efficient computing k-coverage paths in multihop wireless sensor networks xufei mao, shaojie tang,...
TRANSCRIPT
![Page 1: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/1.jpg)
Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks
XuFei Mao, ShaoJie Tang, and Xiang-Yang LiDept. of Computer Science, Illinois Institute of Technology, Chicago, IL
IEEE Transactions on Parallel and Distributed Systems 2009
![Page 2: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/2.jpg)
Outline • Introduction • Problem Formulation• The k-th Nearest Point Voronoi Diagram• Best case coverage: Minimum k-Support Path• Distributed Algorithm for Compute the Minimum k-Support Path• Worst case coverage: Maximum k-Breach Path• Simulation • Conclusion
![Page 3: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/3.jpg)
Introduction• Coverage is a measure of quality of service (QoS) of a
sensor network to some extend• In many wireless sensor network applications, we are often
required to find a path from a source point to a destination point such that the found path is the optimum one under a certain quality measurement
• For example, when some emergency happens, the sensor network should provide safe path(s) which can guide the users leaving from the working place to some safe exit(s)
• In this scenario, the path should be close to some sensor(s) such that the situation along the path can be monitored well
![Page 4: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/4.jpg)
Goal • (1) Finding a path connecting a source point S and a destination point D
inside the given area, which maximizes the smallest observability of all points along the path. This is called best coverage problem
• (2) Finding a path connecting a source point S and a destination point D inside the given area, which minimizes the largest observability of all points on the path. This is called worst coverage problem
![Page 5: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/5.jpg)
Problem Formulation• we assume all sensor nodes have enough sensing range such that it can
sense any point in wireless sensor network
• However, the sensing ability(observability) of a sensor node for a point depends on the Euclidean distance between them
• We use Euclidean distance as the measurement of QoS.
![Page 6: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/6.jpg)
Problem Formulation• Definition 1: Given a point p in the field Ω and the set of sensors U, the k-th distance of p, with respect to U, denoted as , is defined as the
Euclidian distance from p to its k-th nearest sensor node in U.
p
![Page 7: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/7.jpg)
Problem Formulation• Definition 2: Given a path P connecting a source point S and a destination
point D, the k-support of P, denoted by Sk(P), is defined as the maximum
k-th distance of all points on P. In other words, where p is a point on path P
S
DP
![Page 8: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/8.jpg)
Problem Formulation• Problem 1: Optimal k-support Path (Best Case Coverage) Problem:
Given a source point S and destination point D, find a path P in the field to connect S and D such that Sk(P) is minimized
• Problem 2: Optimal k-breach Path (Worst Case Coverage) Problem: Given a source point S and destination point D, find a path P in the field to connect S and D such that Bk(P) is maximized.
S
DP
S
D
P
![Page 9: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/9.jpg)
The k-th Nearest Point Voronoi Diagram• • we call each independent polygon k-th nearest-point Voronoi cell of node ui
and use Ck(ui) to denote it
• we simply call ui is the owner of Ck(ui)
C2(u3) and its owner is u3
KNP Voronoi edge
KNP Voronoi vertex
![Page 10: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/10.jpg)
Compute The kNP Voronoi Diagram • (1) Compute the order-k Voronoi diagram of given sensor nodes set U
using the algorithm given in [7]
• (2) Compute the farthest Voronoi diagram of its corresponding k sensor nodes [14]▫ It is a partition of the plane into polygons such that points in a
polygon have the same farthest sensor node in U.
![Page 11: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/11.jpg)
Compute The kNP Voronoi Diagram
• (3) For each sensor node ui, we merge the partial cells computed above into one KNP Voronoi cell if they share one edge.
![Page 12: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/12.jpg)
Best Case Coverage• Problem 1: Optimal k-support Path (Best Case Coverage) Problem:
Given a source point S and destination point D, find a path P in the field to connect S and D such that Sk(P) is minimized
S
DP
S
D
P
![Page 13: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/13.jpg)
Best case coverage: Optimal k-Support Path-Preliminaries
• Theorem 2: Based on any given path P1 connecting source node S and destination node D, we can always construct another (maybe same) path P2 composed by only a finite number of line segments such that
kNP Voronoi Diagram
![Page 14: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/14.jpg)
(1) Sk(pab)
line segment ab is entirely contained in a disk centered at ui with radius
(2) Sk(ab)
(3) Sk(ab) Sk(pab)
![Page 15: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/15.jpg)
Best case coverage: Optimal k-Support Path-Preliminaries
• Theorem 3: Based on any given path P1 connecting source node S and destination node D, we can construct another path P3 consisting of only line segments whose end points are perfect support location of the KNP Voronoi edges such that
• Definition 4 (Perfect Support Location): The perfect support location of a KNP Voronoi edge is defined as the point (on this edge) which has the minimum Euclidean distance to its owner (k-th nearest sensor node)
![Page 16: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/16.jpg)
perfect support location
We use to denote this part of path P1
![Page 17: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/17.jpg)
Best case coverage: Optimal k-Support Path-Preliminaries• Theorem 4: There is one optimal k-support path consisting of only line
segments whose end points are located at the perfect support locations of the KNP Voronoi edges.
This theorem is straightforward from Theorem 2 and 3.
![Page 18: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/18.jpg)
Compute the Minimum k-Support Path• After getting the KNP Voronoi diagram G with respect to U by Algorithm
1, we present our algorithm to compute the optimal k-support path based on Theorem 4
• As shown in Theorem 4, there must exist one minimum k-support path consisting of only line segments and all of these line segments’ end points are located on the perfect support location of some KNP Voronoi edges.
• Clearly we only need to consider all the paths that using only line segments connecting the perfect support locations of the KNP Voronoi edges
![Page 19: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/19.jpg)
Compute the Minimum k-Support Path• First, we construct a new graph G’ based on KNP Voronoi diagram G as
follows:
S’D’
w(v’) is equal to the k-th distance of the perfect support location of edge
perfect support location
![Page 20: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/20.jpg)
Compute the Minimum k-Support Path• First, we construct a new graph G’ based on KNP Voronoi diagram G as
follows:
S’D’
1
2
3
45
6
![Page 21: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/21.jpg)
[15] Shaojie Tang, Xufei Mao, and Xiang-Yang Li. Optimal k-support coverage paths in wireless sensor networks. In IQ2S Workshop of PerCom 2009, 2009.
S’D’
1
2
3
45
6
adding an edge between any two nodes u’ and v’ in G’ if and only if their corresponding KNP-Voronoi edges belong to the same KNP-Voronoi cell in G
Compare
![Page 22: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/22.jpg)
Compute the Minimum k-Support Path• Next, we use Algorithm 2, which originates from Dijkstra’s shortest path
algorithm, to find a minimum weight path P’ in G’ to connect S’ and D’
![Page 23: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/23.jpg)
5/5
3/5
3/5
6/6
minimum weight path
![Page 24: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/24.jpg)
Distributed Algorithm for Compute the Optimal k-Support Path• we present our distributed algorithm to compute the optimal k- support
path after getting the KNP Voronoi diagram with respect to U by Algorithm 1
• First, we construct a new graph G’ based on KNP Voronoi diagram G in a distributed manner
• we let each sensor node record its owned KNP-Voronoi cells
• Next, we present our distributed algorithm to compute the optimal k-support path based on Theorem 4
![Page 25: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/25.jpg)
![Page 26: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/26.jpg)
![Page 27: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/27.jpg)
Worst Case Coverage• Problem 2: Optimal k-breach Path (Worst Case Coverage) Problem:
Given a source point S and destination point D, find a path P in the field to connect S and D such that Bk(P) is maximized
• Definition 3: Given a path P which is connecting source point S and destination point D, the k-breach of P, denoted by Bk(P),
is defined as the minimum k-th distance of all points on P,
S
DP
S
DP
![Page 28: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/28.jpg)
Worst case coverage: Optimal k-Breach Path -Preliminaries
• Theorem 10: Based on any given path P1 connecting source node S and destination node D, we can always construct another (maybe same) path P4 which only use KNP Voronoi edges such that
![Page 29: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/29.jpg)
Bk(pab) has upper bound
Bk(p’) Bk(pab)
![Page 30: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/30.jpg)
compute the Maximum k-breach path• Theorem 11: There is one maximum k-breach path which lies along the KNP Voronoi edges
(except the first edge or last edge when S or D is not on some Voronoi edge)• (1) Use Algorithm 1 to generate KNP Voronoi Diagram G of U• (2) Each KNP Voronoi vertex v G is assigned a weight w(v)• (3) We add an edge between S (resp. D) and a• (4) We let the weight of (u, v) be equal to the minimum k-th distance among all points on (u, v)
ui
a
s
![Page 31: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/31.jpg)
maximum weight path
![Page 32: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/32.jpg)
Simulation• In our simulation, a set of n wireless sensors is randomly and uniformly
deployed in the target square region with size 500 * 500 meter2
![Page 33: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/33.jpg)
Simulation
![Page 34: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/34.jpg)
Simulation • This result can be used to estimate the coverage quality if the number of
sensors and required coverage degree are given.
![Page 35: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/35.jpg)
Conclusion• In this paper, we proposed polynomial time algorithms (both
centralized and distributed) for two k-coverage problems in wireless sensor networks
• An interesting future work, we would like to design algorithms that can address the coverage problem when the sensing abilities of sensors are heterogeneous
![Page 36: Efficient Computing k-Coverage Paths in Multihop Wireless Sensor Networks XuFei Mao, ShaoJie Tang, and Xiang-Yang Li Dept. of Computer Science, Illinois](https://reader035.vdocuments.mx/reader035/viewer/2022062804/5697bf881a28abf838c8998d/html5/thumbnails/36.jpg)
Thank You!