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The Throughput Order of Ad Hoc Networks with Physical-layer Network Coding and Analog Network Coding Chen Chen

Haige Xiang

School of Electronic Engineering and Computer Science Peking University Beijing, P.R.China Email: [email protected]

School of Electronic Engineering and Computer Science Peking University Beijing, P.R.China Email: [email protected]

Abstract—Based on the result of Gupta and Kumar’s seminal work, it has been proven that the scheme which allows multipacket reception (MPR) can improve the throughput capacity by Θ (log n), while the improvement of network coding (NC) scheme is only upper bounded by a constant. On the development of NC, however, the most recently proposed schemes, physicallayer network coding (PNC) and analog network coding (ANC), advanced the innovation of increasing network capacity. They embrace the interference and allow simultaneous reception. Motivated by these two new approaches, in this paper, we study their throughput order. The results of our paper are as follows. The upper bound
of the throughput order with the PNC/ANC scheme is Θ

log n n

, which is the same as the

MPR scheme. lower bound of the PNC/ANC  The throughput  1 √ scheme is Θ , which is the same as the traditional and n log n

NC schemes. Moreover, both the upper bound and the lower bound can be achieved. The throughput varies with the locations of the nodes between these two bounds.

I. I NTRODUCTION Gupta and Kumar [1] demonstrated that the per node throughput of random ad hoc networks is λ(n) = √ Θ(1/ n log n). This indicates that the capacity cannot be scaled very well when the number of nodes n increases. Therefore, there are many challenges involved in the design of the ad hoc networks to increase the network capacity. Many researchers have attempted to find new approaches to boost network capacity. The work of Ahlswede et al. [2] proposed the concept of network coding (NC), which allows nodes to combine several input packets into one or several output packets instead of simply forwarding them. It showed that NC can achieve min-cut throughput for a directed graph in a multicast case. However, Liu et al. [3] recently obtained the result that NC cannot increase the order of throughput for a multi-pair unicasts case when the nodes are half-duplex. Furthermore, Li and Li [4] showed that NC does not gain capacity for unicast and broadcast cases, and it can provide at most twice the throughput with no NC in an undirected graph. Therefore, although the capacity of the networks may possibly be improved by a big constant, using NC alone cannot solve

the scaling problem as in case [1] while the number of the nodes goes up to infinity. Wang et al. [5] recently proved that the scheme which allows multi-packet reception (MPR) and successive interference cancelation (SIC) techniques in any receiver can improve the throughput capacity by Θ(log n) for the protocol model as Gupta and Kumar proposed. In addition, GarciaLuna-Aceves, Sadjadpour and Wang [6] demonstrated that the protocol architectures that exploit MPR provide a better capacity improvement of random wireless ad hoc networks than NC in the case of a single-source multicast and multipair unicasts. Thus, involving MPR will make ad hoc networks more scalable. MPR techniques, however, have not been fully resolved yet in today’s field of wireless communication. Returning to the approach of coding, Zhang, Liew and Lam [7] proposed a new scheme named physical-layer network coding (PNC), which makes use of the additive nature of electromagnetic (EM) waves and codes with the mixed signals. This approach is able to save more time slots than the straightforward network coding because it allows the receivers to deal with simultaneously received signals. Moreover, in the most recently related work, Katti, Gollakota and Katabi [8] implied that the analog network coding (ANC), in which the router just simply amplifies and forwards the mixed signals, has the same ability to improve the network capacity in the same way as PNC. Furthermore, Hao et al. [9] presented another physical-layer network coding scheme inspired by TomlinsonHarashima precoding (THP), which is similar to ANC scheme and may have better performance (in this paper, we consider all kinds of these schemes as the PNC/ANC scheme). Hence, the following questions arise: “how much do they improve?” and “can PNC and ANC improve the order capacity of ad hoc networks in Gupta and Kumar’s model?” To the best of our knowledge, the authors who proposed PNC did not provide the answer. As for ANC and THP-PNC, Katti et al. [8] and Hao et al. [9] only performed capacity analysis against the signal to noise ratio (SNR), in which the potential capacity boost when ANC and THP-PNC are used in the ad hoc networks has not

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(a) Traditional routing scheme

(b) Network coding scheme Fig. 1.

Θ(log n). This is similar to the MPR scheme, because the capacity is not limited by the multiple access interference (MAI). Different from the MPR scheme, however, we found that the uniform convergence  lower bound of a feasible PNC/ANC
√

1 n log n

. This is essentially due to the follow-

ing: In the MPR scheme, each receiver node can get all the information sent from the area of its transmission radius. • In the PNC/ANC scheme, however, the receivers will decode all the data in a system of linear equations and get k sets of new information, where k is the rank. When the network is crowded, we cannot guarantee with high probability (w.h.p.) all the equations generated by the simultaneously receptions will be linearly independent. Thus, as compared to the MPR scheme, which allows the transmissions across the sparsity-cut to initiate from any location in an area, the capacity of the PNC/ANC scheme varies with the locations of the nodes between the two bounds. Moreover, we show that both the upper bound and the lower bound can be achieved by some node distribution patterns, and there exists a pattern of the PNC/ANC scheme whose throughput order equals to the lower bound 
maximum •

Θ

√

1 n log n

(d) PNC/ANC scheme

Transmission models of traditional, NC, MPR and PNC/ANC schemes

been analyzed. In this paper, we demonstrate that when comparing with the result of traditional routing as the network experiences multipair unicasts [1], the upper
bound of the order capacity of  log n and the order gain is the PNC/ANC scheme is Θ n

scheme is Θ

(c) MPR scheme

.

Although the lower bound of PNC/ANC scheme equals to the schemes of traditional routing and NC, it still has potential because it embraces the interference. Up until now, our knowledge of the PNC/ANC scheme is localized by the basic three-node-transmission model because it is new. We hope that this paper motivates the design and analysis of the architecture and protocol, which expand the existing PNC/ANC scheme to become adaptive to ad hoc networks. II. M ODEL AND P RELIMINARIES A. Transmission Strategies of the Traditional, NC, MPR, and PNC/ANC Schemes The transmission models of the different schemes are shown in Fig. 1, where S1 and S2 are the two sources, while X and

Y are the two nodes in the network which could as well be S1 and S2 themselves. The two sources transmit two bits of data a and b where the expression a(i) , b(i) means that the bit a or b is transmitted in the time slot i of the channel. First, by the traditional scheme as shown in Fig. 1, the two bits are transmitted successfully in four time slots. Second, by the NC and MPR schemes, this transmission consumes three time slots. The router in the NC scheme sends the combination of the two bits to both of the destinations simultaneously, while the router in the MPR scheme is able to receive the two bits simultaneously and send them to the two destinations in different time slots. Finally, by the PNC/ANC scheme, only two time slots are needed to finish this two-bit transmission, in which the router can receive and send from/to the two channels simultaneously by employing the physical-layer processing techniques on the node. In particular, in the PNC scheme, the router broadcasts the re-mapping result of the two simultaneously received signals, while in the ANC scheme, the router only amplifies and rebroadcasts the mixed signals. The receiver nodes in both schemes of NC and PNC/ANC get the information they need by decoding all the combinations that they have received. B. Network Model Our network model is based on the model of Gupta and Kumar [1], where n nodes are randomly, independently, and uniformly located in a region of a fixed area. There are n randomly chosen source-destination pairs in the network, and the data are transmitted through a multi-hop manner from the source to its destination (every node is a transmitter and a receiver). We consider the network represented with an undirected graph such that nodes are the vertices, and there exists an edge between the two vertices if they are within a distance of the transmission radius r(n). Each node can communicate with each other only through the edges while it is a half-duplex work so that the nodes cannot transmit and receive at the same time. Furthermore, in this paper, our analysis is focused on the 2-D dense networks. There are some existing results as follows: The expected number of the nodes in an area of S is represented as E(NS ) in [1], [5] E(NS ) = n|S|

(1)

where |S| is the area of S.

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r ( n)

r ( n)

r ( n)

(1  ')r (n)

(a) Traditional routing scheme Fig. 2.

2(1  ')r (n)

2(1  ')r (n)

(b) NC/broadcasting scheme

(c) MPR scheme

Protocol models for the traditional, NC, MPR, and PNA/ANC schemes

To ensure the connectivity of the nodes, the lower bound of the transmission radius is given by [1], [5]   log n (2) r(n) ≥ Θ n

r ( n)

Let Nj be a random variable representing the number of nodes in a region Aj . Then, we have the following result as the Chernoff bounds [5], [10]: eδ n|Aj | 1. For any δ > 0, P [Nj > (1+δ)n|Aj |] < ( (1+δ) 1+δ ) 1

2

2. For any 0 < δ < 1, P [Nj < (1−δ)n|Aj |] < e− 2 n|Aj |δ Combining the two inequalities we can get for any 0 < δ < 1 [6]: (3) P [|Nj − n|Aj || > δn|Aj |] < e−θn|Aj | where θ = (1 + δ) ln(1 + δ) − δ in the case of the first bound, and θ = 12 δ 2 in the case of the second bound. In this paper, we make use of the Chernoff bounds to derive the lower capacity bound based on the method of Wang et at. [5] and GarciaLuna-Aceves et al. [6]. C. Protocol Models of the Traditional, NC, MPR, and PNC/ANC Schemes In this paper, our result is obtained under the protocol model. In order to get the model of the PNC/ANC scheme, we first analyze the models of the traditional routing, NC, and MPR schemes, which are shown in Fig. 2. Gupta and Kumar’s protocol model [1] for the traditional routing scheme is shown in Fig. 2(a). A transmission from transmitter a to receiver i is successful if the distance between them satisfies |Xa − Xi | ≤ r(n) and any other simultaneously transmitting node b satisfies |Xb − Xi | ≥ (1 + ∆)r(n), where ∆ > 0 is a constant, which depends on the properties of the wireless medium that ensures a guard zone to limit the interference, and Xa , Xb , and Xi are the locations of node a, b, and i, respectively. Based on the same protocol model which Gupta and Kumar [1] employed, Liu et al. [3] involved broadcasting when the nodes are transmitting the combination of the data with the NC centered at scheme. In their model, the disks of radius ∆r(n) 2 each receiver are disjoint because a sender cannot be within (1 + ∆)r(n) of the other senders’ receiver; otherwise, it will cause a collision of multiple packet reception which is not allowed in the scheme of NC. That is to say, the distance

Fig. 3.

Protocol model of PNC/ANC scheme

between two concurrent transmitters (e.g., the transmitter a and b in Fig. 2(b)) should be at least 2(1 + ∆)r(n). In the scheme that supports MPR, although the nodes in the network are able to perform MPR, each node can only transmit one packet at most at a time because multi-packet transmission (MPT) is not allowed. Hence, in the protocol model of the MPR scheme, the distance between the two concurrent receivers should be at least 2(1 + ∆)r(n) (see [5], [6]). Otherwise there will be a transmitter located in the overlapped area of the disks (shown in Fig. 2(c), node i and j in the center of the disks are the two receivers) which performs MPT. In the scheme of PNC/ANC, however, the restrictions of node distance of the NC and MPR schemes no longer exist. The nodes in the PNC/ANC scheme are not only able to receive from concurrent transmitters but also broadcast the combination of the received data to all their neighbors within the transmission radius, as shown in Fig. 3. Hence, the protocol model of the PNC/ANC scheme is more complex than other models. In the next section, we will display some properties of this model and analyze the throughput order of the PNC/ANC scheme. III. M AIN R ESULTS A. Observations for the PNC/ANC Scheme Definition 3.1: (Sparsity-Cut) Sparsity-cut Γ is a cut that separates a graph into two sets. The sum of bandwidth of the edges crossing cut Γ is the minimum of all cuts in the graph. Fig. 4 displays the sparsity-cut for the area of the 2-D case, which is induced by a line segment ΓAB with length lΓ . By

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denoting the two subregions of the area divided by the sparsitycut as Γ1 and Γ2 , this sparsity-cut captures the average traffic bottleneck of such a random network. Hence, to analyze the capacity bound of the network, we only need to consider the transmissions across the sparsity-cut, whose sources are in Γ1 and whose destinations are in Γ2 . The number of sources in Γ1 whose destinations are in Γ2 is denoted as nΓ1,2 . The capacity of the sparsity-cut is bounded by the maximum number of simultaneous transmissions across the cut. As shown in Fig. 4, we suppose that the transmitters are located in the region on the left side of ΓAB (we call it as transmitter region RT ), with width is r(n), while the receivers are in the region on the right side of ΓAB with width r(n) (we call it as receiver region RR ). In addition, the data rate of each transmitter/receiver pair is a constant value of W . Given that W does not change the order capacity of the network, we normalize its value to 1. Suppose that the transmissions across the sparsity-cut are launched by ns independent sources and there are nd destinations. The data that each destination receives are given by a matrix R, where εds is a 0 or 1 variable representing whether the destination d receives the data from source s.   ε12 · · · ε1ns  ε11         ε21 ε22 · · · ε2ns  (4) R= . .. . ..   ..   . .. .       εnd 1 εnd 2 · · · εnd ns Lemma 3.2: The sparsity-cut capacity of the PNC/ANC scheme is the rank of the corresponding matrix of the source/destination pairs across the sparsity-cut, after removing the columns that contain the known information. Proof: The proof of this lemma is obvious. Although the PNC/ANC transmission strategies often make use of the information which is already known by the destinations, such information is useless for the capacity calculation. Hence, to study the capacity, we only need to concern with how many units of new information have been transmitted across the sparsity-cut, which equals to the rank of the corresponding matrix after removing the columns that contain the known information. As stated above, the rank of the corresponding matrix is important in determining the capacity of the PNC/ANC scheme. Moreover, we should ensure that all the combinations passed through the sparsity-cut can be decoded correctly. Let X = [x1 , x2 , x3 , · · · , xns ] stand for the information transmitted across the sparsity-cut, and B = [b1 , b2 , b3 , · · · , bnd ] stand for the values of combinations that the receiver nodes have received. We get Definition 3.3: (Feasible PNC/ANC scheme) Given that xj is known (the number of xj belongs to [0, nns ]), Xc is the column vector containing every known xj in order, aj = [ε1j ε2j · · · εnd j ]
is a column in the corresponding matrix R, and Ac is the row vector containing every aj in order. A PNC/ANC transmission scheme is feasible if X ∗ (the solution of A∗ X ∗ = B ∗ ) exists, where A∗ is the corresponding matrix

r ( n)

RT

r ( n)

RR

*2

*1

Fig. 4.

r ( n)

RT

Cut capacity

r ( n)

RR

2(1  ')r (n)

Fig. 5. Guarantee each receiver centered at the circles get a combination which is linearly independent with others

R eliminating Ac , B ∗ = B − Xc · Ac , and X ∗ is the vector of the unknown information. Thus, in a feasible PNC/ANC scheme, the receivers can decode all the unknown data in the system of linear equations that they have received. Protocols should be used in the PNC/ANC strategy in order to guarantee the transmission scheme feasible. B. Capacity Upper Bound with Feasible PNC/ANC Scheme Lemma 3.4: The capacity of the sparsity-cut Γ of a random network with the PNC/ANC scheme has an upper bound of nlΓ r(n), where n is the density of the nodes, lΓ is the length of the sparsity-cut, and r(n) is the transmission radius. Proof: Based on the protocol model of PNC/ANC scheme

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stated in Section II, there is no restriction for the location of the transmitters and receivers. From Definition 3.3, we only need to ensure that our PNC/ANC scheme is feasible. Consider the possible locations of the transmitters which are in the transmission radius of each receiver. This region is on the left of the cut ΓAB of the fuscous disks in Fig. 4. The area in which the transmitters are located is S

=

N R 

(Si ∩ RT )

(max)

≤ lΓ r(n)

(5)

where each Si is the transmission area (which is πr2 (n)) of the receiver i, and NR is the number of the receivers. We use Equation (1) and (3) to estimate the average number of nodes located in S as n|S|. Hence, the upper bound of the number of transmitters is (max)

= nlΓ r(n)

(6)

The possible locations of the receivers are similar to the locations of the transmitters. Thus, the upper bound of the number of receivers is (max)

NR

= nlΓ r(n)

(7)

From the observations in the last subsection, in order to get the upper bound, we let the corresponding matrix of the source/destination pairs across the sparsity-cut be a nonsingular matrix. From Lemma 3.2, the capacity of the sparsitycut is equal to the rank of the matrix. Because in a non-singular matrix, the rank equals to the number of the rows, which is the number of the receivers, therefore, the capacity of the sparsitycut has an upper bound of nlΓ r(n). In Equation (5), the equality holds because regions Si ∩ RT can overlap and can be closely crammed into the transmitter region RT one by one (similar with the receiver region). Based on Lemma 3.4, we get the upper bound order capacity with the PNC/ANC scheme. Theorem 3.5: The per source-destination throughput of PNC/ANC scheme is upper bounded by Θ(r(n)). Proof: Regardless of the shape of the unit area, there exists a sparsity-cut for each orientation of the cut line. If lΓ is not a function of n, by the Chernoff bound [10] argument, there are Θ(n) pairs of source-destination nodes crossing Γ in one direction, i.e., nΓ1,2 = nΓ2,1 = Θ(n) w.h.p.. From Lemma 3.4, we get the capacity upper bound Θ(r(n)). If we consider the most simple PNC/ANC  and
scheme, choose the lower bound of r(n) (which is Θ

logn n

from

Equation (2)) to ensure the connectivity of the nodes, we get the following theorem for the order capacity of a wireless ad hoc network with n nodes. Theorem 3.6: The per source-destination
throughput  of the  PNC/ANC scheme is upper bounded by Θ

Theorem 3.7: When n nodes are dropped uniformly over a unit square, the sparsity-cut capacity for  feasible PNC/ANC  lΓ . scheme has a lower bound of Θ r(n) Proof: Based on the method of Wang et al. [5] and Garcia-Luna-Aceves et al. [6], from the Chernoff bound [10] and Equation (3), we can find θ > 0 for any given 0 < δ < 1 such that NR

i=1

NT

C. Capacity Lower Bound for Feasible PNC/ANC Scheme

log n n

.

lim P [

n→∞



|Nj − E(Nj )| < δE(Nj )] = 1

(8)

j=1

Thus, when n nodes are dropped uniformly over a unit (max) circles whose centers are in the square, there are NR receiver region RR in Fig. 4, and each of the circles contains Θ(nr2 (n)) nodes w.h.p.. However, in a feasible PNC/ANC scheme, not all the nodes in each receiver’s circle are able to transmit valid information. In order to get the minimum rank of the corresponding matrix, we consider the case as shown in Fig. 5. If we can guarantee the transmission strategy is feasible, each receiver centered at the disjoint circles in Fig. 5 will receive a combination which is linearly independent with others. Therefore, the rank of the corresponding matrix is at lΓ . From Lemma 3.2, regardless of the constant least 2(1+∆)r(n) coefficient,   the sparsity-cut capacity has a lower bound of lΓ Θ r(n) . Corollary 3.8: The per source-destination throughput of a
feasible PNC/ANC scheme has a lower bound of . The order is the same as the order of traditional Θ √ 1 n log n

routing and NC schemes. Proof: From Theorem 3.7, regardless of the constant coefficient, it equals to the sparsity-cut capacity of the NC scheme. Liu et al. [3] have proven that the
throughput  order of this scheme for multi-pair unicasts is Θ

√

1 n log n

, which

has the same order as the result of Gupta and Kumar’s work [1]. D. Example: A Pattern that Achieves The Lower Bound Fig. 6 illustrates a possible node distribution, where the transmitters are located one by one on the left of the sparsitycut, while the receivers are on the right. Each receiver node belongs and only belongs to the intersection of two adjacent transmitters’ circles, and so does each transmitter node. We assume that the distance between two adjacent transmitters/receivers is cL r(n), where cL is a constant. The corresponding matrix of this pattern is an upper triangular matrix.   1 1 0 · · · 0 0          0 1 1 · · · 0 0       . . . . . . . . .. .. (9) RL = .. .. ..        0 0 0 · · · 1 1         0 0 0 ··· 0 1

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0.5

0.45

0.4

0.35

cL r ( n )

Probability

d

0.3

0.25

0.2

0.15

0.1

0.05

Fig. 6.

Fig. 7.

A pattern that achieve the lower bound

Theorem 3.9: capacity order of the pattern 
The maximum 1 . in Fig. 6 is Θ √ n log n

Proof: The distance between two adjacent receivers in Fig. 6 is cL r(n). Hence, the rank of the corresponding matrix Γ , which is the sparsity-cut capacity. is upper bounded by cLlr(n) Regardless of the constant coefficient, it equals to the sparsitycut capacity of the NC scheme. From the result of Liu et al. [3], we get this theorem. If we add a new source/destination pair to the pattern of Fig. 6, one row and one column will be inserted into the corresponding matrix RL .  1      0     .  .. R
L =  0      0     ε(nd +1)1

1

0

···

0

0

1 .. . 0

1 .. . 0

···

0 .. . 1

0 .. . 1

ε1(ns +1)

           

ε2(ns +1) .. . .  ··· ε(nd −1)(ns +1)      0 0 ··· 0 1 ε(nd )(ns +1)      · · · · · · · · · · · · · · · ε(nd +1)(ns +1) (10) However, if the newly added source node and destination node are located randomly, which is similar to inserting one row and one column with the random element of 0 or 1 distribution to the original matrix, the new matrix cannot be guaranteed to be a non-singular matrix. By Lemma 3.2, the throughput capacity may not increase. Furthermore, there are some restrictions for the newly inserted row and column which are caused by the spatial distribution of the original nodes in the area. For example, if we add a new pair of nodes to Fig. 6 and obtain a new matrix with nd + 1 rows and ns + 1 columns as shown in Equation (10), there cannot exist any 0 between consecutive 1 in the new row from ε(nd +1)1 to ε(nd +1)ns and in the new column from ε1(ns +1) to εnd (ns +1) . This is because the original source nodes and destination nodes are located one by one. Besides, the number of consecutive 1 in the new row ..

1

2

3 Number of new nodes

4

5

The probability of making the new matrix non-singular

c and the new column should be no more than  2r(n)k + 1 lΓ because no more original nodes can be circled within the transmission radius r(n) of the new node, where kc is the rank of the original matrix and lΓ is the length of the sparsity-cut. These kinds of restriction additionally decrease the probability to make the new matrix non-singular. Moreover, it implies that if we add more than one pair of source and destination nodes, the probability of keeping the new matrix non-singular might be smaller. The following sample simulation result convinces our point. Sample simulation: In an area of 1×1, we choose n = 100,  log n and r(n) = n . With the spatial distribution of Fig. 6, we select a scenario where the distance between each source/destination is 0.1, which equals to the mean distance value of a uniform distribution when n = 100. The result of the simulation is shown in Fig. 7, where the horizontal axis represents the number of newly added nodes, and the vertical axis displays the probability of whether the new matrix is nonsingular. Meanwhile, we ensure that the newly added nodes have connectivity to the original network. Fig. 7 shows that the probability decreases rapidly when more than one pair of nodes is randomly added to the network. The above analysis and simulation imply that in a uniform distributed ad hoc network, by randomly adding more nodes, we cannot improve the capacity of the pattern in Fig. 6, w.h.p.. Therefore, the maximum capacity order under this pattern is equal to the order of the lower bound from Corollary 3.8.
That is,this pattern can only achieve the lower bound . This lower bound exists and is reachable. Θ √ 1

n log n

E. Example: A Pattern that Achieves The Upper Bound In the last subsection, we have shown an example that only achieves the lower bound. Because in PNC/ANC scheme, the lower bound is not equal to the upper bound, the next question is, “is there any pattern that can achieve the upper bound?” The answer is “yes”. In this subsection, we display a deterministic

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r ( n)

w.h.p.. Therefore,  the achievable capacity order of this pattern
 log n , which equals to the order of the upper bound is Θ n

r ( n)

RT

RR

of the PNC/ANC scheme. The difference between this pattern and the pattern of Fig. 6 is • In the pattern of Fig. 6, the locations of the source/destination nodes of valid transmissions are restricted to a line, so its capacity order is only
Θ •

√

1 n log n

.

In the pattern of Fig. 8, the nodes can be closely crammed region into the transmitter region RT and the receiver
R,  R so its capacity can achieve the upper bound Θ

a Fig. 8.

b

c

i

j

k

A deterministic flow construction example

example of the pattern which can achieve the upper bound, as shown in Fig. 8. The pattern in Fig. 8 is an extension of the node distribution of Fig. 6, where we do not restrict the distance between two adjacent nodes. The nodes in Column a and Column i, Column b and Column j, and Column c and Column k are the duplications of the pattern in Fig. 6, respectively, where the nodes in Column a, b and c are transmitters, and the nodes in Column i, j and k are receivers. The corresponding matrix (if one column contains four nodes) of this pattern is   . . . . . .. .. ..     . . . .. .. .. .. .. . . .           ε ε ε · · · 1 0 0 0 ε  j c j c j c j c 1 1 1 2 1 3 1 4         · · · 1 1 0 0 ε ε ε ε   j c j c j c j c 2 1 2 2 2 3 2 4          · · · 0 1 1 0 εj3 c1 εj3 c2 εj3 c3 εj3 c4  (11) · · · 0 0 1 1 ε ε ε ε j4 c1 j4 c2 j4 c3 j4 c4         ··· 0 0 0 0 1 1 0 0           · · · 0 0 0 0 0 1 1 0          0 1 1    ··· 0 0 0 0 0       ··· 0 0 0 0 0 0 0 1 Theorem 3.10: The achievable capacity order of the pattern 
 log n , which equals to the order of the in Fig. 8 is Θ n upper bound of the PNC/ANC scheme. Proof: By elementary row transformation, the corresponding matrix of the pattern in Fig. 8 can always be transformed into an upper triangular matrix which is nonsingular (by removing some redundant “1” in (11) out of the upper triangular region). So its rank equals to the number of the transmitters/receivers (the valid sender/receiver matrix is square). By the Chernoff bound [10] and Equation (1), (2), and (3), there exist nlΓ r(n) transmitters in the transmitter region RT and nlΓ r(n) receivers in the receiver region RR ,

log n n

.

If the locations of the nodes deviate from the original locations of the pattern in Fig. 8, it will degenerate to the pattern of Fig. 6. IV. C ONCLUSION We have shown that the upper bound of the throughput order is promoted by the PNC/ANC scheme with Θ (log n) for multi-pair unicasts, which is the same as the MPR scheme. Furthermore, we have demonstrated
that thefeasible , which PNC/ANC scheme has a lower bound of Θ √ 1 n log n

is the same as the routing and NC schemes. Although the gain of the PNC/ANC scheme varies with the locations of the nodes, we still believe that the PNC/ANC scheme has an advantage because the capacity of ad hoc networks with PNC/ANC is no longer limited by the multiple access interference. New design and analysis in the future may help us expand our knowledge of the PNC/ANC scheme and create a more scalable ad hoc network. R EFERENCES [1] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 388–404, 2000. [2] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Transactions on Information Theory, vol. 46, no. 4, pp. 1204–1216, 2000. [3] J. Liu, D. Goeckel, and D. Towsley, “Bounds on the gain of network coding and broadcasting in wireless networks,” in Proc. of IEEE INFOCOM 2007, Anchorage, Alaska, USA., May 6-12 2007. [4] Z. Li and B. Li, “Network coding in undirected networks,” in Proc. of CISS 2004, Princeton, NJ, USA., March 17-19 2004. [5] Z. Wang, H. R. Sadjadpour, and J. Garcia-Luna-Aceves, “On the capacity improvement θ(log n) of ad hoc wireless networks with multipacket reception,” in Information Theory and Application workshop, UCSD, 2007. [6] J. Garcia-Luna-Aceves, H. R. Sadjadpour, and Z. Wang, “Challenges: Towards truly scalable ad hoc networks,” in MobiCom 2007, Montreal, Quebec, Canada, 2007. [7] S. Zhang, S.-C. Liew, , and P. Lam, “Physical-layer network coding,” in MobiCom 2006, California, USA., September 2006. [8] S. Katti, S. Gollakota, and D. Katabi, “Embracing wireless interference: Analog network coding,” in ACM SIGCOMM, 2007, Kyoto, Japan, August 27-31 2007. [9] Y. Hao, D. Goeckel, Z. Ding, D. Towsley, and K. K. Leung, “Achievable rates of physical layer network coding schemes on the exchange channel,” in ITA, 2007, UMUC, Sep 25-27 2007. [10] R. Motwawni and P. Raghavan, Randomized Algorithms, 1995.

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