Asymptotic Critical Transmission Radii for Greedy
Forward Routing in Wireless Ad Hoc Networks
Peng-Jun Wan, Chih-Wei Yi, Member, IEEE, Lixin Wang, Frances Yao, and Xiaohua Jia
Abstract—In wireless ad hoc networks, greedy forward routing
is a localized geographic routing algorithm in which one node discards a packet if none of its neighbors is closer to the destination of the packet than itself, or otherwise forwards the packet to the neighbor closest to the destination. If all nodes have the same transmission radii, the critical transmission radius for greedy forward routing is the smallest transmission radius which ensures packets can be delivered by greedy forward routing through any source-destination pair. In this paper, we study asymptotic critical transmission radii of randomly deployed wireless ad hoc networks. Assume network nodes are represented by a Poisson point process of density n over a unit-area convex compact region whose boundary curvature is bounded. We show that the ratio of critical transmission radii to ln n
πn is
asymptotically almost surely equal to 1/2 3− √ 3 2π ≈ 1.6. Index Terms—Wireless ad hoc networks, greedy forward
routing, critical transmission radii, random deployment.
I. INTRODUCTION
A
wireless ad hoc network is a collection of wireless devices distributed over a geographic region. Each ad hoc device is equipped with an omnidirectional antenna. A communication session is established either through a single-hop radio transmission if the communication party is close enough, or through relaying by intermediate devices otherwise. The selection of intermediate relay nodes is determined by routing algorithms. Greedy forward routing (abbreviated by GFR) is one of the localized geographic routing algorithms proposed in literature.In GFR, one node discards a packet if none of its neigh-bors is closer to the destination of the packet than itself, or otherwise forwards the packet to the neighbor closest to the destination. Therefore, each packet should contain the location of its destination, and each node only needs to maintain the locations of its one-hop neighbors. GFR can be implemented in a localized and memoryless manner. There are some variations of GFR. For example, in [1] and [2], the Paper approved by R. Fantacci, the Editor for Wireless Networks and Systems of the IEEE Communications Society. Manuscript received July 24, 2006; revised March 18, 2007, December 17, 2007, June 14, 2008, and June 21, 2008.
P.-J. Wan and L. Wang are with the Department of Computer Science, Illinois Institute of Technology, Chicago, IL 60616 (e-mail: [email protected], [email protected]).
C-W. Yi is with the Department of Computer Science, National Chiao Tung University, Hsinchu City 30010, Taiwan (e-mail: [email protected]).
F. Yao and X. Jia are with the Department of Computer Science, City University of Hong Kong, Hong Kong (e-mail: {csfyao, csjia}@cityu.edu.hk). A short version of this paper had been presented in the Seventh ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc 2006), Florence, Italy, 22-25 May 2006.
Digital Object Identifier 10.1109/TCOMM.2009.05.070307
v
w
w
3w
2w
1u
4Fig. 1. u is a source node and v is the corresponding destination node.
shortest projected distance to the destination on the straight line joining the current node and the destination node is considered as the greedy metrics. In [1], packets are allowed to be sent backward if there is no forwarding neighbor. In [2], only nodes whose Voronoi cells intersect with the source-destination line segment are eligible for being relay nodes. Here the Voronoi cell of a node is the set of points in the plane that are closer to the node than to any other node [3].
Due to existence of local minima where none of neighbors is closer to the destination than the current node, a packet may be discarded before arriving its destination. To ensure that every packet can arrive its destination, all nodes should have sufficiently large transmission radii to avoid being local minima. For points x, y ∈ R2 and a positive real number r,
let B(x, r) denote the open disk of radius r centered at x,
x denote the Euclidean norm of x, and x − y denote
the Euclidean distance between x and y. Consider Fig. 1. Let u be a source or relay node, v be the corresponding destination node, and wi denote nodes other than u and v.
Nodes that can relay packets for u toward v must be in the region B(u, u − v)∩B (v, u − v) based on the following observations. If wi can relay packets for u toward v, it must
be closer to v than u, i.e.v − wi < v − u or equivalently wi∈ B (v, u − v). w2, w3, w4 satisfy this rule and w1 does
not. On the other hand, if no one can relay packets for u, packets should be directly transmitted from u to v. So, in the worst case, u at most needs to set its transmission radius to u − v. This implies candidates of relay nodes must be
in B(u, u − v). For example, in Fig. 1, w4 can’t be a
candidate of relay nodes. Thus, only w2 and w3 can relay packets for u toward v. In addition, if the transmission radius is set tomin (w2− u , w3− u), u has at least one neighbor
to relay packets. The procedure of selecting the minimal transmission radii to ensure either u can send packets directly to v or there exists at least one node to relay packets for u 0090-6778/09$25.00 c 2009 IEEE
toward v can be expressed asminwi∈B(v,u−v)wi− u. For
a given point set V in the plane, let
ρ(V ) = max (u,v)∈V2 u=v min w∈B(v,u−v)∩Vw − u . (1)
It is the the maximum ofminw∈B(v,u−v)∩Vw − u over
all(u, v) pairs of nodes.
To eliminate local minima in the network, we choose
ρ(V ) as the transmission radius. According to the previous
discussion, any node u always can deliver packets toward any other node. However, is ρ(V ) the optimal (smallest) transmission radius for local-minimum-free? The answer is positive. Consider the pair of nodes(u, v) that gives the value
ρ(V ). If the transmission radius is set less than ρ (V ), u
can’t directly send packets to v and there is no other node that can relay packets for u toward v. So, u is a local minimum w.r.t. v. So, ρ(V ) is the optimal one and called the critical transmission radius for (local-minimum-free) GFR that guarantees the deliverability of packets. In the rest of this paper, the critical transmission radius for GFR is simply written as the critical transmission radius and abbreviated as CTR.
The analytic work of GFR can be dated back to 1984 by Takagi and Kleinrock [1]. They studied the optimal trans-mission radius to maximize the expected progress of packets based on most forward and least backward routing strategy in which every node delivers each packet to the neighbor (not including itself) with the shortest projected distance to the des-tination on the straight line joining the current node. However, the deliverability of packets is not considered. Recently, Xing
et al. [2] (2004) show that in a fully covered homogeneous
wireless sensor network, if the transmission radius is larger than 2 times of the sensing radius, the deliverability can be guaranteed between any source-destination pair by greedy forwarding schemes in which a packet is sent to the neighbor either with the shortest Euclidean distance to the destination [4] [5] or with the shortest projected distance to the destination on the straight line joining the current node and the destination node [1] and by bounded Voronoi greedy forwarding scheme in which only those nodes whose Voronoi cells intersect with the line segment between the source and destination are eligible to relay the packet.
Another related and interesting problem in literature is the longest edge of connected geometric graphs. Penrose [6] (1997) [7] (1999) studied the longest edge of a minimal span-ning tree which is corresponding to the critical transmission radius for connectivity in random geometric graphs. Later, by applying the percolation theory, Gupta and Kumar [8] had sim-ilar results for wireless networks. Recently, Baccelli and Bor-denave [9] (2007) introduced a structure called radial spanning trees (RSTs) in which each node, excluding the root s at the origin of the plane, has an edge to its closest neighbor among nodes closer to the root s. The length of the longest edge
of RSTs can be given by max
u∈V,u=sw∈B(s,s−u)∩Vmin w − u.
If s is the only destination, then the value is the critical transmission radius for local-minimum-free GFR.
In this paper, we study the deliverability by giving the asymptotics of ρ(V ) where V is a Poisson point process.
Assume that the deployment region D is a convex compact region whose boundary has bounded curvature. By scaling, we assume D have unit area. Let Pn denote a Poisson point
process of density n overD. The ratio of ρ (Pn) to
ln n πn is
asymptotically almost surely equal to 1/2 3− √ 3 2π ≈ 1.6.
The rest of this paper is organized as follows. In Section II, we present our main results and show some possible applications. In Section III, the proof of main results is given, but most calculation details and related geometric and probabilistic lemmas are left in the appendix. In Section IV, simulation results were given to evidence our asymptotic analysis. Our conclusions are in Section V.
II. MAINRESULTS
LetD be a unit-area convex compact region with a bounded-curvature boundary, andPndenote a Poisson point process of
density n over D. Let β0= 1/
2 3− √ 3 2π ≈ 1.62. The main
result of this paper is the following theorem.
Theorem 1: For any ε >0,
lim n→∞Pr (1 − ε) β0ln n nπ ≤ ρ (Pn) ≤ (1 + ε) β0ln n nπ = 1.
Since the converge is in probability, we remark Theorem 1 can’t be simplified to limn→∞ρ(Pn) =
β0ln n
nπ . Based on
Theorem 1, we have the following corollary.
Corollary 2: If the transmission radius is set toβ ln n πn for
some constant β, we have
1) If β > β0, it is asymptotic almost sure that packets can be delivered by GFR between any pair of nodes.1
2) If β < β0, it is asymptotic almost sure that packets can’t be delivered by GFR between some pairs of nodes.
Possible Applications: In the rest of this section, we show
some possible applications of Theorem 1. Due to harsh de-ployment environment coupled with a large amount of sensors to be deployed, random deployment is unavoidable in many applications of wireless ad hoc and sensor networks. At the same time, owing to the constraint on the maximal transmis-sion power, each wireless device can only communicate with nearby nodes, and therefore connectivity of network topology and deliverability of routing protocols are the most important issue of randomly deployed networks. Our asymptotic research results associated with simulation data can be a good reference to the following problems and help us to improve energy efficiency.
• Maximal transmission power: According to path loss
models of wireless communications, the maximal mission power is strongly related to the maximal trans-mission radius and is a key parameter during the design phase of wireless devices. The choosing of the maximal transmission power can base on the maximal transmission radius. Our results show that Θln n
n
is a good reference for choosing the maximal transmission radius.2 1An event is said to be asymptotic almost sure (abbreviated by a.a.s.) if it
occurs with a probability converges to one as n→ ∞.
2For two sequences fn and gn, we write fn = Θ (gn) if there exist
constants c1 > 0, c2 and n0 such that c1|gn| ≤ |fn| ≤ c2|gn| for all
u
w
v
Fig. 2. w is the intersection point of the segment uv and the circle B(u, rn). The shaded area is B(u, rn) ∩ B (w, rn) which is contained in B (u, rn) ∩
B (v, u − v).
• The critical number of nodes: To deploy a WSN over a
region, if the transmission range of nodes is known, we need to decide how many sensor nodes are enough such that the network can be connected by routing algorithms. By scaling the deployment region to unit-area and also scaling the transmission radius by the same ratio, we can have a critical number of nodes based on the theoretical formula or simulation data.
• Light-weight routing algorithms: If geographic
informa-tion is available, greedy forward routing is easy to imple-ment and requires few resources, but suffers from local minimum problems. Therefore, some relatively complex compensatory algorithms are needed to handle such ex-ceptional situations. If the delivery rate can be predicted and controlled above tolerable level or even more the deliverability can be guaranteed, the pure greedy forward routing is enough, and complex compensatory algorithms are not necessary.
III. OUTLINE OFPROOF
This section is dedicated to the proof of Theorem 1.
A. Upper Bounds for the Critical Transmission Radius
For a given ε > 0, let β = (1 + ε)2β0. The
up-per bound for ρ(Pn) given in Theorem 1, i.e. ρ (Pn) ≤
(1 + ε)β0ln n
nπ , can be proved by showing that if rn =
β ln n
πn = (1 + ε)
β0ln n
nπ , there a.a.s. don’t exist local
minima. For a pair of nodes(u, v), u is a local minimum w.r.t.
v if and only if u − v > rn and there are no other nodes
in B(u, rn) ∩ B (v, u − v). Now, assume u − v > rn
and let w be the intersection point of the segment uv and the circle ∂B(u, rn). See Fig. 2. For convenience, for any two
points x, y∈ R2, the region B(x, x − y) ∩ B (y, x − y),
denoted by Lxy, is called the lune associated with x and
y, and the segment xy is called the waist of Lxy. Since Luw⊂ B (u, rn)∩B (v, u − v), "there exist nodes in Luw"
implies "u is not a local minimum w.r.t. v". We shall show that any lune whose waist is of length rn, e.g. like Luw, a.a.s.
covers some nodes. Thus, the network is local-minimum-free.
0 5 10 15 0 1 2 3 4 5 6 7 8 9 10 β L (β)
Fig. 3. The graph ofL (β).
We use # (S) to denote the cardinality of a countable set
S. For any finite point set V ⊂ D and any r > 0, define S (V, r) = min
u,v∈D,u−v=r# (V ∩ Luv) .
S (V, r), called the minimal scan statistics, is the minimal
number of nodes of V that can be covered by a lune whose waist is fully contained in D and with length r. So, the event
S (Pn, rn) > 0 implies the event ρ (Pn) ≤ rn. An a.a.s. lower
bound forS (Pn, rn) will be given in Lemma 3 and implies
that if β > β0,S (Pn, rn) > 0 is a.a.s..
Let φ(μ) denote the function φ (μ) = 1 − μ + μ ln μ over
μ∈ (0, ∞). φ is strictly convex and has the unique minimum
zero at μ= 1. Let φ−1: [0, 1) → (0, 1] be the inverse of the
restriction of φ to (0, 1]. We define a function L over (0, ∞) by
L (β) =
βφ−1(1/β) if β > 1,
0 otherwise.
The graph of L (β) is illustrated in Fig. 3. We have the
following lemma.
Lemma 3: Suppose that nπr2
n = (β + o (1)) ln n for some β > β0.3 Then for any constant β1∈ (β0, β), it is a.a.s. that
S (Pn, rn) > 12L β1 β0 ln n > 0.
A proof of Lemma 3 is given in the appendix and also can be found in [10]. According to Lemma 3, we have ρ(Pn) ≤ rn= (1 + ε)
β0ln n
πn is a.a.s..
B. Lower Bounds for the Critical Transmission Radius
The lower bound for ρ(Pn) given in Theorem 1, i.e.
(1 − ε)β0ln n
nπ ≤ ρ (Pn), will be proved in this
subsec-tion. For a given ε > 0, let β = (1 − ε)2β0. The lower
bound can be proved by showing that if rn = β ln n πn =
(1 − ε)β0ln n
nπ , there a.a.s. exist local minima. The plane is
going to be tessellated into equal-size square cells. For each cell, an event that implies existence of local minima within the cell is introduced, and a lower bound for the probability of the event is derived. Since these events are identical and
3For two sequences fnand gn, we write fn= o (gn) if limn→∞fn gn= 0.
independent among cells, we can estimate an low bound for the probability of existence of local minima in the network, and prove the lower bound is a.a.s. equal to 1.
Let β1 and β2 be two positive constants such that max 1 4β0, β < β1< β2< β0, and π2 c2 1 − √ β1 √ β2 <1. (2)
Here c is the constant in Lemma 6 that is given in Appendix. Let R1(n) and R2(n) be given by
nπ(R1(n))2= β1ln n and nπ (R2(n))2= β2ln n. (3) Divide D by a 4ln n nπ -tessellation.4 Let I n denote the
number of cells fully contained inD, and we have
In= Θ
n
ln n
. (4)
For each cell fully contained inD, we draw a disk with radius
1 2
ln n
nπ at the center of the cell. For1 ≤ i ≤ In, let Ei be
the event that there exist two nodes X, Y ∈ Pn such that
their midpoint is in the i-th disk and their distance is between
R1(n) and R2(n), and there is no other node in the lune LXY. For any two nodes u and v withu − v > rn, if there
is no other node in Luv, u and v are local minima w.r.t. each other. So, Ei implies existence of local minima and
Pr [ρ (Pn) > rn] ≥ Pr [at least one Ei occurs] . (5)
Let oidenote the center of the i-th disk, and u, v be two points
such that their midpoint is on the disk and their distance is between R1(n) and R2(n). (See Fig. 4.) Since the middle
point of u and v, called z, is in the disk, we haveoi− z ≤ 1
2
ln n
nπ. For any point w∈ Luv, the distance between w and z, i.e. w − z, is at most √23u − v ≤ √23
β0ln n nπ . For
any point w∈ Luv, applying triangle inequality, we have w − oi ≤ w − z + oi− z < √ 3β0 2 ln n nπ + 1 2 ln n nπ ≈ 1.885 ln n nπ <2 ln n nπ.
Since the cell width is4ln n
nπ, u, v and Luv are contained
in the i-th cell. Therefore, E1,· · · , EIn are independent. In
addition, E1,· · · , EIn are identical. Then,
Pr [none of Ei occurs] = (1 − Pr [E1])In≤ e−InPr(E1).
If InPr (E1) → ∞, then Pr [ρ (Pn) > rn] → 1 follows, and
from Eq. (5), the lower bound for ρ(Pn) in Theorem 1 is
obtained. So, we only need to prove the following lemma.
Lemma 4: InPr (E1) → ∞.
The proof of Lemma 4 is given in the appendix and also can be found in [10].
4An ε-tessellation is a technique that divides the plane by vertical and
horizontal lines into a grid in which each grid cell has width ε.
0000
0000
0000
0000
0000
1111
1111
1111
1111
1111
v
iz
w
u
o
Fig. 4. The cell width is4ln n
nπ, oiis the center of the cell, and R1(n) <
u − v < R2(n). The disk is centered at oiand with radius 12
ln n
nπ, and
z is the middle point of u and v. Luv is fully contained in the cell.
CDFs of Normalized CTR s 0 0.2 0.4 0.6 0.8 1 1.2 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Normalized CTR Pr ob. n=200 n=400 n=800
Fig. 5. The cumulative distributed functions of normalized CT Rs for n= 200, 400, and 800.
IV. SIMULATIONS
In the simulation, networks are composed of 200, 400, or 800 nodes distributed over a unit-area disk. Let n denote the network size, i.e. the number of nodes in a network. For each network size,400 topologies are generated by uniform random point processes. For each network topology, the actual critical transmission radius, denoted by CT R, is computed according to Eq. (1). To avoid ambiguity, the estimated (or theoretical) critical transmission radius given by Theorem 1 is denoted by
ρn.
First, we would like to observe the trend of convergence of CT Rs. For n = 200, 400, and 800 respectively, the average CT Rs are 0.1808, 0.1332, and 0.1000, and the theoretical radius ρn are0.1469, 0.1104, and 0.0825. To have a fair comparison over different network sizes, CT Rs are normalized by being divided by the corresponding ρn. The
Ratios of Deliverable Source-Destination Pairs 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 0.75 0.8 0.85 0.9 0.95 1 1.05
Transmission Radius Factor s
Ratio
n=200 n=400 n=800
Fig. 6. Average percentage of deliverable source-destination pairs in networks with n= 200, n = 400, and n = 800.
bold green dotted line marked by triangles is the CDF of normalized CT Rs for n = 200, the bold solid purple line marked by squares is for n = 400, and the fine solid red line marked by circles is for n = 800. For each network size, the transition width is the difference between the largest and smallest CT Rs among 400 network topologies. The normalized transition width for n = 200 (respectively, 400 and800) is 0.9168 (respectively, 0.7591 and 0.6419) that is the horizontal distance between the right most and left most triangle (respectively, square and circle) markers in Fig. 5. The decreasing of the normalized transition width agrees with the trend of convergence.
Next, if transmission radii are set below CT Rs, we would like to investigate the impact on the deliverability of GFR. Since CT Rs usually are different from one topology to another, to have a comparison basis, for each network topol-ogy, the CT R is first computed according to Eq. (1), and then transmission radii are set to s times of the CT R for
s = 0.8, 0.85, 0.9, 0.95, or 0.99. In other words, for each
network topology, according to its CT R, transmission radii are set to 0.8 · CT R, 0.85 · CT R, 0.9 · CT R, 0.95 · CT R, or0.99 · CT R. The number of deliverable source-destination pairs in each network is counted. For each transmission radius factor s, the average ratio of deliverable source-destination pairs are calculated over 400 network topologies. In Fig. 6, the x-axis represents the transmission radius factor s, and the
y-axis is the average ratio of deliverable source-destination
pairs. We can see that transmission radii have larger impact on deliverability in sparse networks than in dense ones.
Last, we investigate the delivery efficiency of GFR. The effective progress ratio (EPR) of a routing path is defined as the ratio of the Euclidean source-destination distance to the total Euclidean path length. The ratio can be an indicator of delivery efficiency. In the simulation, we calculated average EPRs under various transmission radii and node densities. Similarly, for each network topology, the CT R was first calculated, and then transmission radii are set to s times of the
CT R. Here s are0.8, 0.9, 1, 1.1, 1.2, and 1.3. In Fig 7, the
x-axis represents the transmission radius factor s, and the y-x-axis
Average Effective Progress Ratios
0.75 0.8 0.85 0.9 0.95 1 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
Transmission Radius Factor s
EPR
n=200 n=400 n=800
Fig. 7. Effective progress ratios (EPRs) under various transmission radii and network sizes.
is the average EPR over all deliverable source-destination pairs in 400 network topologies. We can see that the EPR mainly depends on the transmission radius factor s but is insensitive to the network size. If the EPR is a major concern, transmission radii will be one of the primary parameters to tune the system.
V. CONCLUSIONS
Greedy forward routing is a localized and memoryless geographic routing algorithm. However, it cannot guarantee the deliverability of packets if transmission radii of nodes are not large enough. If all nodes have the same transmission radii, the smallest transmission radius that ensures the deliv-erability of packets is referred to as the critical transmission radius. In this paper, we provides tight a.a.s. bounds for the critical transmission radius of randomly deployed wireless ad hoc networks in which nodes are represented by a Poisson point process. We also investigated a number of parameters related to GFR by simulations, including the average of one-hop progress, the expected number of one-hops between source-destination pairs, and the effective hop progress. As a future work, it is interesting to study the asymptotics of other localized geographic routing protocols.
APPENDIX
In the appendix, we give the proof of Lemma 3 and 4. In what follows, |A| is shorthand for 2-dimensional Lebesgue
measure (or area) of a measurable set A⊂ R2. All integrals
considered will be Lebesgue integrals. The diameter of a set
A⊂ R2is denoted by diam(A). The topological boundary of
a set A⊂ R2 is denoted by ∂A. P o(λ) represents a Poisson
RV with mean λ. The symbols O,Θ, Ω, o, ∼ always refer to the limit n→ ∞. To avoid trivialities, we tacitly assume n to
be sufficiently large if necessary. For simplicity of notation, the dependence of sets and random variables on n will be frequently suppressed.
A. Geometric Preliminaries
The lemmas given in this subsection are from [10], and we will skip their proof. If u − v = 1/√π, a straightforward
Fig. 8. The cells intersecting with the polygon form a polyquadrate.
calculation yields that|Luv| = 23 − √
3
2π = β10. Let R0 denote
the minimum of the radius of curvature over ∂D. We have the following lemma.
Lemma 5: For any u, v∈ D, if u − v ≤ R0 then |Luv∩ D| ≥ |Luv| /2.
For two nearby lunes, we use the following lemma to estimation their areas.
Lemma 6: Assume c = 0.039, R > 0, and a1, b1, a2, b2∈
R2. Let z
1=12(a1+ b1), r1= a1− b1, z2=12(a2+ b2) ,
and r2= a2− b2. If r1, r2∈ 12R, R ,z1− z2 ≤√3R, a1, b1∈ L/ a2b2, and a2, b2∈ L/ a1b1, then
|La1b1∪ La2b2| − |La1b1| ≥ cR z1− z2 .
For any convex compact set C⊂ R2, we use C
−rto denote
the set of points of C that are away from ∂C by at least r.
Lemma 7: Suppose that C ⊂ R2 is a convex compact set
with diameter at most d. Then,
|C−r| ≥ |C| − πdr.
An ε-tessellation divides the plane by vertical and horizontal lines into a grid in which each grid cell has width ε. Without loss of generality, we assume the origin is a corner of cells. In a tesselation, a polyquadrate is a collection of cells intersecting with a convex compact set. For example, in Fig. 8, the shaded cells form a polyquadrate induced by a polygon. The horizontal span of a polyquadrate is the horizontal distance measured in the number of cells from the left to the right. The vertical span of a polyquadrate is defined similarly but in the vertical direction. If the span of a convex compact set is s and the width of each cell is l, the span of the corresponding polyquadrate is at mosts/l + 1.
Lemma 8: If S consists of m cells and τ is a positive
integer constant, the number of polyquadrates with span at most τ and intersecting with S isΘ (m).
Now, we introduce a technique to obtain the Jacobian determinant in the change of variables that will be implicitly used in the proof of Lemma 4. Assume a tree topology is fixed over x1, x2,· · · , xk ∈ R2. Without loss of generality,
we may assume (xk−1, xk) is one of edges. Let zk−1 = 1
2(xk−1+ xk), r = 12xk− xk−1, and θ be the slope of
xk−1xk. For 1 ≤ i ≤ k − 2, we use p (xi) to denote xi’s
parent in the tree rooted at xk, and let zi = 1
2(xi+ p (xi)).
Let I2denote a2×2 identity matrix and 0 denote a 2×2 zero matrix. Then, the Jacobian determinant for changing variables
x1,· · · , xk−1, xk by z1,· · · , zk−1,(r, θ) is ∂(x1,· · · , xk−1, xk) ∂(z1,· · · , zk−1, r, θ) = ∂(x1+ p (x1) , · · · , xk−1+ p (xk−1) , xk) ∂(z1,· · · , zk−1, r, θ) = 4k−1 ∂ x1+p(x1) 2 ,· · · ,xk−1+p(x2 k−1), xk ∂(z1,· · · , zk−1, r, θ) = 4k−1 ∂(z1,· · · , zk−1, xk− zk−1) ∂(z1,· · · , zk−1, r, θ) = 4k−1 I2 · · · 0 0 .. . ... ... ... 0 · · · I2 0
0 · · · 0 cos θ −r sin θsin θ rcos θ
= 4k−1r.
In the first equality, each non-root variable is added by its parent variable. The equality stands since the Jacobian determinant is equal to 1 as we add one variable to another. We remark if the function to be integrated is independent of the variable θ, then after changing variables, the integral over
θ is equal to2π. Actually, this is the most case in this paper. B. Preliminaries of Poisson RVs
We first present an estimation of the lower-tail distribution of Poisson RVs.
Lemma 9: For any μ∈ (0, 1),
lim λ→∞Pr (P o (λ) ≤ μλ) = 1 √ 2π 1 √ μ(1 − μ) 1 √ λe −λφ(μ).
Proof: In this proof, the symbol ∼ refers to the limit λ→ ∞. First, for any μ ∈ (0, 1), we show that the lower tail
distribution of a Poisson RV can be given by
Pr (P o (λ) ≤ μλ) ∼ 1 − μ1 Pr(P o (λ) = μλ). Since Pr (P o (λ) = k − 1) Pr (P o (λ) = k) = λk−1 (k−1)!e−λ λk k!e−λ = k λ, we have Pr (P o (λ) ≤ μλ) = 0 k=μλ Pr (P o (λ) = k) = μλ k=0 k!μλk λk Pr (P o (λ) = μλ) ∼ μλ k=0 (μλ)k λk Pr (P o (λ) = μλ) ∼ 1 1 − μPr (P o (λ) = μλ) .
By Sterling’s formula, we have Pr (P o (λ) ≤ μλ) ∼ 1 1 − μ λμλ (μλ)!e−λ ∼1 − μ1 √ λμλ 2πμλ (μλ)μλe−μλe−λ =1 − μ1 √ 1 2πμλμμλe −λ+μλ =1 − μ1 √ 1 2πμλe−λ+μλ−μλ ln μ =√1 2π 1 √ μ(1 − μ) 1 √ λe −λ(1−μ+μ ln μ) =√1 2π 1 √ μ(1 − μ) 1 √ λe −λφ(μ).
Thus, the lemma is proved.
Assume Y is a Poisson RV with large mean. If Y generates an output, the outcome should be close to the mean with high probability. But as Y generates more outputs, the outcomes are more diverse, and the minimum over the outcomes become smaller. Corresponding to this simple observation, the follow-ing lemma gives an quantitative result about the minimum over a collection of Poisson RVs and it will be used in the proof of Lemma 3.
Lemma 10: Assume that limn→∞ln nλn = β for some β >
1. Let Y1, Y2,· · · , YIn be In Poisson RVs with means at least λn.
1) If In = on√ln n
, then for any 1 < β < β,
minIn
i=1Yi>L (β ) ln n a.a.s..
2) If In = O n
ln n
, then for any 1 < β < β,
minIn
i=1Yi> 12L (2β ) ln n a.a.s..5
Proof: We first assume that Y1, Y2,· · · , YIn all have
means λn. Let Y be a Poisson RV with mean λn. We claim
that for any μ >0, Pr In min i=1 Yi≤ μλn ≤ InPr [Y ≤ μλn] .
To prove that this holds, let Xi be the indicator of the event Yi ≤ μλn. Then Xi is a Bernoulli RV with probability
Pr [Y ≤ μλn]. Let X = X1+ · · · + XIn. Then,minIi=1n Yi ≤ μλn if and only if X≥ 1. By Markov’s inequality,
Pr In min i=1 Yi≤ μλn = Pr [X ≥ 1] ≤ E [X] = In i=1 E[Xi] = InPr [Y ≤ μλn] .
Now, assume that In = on√ln n
. Since L (β ) < L (β) = βφ−1(1/β), we have L (β ) /β < φ−1(1/β).
We choose a constant μ ∈ L (β ) /β, φ−1(1/β). Then, μ∈ (0, 1) , μβ > L (β ) and βφ (μ) > 1. Thus, for sufficiently
large n, μλn≥ L (β ) ln n, which implies that
Pr In min i=1 Yi≤ L (β ) ln n≤ PrminIn i=1Yi≤ μλn ≤ InPr [Y ≤ μλn] . 5For two sequences fn and gn, we write fn = O (gn) if there exist
constants c and n0such that|fn| ≤ c |gn| for all n ≥ n0.
By Lemma 9, Pr In min i=1 Yi≤ L (β ) ln n √1 2πβ 1 √ μ(1 − μ) In n√ln nn 1−(λn/ ln n)φ(μ). Since 1 − (λn/ln n) φ (μ) → 1 − βφ (μ) < 0, we have Pr I n min i=1Yi≤ L (β ) ln n= o (1) . Hence minIn i=1Yi>L (β ) ln n a.a.s..
Next, assume that In = O
n ln n . Since L (2β ) < L (2β), we have L (2β ) / (2β) < φ−1(1/ (2β)). We choose a constant μ ∈ L (2β ) / (2β) , φ−1(1/ (2β)). Thus, μ ∈ (0, 1) , μβ > 1
2L (2β ) and βφ (μ) > 1/2. Thus, for
suffi-ciently large n, μλn≥ 12L (2β ) ln n, which implies that
Pr In min i=1Yi≤ 1 2L (2β ) ln n ≤ Pr In min i=1Yi≤ μλn ≤ InPr [Y ≤ μλn] . By Lemma 9, Pr I n min i=1 Yi ≤ 1 2L (2β ) ln n √1 2πβ 1 √ μ(1 − μ) In √ nln nn 1/2−(λn/ ln n)φ(μ). Since 1/2 − (λn/ln n) φ (μ) → 1/2 − βφ (μ) < 0, we have Pr In min i=1Yi≤ 1 2L (2β ) ln n = o (1) . Hence minIn i=1Yi> 12L (2β ) ln n a.a.s..
Finally, we consider that general case that Y1, Y2,· · · , YIn
have means λn,1, λn,2,· · · , λn,In respectively with λn,i≥ λn
for each 1 ≤ i ≤ In. Let Y1 , Y2 ,· · · , YI n be In Poisson
RVs with means λn. For each 1 ≤ i ≤ In, let Yi be a
Poisson RV with mean λn,i− λn which is independent with Yi . Then by the superposition property of Poisson RVs, Yi= Yi + Yi . Therefore,minIn
i=1Yi ≥ minIi=1n Yi > μλn. By the
above argument, the lemma also holds in this general case. At the end of this subsection, we state the Palm theory [11] on the Poisson process.
Theorem 11: Let n > 0. Suppose k ∈ N, and h (Y, X ) is a bounded measurable function defined on all pairs of the form (Y, X ) with X ⊂ R2 being a finite subset andY being
a subset of X , satisfying h (Y, X ) = 0 except when Y has k
elements. Then E ⎡ ⎣ Y⊆Pn h(Y, Pn) ⎤ ⎦ =nk k!E [h (Xk,Xk∪Pn)]
where the sum on the left-hand side is over all subsets Y of
the random Poisson point setPn, and on the right hand side
the setXk is a binomial process with k nodes, independent of Pn.
We need to estimate the number of subsets with some specified topology, for example, two nodes are local minima w.r.t. each other. But it is not so easy to estimate this among Poisson point processes. The Palm theory allows us to place a set of random points first and then estimate the expectation over the Poisson point process. This technique will be used in the proof of Lemma 4.
C. Proof of Lemma 3
To have the lower bound for minimal scan statistics, we ap-ply the tessellation technique to discrete the scanning process. The deployment region is tessellated into equal-size square cells by properly choosing the cell size such that: (1) each copy of the lune contains a polyquadrates with area at least
cln nn for some c >1 (or 12cln nn if the copy crosses ∂D), and
(2) the number of polyquadrates is O n ln n (or O n ln n if the copy crosses ∂D). Then, the lemma follows Lemma 10. The detail is given below.
Proof: For a given β1, choose a constant β2 ∈ (β1, β).
Let ε = 1 6√2β0 1 − β2 β , d = √3rn, and consider an
εd-tessellation. (Note that ε is chosen such that each copy of the lune contains a polyquadrate with area at least cln n for some
c >1.) Let In denote the number of polyquadrates inD with
span at most 1ε and area at least β2 β0 πrn2 β = β2 β0 + o (1) ln n n ,
and Yibe the number of nodes on the i-th polyquadrate. Then Yi is a Poisson RV with rate at least
β2 β0+ o (1)
ln n. Since the number of cells inD is O n
ln n , by Lemma 8, In= O 1 εd 2 = Oln nn .
By Lemma 10, it is a.a.s. that minIn i=1Yi ln n ≥ L β2 β0 >L β1 β0 . Now, let I
n denote the number of polyquadrates inD \ D−d
with span at most 1
ε and area at least 12 β2 β0 πr2n β = 1 2 β2 β0 + o (1) ln n
n , and Yi be the number of nodes on the i-th polyquadrate. Then Yi is a Poisson RV with rate at least 1 2 β2 β0 + o (1)
ln n. Since the number of cells in D \ D−dis
Oln nn , by Lemma 8, In = O 1 εd = O n ln n .
By Lemma 10, it is a.a.s. that minIn i=1Yi ln n ≥ 1 2L β2 β0 >1 2L β1 β0 .
Therefore, it is a.a.s. that minminIn i=1Yi,minI n i=1Yi ln n > 1 2L β1 β0 .
Thus, the lemma follows if we can show that
S (Pn, rn) ≥ min In min i=1 Yi, In min i=1Y i .
To prove this inequality, it is sufficient to show that for any lune L of two points inD which are separated by a distance of
rn, it either contains a polyquadrate in D with span at most 1
ε and area at least β2 β0
πrn2
β , or contains a polyquadrate in
D \ D−d with span at most 1ε and area at least 12β2 β0
πrn2 β . We
shall prove this in two cases.
Case 1: L is contained inD. Let P denote the polyquadrate induced by L−√
2εd. Then, P ⊆ L ⊆ D, and the span of P is
at mostd−2√2εd εd
+ 1 ≤ 1
ε. By Lemma 7 and using the fact
that|L| = πr2 n/β0= πd2/(3β0), we have |P | ≥ L−√2εd ≥ |L| − πd√2εd = |L| −√2επd2 = |L|1 − 3√2β0ε >|L| 1 − 6√2β0ε = β2 β |L| =β2 β0 πrn2 β .
Case 2: L is not contained inD. Then L must be disjoint withD−d. Let L = L ∩D and let P denote the polyquadrate
induced by L
−√2εd. Then P ⊆ L ⊆ D \ D−d and the the
span of P is also at most 1
ε. By Lemma 7 and Lemma 5, we
have |P | ≥ L −√2εd ≥ |L | − πd √ 2εd≥ 1 2|L| − √ 2πεd2 =12|L|1 − 6√2β0ε = 12β2 β |L| = 1 2 β2 β0 πr2n β .
Thus, the lemma is proved.
D. Proof of Lemma 4
We introduce several relevant events and derive their prob-abilities. For convenience, we use R1and R2as shorthand for
R1(n) and R2(n), respectively. Note that 12R2 ≤ R1 ≤ R2
and π2 c2 1 − R1 R2
< 1. Let A denote the disk with radius 1
2
ln n
nπ at the center of the first cell. Assume V is a point
set and T ⊂ V . Let h1(T, V ) denote a function such
that h1(T = {x1, x2} , V ) = 1 only if 12(x1+ x2) ∈ A, R1 ≤ x1− x2 ≤ R2, and there is no other node of V
in the lune area Lx1x2; otherwise, h1(T, V ) = 0. Then, E1
is the event that there exist two nodes X, Y ∈ Pn such that h1({X, Y } , Pn) = 1. In addition, under Boolean addition,
for any {x1, x2, x3} ⊆ V , let
h2({x1, x2, x3} , V ) = h1({x1, x2} , V ) · h1({x1, x3} , V ) + h1({x2, x1} , V ) · h1({x2, x3} , V ) + h1({x3, x1} , V ) · h1({x3, x2} , V ) ;
for any {x1, x2, x3, x4} ⊆ V , let
h3({x1, x2, x3, x4} , V ) = h1({x1, x2} , V ) · h1({x3, x4} , V ) + h1({x1, x3} , V ) · h1({x2, x4} , V ) + h1({x1, x4} , V ) · h1({x2, x3} , V ) .
For the sake of clarity, in the remaining of this sub-section, we use X1, X2, X3 and X4 to denote inde-pendent random points with uniform distribution over D and independent of Pn, and X1 , X2 , X3 and X4 to
de-note elements of Pn. Let F1 ({X1 , X2 }) be the event that h1({X1 , X2 } , Pn) = 1; F2 ({X1 , X2 , X3 }) be the event
that h2({X
1, X2 , X3 } , Pn) = 1; and F3 ({X1 , X2 , X3 , X4 })
be the event that h3({X
1, X2 , X3 , X4 } , Pn) = 1. Applying
Boole’s inequalities which is a special case of the inclusion-exclusion principle, we have
Pr [E1] ≥ {X 1,X2}⊆Pn Pr F1 X1, X2 − {X 1,X2,X3}⊆Pn Pr F2 X1, X2, X3 − {X 1,X2,X3,X4}⊆Pn Pr F3 X1, X2, X3, X4 . (6)
Let F1 be the event that
h1({X1, X2} , {X1, X2} ∪ Pn) = 1, F2 be the event that h2({X1, X2, X3} , {X1, X2, X3} ∪ Pn) = 1, and F3 be the
event that h3({X1, X2, X3, X4} , {X1, X2, X3, X4} ∪ Pn) =
1. According to the Palm theory (Theorem 11), we have {X1,X2}⊆Pn Pr [F 1({X1 , X2 })] = E ⎡ ⎢ ⎣ {X1,X2}⊆Pn h1({X1 , X2 } , Pn) ⎤ ⎥ ⎦ = n2!2E [h1({X1, X2} , {X1, X2} ∪ Pn)] = n22Pr [F1] ; (7) {X1,X2,X3}⊆Pn Pr [F 2({X1 , X2 , X3 })] = E ⎡ ⎢ ⎣ {X1,X2,X3}⊆Pn h2({X1 , X2 , X3 } , Pn) ⎤ ⎥ ⎦ =n3!3E [h2({X1, X2, X3} , {X1, X2, X3} ∪ Pn)] = 3n3!3Pr [F2] =n 3 2 Pr [F2] ; (8) and {X1,X2,X3,X4}⊆Pn Pr [F 3({X1 , X2 , X3 , X4 })] = E ⎡ ⎢ ⎣ {X1,X2,X3,X4}⊆Pn h3({X1 , X2 , X3 , X4 } , Pn) ⎤ ⎥ ⎦ =n4!4E [h3({X1, X2, X3, X4} , {X1, X2, X3, X4} ∪ Pn)] = 3n4!4Pr [F3] = n 4 8 Pr [F3] . (9)
From Eq. (6), (7), (8), and (9), we have Pr [E1] ≥ n 2 2 Pr [F1] − n3 2 Pr [F2] − n4 8 Pr [F3] . (10) In the next, we derive the probabilities of F1, F2, and F3. Let S1(R1, R2) denote the set
(x1, x2) 12(x1+ x2) ∈ A, R1≤ x1− x2 ≤ R2 .
For simplicity, S1 is shorthand for S1(R1, R2). We have
Pr [F1] = ! ! S1 Pr [F1| X1= x1, X2= x2] dx1dx2 = ! ! S1 e−n|Lx1x2|dx1dx2 = ! ! S1 e−nβ01πx1−x22dx 1dx2. Let z= x1+x2 2 and r= 12x1− x2. Then, Pr [F1] = ! z∈A ! R2 2 r=R12 e−β04 nπr28πrdrdz = 4 ! z∈A ! R2 2 r=R12 e−β04nπr22πrdrdz = 4 ! z∈A ! R2 2 r=R12 e−β04nπr2dπr2dz = − ⎛ ⎝ β0 ne − 4 β0nπr2 R2 2 r=R12 ⎞ ⎠ |A| = β0 4n2 n−β1β0 − n−β2β0 ln n. (11)
Let S2(R1, R2) denote the set
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩(x1 , x2, x3) x1+x2 2 ,x1+x2 3 ∈ A; R1≤ x1− x2 ≤ R2; R1≤ x1− x3 ≤ R2; x1, x2∈ L/ x1x3; x1, x3∈ L/ x1x2 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ .
Again, for simplicity, S2is shorthand for S2(R1, R2).
Apply-ing Lemma 6, if(x1, x2, x3) ∈ S2, we have
Pr [F2|X1= x1, X2= x2, X3= x3] ≤ e−n|Lx1x2∪Lx1x3| ≤ e−n 1 β0πx1−x22+cR2x1+x22 −x1+x32 . Therefore, Pr [F2] = ! ! ! S2Pr [F2|X1 = x1, X2= x2, X3= x3] dx1dx2dx3 ≤ ! ! ! S2e −n 1 β0πx1−x22+cR2---x1+x22 −x1+x32 -- - dx1dx2dx3. Let z1 = x1+x2 2 , r1 = 12x1− x2, z2 = x1+x2 3, and ρ= z1− z2. Then, Pr [F2] ≤ 16 ! z1∈A ! R2 2 r1=R12 ! z2∈A e−n 4 β0πr21+cR2z1−z2 2πr1 · dr1dz1dz2 ≤ 16 ! z1∈A ! R2 2 r1=R12 e−β04nπr212πr 1dr1dz1 · ! z2∈A e−cnR2z1−z2dz 2 ≤ 16 ! z1∈A ! R2 2 r1=R12 e−β04nπr21dπr2 1 dz1
· ! ∞ ρ=0 e−cnR2ρ2πρdρ = − ⎛ ⎝ 4β0 n e − 4 β0nπr2 R2 2 r=R12 ⎞ ⎠ |A| 2π (cnR2)2 = 2πβ0 c2(nR22) n3 n−β1β0 − n−β2β0ln n. (12)
Let S3(R1, R2) denote the set
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩(x1 , x2, x3, x4) x1+x2 2 ,x3+x2 4 ∈ A; R1≤ x1− x2 ≤ R2; R1≤ x3− x4 ≤ R2; x1, x2∈ L/ x3x4; x3, x4∈ L/ x1x2 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ .
Again, for simplicity, S3is shorthand for S3(R1, R2).
Apply-ing Lemma 6, if(x1, x2, x3, x4) ∈ S3, we have
Pr [F3|X1= x1, X2= x2, X3= x3, X4= x4] ≤ e−n|Lx1x2∪Lx3x4| ≤ e−n 1 β0πx1−x22+cR2x1+x22 −x3+x42 . Therefore, Pr [F3] = ! ! ! ! S3Pr [F3|X1= x1, X2= x2, X3= x3, X4 = x4] · dx1dx2dx3dx4 ≤ ! ! ! ! S3e −n 1 β0πx1−x22+cR2---x1+x22 −x3+x42 -- - · dx1dx2dx3dx4. Let z1 = x1+x2 2, r1 = 12x1− x2, z2 = x3+x2 4, r2 = 1 2x3− x4, and ρ = z1− z2. Then, Pr [F3] ≤ ! z1∈A ! R2 2 r1=R12 ! z2∈A ! R2 2 r2=R12 e −n 4 β0πr21+cR2z1−z2 · (8πr1dr1dz1) (8πr2dr2dz2) ≤ 4 ! z1∈A ! R2 2 r1=R12 e −4 β0nπr212πdr1dz1 · 8πR22 R2 2 − R1 2 ! z2∈Ae −cnR2z1−z2dz2 ≤ 4 ! z1∈A ! R2 2 r1=R12 e −β04nπr21 dπr12 dz1 · 8π R22 R2 2 − R21 ! ∞ ρ=0e −cnR2ρ2πρdρ = β0ln n 4n2 n−β1β0 − n−β2β0 4π2 (cnR2)2R2(R2− R1) = π2β0 c2n4 1 − R1 R2 n−β1β0 − n−β2β0 ln n. (13)
Put Eq. (10), (11), (12) and (13) together. We have
Pr [E1] ≥ β0 8 − πβ0 c2(nR22)− π2β0 8c2 1 −R1 R2 n−β1β0 − n−β2β0 ln n ∼ β80 1 − π2 c2 1 − R1 R2 n−β1β0 − n−β2β0 ln n.
Recall that for a given β, β1 and β2 are constants, and so are β1 β0 and β2 β0. According to Eq.(3), R1 R2 also is a constant.
We write fn = Ω (gn) for two sequences fn and gn if there
exist constant c1 >0 and n0 such that |fn| ≥ c1|gn| for all n≥ n0. From Eq. (2), we have
Pr [E1] = Ω n−β1β0 − n−β2β0 ln n. Since In = Ω n ln n
from Eq. (4), we have
InPr [E1] = Ω
n1−β1β0
→ ∞.
This complete the proof of Lemma 4.
ACKNOWLEDGMENT
The work of P.-J. Wan is supported in part by the NSF under Grant 557904 and CityU of Hong Kong under Grant 7200031.
This work of C.-W. Yi is supported in part by the NSC under Grant No. NSC95-2221-E-009-059-MY3 and NSC97-2221-E-009-052-MY3, by the ITRI under Grant No. 7352B12100, and by the MoE ATU plan.
The work of F. Yao described in this paper was partially supported by grants from the Research Grants Council of the Hong Kong SAR, China, under Project No. 122105, 122807, and the National Basic Research Program of China Grant 2007CB807900, 2007CB807901.
The work of X. Jia is partially supported by a grant from Research Grants Council of Hong Kong [Project No. CityU 114006].
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Peng-Jun Wan received his PhD degree from
Uni-versity of Minnesota, MS degree from The Chinese Academy of Science, and BS degree from Tsinghua University. He is currently an Associate Professor in Computer Science at Illinois Institute of Tech-nology, and at City University of Hong Kong. His research interests include wireless networks, optical networks, and algorithm design and analysis.
Chih-Wei Yi received his Ph.D. degree from the
Illinois Institute of Technology, and MS and BS degrees from the National Taiwan University. He is currently an Assistant Professor in Computer Science at the National Chiao Tung University. His research focuses on wireless ad hoc and sensor networks.
Lixin Wang received the M.S. degree in CS from
the University of Houston at Clear Lake, the M.S. degree in Applied Math from the University of Houston and the M.S. degree in Math from the Fudan University, Shanghai, China. He is currently a Ph.D. student in Computer Science at the Illinois Institute of Technology, Chicago. His research is on wireless networks, and algorithm design and analysis.
Frances Yao received her BSc (1969) from
Na-tional Taiwan University and her Ph.D. (1973) in Mathematics from Massachusetts Institute of Tech-nology. She is currently Head of the Department of Computer Science at City University of Hong Kong. Her research interests include combinatorial and ge-ometric algorithms, energy-efficient computing and sensor networks. She is a Fellow of AAAS.
Xiaohua Jia received his BSc (1984) and MEng
(1987) from the Univ of Science and Technolog of China, and obtained his DSc (1991) in Information Science from the Univ. of Tokyo, Japan. Prof. Jia is currently associated with Dept of Computer Science at City Univ of Hong Kong. His research inter-ests include distributed systems, computer networks, WDM optical networks, and Internet and mobile computing.
