TCP VON: Joint Congestion Control and Online Network Coding for Wireless Networks

Transcription

1 TCP VON: Joint Congestion Control and Online Network Coding for Wireless Networks Wei Bao, Vahid Shah-Mansouri, Vincent W.S. Wong, and Victor C.M. Leung Deartment of Electrical and Comuter Engineering The University of British Columbia, Vancouver, Canada {weib, vahids, vincentw, Abstract In this aer, we roose TCP Vegas with online network coding (TCP VON), which incororates online network coding into TCP. It is shown that the use of online network coding in transort layer can imrove the throughut and reliability of the end-to-end communication. Comared to generation based network coding, in online network coding, ackets can be decoded consecutively instead of generation by generation. Thus, online network coding incurs a low decoding delay. In TCP VON, the sender transmits redundant coded ackets when it detects acket losses from acknowledgement. Otherwise, it transmits innovative coded ackets. We establish a Markov chain to analytically model the average decoding delay of TCP VON. We also conduct ns-2 simulations to validate the roosed analytical model. Finally, we comare the delay and throughut erformance of TCP VON and automatic reeat request (ARQ) network coding based TCP (TCP ARQNC). Simulation results show that TCP VON outerforms TCP ARQNC in terms of the average decoding delay and network throughut. I. INTRODUCTION It has been shown that network coding can increase the throughut and imrove the reliability of the end-to-end communication [] [3]. There are several related works for network coding in a ractical setting [4] [6]. In these works, the source divides native data ackets into grous called generations. Network coding is erformed within the same generation at the source and intermediate nodes. The source starts to transmit the next generation of the ackets only after being acknowledged by the destination that the current generation of the ackets have been decoded. The decoding delay of generation based network coding can be large since the destination has to receive enough indeendent coded ackets before decoding a generation. The concet of online network coding was roosed by Sundararajan et al. [7] and discussed further in [8], [9]. For online network coding, ackets can be decoded consecutively instead of generation by generation. In [8], a new encoding rule was roosed to guarantee small decoding delay by recovering acket losses within reasonable time. In [9], the erformance of online random linear network coding aroach for time division dulexing channels under Poisson arrivals was studied. SlideOR [] is an online network coding scheme, which uses a moving sliding window at the source to determine the set of native ackets to be coded. Recently, the joint roblem of network coding and transort layer design has received attention. Hassayoun et al. [] analyzed the erformance of TCP in coded wireless mesh networks with random acket loss. Chen et al. [2] roosed a distributed rate control algorithm for network coding based on the utility maximization model. An automatic reeat request (ARQ) network coding based TCP rotocol (referred to as TCP ARQNC in this aer) was roosed in [3] and extended in [4]. This aroach gives a new solution for the sliding window based TCP algorithm. However, ackets of one generation are decoded when the receiver receives enough indeendent coded ackets. This can otentially lead to a large decoding delay. In this aer, our goal is to design an online network coding based TCP for real time alications with a small decoding delay. Our roosed algorithm includes a congestion control art and an online network coding control art. For congestion control, we use TCP Vegas [5]. We can distinguish random acket losses from the congestion losses using the difference between the time that a acket is transmitted and the time that the sender is notified the acket is lost. The online network coding is used to address the effect of random acket loss for wireless links. The sender chooses to transmit either a acket with new information or a redundant acket according to the feedback information. Meanwhile, ackets are decoded consecutively instead of generation by generation so that the average decoding delay is small. Another advantage of our rotocol is that there is no need to change the rotocol stack at the intermediate nodes. We only need to modify TCP in the transort layer at the sender and receiver. In summary, the main contributions of our work are as follows: We roose TCP Vegas with online network coding (TCP VON), which is a combination of TCP Vegas and online network coding with low decoding delay. We establish an analytical framework to model the TCP VON and derive the average decoding delay. We conduct ns-2 simulations to validate the analytical model and comare the erformance of TCP VON with TCP ARQNC [3]. The results show that TCP VON outerforms TCP ARQNC in terms of network throughut and average decoding delay. This aer is organized as follows: Section II describes the background. Our roosed TCP VON algorithm is resented in Section III. The analytical model of our TCP VON is resented in Section IV. The erformance comarison is resented in Section V. Conclusions are given in Section VI.

2 II. PRELIMINARIES AND BASIC DEFINITIONS In this section, we first define different tyes of ackets and then resent the basic idea of how ackets are coded in an online manner. According to the revious works in [4], [5], [3], there are two tyes of data ackets in the network, namely native ackets and coded ackets. Native ackets are the original ackets generated by the sender, which are treated as vectors over a finite field F q of size q. Each native acket has a unique index number k corresonding to the order it is generated. Let k denote the kth native acket. A coded acket is a linear combination of several native ackets. For examle, if a coded acket q is a linear combination of ackets, 2,, m, then q = P m i= i i, where i is a non-zero multilicative coefficient randomly selected from the field F q. A native acket k is called a sent acket when the sender has transmitted a coded acket of the form q = k k + P l6=k l l. In this aer, we use the term transmit for coded ackets and the term send for native ackets. A native acket k is sensed when the receiver has received a coded acket q = k k + P l6=k l l. That is, k is combined in a coded acket q, and q is received at the receiver. Otherwise, k is unsensed. As discussed in [3], a native acket k is defined as seen, when the receiver has enough information to comute a linear combination of the form ( k + r k ), where r k = P i>k i i and i 2 F q. Otherwise, k is unseen. A native acket k is called decoded when the receiver has enough information to decode k. Otherwise, k is undecoded. The decoding mechanism at the destination is based on Gaussian elimination [] [3] [4]. If we consider the native ackets as unknown variables, then the set of received ackets comoses a system of linear equations. The decoding rocess is equivalent to solving a set of linear equations. The receiver can form a decoding matrix using the coefficients of coded ackets. Gaussian elimination can be alied to convert this matrix to a reduced row echelon form and solve the linear equations. Linear oerations corresonding to the Gaussian elimination are erformed on the received ackets. At the sender side, let i, i q and i r denote the smallest index among the ackets that have not been decoded, seen, and sensed, resectively. Let i s denote the smallest index of the acket that has not been sent. Note that i ale i q ale i r ale i s. To transmit a new coded acket, the sender combines native ackets with consecutive indices within a coding window. Let i min and i max denote the minimum and maximum index of the native ackets combined in the new coded acket, resectively. That is, the coding window begins at i min and ends at i max. To create a new coded acket, it is required that i min = i q. However, i max can be equal to either i s or i s for two reasons: First, it is not necessary to have i min <i q. Even if acket l (for l<i q ) is combined in the coded acket, since l has already been seen at the receiver, the Gaussian elimination will eliminate the term of l. Because iq may not be seen at the receiver, the sender must combine iq in the new coded acket. Therefore, i min = i q. We call the acket with index Next decoded i i Code base iq Next sensed ir imin iq imin Coding window ir Coding window imax Next sent (a) Redundant acket: (b) Innovative acket: is is imax decoded seen but not decoded sensed but not seen Fig.. Examle of coded ackets. i =4, i q =6and i r =8. The sender can transmit either (a) a redundant acket or (b) an innovative acket. i q the code base. Second, if i max >i s, the transmitted acket contains at least two new variables for the receiver. However, it adds only one equation to the system of linear equations at the receiver. Therefore, to rovide decoding oortunity at the receiver, i max ale i s. Since acket is has already been sent, we have i max i s. Consider the examle in Fig.. If i max is equal to i s, then the transmitted acket is a linear combination of ackets iq, iq+,, is. This acket contains information of a new native acket for the receiver and increases the number of variables at the receiver by one. We call such a acket an innovative acket. If i max is equal to i s, then the coded acket is a linear combination of native ackets iq,, is, which is called a redundant acket. For the case that there is no acket loss, the receiver can decode one native acket each time it receives an innovative acket. On the other hand, for the case that there are acket losses in the network, the sender should transmit redundant ackets to comensate the effect of acket losses. Let I, I q and I r denote the index of the next acket to be decoded, seen and sensed at the receiver, resectively. Note that i, i q and i r are variables at the sender while I, I q and I r are variables at the receiver and in the ACK ackets. When an ACK arrives at the sender, i, i q and i r are udated by the values of I, I q and I r, resectively. III. TCP VON ALGORITHM In this section, we resent the TCP VON algorithm for the sender and receiver, searately. A. Sender Side Oeration ) Congestion and Flow Control: For the congestion and flow control, we use the TCP Vegas algorithm [5], which has two distinct advantages. First, TCP Vegas detects and reacts to congestion before it haens by monitoring the round tri time instead of detecting acket loss. Second, acket loss occurred on wireless links are not considered as a sign of congestion. The sender maintains several variables including congestion window size cwnd, measured round tri time RT T, the minimum of all measured round tri times BaseRT T, standard deviation of the measured round tri sent

3 time, and the receive window size rwnd. The number of ackets on the flight is aroximated by the number of sent but unseen ackets (i.e., i s i q ), which is limited by the congestion window size cwnd and receive window size rwnd i s i q ale min{cwnd, rwnd}. () In order to focus on congestion control, we assume that the buffer size at the receiver is large enough so that the effect of the receive window rwnd can be ignored [5], [6]. The congestion window size cwnd is adjusted according to the algorithm from TCP Vegas [5]. We set two ( redetermined) threshold values a and b, a < b. If cwnd BaseRT T < a, then cwnd is linearly increased. If ( ) BaseRT T > b, then cwnd is linearly decreased. In our algorithm, we select the default values of a =acket and b =3ackets. 2) Online Network Coding Control: The core of our online network coding control is to decide whether to transmit an innovative acket or a redundant acket such that the receiver can decode the ackets with small decoding delay. We start by introducing the variables being used. The real ga RealGa denotes the number of native ackets sensed but unseen at the sender. That is, RealGa = i r i q. The value of RealGa is udated whenever the sender receives an ACK acket. RealGa shows the number of lost innovative ackets. If the RealGa is zero, it indicates that all ackets have been decoded at the receiver. If the RealGa is greater than zero, it imlies that there are acket losses in the system. The old real ga OldRealGa is the variable to store the revious value of RealGa. Whenever an ACK is received, OldRealGa is set to be the value of RealGa just before RealGa is being udated. RealGaDif f is the difference between RealGa and OldRealGa (i.e., RealGaDif f = RealGa OldRealGa), which shows the decrease in real ga uon receiving an ACK acket. RealGaDif f is calculated right after RealGa has been udated. The ExGa is the maximum number of lost ackets which can be tolerated at the sender. It is udated whenever a redundant acket is transmitted or RealGa is decreased. State is a variable taking either REDUNDANT or INNOVATIVE to instruct whether the next acket to transmit is a redundant acket or an innovative one. If RealGa > ExGa, then the number of lost ackets exceeds the maximum number that the sender can tolerate and the State will change into REDUNDANT. The sender transmits a redundant acket. If RealGa ale ExGa, State is INNOVATIVE and the sender transmits an innovative acket. We now exlain the use of ExGa. First, both RealGa and ExGa are set to zero. During oeration, the RealGa is equal to one, which indicates a acket loss. The sender transmits a redundant acket to comensate the acket loss and ExGa is increased by one. It causes RealGa to be equal to ExGa, and the sender does not transmit more redundant ackets during the eriod when RealGa is equal to ExGa. If the transmitted redundant acket gets lost in the network, the sender should be able to detect the loss of the redundant BaseRT T cwnd RT T cwnd BaseRT T cwnd RT T acket. Therefore, as soon as the sender transmits a redundant acket, it starts a timer with value RT T +4, where 4 is used to rovide some time margin. When the sender receives an ACK indicating that the redundant acket is successfully received by the receiver, the RealGa is decreased by one before the timer exires. If RealGa has not been decreased when the timer exires, it indicates that the redundant acket has got lost. In this case, the sender decreases ExGa by one, which causes RealGa to be greater than ExGa, so that the sender transmits another redundant acket. Note that one timer starts whenever a redundant acket is transmitted, and each redundant acket has its own timer. If the transmitted redundant acket is received successfully at the receiver, then the sender is informed by an ACK acket indicating the decrease of RealGa by one (i.e., RealGaDif f =). The sender decreases ExGa accordingly. The timer corresonding to the transmission of the redundant acket is stoed. RealGaDif f can be larger than one, which indicates that more than one redundant ackets have been received. In this case, ExGa is decreased by RealGaDif f. RealGaDif f oldest timers are stoed. 3) Algorithm of Sender Side Oeration: Algorithm shows TCP VON algorithm at the sender. Lines to 3 are used for initialization. Let i m denote the total number of native ackets created. As shown in Lines 5 to 8, if new data with X bytes arrives from uer layer, n s = d X L e native ackets are created, with length of L bytes (the last acket is added with if necessary). The total number of native ackets is increased by n s. Lines 9 to 22 show the online network coding algorithm. When RealGa is greater than ExGa, a redundant acket is transmitted, ExGa is increased by one, and a timer is started. Otherwise, an innovative acket is transmitted. When an ACK acket received (Line 23), the sender udates BaseRT T,, i, i q and i r (Lines 24 to 25). Then, it calculates the value of OldRealGa, RealGa and RealGaDif f (Lines 26 to 28). If the RealGaDif f is larger than zero, the sender will decrease ExGa accordingly and RealGaDif f timers will be stoed. Then, the congestion control is erformed in Lines 33 to 37. Lines 39 to 4 show the timeout event. The ExGa will be decreased by one when a timeout event occurs. B. Receiver Side Oeration Algorithm 2 shows TCP VON algorithm at the receiver. I, I q and I r are initialized with value and decoding matrix D is initialized with an emty matrix (Line ). When the receiver receives a correct coded acket, it first checks the acket header and retrieves i min, i max and the coding coefficients of native ackets combined (Line 5). The coefficients are added as a new row to the decoding matrix and Gaussian elimination is executed (Lines 6 to 7). Oerations corresonding to the Gaussian elimination are then erformed on the received ackets so far (Line ). The receiver finds the next acket to be decoded in Lines 2 to 5. The sender increases the value of I by one until it finds that the native acket with index I is not decoded. Thus, I is equal to the index of

4 Algorithm Algorithm of TCP VON at the Sender. : Set i, i q, i r, i s, i min, and i max to one and i m to zero 2: Set RealGa and ExGa to zero 3: Set State to INNOV ATIV E and cwnd to one 4: while the TCP connection is established 5: if data with length X bytes is received from uer layer 6: 7: Segment data in n s := d X e ackets L i m+,, im+ns Set i m := i m + n s 8: end if 9: while i s i q ale cwnd and i max ale i m : if RealGa > ExGa : Set State := REDUNDANT 2: Set i min := i q, i max := i s 3: Set ExGa := ExGa + 4: Start a countdown timer with value RT T +4 5: else 6: Set State := INNOV ATIV E 7: Set i min := i q, i max := i s, i s := i max + 8: end if 9: Create coded acket q = P i max j=i min j j 2: Include i min, i max and imin,, imax in header 2: Transmit the coded acket 22: end while 23: if an ACK is received 24: Set i := I, i q := I q, i r := I r 25: Comute BaseRT T and 26: Set OldRealGa := RealGa 27: Set RealGa := i r i q 28: Set RealGaDiff := RealGa OldRealGa 29: if RealGaDiff > 3: Set ExGa := ExGa RealGaDiff 3: Sto RealGaDiff oldest timers. 32: end if 33: if ( cwnd ) cwnd BaseRT T RT T BaseRT T > 3 34: Set cwnd := cwnd in the next RTT 35: elseif ( cwnd ) cwnd BaseRT T RT T BaseRT T < 36: Set cwnd := cwnd +in the next RTT 37: end if 38: end if 39: if a timer timeouts 4: ExGa := ExGa 4: end if 42: end while the first undecoded acket when the while loo is terminated. Similarly, the receiver will find the first acket unseen in Lines 6 to 8, and the first acket unsensed in Lines 9 to 2. Finally, an ACK acket indicating the indices I, I q and I r is generated and sent at the receiver (Line 22). IV. DECODING DELAY ANALYSIS OF TCP VON In this section, we establish analytical model to determine the decoding delay of TCP VON. The decoding delay is the difference between the time that a acket is transmitted by the sender and the time it is decoded at the receiver. We consider a toology where the last ho is a wireless bottleneck link with caacity C. The sender has a large file to send. The acket size is L bits. The roagation delay from the sender to the receiver is T. Due to the congestion control mechanism of TCP Vegas, the transmission rate at the sender is C, and there is no congestion loss. The time to transmit a acket is = L/C. The end-to-end delay from the sender to the Algorithm 2 Algorithm of TCP VON at the Receiver. : Set I :=, I q :=, I r :=, D := [ ] 2: while TCP connection is established 3: if a acket is received 4: if acket is not corruted 5: Retrieve i min, i max, and the coding coefficients 6: Add coding coefficients as a new row to matrix D 7: Perform Gaussian elimination on D 8: if an all-zero row aears 9: Discard the acket and eliminate the row in D : else : Perform linear oerations to ackets corresonding to the Gaussian elimination 2: while native acket with index I is decoded 3: Deliver data in I to uer layer 4: I := I + 5: end while 6: while native acket with index I q is seen 7: I q := I q + 8: end while 9: while native acket with index I r is sensed 2: I r := I r + 2: end while 22: Send an ACK acket indicating I, I q and I r 23: end if 24: else 25: Discard the corruted acket 26: end if 27: end if 28: end while receiver and from the receiver to the sender are aroximately T = T +3L/C and T 2 = T, resectively. The round tri time is RT T = T +T 2. We use K to denote the ratio of RT T over. We aroximate K with drt T/ e. Since RT T is much larger than, the error of aroximation is negligible. Each acket exeriences random loss at the wireless link indeendently with robability. Equivalently, we assume that ackets are corruted indeendently with robability. Let q k denote the kth coded acket received. A grou of ackets is a set of ackets which are decoded together. The exected decoding time of a acket q k, denoted as Γ k, is the difference between its arrival time and the time it is exected to be decoded if no further acket corrutions haen. If there are no corruted acket arrivals before q k and q k is correctly received, Γ k is equal to. However, if q k is corruted, Γ k will be delayed u to RT T +. It is because the receiver has to wait for one redundant acket which is exected to arrive after RT T + seconds. There will be several ackets in one grou in this case. In Fig. 2(a), if there is no corruted acket arrival at the receiver, each acket can be decoded immediately with decoding delay T. If a corruted acket arrives, the consecutive innovative ackets cannot be decoded until a redundant acket arrives. In Fig. 2(b), when a corruted acket q x arrives, the exected decoding time is RT T + =(K + ). If it receives a correct acket (e.g., acket q x+ or q x+2 ), the exected decoding time decreases by. However, if there is another corruted acket q x+5, the exected decoding time is

5 Fig. 2. Redundant Packet No corrution, decoding delay is T One grou One corruted acket, exected decoding time is delayed to RTT+ = (K+) One grou (a) Zero or one acket corrution in one grou. qx qx+ qx+2 qx+5 Redundant Packet Redundant Packet x =(K+) Sx = K+ x+ = K Sx+ = K x+2 =(K ) Sx+2 = K x+5 =(K+) Sx+5 = K+ One grou (b) Multile acket corrutions in one grou. Grous of ackets with different number of corruted ackets. (K + ). The exected decoding time Γ k+ is determined based on Γ k and whether q k+ is corruted or correct. Thus, Γ k+ is indeendent of Γ k,,γ. We use S k to denote the ratio of Γ k to (i.e., S k = Γ k ). Therefore, S k+ is indeendent of S k,,s. The sequence S,S 2,constructs a discrete Markov chain. If a corruted acket arrives, the exected decoding time becomes (K +) and the state becomes K +. If a acket is correctly received, the exected decoding time is decreased by and the state decreases by. Fig. 3 shows the state transition of the Markov chain. Let M = {m jk } (K+2) (K+2),j =,, K +, k =,, K + denote the transition robability matrix of the Markov chain. m j(k+) = (j =,,,K + ), m (j+)j = (j =,,,K), m = and all the other entries of M are. When the state is, all the ackets can be decoded. If there are i ackets in one grou, there will be i transitions between two consecutive visits of state accordingly. Let P i denote the robability that there are i ackets in one grou. Matrix M can be divided into four submatrices as M M M = (2) M M m m m (K+) m m m (K+) = C. A. m (K+) m (K+) m (K+)(K+) K K+ Fig. 3. State transition of Markov chain. We have P = m and P 2 = P K+ j= m jm j = M M. In general, we have P i = K+ X K+ X j = j 2= K+ X j i = m j m jj 2 m ji 2j i m ji = M M i 2 M, i 3. (3) The decoding delay of the ith (last) acket in the grou with i ackets is equal to the roagation delay T. The decoding delay of the (i )th acket in the grou is T +. The decoding delay of the (i j)th (j =,,,i ) the acket in the grou is T + j. The average decoding delay of a grou with i ackets is i = P i j= (T + j ) /i = T +(i ) /2, i. (4) Let t denote the total transmission time, T (t) denote the total decoding delay of all the ackets during time t and N(t) denote the total number of ackets transmitted during time t. We use n i (t) to denote the number of grous with i ackets in total during time t. We have N(t) = P i= in i(t) and T (t) = P i= in i(t) i. For large values of t, by the law of large numbers, the limit of n i (t)/ P j= n j(t) aroaches P i, the robability that there are i ackets in one grou. Thus, the average decoding delay can be comuted as P T (t) = lim = lim Pi= in i(t) i t! N(t) t! i= in i(t) = lim t! P i= i n i (t)/ P i= i n i (t)/ P j= n j(t) i P = j= n j(t) P Pi= ip i i i= ip. i V. PERFORMANCE ANALYSIS In this section, we analyze the erformance of TCP VON via ns-2 simulations. We first validate our analytical model by comaring the analytical and simulation results. Then, we comare the throughut and delay erformance of TCP VON and TCP ARQNC [3], [4]. Unless stated otherwise, the acket size L is set to 25 bytes. The results are average on 5 simulation runs and each simulation lasts seconds. We consider a single-ho wireless network with one sender and one receiver. C is the caacity of the wireless link. Fig. 4 shows the comarison between the analytical and simulation results under different loss robabilities. The roagation delay from the sender to the receiver T is 5 ms. The random acket loss robability varies from to. with ste of.. The results show that the simulation results match the analytical results, validating the correctness of our analytical model.

6 Average decoding delay (s) Analytical, C = 3 Mb/s Simulation, C = 3 Mb/s Analytical, C = 2 Mb/s Simulation, C = 2 Mb/s Analytical, C = Mb/s Simulation, C = Mb/s Loss robability Fig. 4. Analytical and simulation results of average decoding delay for single ho toology under different values of (T =5ms, and L = 25 bytes). Next, we comare the erformance of TCP ARQNC [3], [4] and TCP VON. For TCP ARQNC, the coded ackets are transmitted in a generation by generation fashion. There are N native ackets in one generation and each coded acket is a linear combination of the N native ackets. For each generation, the sender first transmits N G = N + R coded ackets, where R is the redundant factor. The receiver can decode all the native ackets if N or more coded ackets are successfully received at the receiver. If more than R ackets are lost, the sender has to transmit more linear combinations until the receiver receives enough indeendent coded ackets. We consider a three-ho tandem toology for simulations. The first two hos are wired links with caacity of 5 Mb/s and roagation delay of 5 ms. The last ho is wireless link with caacity of Mb/s, roagation delay of 2 ms, and random acket loss robability. There is a TCP session between the source and destination. We investigate the joint throughut and delay erformance of the different TCP schemes using ns-2 simulation. Fig. 5 shows the throughut and delay erformance of TCP VON and TCP ARQNC when acket loss robability changes from to. with ste of.. The throughut erformance of TCP ARQNC is much worse than that of TCP VON when N and R are small. When we increase N and R, although the throughut of TCP ARQNC is increased, the decoding delay erformance degrades a lot. Both of the throughut and delay erformance of TCP VON are better than that of TCP ARQNC when is small (e.g., <.5). VI. CONCLUSIONS In this aer, we roosed TCP VON, which incororates online network coding into TCP. By executing online network coding control, ackets can be decoded consecutively and the decoding delay is small. Then, we established a Markov chain model to comute the analytical delay erformance of TCP VON. We also conducted ns-2 simulations to validate the correctness of our analytical models. Finally, we comared the delay and throughut erformance of TCP VON and TCP ARQNC. Results showed that TCP VON outerforms TCP Average throughut (Mb/s) = =. 3 TCP VON =. TCP ARQNC, N = 2, R = 2 2 TCP ARQNC, N =, R = TCP ARQNC, N = 2, R = 2 TCP ARQNC, N =, R = Average delay (s) Fig. 5. Performance comarison for three-ho tandem toology under different values of acket loss robability. ARQNC. For future work, we lan to carry out emirical study of TCP VON and aly it in real networks. REFERENCES [] R. Ahlswede, N. Cai, S. Y. R. Li, and R. W. Yeung, Network information flow, IEEE Trans. on Information Theory, vol. 46, no. 4, , Jul. 2. [2] S. Y. R. Li, R. W. Yeung, and N. 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