Probabilistic Model of Triangulation Xiaoyun Li Department of Computing and Electronic Systems University of Essex Colchester, UK CO4 3SQ
[email protected]
Abstract— This paper analyses the probability that randomly deployed sensor nodes triangulate any point within the target area. Its major result is the probability of triangulation for any point given the number of nodes lying up to a specific distance (2 units) from it, employing a graph representation where an edge exists between any two nodes close than 2 units from one another. The expected number of un-triangulated coverage holes, i.e. uncovered areas which cannot be triangulated by adjacent nodes, in a finite target area is derived. Simulation results corroborate the probabilistic analysis with low error, for any node density. These results will find applications in triangulation-based or trilateration-based pointing analysis, or any computational geometry application within the context of triangulation. Categories and Subject Descriptors F.2.2 [Nonnumerical Algorithms and Problems]: Geometrical problems and computations F.1.2 [Modes of Computation]: Probabilistic computation G.3 [PROBABILITY AND STATISTICS]: Probabilistic algorithms; Distribution functions; Statistical computing; Stochastic processes G.2.1 [Combinatorics]: Counting problems
General Terms Algorithms, Design, Experimentation.
Keywords Computational geometry; coverage; Voronoi diagram; Delaunay triangulation.
1. INTRODUCTION
T
riangulation in computational geometry is widely used in coverage and localization algorithms. A triangulation-based pointing protocol [1] (point estimation based upon angles to three landmark points), and trilateration based localization [2] (point estimation based upon distance to three landmark points) have both been proposed for accurate target localization. It has been shown that sensors located at the boundary of a target area projecting laser beams to form a triangle grid can track a mobile object, with the error localized to a small neighborhood after the object moves a short trace length [3]. In graph theory, a planar graph is called a triangulation if adding any edge would destroy the property of planarity (no edge
David K. Hunter Department of Computing and Electronic Systems University of Essex Colchester, UK CO4 3SQ
[email protected] intersections) [4]. Then each face of the graph is a triangle bounded by three edges. Triangulation in graph theory is a fundamental tool for hole detection and recovery using connectivity information as proposed in the coordinate-free 3MeSH hole detection and recovery algorithms [5-6]. The Delaunay triangulation and the Voronoi diagram are both important constructions in computational geometry [7-9]. They are used widely in the sensing coverage area to implement hole detection, if the location of each sensor node (or just “node”) is provided. The Delaunay triangulation may be obtained by connecting those sites in the corresponding Voronoi diagram whose polygons share a common edge. The largest empty circle from the Delaunay triangulation of given nodes can be found; this constitutes the largest void area. Therefore the centre point of this largest void area is least likely to be monitored adequately by the currently available nodes. A node may then be placed there to maximize coverage [10]. The Voronoi diagram can also be used to discover coverage holes [11]. The sensing area is partitioned into Voronoi polygons where each polygon contains exactly one node. If the coverage region of a node cannot cover its corresponding polygon, a coverage hole is detected, with this node being the boundary node. Algorithms employing the Voronoi diagram and Delaunay triangulation have also been proposed to find a path between two predefined points [12-13], using either the Maximal Breach Path (worst-case coverage) or the Maximal Support Path (best-case coverage). These algorithms must be provided with all relevant node locations in order to calculate the minimum sensing range required in order to compute both paths. This is calculated by considering the cells in the corresponding Voronoi diagram, and computing the minimum radius necessary to cover each cell along the path. The best-case coverage problem has been further explored through proposing a distributed algorithm [14] to improve an earlier algorithm [12] by using the relative neighborhood graph. Those algorithms could ensure coverage and guide node deployment with isotropic node coverage in wireless sensor networks, if all node locations and sensing ranges were provided. However, the following question still remains to be answered. For a given node or node density, with specified coverage radii (1 unit), what is the probability that any point in the target area can be triangulated with links, where a link is said to exist between a pair of nodes no more than 2 units apart?
Relevant Work to Probability of Coverage The probability of triangulation is closely related to the probability of existence of non-triangulated coverage holes and the probability of coverage. Unfortunately, such questions have not been discussed previously to any extent, although a few
papers [15-18] provide mathematical methods for the calculation of coverage probability. Assume nodes are randomly deployed in the target area according to a two-dimensional Poisson process, with node density λ. In many applications, sensor nodes are assumed to be deployed randomly in the target area. In such cases, the position of each node may be independent and the node density at any point of the target area may also be independent. Then the node density distribution is a Poisson process with node density λ. For other deployment distributions the mean node density at any position may be different or dependent on other nodes in the immediate vicinity, therefore the following integral calculations would be more complex and are not discussed here. Throughout this paper, λ is the mean number of nodes lying 2 inside a circular area of πR , with each node having a circular coverage area of radius R. A method has been introduced to calculate the probability of coverage (Pc) for any single point not located near the boundary of the area S [16], which is defined by the probability that at least one node lies inside a circle of unit area centered there. Pc is defined in terms of the Poisson distribution with node density λ: ∞ λk e −λ −λ Pc = ∑ = 1− e k ! k =1 All the above calculations assume πR2 = 1, and hence R = 1/√π, although it is assumed below that R = 1.
Motivation This paper determines the triangulation probability of any point in the target area, and hence derives the probability of full sensing coverage without un-triangulated holes. The probability of triangulation for any point in the target area is described below. Assume that each node has a circular coverage area of radius R = 1, and that the point in question is further than 2 units from the boundary of the target area, where a unit is equal to a node’s coverage radius. An edge is said to exist between any two nodes less than 2R apart, which implies that their sensing areas must overlap. In graph theory, a planar graph is called a triangulation if adding any edge would destroy the property of planarity (no edge intersections) [4]. Then each face of the graph is a triangle bounded by three edges, with no polygon with more than three edges (or three vertices – Figure 1).
probability of existence of a non-triangulated hole and the probability of full coverage may be derived. The definition of coverage probability employed here assumes that each node has isotropic coverage, which is not generally true in reality. Despite this assumption, the following results are useful for bounding the probability that a point lies inside a void area untriangulated by links of length up to 2 units, even if the coverage area shape is irregular. It also has practical significance when determining the node density required for a given probability of coverage, or when determining whether various types of hole exist. It can be difficult for a node to determine its position within a non-triangulated area enclosed by a polygon, regardless of whether the position is covered by a node. For example, when tracking the position of an object using triangulation-based pointing [1] or trilateration based localization [2], the error rate for localization is higher when estimating a position inside a non-triangulated polygon compared to that of a triangulated position. If the full radio transmission range is available for triangulation, the area inside a non-triangulated polygon may create a routing hole [19], causing difficulty in data forwarding. Routing holes can exist due to incomplete routing information when a node X attempts to communicate with node D (the destination) outside its transmission range through a closer intermediate node. The larger bold circle in Figure 2 is the transmission range of X. Because of a routing hole, all the intermediate nodes are further away from D than from X, therefore X cannot find any intermediate node closer to D. Transmission therefore fails.
Figure 2. Routing hole
2. PROBABILITY OF TRIANGULATION Probabilistic Model of Triangulation Figure 1(a). Non-triangulated polygon (hole).
Figure 1(b). Triangulated area.
Any point A inside a triangulated area enclosed by three nodes that are mutually connected by links is also said to be triangulated (Figure 3). If nodes are randomly deployed in a target area, the probability of triangulation depends on the node density and distribution there. If a point is non-triangulated, it is located inside a polygon with four or more vertices, which is not itself subdivided by links into triangles. This polygon is named a non-triangulated hole or large hole, with an area in the centre that is not covered by the sensing areas of any nodes (Figure 1(a)). Hence, using the probability of triangulation, the
If a point A lies within the area being studied, the probability of triangulation can be calculated, namely the probability that A has three neighbors (each lying within 2 units of A) connected to each other by links which form a triangle around A. These neighbors are called N0, N1, and N2. It is assumed that the closest neighbor is called N0, and it is a vertex of this triangle because the closest node is most likely to triangulate the point. 106 simulations with varied numbers of neighbors showed that if the closest node cannot triangulate A, its probability of being triangulated by any other three nodes is less than 2%, which can be neglected in order to simplify the calculation. The distance between N0 and A is x0, with 0 < x0 < 2, assuming each node has circular sensing area of radius R = 1. N1 and N2 are further than x0 units from A. It is necessary that 0 < x0 < 2/√3, in order for a suitable triangle to exist.
Figures 4 and 5 show the areas SN1 (inside which node N1 must lie) and SN2 (inside which node N2 must lie). Any point in SN1 must be less than 2 units from N0 to ensure that N1 and N0 are connected. Moreover, regardless of what point N1 occupies within SN1, the area of SN2 (defined by the condition which follows) must be greater than zero. If N1 is located in one semicircle (left or right semicircle), SN2 must be a sub-area of SN1 in the other semicircle with each point within it closer than 2 units from N1, so that N2 is connected to both N0 and N1. Point A is therefore triangulated by N0, N1 and N2. Now that SN1 and SN2 have both been determined, the probability may be found that at least one node falls inside SN1 and the other inside SN2, namely the probability of triangulation for some specified value of x0. The integral of this over the range 0 < x0 < 2/√3 is the probability of triangulation by the closest node and two other neighbors. 2/√3 is the longest possible distance to the closest node which triangulates the point; the furthest possible distance to the other two nodes triangulating the point is 2 units.
Figure 4(a).The area in which N1 may lie (x0 ≤ 1).
Figure 3. Triangulation of point A. If both N1 and N2 lie in the same semicircle in Figure 4(a), N0, N1 and N2 cannot form a triangle enclosing A. Similarly, N0 and N2 (or N1) must not lie to the same side of line AN1 (or AN2 – Figure 3), otherwise A would lie outside the triangle formed by N0, N1 and N2. The node in the right semicircle is designated N1, while N2 is in the left semicircle.
Figure 4(b).The area in which N2 may lie (x0 ≤ 1).
Methodology N1 must be within 2 units of N0 in order to connect to it, and should be further than x0 units from A because the distance between the closest node N0 and A is x0. Therefore N1 must lie within SN1, which is the intersection of the circles centered on N0 and A respectively, each with radius 2 units. However, the circle centered on A with radius x0 is excluded. Hence N1 is within 2 units of both A and N0. If N1 is located in the right (left) side of SN1, then N2 should lie in the left (right) side of SN1 in order to enclose A as mentioned in the last paragraph. Similarly, if N1 is located to the same side of the y-axis as N0 (under point A in Figures 4 and 5), then N2 should be on the other side. Therefore the area SN2 (containing N2) is the intersection of SN1 and the circle centered on N1 with radius 2. SN2 lies on the opposite side of the y-axis from N1. Unfortunately, for some points in SN1, SN2 is the empty set because both N1 and N2 are located at the left (right) side, or there is no intersecting area above A when N1 is located beneath it. In order to ensure that SN2 is non-empty, it should include at least one point (Cleft and Cright) within the left and right semicircles respectively, also within SN1, that are closest to both N0 and the y-axis (Figure 5(a)).
Figure 5(a). The area in which N1 may lie (1 < x0 < 2/√3).
Figure 5(b). The area in which N2 may lie (1 < x0 < 2/√3).
Cright (Cleft) is the point in the right (left) side of SN1 with minimum mean distance from any point in the left (right) side of SN1, therefore it is the closest point to the y-axis (Figure 5(a)). If there is more than one point closest to the y-axis, then Cright (Cleft) is the closest point to both A and N0 (point C in Figure 3(a)) but Cright (Cleft) should lie on the opposite side of the x-axis from N0 in order to triangulate A. In order to ensure that SN2 is non-empty, SN1 is in fact the intersection of the initially defined area of SN1 (as defined above) and the circles centered on Cright and Cleft, both with a radius of 2 units (Figure 5(a)). Hence SN2 is non-empty because for any point located in the right (left) semicircle of SN1, at least point Cleft (Cright), is within 2 units of N1. Meanwhile SN1 is maximized because Cright (Cleft) is the closest point (mean distance) to SN1 in the left (right) semicircle. For x0 ≤ 1 (Figure 4), the two points Cleft and Cright are the same point named C. Hence there are separate cases for the calculation of SN1 and SN2 for x0 ≤ 1 and x0 > 1. The coordinates of Cleft and Cright are: (0, x0) (±2sinα1, 2cosα1 − x0)
2 − x0
∫ 0
S N 2 ( x1 , y1 ) = 1 2
+
2 − x1
∫ 0
1 2
⎛ ⎜2 ⎜⎜ ⎝
⎞ S N 2 ( x1 , y1 )dx1 ⎟⎟dy1 ⎟ max (x02 − y12 , 0 ) ⎠ 4−( x0 + y1 )2
∫
2 max ⎡min ⎛⎜ 4 − ( x1 + x 2 ) + y1 , 4 − x 22 − x 0 ⎞⎟ − max(0, x 02 − x 22 ) ,0⎤ dx 2 ⎢⎣ ⎥⎦ ⎝ ⎠
2 − x1
∫ 0
2 max ⎡ min ⎛⎜ 4 − (x1 + x2 ) − y1 , 4 − x22 − x0 ⎞⎟ − max(0, x02 − x22 ) ,0⎤ dx2 ⎢⎣ ⎥⎦ ⎝ ⎠
For 1 < x0 ≤ 2/√3 (Figure 5(b)): ⎛ min⎛⎜⎝ ⎜ S N 2 ( x0 ) = ∫ ⎜ − x0 cos 4α1 ⎜ ⎝ S N 2 ( x1 , y1 ) = 2 x0 − x0
− 2 sin α1
∫
x0 cos(α 2 +α 3 )
4 − ( y1 + x0 )2 , 4 − ( 2 x0 − x0 − y1 )2 − 2 sin α1 ⎞⎟ ⎠
∫
x02 − y12
⎞ ⎟ S N 2 ( x1 , y1 ) dx1 ⎟dy1 S N1 ( x0 ) ⎟ ⎠
⎡min⎛⎜ 4 − ( x − x )2 + y , 4 − x 2 − x ⎞⎟ − x 2 − x 2 ⎤ dx 1 2 1 2 0 0 2 ⎥ 2 ⎢⎣ ⎝ ⎠ ⎦
−1 y1 1 ; x2 + x2 + y2 − 4 α 2 = cos −1 0 1 2 1 2 ; α 3 = tan x1 x0 2 x0 x1 + y1 P(x0, n) is the probability that the closest node N0 to A is between x0 and x0 + dx0 units away where dx0 → 0, and that all the other n – 1 neighboring nodes are between x0 and 2 units from A:
α1 = cos−1
for x0 ≤ 1 for x0 > 1
As discussed above, N1 and N2 must lie on opposite sides of the y-axis, in order to ensure that with N0, they form a triangle enclosing A (Figures 4(b) and 5(b)). For x0 ≤ 1 and some specified point N1(x1, y1), it is possible that N0 and N2 lie on the same side of line AN1, so that N2 falls within the area SN2′ (Figure 4(b)). In this case, A is not located inside the triangle formed by N0, N1 and N2. If we consider N1′, located at (x1, –y1), a similar situation occurs when N2 falls inside SN2. Therefore the mean area of SN2 for N1(x1, y1) and N1′(x1, –y1) is (SN2 + SN2′)/2, as shown in Figure 4(b); this result is used in later calculations. SN1(x0) and SN2(x0) are the sizes of the areas in which N1 and N2 respectively may each lie for any x0 (distance between N0 and A). For the purposes of the calculation, N1 and N2 are assumed to lie on the left and right semicircles respectively in order to triangulate point A. Therefore SN1(x0) and SN2(x0) are the areas of each region coinciding with only one semicircle. For x0 ≥ 2/√3:
1 S N1 ( x0 )
S N 2 ( x0 ) =
P(x0 , n ) ≈
2nπx0 dx0 ⎛ 4π − πx02 ⎞ ⎜⎜ ⎟⎟ 4π ⎝ 4π ⎠
n −1
=
nx0 ⎛ 4 − x02 ⎞ ⎜ ⎟ 2 ⎜⎝ 4 ⎟⎠
n −1
dx0
SN1(x0) = 0
For 0 < x0 ≤ 1 (Figure 4(a)): S N1 (x0 ) = 2
2 − x0
∫ 0
⎛ ⎜ ⎜⎜ ⎝
2 − x0 ⎞ 2 ⎟ dx1 ⎟dy1 = 2 ∫ ⎛⎜ 4 − (x0 + y1 ) − max x02 − y12 ,0 ⎞⎟dy1 ⎝ ⎠ ⎟ 2 2 0 max(x0 − y1 , 0 ) ⎠ 4 − ( x0 + y1 )2
∫
(
)
N1(x1, y1) is assumed to lie above the x-axis and to the right of the y-axis only, because SN1 is symmetrical about both the x-axis and the y-axis. For 1 < x0 ≤ 2/√3, the coordinates of D (point with lowest y-coordinate in SN1) in Figure 5(a) are (x0sin4α1, –x0cos4α1) while the coordinates of E (point with lowest ycoordinate in SN2) in Figure 5(b) are (x0cos(α2 + α3), x0sin(α2 + α3) ). The area of SN1 is the integral of y1 between points Cleft and D: min⎛⎜ 4−( y1 + x0 )2 , 4−[2 cosα1 − x0 − y1 ]2 −2 sin α1 ⎞⎟ ⎞ 2 cosα1 − x0 ⎛ ⎠ ⎟ ⎜ ⎝ S N1 ( x0 ) = ∫ ⎜ dx 1 ⎟dy1 ∫ − x0 cos 4α1 ⎜ ⎟ x02 − y12 ⎠ ⎝ =
2 x0 − x0
∫
− x0 cos 4α1
⎡min⎛⎜ 4 − ( y + x )2 , 4 − (2 x − x − y )2 − 2 sin α ⎞⎟ − x 2 − y 2 ⎤dy 1 0 0 0 1 1 0 1 ⎥ 1 ⎢⎣ ⎝ ⎠ ⎦
SN2(x0) is the integral over x1 and y1 of the area SN2(x1, y1) which results when N1 lies at (x1, y1). For 0 < x0 ≤ 1:
Figure 6. Percentage of area (4π) in which N1 and N2 may each lie. Figure 6 shows the percentage area of N1 (SN1(x0)) and N2 (SN2(x0)) inside a circle of area 4π and radius 2. It shows that SN1(x0) drops almost linearly from 50% to 7.5% of 4π for 1 ≥ x0 > 0. For 2/√3 > x0 > 1, SN1(x0) drops more quickly from 7.5% to zero. SN2(x0) drops slower (from 10% to 0.7%) compared to SN1(x0) for 1 ≥ x0 > 0. Pn,3 is the probability that three out of a total of n neighbors (always including N0, the closest neighbor) triangulate A. This is the probability that: • At least two nodes (N1) lie inside SN1 in both the left and right semicircles (Figure 4 or 5), and • At least one node (N2) lies inside SN2 in either the left or right semicircle. This node could be either of the two nodes mentioned above or any other node. This ensures that at least one node (N1) is situated in the left semicircle and one (N2) in the right semicircle or vice versa, and they are both fully connected to the closest node N0.
⎛ ⎡ S N1 ( x0 ) ⎤ Let β = ⎜1 − ⎢1 − ⎜ ⎣ π 2 2 − x02 ⎥⎦ ⎝
(
⎛
⎡
⎜ ⎝
⎣
γ 1 = ⎜1 − ⎢1 −
n−1
)
2 × S N2 ( x0 ) ⎤ π 22 − x02 ⎥⎦
(
)
⎡
⎛
⎣⎢
⎝
γ = γ 1 + (1 − γ 1 ) × ⎢1 − ⎜⎜1 −
n− 2 ⎞ ⎛ ⎡ ⎞ ⎟ × ⎜1 − 1 − S N1 ( x0 ) ⎤ ⎟ ⎢ ⎥ 2 2 ⎟ ⎜ ⎣ π 2 − x0 ⎦ ⎟ ⎠ ⎝ ⎠
(
n −3
)
⎞ ⎟ ⎟ ⎠
2 S N2 ( x0 ) ⎞ ⎤ ⎟⎟ ⎥ S N 1 ( x0 ) ⎠ ⎥ ⎦
2
Pn ,3 = ∫ β × γ × P ( x0 , n) 0
2 3
n −1
nx0 ⎛ 4 − x02 ⎞ ⎜ ⎟ dx0 2 ⎜⎝ 4 ⎟⎠ 0 β is the probability that at least two nodes (N1) lie in the left and right semicircles respectively of SN1 where n – 1 neighbors lie between x0 and 2 units of A (excluding N0). γ1 is the probability that at least one node (N2) is in either the left or right semicircle of SN2 with n – 3 neighbors closer than 2 units from A (excluding the two nodes considered by β). γ is the probability that at least one node (N2) is in either the left or right semicircle of SN2 with n – 1 neighbors (including the two nodes considered by β but excluding N0). Using Pn,3, the probability of triangulation Pt for a specified point (assuming a mean node density of λ in a two-dimensional Poisson process) may be calculated as follows: n −λ ∞ P n ,3 λ e Pt (λ ) = ∑ n! n=0 The probability of triangulation not occurring at a specified point is Pnt(λ): n −λ ∞ P n ,3 λ e Pnt (λ ) = 1 − Pt (λ ) = 1 − ∑ n! n=0 For target area of area S, the total un-triangulated area Snt is: =
∫ β ×γ ×
Snt = S ×Pnt(λ) If the mean un-triangulated area of an un-triangulated hole can be calculated, then the expected number of un-triangulated holes and the probability of no hole may also be derived. Details of this analysis will be published the future and are not presented here.
Figure 7. Probabilities of triangulation
3. CONCLUSIONS The calculations in this paper have quantified the probability of triangulation at any specified point that has K neighboring nodes within a distance of 2 units, and the probability of triangulation was derived where node deployment follows a two-dimensional Poisson process. The calculation results agree with the simulations very well, with a difference of less than 1% for node density λ ≥ 5. The probability of no triangulation is much higher than the probability of no coverage for node density λ > 1. This is useful when calculating the expected number of nontriangulated holes in a finite target area, because the expected non-triangulated area can be calculated via the above calculation, and the mean non-triangulated area inside a hole (a polygon) is always non-zero, although the uncovered area inside a hole approaches zero for high node densities. To the best of our knowledge, this is the first time that the probability of triangulation for any point, given the number of neighboring nodes, has been calculated analytically. It provides a fundamental method for probabilistic analysis in triangulation based applications using computational geometry, especially in the fields of coverage quality control and un-triangulated coverage hole prediction.
Simulation results Ten thousand simulations with varied node densities (λ) were run to confirm the analysis. For each simulation, 4λ nodes are randomly deployed inside a circle with radius 2 centered on point A (Figure 3). If A is located within any triangle formed the by closest node N0 and any other two nodes, all closer than 2 units to each other, then A is triangulated. Figure 7 shows that the simulation results agree with calculations very well for the probability of triangulation at a specified point with exactly 4λ neighbors, with a maximum difference of less than 1% for λ ≥ 5. Furthermore, in contrast to the point in question having a fixed number of neighbors, one thousand simulations with random nodal deployment (two-dimensional Poisson process) for each mean node density λ (12 ≥ λ ≥ 1) were also carried out. Hence the number of neighbors is not necessarily exactly 4λ due to the use of a Poisson process. With λ > 4, the analytical results agree with simulation to within 5% (Figure 7).
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