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

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

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

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

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.

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

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

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

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

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.

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.

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

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

(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)

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)

perfect support location

We use to denote this part of path P1

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.

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

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

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

[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

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’

5/5

3/5

3/5

6/6

minimum weight path

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

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

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

Bk(pab) has upper bound

Bk(p’) Bk(pab)

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

maximum weight path

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

Simulation

Simulation • This result can be used to estimate the coverage quality if the number of

sensors and required coverage degree are given.

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

Thank You!