𝑛

Sparse* 200 nodes in 800m × 600m R=70, k=30 and 50 5~7 2.89% Average Degree 𝐸 |𝑉| 2 G 200 nodes in 1200m × 1000m R=80 6.56 3.30% Vary Sparse* 200 nodes in 1200m × 1000m R=70, k=100 and 120 2~3 1.19% Sparse* 200 nodes in 800m × 600m R=70, k=30 and 50 5~7 2.89% Dense* 200 nodes in 800m × 600m R=100, k=5 and 10 10~14 5.71% *200 graphs for each

G 圖片來源:Minimum cost localization problem in wireless sensor networks

*𝐼𝐴 𝐹 𝑆𝑤𝑒𝑒𝑝2+𝐻𝑢𝑛𝑎𝑔𝐻ℎ𝑜𝑜𝑠𝑒 𝐺 =33 𝐼𝐴 𝐹 𝑆𝑤𝑒𝑒𝑝2+𝐴𝑑𝑎𝑝𝑡𝑖𝑣𝑒𝐶ℎ𝑜𝑜𝑠𝑒 𝐺 =31 G 𝐼𝐴 𝐹 𝐴 𝐺 =|𝑆| *𝐼𝐴 𝐹 𝑇𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛+𝐻𝑢𝑎𝑛𝑔𝐶ℎ𝑜𝑜𝑠𝑒 𝐺 =42 𝐼𝐴 𝐹 𝑇𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛+𝐴𝑑𝑎𝑝𝑡𝑖𝑣𝑒𝐶ℎ𝑜𝑜𝑠𝑒 𝐺 =40 *𝐼𝐴 𝐹 𝑆𝑤𝑒𝑒𝑝2+𝐻𝑢𝑛𝑎𝑔𝐻ℎ𝑜𝑜𝑠𝑒 𝐺 =33 𝐼𝐴 𝐹 𝑆𝑤𝑒𝑒𝑝2+𝐴𝑑𝑎𝑝𝑡𝑖𝑣𝑒𝐶ℎ𝑜𝑜𝑠𝑒 𝐺 =31 *Referred to [8] 2011, Huang et al.

Very Sparse Graphs IAF: cardinality of an initial-anchor set Average Blue:131.12 Red:134.63 Green:156 IAF: cardinality of an initial-anchor set

Very Sparse Graphs IAF: cardinality of an initial-anchor set Average Blue:119.56 Red:122.47 Green:146.995 IAF: cardinality of an initial-anchor set

Sparse Graphs IAF: cardinality of an initial-anchor set Average Blue:62.26 Red:61.585 Green: 61.625 IAF: cardinality of an initial-anchor set

Sparse Graphs IAF: cardinality of an initial-anchor set Average Blue:49.855 Red:50.36 Green:51.005 IAF: cardinality of an initial-anchor set

Dense Graphs IAF: cardinality of an initial-anchor set Average Blue:11.72 Red:6.915 Green:6.875 IAF: cardinality of an initial-anchor set

Dense Graphs IAF: cardinality of an initial-anchor set Average Blue:6.79 Red:5.81 Green:5.795 IAF: cardinality of an initial-anchor set

Very Sparse Graphs Average Blue:57.73% Red:57.80% Green:57.80% 𝑘=100

Very Sparse Graphs Average Blue:57.92% Red:57.87% Green:57.87% 𝑘=120

Very Sparse Graphs Average Blue:65.64% Red:68.06% Green:68.10% 𝑘=100

Very Sparse Graphs Average Blue:71.44% Red:71.37% Green:71.43% 𝑘=120

Sparse Graphs Average Blue:24.26% Red:48.66% Green:48.75% 𝑘=30

Sparse Graphs Average Blue:76.64% Red:88.58% Green:88.70% 𝑘=50

Sparse Graphs Average Blue:49.69% Red:60.23% Green:60.47% 𝑘=30

Sparse Graphs Average Blue:97.08% Red:97.78% Green:97.58% 𝑘=50

Dense Graphs Average Blue:12.19% Red:80.49% Green:82.09% 𝑘=5

Dense Graphs Average Blue:70.36% Red:99.92% Green:99.88% 𝑘=10

Dense Graphs Average Blue:68.59% Red:85.50% Green:87.31% 𝑘=5

Dense Graphs Average Blue:99.69% Red:99.99% Green:99.97% 𝑘=10

A=LocalTri 𝐺 No data Vary Sparse The same Sparse Dense AdaptiveChoose IAF COVERAGE 𝐺 AdaptiveChoose No data Vary Sparse HuangChoose ≈AdaptiveChoose The same Sparse MaxDegree ≈AdaptiveChoose Dense

𝐺 No data Vary Sparse The same Sparse Dense A=Sweep2 AdaptiveChoose IAF COVERAGE 𝐺 AdaptiveChoose No data Vary Sparse HuangChoose ≈AdaptiveChoose The same Sparse ≈MaxDegree Dense MaxDegree ≈AdaptiveChoose

Concluding remarks Sweep2 are simpler than Sweep ([8] 2005, Huang) but cover all the cases. A new algorithm for rigid in Localization-Phase

Future works A much powerful Greedy algorithms to choose anchors. Combine AdpativeChoose and HuangChoose to obtain better result. Given a certain initial-anchor set, determine what kind of graphs are localizable. Design a distributed version of AdaptiveChoose.

Thank you for your attention!