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# Trajectory-Based Multi-hop Relay Deployment in Wireless Networks

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## Abstract

In this paper, we identify a novel problem Trajectory-Based Relay Deployment (TBRD) which aims at maximizing user connection time as the users roam through the target area while complying with relay resource constraints. To solve the TBRD, we first propose the concept Demand Nodes (DNs). Next, we design a Demand Node Generation (DNG) algorithm that transforms the continuous historical user trajectory into a number of discrete DNs. By generating DNs, we convert the TBRD problem into a Demand Node Coverage (DNC) problem, which is NP-complete. After that, we design an approximation algorithm, named Submodular Iterative Deployment Algorithm (SIDA), to solve the DNC problem with the approximation factor $$1-\frac{1}{\sqrt{e\cdot (1-1/k)}}$$. The simulation on five real datasets shows that our algorithm can obtain high coverage for users in motion, leading to better user experience.
Trajectory-Based Multi-Hop Relay Deployment
in Wireless Networks
Shilei Tian, Haotian Wang, Sha Li, Fan Wu, and Guihai Chen?
Shanghai Key Laboratory of Scalable Computing and Systems
Department of Computer Science and Engineering, Shanghai Jiao Tong University,
Shanghai 200240, China
{tianshilei,ltwbwht,Zoey.Lee}@sjtu.edu.cn, {fwu,gchen}@cs.sjtu.edu.cn
Abstract. In this paper, we identify a novel problem Trajectory-Based
Relay Deployment (TBRD) which aims at maximizing user connection
time as the users roam through the target area while complying with
relay resource constraints. To solve the TBRD, we ﬁrst propose the con-
cept Demand Nodes (DNs). Next, we design a Demand Node Generation
(DNG) algorithm that transforms the continuous historical user trajec-
tory into a number of discrete DNs. By generating DNs, we convert the
TBRD problem into a Demand Node Coverage (DNC) problem, which is
NP-complete. After that, we design an approximation algorithm, named
Submodular Iterative Deployment Algorithm (SIDA), to solve the DNC
problem with the approximation factor 1 1
e·(11/k). The simulation
on ﬁve real datasets shows that our algorithm can obtain high coverage
for users in motion, leading to better user experience.
1 Introduction
With the explosive growth of mobile users, wireless coverage has become an
increasingly challenging problem. However, due to transmission distance, limited
coverage of Access Point (AP), path loss, and so forth, the signal quality at
some locations fails to provide satisfactory Internet access [3]. Deploying relays
in multi-hop networks has become an eﬀective method to improve the wireless
coverage and service quality [6].
In this paper, we investigate the relay deployment problem under the non-
stationary user setting. Due to limited transmission power and path loss, the
base station (BS) may fail to cover all users at all times. We hope to deploy
a limited number of relays to keep users connected to the Internet as long as
possible when they are wandering.
?G. Chen is the corresponding author.
This work was supported in part by Program of International S&T Cooperation
(2016YFE0100300), the State Key Development Program for Basic Research of
China (973 project 2014CB340303), China NSF Projects (Nos. 61672348, 61672353,
61422208, and 61472252), and CCF-Tencent Open Research Fund.
2
Existing works designed algorithms according to the exact user locations,
they all assumed that users are stationary, which is not realistic in practice. As
a result, once a user location changes, the network performance will be aﬀected.
In fact, the movements of users within an area are not completely random.
They are strongly aﬀected by people’s social demands [4]. Therefore, some hot
spots, which mean locations where users often pass, or linger around, can be
inferred from the user trajectory. Therefore, we consider utilizing the historical
user trajectory to infer the tendency of the user movement and deploy the relays.
In this paper, we ﬁrst deﬁne the connectivity of the network. Since relays
cannot access the Internet directly, each relay must have a path to the BS.
Then we deﬁne the Trajectory-Based Relay Deployment (TBRD) problem, which
aims at maximizing user connection time as the users roam through the target
area while complying with relay resource constraints. We introduce a concept
Demand Nodes (DNs), which are virtual weighted nodes representing locations
where users often pass or stay for a long time. Next, we propose a matrix-
based trajectory representation and design the Demand Node Generation (DNG)
algorithm. After that, the original TBRD problem is converted to a new problem
called Demand Node Coverage (DNC). We claim that a DN is covered if its
distance to an AP is less than the coverage radius of the AP. The DNC problem
is to maximize the total weight of DNs covered by deployed relays and BS.
The DNC is NP-complete, which can be reduced from a known NP-complete
problem named budget set cover (BSC) [2]. To tackle this problem, we propose
an approximation algorithm, named Submodular Iterative Deployment Algorithm
(SIDA), which has an approximation ratio of 1 1
e·(11/k), where eis the
mathematical constant, and kis the relay number constraint. Finally, we use
real datasets to evaluate our algorithm. The simulation results indicate that our
algorithm can perform well.
The paper is organized as follows. The problem statement is given in Section
2. Section 3 describes the DNG algorithm. Section 4 presents the SIDA. Simu-
lations are demonstrated in Section 5. Finally, Section 6 concludes this paper.
2 Problem Statement
2.1 System Model
In this model, the user can either communicate with BS directly, or connect to
BS with the help of relays. Since too many hops will lead to a high delay, we
limit the number of communication hops to 2.
2.2 Problem Deﬁnition
The user trajectory set is denoted by T.PBand PRrepresent the set of BS and
relay candidate positions respectively. kis the number of relays we can deploy.
Deﬁnition 1 (Communication Radius). Two APs can communicate with
each other within a communication radius. We use dBand dRto denote the
communication radius of BS and relay respectively.
3
Deﬁnition 2 (2-Hop Relay Connectivity). Given the AP candidate position
set P=PBPR, we generate a weighted graph G= (P, E), where (pi, pj)E
if the distance between these two locations is less than the corresponding com-
munication radius. The weight of each edge is set to 1.2-hop relay connectivity
means that in the induced graph G[F], there always exists a path between any
selected relay node and the selected BS node, while its distance is less than or
equal to 2.
Deﬁnition 3 (TBRD Problem). Given a set of trajectories T, BS candidate
locations PB, relay candidate locations PR, relay number constraint k, the TBRD
problem is to ﬁnd a BS location pBPBand relay locations PSPRto
maximize user connection time. PSmust be subject to |PS|=k, and the induced
subgraph G[{pB} ∪ PS]has 2-hop relay connectivity.
As we mentioned before, hot spots can be inferred from historical user trajectory.
We introduce a novel concept called Demand Node (DN) to represent them.
Deﬁnition 4 (Demand Node). Demand Nodes (DNs) are virtual weighted
nodes representing the locations where users often pass or stay for a long time.
They are at the center of the grids which are generated by the division of the target
area. The weight is the probability of user’s appearance in the corresponding
location. The larger the weight is, it is more possible that users will pass through
or stay at the corresponding location.
rRfor a relay, is the distance threshold for the BS or relay. Only DNs whose
distance to an AP is less than its coverage radius can ensure Internet connection
for users. We say that the DN is covered by the corresponding AP.
Before we introduce the Demand Node Coverage (DNC) problem, we ﬁrst give
some deﬁnitions that are used throughout this paper. We use Dto denote the
DNs set, and Wfor the weight set of DNs.
Deﬁnition 6 (Covered DNs Set). The covered DNs set C(·)is the set of
DNs covered by a given AP. For a BS candidate location pi
BPB,C(pi
B) =
{dj|dist(dj, pi
B)rB}where dist(·)denotes the Euclidean distance. For a relay
candidate location pi
RPR,C(pi
R) = {dj|dist(dj, pi
R)rR}.
Deﬁnition 7 (Weight Function). The weight function w(·)is the sum of
weights of the covered DNs set. For an AP candidate location pPBPR,
w(p) = PsiC(p)wsi. For an AP candidate location set P, the DNs covered by
Pare represented as DC=piPC(pi),w(P) = PsiDCwsi.
Deﬁnition 8 (Residual Weight). Considering a selected AP candidate loca-
tion set SA, when we continue to select a AP candidate location set SB, the resid-
ual weight of SBbased on SAis deﬁned as wR(SA, SB) = w(SB)w(SASB).
Assume the width of the target area is w, and the height is h. There is also a
ﬁlter threshold θ, which constrains the weight of each generated DN to be larger
than θ. Now we can deﬁne the DNG problem.
4
Deﬁnition 9 (DNG Problem). Given a user trajectory set T, the width w
and height hof the target area, and a ﬁlter threshold parameter θ, the DNG
problem is to generate a set of DNs Dand a relative weight set W. The weight
of each DN is in the range of [θ, 1].
Now we can deﬁne the Demand Node Coverage (DNC) problem.
Deﬁnition 10 (DNC Problem). Given a set of DNs Dand the corresponding
weight set W, BS candidate locations PB, relay candidate locations PR, relay
number constraint k, the DNC problem is to ﬁnd a location pBPB, and relay
candidate locations subset PSPRto maximize w(F), where F={pB} ∪ PS
while |PS|=k. The induced subgraph complies with the 2-hop relay connectivity
constraint.
3 Demand Node Generation
In this section, we show how to extract “hotspots” which we refer to as Demand
Nodes (DNs) from user trajectories. The Demand Node Generation (DNG) al-
gorithm consists of three major steps: (1) trajectory matrix generation; (2) pre-
diction matrix; (3) ﬁltering.
3.1 Trajectory Matrix Generation
Since the DNs depend on both the temporal and spatial information of the user
trajectory, we segment each trajectory according to a ﬁxed time span tand
record the location of each segment where the user appears in the target area
by a binary matrix. Fig. 1 illustrates the details of converting a trajectory into
a binary matrix. Fig. 1(a) shows a trajectory in the area.
(a) Original (b) Segments (c) 6 ×6 grids
10
0
0
0
1 1 1
1 1
0 00 0
0
0
0
0
0
0
000
0
00 0
0
0
0
000
000
(d) Matrix
Fig. 1. An illustration of the process of a trajectory.
Firstly, we divide the trajectory into a number of segments, and each segment
shows the trajectory of a user at the corresponding time span t, as shown in
Fig. 1(b). Then, the target area is further partitioned into small sizes of grids
which are the candidate locations for the demand nodes. Fig. 1(c) shows the
distribution of the upper left segment. Lastly, Fig. 1(d) shows the binary matrix
5
of the trajectory at one time segment. The whole trajectory area is seen as a
matrix and entries of the matrix represent the partitioned grids. If the trajectory
passes through the grid, the corresponding entry of the matrix is set to 1.
After the conversion, we obtain numerous binary matrices for the target area.
3.2 Prediction Matrix
Since the value of a grid xij is 0 or 1, we assume the probability distributions
of these grids are independent Bernoulli distributions, which can be written as
xij p(xij |µij ) = µxij
ij (1 µij )1xij , where the parameter µij [0,1] is the
probability of xij = 1.
We can estimate the µij by maximizing likelihood estimation. However, this
may lead to over-ﬁtted results for small datasets [1]. In order to alleviate this
problem, we ﬁrst introduce a prior distribution p(µij|aij , bij ), beta distribution,
over the parameter µij, which is easy to interpret while having some properties.
The posterior distribution of µij is now obtained by Bayesian theorem
p(µij |Xij ) = p(Xij |µij )p(µij |aij , bij )
Rp(Xij |µij )p(µij |aij , bij )ij
.(1)
Then, we estimate the value of µij by maximizing the posterior distribution
p(µij |xij ). We see that this posterior distribution has the form
p(µij |Xij )µm+aij 1
ij (1 µij )nm+bij 1.(2)
Finally, maximizing Eq. (2) with respect to µij , we obtain the maximum
posterior solution given by µij =m+aij
n+aij +bij .
3.3 Filtering
After the prediction matrix of the target area is determined, the DNs are at the
center of those grids with higher probabilities for 1. In our model, a threshold θ
is set, and the grids whose probabilities for 1 are not less than θare DNs.
4 Submodular Iterative Deployment Algorithm (SIDA)
We now focus on selecting the locations for APs from the candidate location set.
It is clear that the weight function w(·) is a submodular function.
4.1 The SIDA
The main idea of SIDA is as follows. First, we construct an undirected graph
G= (P, E), where P=PBPR. For any two nodes pi, pjP, (pi, pj)Eif
dist(pi, pj) is less than the corresponding communication radius. Then, we scan
each pi
Bsequentially, and generate a subgraph with its 2-hop neighbors. The
following operations are taken within this subgraph.
6
Algorithm 1: SIDA
Input: An instance of DNC problem, hPB, PR, k, w(·), wR(·)i
Output: The ﬁnal solution F
1D← ∅;
2for bPBdo
3k0k;Sb;Vt← {v:hop(v, b)2, v PR};// hop(v, b)is the
least hop number from vto b, the same as below.
4while k0>0and Vt6=Sdo
5j0;
6while j≤ bk0/2cdo
7Find max{wR(S, S ∪ {v}) : vVt};SS∪ {v};jj+ 1;
8for vSdo
9if vis not connected with bthen
10 Vd← {u|uis one hop neighbor of vthat also one hop neighbor
of b}; Find max{wR(S, S ∪ {u}) : uVd};SS∪ {u};
11 k0k− |S|;
12 if w(F)w(S)then
13 FS;
14 return F;
Next, we repeatedly select bk/2ccandidate locations with maximum residual
weight in the subgraph. For each selected candidate location pi, check whether
it is the 1-hop or 2-hop neighbor of the BS. If it is a 2-hop neighbor, then we
check whether those selected locations can construct a path from pito the BS. If
not, we need to select another one pjfrom the 1-hop neighbors of pithat brings
the maximum residual weight while ensuring that pipjBS is a path. In
this way, the number of all selected locations is at most bk/2c×2k. It is very
likely that we still have available relays. Therefore, assume that we have selected
grelays, and g < k, then we run the same procedure on this subgraph with
k=kg. Repeat this procedure and use Sto record all the selected locations,
and it will terminate once |S|=kand Sis a feasible solution.
Finally, choose the solution with the maximum total weight. The details of
SIDA are shown in Algorithm 1.
4.2 Performance Analysis
In this subsection, we analyze the performance guarantee of SIDA. We consider
a BS location, its 2-hop neighbors and the generated subgraph. We propose two
lemma for this subgraph.
Lemma 1. After each greedy iteration li,i= 2, . . . , t,t≤ bk/2c, the inequality
w(Gi)w(Gi1)1
kw(OP T 0)w(Gi1)holds, where Giis the selected set
after i-th iteration, and OP T 0is the optimal solution within the current subgraph.
7
Proof. First, we denote w(Gi)w(Gi1) as W0
i, which is the maximum residual
weight in ith iteration according to the greedy strategy. Clearly, w(O P T 0)
w(Gi1) is no more than the weight of the elements covered by OP T 0, but
not covered by Gi1, i.e. w(O P T 0)w(Gi1)w(OP T 0\Gi1). Since the size
of the set OP T 0\Gi1is bounded by the budget k, the total weight of DNs
covered by O P T 0\Gi1and not covered by Gi1, is at most kW 0
i. Hence we
get w(OP T 0)w(Gi1)kW 0
i. Substituting w(Gi)w(Gi1) for W0
i, and
multiplying both sides by 1/k, we get the required inequality.
Lemma 2. After each iteration li,i= 2, . . . , t,t≤ bk/2c, the inequality w(Gi)
[1 (1 1/k)i]w(OP T 0)holds.
Proof. According to Lemma 1, we have:
k(w(Gi)w(Gi1)) w(OP T 0)w(Gi1)w(Gi)w(OP T 0)
w(Gi1)w(OP T 0)11/k.
Therefore, let j= 1,2, . . . , i, and multiply those inqualities, we can get:
i
Y
j=1
w(Gj)w(OP T 0)
w(Gj1)w(OP T 0)(1 1/k)iw(Gi)w(O P T 0)
w(G0)w(OP T 0)(1 1/k)i.
Since G0=, thus w(G0) = 0, then we have w(Gi)[1 (1 1/k)i]w(O P T 0).
Theorem 1. SIDA achieves an approximation factor of 11
e·(11/k)for the
DNC problem.
Proof. For each BS candidate location, the algorithm iterates for at least bk/2c
times. We suppose OP T is the optimal solution of the DNC problem. For the
subgraph which contains OP T , we denote the set of locations selected by SIDA
as FOP T . Then in the light of Lemma 2, we could get:
wFOP T [1 (1 1/k)bk/2c]w(OP T )h11
pe·(1 1/k)iw(OP T ).
5 Simulations
In this section, we conduct extensive simulation experiments to evaluate our
algorithm via C++. We evaluate our entire procedure including trajectory pro-
cessing, DNG, and SIDA on ﬁve real GPS data [5] from CRAWDAD: NCSU and
KAIST, New York City, Orlando, and North Carolina state fair. We randomly
divide each dataset into training and validation group.
The parameters of each dataset are shown in Table 1. We divide the map
into grids of gB×gBand set the candidate BS locations to the center of these
grids. Similarity, the candidate relay locations are set to the center of gR×gR
grids. dBis set to rR+rB, and dRis set to 2 ×rR. The number of relays we
8
Table 1. Parameters of each dataset
Dataset s rBrRgBgRθ
KAIST 200 1200 600 3000 500 0.14
NCSU 200 1200 600 3000 500 0.21
New York 400 2400 1200 6000 1000 0.21
Orlando 300 1500 1000 5000 1000 0.20
Statefair 20 150 75 350 50 0.35
can deploy is k= 5. For simplicity, both the two parameters aij and bij of beta
distribution are set to 5. The time slot is set to t= 200.
We repeated the partition of the validation set and ran the procedure for
1000 times, and then took the average. The coverage performance for the ﬁve
datasets are 95.10%, 85.83%, 62.52%, 85.44%, and 60.70%, respectively.
6 Conclusion
In this work, we have proposed the Trajectory-Based Relay Deployment (TBRD)
problem in wireless networks, which aims at maximizing user connection time
as the users roam through the target area while complying with relay resource
constraints. We ﬁrst transform the trajectories into a number of virtual weighted
discrete Demand Nodes (DNs). In this way, the original TBRD problem is con-
verted to an NP-complete problem called Demand Node Coverage (DNC) prob-
lem, which is to maximize total covered DN weight. Then, we design an approx-
imation algorithm named Submodular Iterative Deployment Algorithm (SIDA)
to solve the DNC problem, with an approximation ratio of 1 1
e·(11/k). The
simulation on ﬁve real datasets results show that our algorithm can obtain high
coverage performance and thus signiﬁcantly improve the user experience. To the
best of our knowledge, we are the ﬁrst to consider user trajectories for relay
deployment.
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Wireless relay network have been widely used in many application to improve the wireless service. In this paper, we aim to maximize users' satisfaction by deploying limited number of relays in a target region to form a wireless relay network, and define the Deployment of Cooperative Relay (DoCR) problem, which is proved to be NP-complete. We first propose two approximation algorithms, an $O(\log \;n)$ algorithm that utilizes the algorithms for budget weighted Steiner tree problem with novel position weighting assignment, and an $O(\sqrt k)$ algorithm that iteratively scans potential positions and determines relay placement plan with the help of submodular function theory, partition technique, and greedy strategy. We name them Relay Effective Deployment Algoirthm (REDA) and Submodular Iterative Deployment Algorithm(SIDA) respectively. We further propose Gradient-Descent Based Algorithm (GDBA), a heuristic method, to solve the DoCR problem releasing potential location constraints. Our extensive experiments indicate that the algorithms we propose can significantly improve the total satisfaction of the network. Furthermore, we establish a testbed using USRP to showcase our designs in real scenarios. To the best of our knowledge, we are the first to propose approximation algorithms for relay placement problem to maximize user satisfaction, which has both theoretical and practical significance in the related area.