DAROS: Distributed User-Server Assignment And Replication For Online Social Networking Applications

Transcription

1 DAROS: Ditributed Uer-Server Aignment And Replication For Online Social Networking Application Thuan Duong-Ba School of EECS Oregon State Univerity Corvalli, OR 97330, USA Thinh Nguyen School of EECS Oregon State Univerity Corvalli, OR 97330, USA Duc A. Tran Department of Computer Science Univerity of Maachuett at Boton Boton, MA 02125, USA Abtract In thi paper we tudy the problem of aigning uer to erver and data replication in a ditributed manner for online ocial networking OSN) application. Typical OSN application uch a Facebook and Twitter are built on top of an infratructure of erver, which handle the uer data torage and communication. Thu, for a given uer communication pattern, the load of the erver depend critically on the aignment of uer to erver. A good aignment will reduce the overall load of the ytem. Furthermore, by replicating data acro the erver judiciouly, the overall load can alo be further reduced. Unfortunately, thi optimal aignment and data replication problem i NP-hard. Therefore, we introduce a ditributed heuritic algorithm in which the erver perform local computation and exchange information among each other iteratively in uch a way that the algorithm converge to a good aignment and replication in term of reducing the overall ytem load a well a balancing the load among the erver. In contrat with a centralized algorithm, a ditributed algorithm offer the advantage of balancing the computation among all the erver a well a the ability to naturally adapt to timevarying uer communication pattern. Simulation reult how promiing performance for the propoed algorithm. Index Term Ditributed optimization, ocial network, large cale ditributed ytem, data replication I. INTRODUCTION For calability, online ocial network OSN) uch a Facebook, Twitter, Linkedin, etc. employ an underlying infratructure of erver which handle the communication and data torage on behalf of the uer. Specifically, each uer i aigned to a primary erver. When data i tranferred from one uer to another, the uer do not et up a network connection directly with each other. Intead, their primary erver will perform the data tranfer on their behalf. If the two uer are aigned to the ame primary erver, then intercommunication between two erver are not needed. A a reult, for a given uer communication pattern, the load of the erver depend critically on the aignment of uer to erver. To be concrete, Thi work wa upported in part by the National Science Foundation under grant CNS and CNS in thi paper we conider the following protocol ued in a typical cenario of OSN application wall pot in FaceBook. When a uer u goe online, it will 1) iue a read requet to it primary erver u for tored peronal data and then requet for it friend information; 2) If it friend are aigned on the ame u, the read requet for friend meage i imply reading data from the local torage; 3) However, if ome of it friend are aigned to different erver, then the primary erver forward the read requet to every primary erver of it friend. In repone to the requet, the friend primary erver read the data e.g., the previouly poted meage by it friend) from their databae, and end them back to u. u then return the data to uer u. In thi cenario, the erver load i incurred by 1) reading data from the databae and 2) communication between erver. A an example, let u conider a uer u, it two friend, uer v and w with their different primary erver u, v, and w, repectively. When uer u goe on line, it will read data from it primary erver S u which incur a read load. Next, S u end two requet to v and w to requet the updated data from v and w on behalf on u. Thi incur a communication load on three erver. v and w then read the updated data from their databae which incur a read load on them. v and w then return the updated data back to u which further incur a communication cot. We note that the read cot i primarily due to time and computation involved with reading data from dik while the communication cot i primarily due to the bandwidth cot. The communication cot i reduced if uer v and w reide on the ame primary erver u. In that cae, no inter-erver communication i needed, thu no communication load i incurred. On the other hand, the read load remain the ame ince three read from the databae are till required. Thu, to reduce the overall load, it make ene to aign friend to the ame erver. However, thi i not alway poible. One way to reduce the load i to replicate the uer data acro multiple erver. Replication. If the data of uer v i replicated on the primary erver of it friend uer u. Then when u goe online,

2 no inter-erver communication load i incurred. Furthermore, replicating uer data on different erver alo make the ytem le uceptible to bottleneck failure. However, data replication incur a load of writing additional data to the databae. Thu, there i a trade-off in term of the communication load and writing load which in turn depend on the frequency of reading veru writing for different uer. Load Balance. Another important apect affecting the ytem performance i to balance the load among the erver, i.e., to prevent the cenario where ome erver are overloaded while other are idle. It i well-known that thee cenario can ubtantially degrade the performance of a ditributed ytem. Baed on the dicuion above, for a given uer communication pattern, the load of the erver depend critically on the aignment of uer to erver. The uer communication pattern are defined by not only the ocial network of friend typically repreented a a relationhip graph among friend), but alo how often uer log in Facebook read), pot meage write), and the length of the poted meage. A good aignment will reduce the overall load and balance the individual load among the erver. Furthermore, by replicating data acro the erver judiciouly, the overall load can alo be further reduced. II. BACKGROUND There are two main tak of data management in OSN: aignment uer to erver and data replication. There are much work focuing on thee uch problem. In [1], author propoed a centralized algorithm for partitioning the network of uer baed on a relaxation of an integer program. The dynamical change of the network a well a replication problem were not conidered in thi work. Algorithm propoed in [2] and are baed on the community tructure of the network. Modularity i a very good parameter which repreent the tructure of the network a well a how uer are related among themelve. However, computing modularity for the whole network i computationally expenive. Replication cheme propoed in [3] and [4] relie on a fixed partition cheme and a fixed number of replica i applied for all uer data. When network tructure change, i.e node addition/deletion, a new aignment cheme might be needed to achieved before changing replication cheme would lead to better ytem performance. Moreover, ocial network are very kew where mot uer having low degree of friend; and with a good aignment cheme the required replica i minimal and the redundant replica merely upport load balance among erver. Our work focue on ditributed algorithm which adapt to the dynamical change of the network and minimize the global objective function value which i the um of load incurred by uer activitie. Unfortunately, thi optimal aignment and data replication problem i NP-hard. Therefore, we introduce a ditributed heuritic algorithm in which the erver perform local computation and exchange information among each other iteratively in uch a way that the algorithm converge to a good aignment and replication in term of 1) reducing the overall ytem load a well a 2) balancing the load among the erver. In contrat with a centralized algorithm, a ditributed algorithm offer the advantage of balancing the computation among all the erver and the ability to naturally adapt to the time-varying uer communication pattern which i often the cae for OSN application. III. PROBLEM FORMULATION We firt define the following notation and derive the quantitie of interet: N, M: Number of uer and erver, repectively. V, S: A et of all uer and erver, repectively. Nv): et of uer who ha relationhip with uer v v neighbor). c r v, c w v, c t v: average cot of reading/writing and tranferring v data repectively. Thee cot are function of v total load amount. P {0, 1} N M : An aignment matrix where p u = 1 if uer u i aigned to a primary erver, and p u = 0 otherwie. Since a uer i aigned exactly to one primary M erver, p u = 1 and 0 N p u N, i. =1 u=1 X {0, 1} N M : A replication matrix where x up = 1 if uer u data i replicated on erver p, and x u = 0 otherwie. Since uer data i never replicated on it primary erver, x u p u.if uer u data i replicated M on K erver, x u = K. =1 r v, w v : the uer v average rate of reading and writing, repectively. In thi ection, we firt derive different type of load for a given uer-erver aignment without replication. Thee are neceary for the mathematical formulation and are the building block for the propoed ditributed algorithm. Replication will be conidered hortly. We now tart with the read and communication load. Read Load. Uer v data i intereted in by v itelf and it friend. Then the total read load on a erver i: L r = v V p v rv c r v + r u c r ) v Communication Load. If v and u are aigned to the ame primary erver, then there i no communication load. On the other hand, data need to be tranferred between the two, and thu the total communication load of i computed a: L c = v V p v 1) r u c t v1 p u ) ). 2) where p u [0, 1]. Write Load. Whenever a uer v update it data, a write load i incurred at it primary erver, and total write load at erver i: L w = v V w v c w v p v. 3)

3 From all the above, the total load of all the erver without replication L wor i computed a: L wor = S L r + L c + L w. 4) Data Replication. Up until now we dicu the communication, read, and write load auming no data replication. Now if uer data i replicated on erver, a dicued previouly, the communication load will be reduced. Therefore, the total change in the ytem load due to replication i: L Rep = v V x vn wv c w v c t v n S r u p un ) The total load with replication L can be now rewritten a: Let 5) L = L wo + L Rep. 6) L = L M and d = L L denote the average erver load and the imbalance load of erver, repectively. We define our objective function a: F = L + α max d, 7) where α, 1) i the weighted factor for controlling the trade-off between load imbalance and the total erver load. The problem of aigning uer to erver and replication data i now cated a an integer programming problem with the optimization variable being p u and x u, denoting the primary uer-erver aignment and the replication cheme, repectively. Formally, the optimization problem i: Minimize: F Subject to: p u = 1 u 8) p u + x u 1 u 9) x u K 10) p u {0, 1} 11) x u {0, 1} 12) The contraint 8) enure that every uer ha exactly one primary erver; contraint 9) guarantee that uer u data i never replicated on the ame primary erver; and contraint 10) retrict the maximum number of replica within ytem. Thi optimization problem i unfortunately NP-hard. The proof i omitted due to limited pace. Therefore, in the next ection we will preent a ditributed algorithm that produce an approximate olution. Propoition 1: The global objective function i bounded by: F L F < F U 13) and Where F L = v V wv c w v + r v c r v + { F U = F L + max L c max, L Rep max with L c max = v V } r u c r ) v 14) + α N 1 N F L 15) r u c t v1 p u ) L Rep max = v V w v c w v min{ Nv), N} Proof: The detailed proof i omitted due to the lack of pace. IV. DISTRIBUTED ALGORITHM/PROTOCOL In thi ection, we propoe an approximate ditributed algorithm for olving the optimization problem above. The algorithm conit of two part. The firt part i a greedy method to provide quick convergence to a local olution. The next part employ imulated annealing method to overcome the local olution. For the firt part, the main idea i: at each time tep, the mot overloaded erver i elected to 1) reaign ome of it uer to other erver or 2) replicate it uer data to other erver in order to reduce it load. By appropriate taking either one or two action above, both the total load and and load imbalance will be imultaneouly reduced over time. For uch an algorithm to work, ome mall amount of information mut be kept and exchanged among the erver to allow the erver to know immediately whether 1) it i the mot overloaded erver and 2) whether it hould take action 1) or 2). The econd part, imulated annealing method will alo be computed locally. In what follow, we how the following information and local computation are ufficient to perform uch a ditributed algorithm. Local information and computation. Each erver i aumed to handle a number of uer, and thu it maintain all information pertained to it local uer. Specifically, thi information include a table of total read load incurred by each uer due to it friend from other uer. Let u define the following locally tored information at erver : N, V : The number and the et of uer primarily aigned to erver, repectively. L N M 1) : A matrix repreent the average load due to v data requet from other erver whoe entrie L v, n), n S \ denote the load of reading and tranferring v data to it friend located on erver n. Recall that L v, n) = p ut1 x vn )r u c r v. N r, V r : : The number and the et of uer whoe data are replicated on erver. kv): number of v replica. Given the information above, the total read and communication load at erver can be computed locally a: L r = v V rv c r v + n L v, n) ) 16)

4 Similarly, the total write load at erver can be computed locally a: L w = v V w v c w v + u V r w u c w u 17) The total load that erver contribute to the overall ytem load i: L = L r + L w 18) Now, each erver exchange their total load among each other. Thu, every erver will be able to determine whether it i the maximum load erver. Next, given that a erver i the maximum load erver, it will either 1) reaign ome of it uer to other erver or 2) replicate it uer data to other erver in order to reduce it load. To chooe either of the action, the maximum load erver will ue a greedy approach that take the action out of two that reult in the maximum reduction of the objective value F. The reduction of the objective value due to each of the action can be etimated locally a follow. Load Change due to Replication. If uer v current data on, i replicated on erver n, then the local read and communication load at erver will reduce by L v, n) ince L v, n) = 0 after the replication action. The write load at erver t will increae by w r. The load at erver will decreae by L v, n), the load at erver n will increae by w v c w v and the overall ytem load will change by: L = w v c w v L v, d) 19) Load Change due to Reaignment of Primary Server. Aume i the primary erver of v, then by reaigning uer v to another erver n, the load at erver will decreae by L v, n) + w v c w v, the load at erver n will increae by p ur u c r v + 1 x vn )w v c w v, and the change in the overall ytem load i: L = p u r u c r v L v, n) x vn w v c w 20) Load Change due to Replication Removal. Often, due to greedy move, it i poible that one get a lower objective value by backtracking via removing a replication. Therefore, we will calculate the reduced load due to removing a replication. If we remove u replication on erver, the load at erver will increae by v Nu) p vr v c t u w u c w u, and the change in overall ytem load i: L = p v r v c t u w u c w u 21) v Nu) For each of the change in load, we will alo be able to compute the change in F which include the load imbalance ince the load from all other erver are available. It i important to note that 19),20),21) can be computed locally. Thu, we propoe the following ditributed algorithm DAROS) for uer-erver aignment and replication. There TABLE I: Simulation reult M,K) % Global cot aved %Inter-erver % Repl. cot 8,1) ,3) ,7) ,1) ,3) ,7) are two tage in DAROS: optimization mode and convergence mode. In optimization mode, only overload erver involve in the optimization proce. Each overload erver elect each currently aigned uer and compute the gain of reaigning uer to a new erver or replicating uer data only and/or removing current replication. The deciion are made baed on which action will reult in better gain. The proce of replicating and reaigning uer in the optimization mode will be iterated until no more change in uer-aignment i oberved, i.e., the algorithm probably converge to a local olution. We call thi the convergence mode. To overcome the local olution, in the convergence mode, all erver will compute the gain of each tentative action reaign, replicate or remove replication) and make a deciion with probability of min{1, exp F/ where F i the etimated gain and i an experimental coefficient for controlling the convergence rate of the overall DAROS. We ued T B a a function of global objective value in uch a way that if the global objective function value i near the global optimal point, the probability of perturbation i low, i.e = β F F L F U F L. Thi i the bai of the imulated annealing method. We now decribe with the firt tage of the algorithm. Algorithm 1 DAROS 1: Input: {V }, {L }, K 2: Output: p u, x u 3: Aign equal number of uer to each erver uniformly at random. 4: Set mode to optimization 5: repeat 6: if mode i optimization then 7: Run algorithm 2 8: if No action ha been made then 9: Switch to convegence mode 10: end if 11: ele//convergence mode 12: Run algorithm 3 13: Switch to Optimization mode 14: end if 15: until Freeze V. SIMULATION RESULT In thi ection, we how our preliminary imulation reult. In particular, we ue the Facebook data [5] which ha N = uer and 817,090 relationhip. Simulation parame-

5 Algorithm 2 Optimization mode 1: At each time tep, the erver exchange their load among each other. 2: if i the mot overloaded erver then 3: for each v in V do 4: Reaign uer baed on F Reaign 20) 5: Replicate uer data baed on F Repl 19) 6: end for 7: % Now we perform backtrack ince during the previou greedy move might be uboptimal. 8: for each v in V r do 9: Let F removing = change in objective value by removing v replication uing 21) 10: if F removing < 0 then 11: Removing v replication on 12: Decreae kv) 13: end if 14: end for 15: end if Algorithm 3 Convergence mode 1: Set T B = β F F L F U F L 2: for each v at every erver do 3: for each n S \ do 4: Compute F Reaign by 20) 5: Reaign v to t with 6: probability min{1, exp F Reaign 7: if There i no v replication on t then 8: F Repl by 19) 9: Replicate v data on t 10: with probability min{1, exp F Repl 11: end if 12: for each v in V r do 13: F Remove by 21) 14: Remove v data replication 15: with probability min{1, exp F Remove 16: end for 17: end for 18: end for ter are: ReadRate=1, WriteRate=0.1, ReadCot=0.01, Write- Cot=1, TranferCot=0.02, α = 0.5; M,K) = 8,1),8,3),8,7). Fig. 1a), Fig. 1b), and 1c) how the change in global objective function value, the inter-erver communication cot, and the replication cot, repectively. Initially, the erver are aigned an equal number of uer uniformly at random. Thu, a imple algorithm of aigning equal number of uer to each erver would have the performance imilar to that of the propoed algorithm at iteration = 0. A een, the objective function decreae ignificantly over time, howing the ignificant benefit of our propoed algorithm over the naive method. Each pike in the graph correpond to the imulated annealing method to overcome the local minimum. Once the uer are perturbed in convergence mode, the communication and replication cot increae dramatically. However, at next iteration, thee cot reduce ignificantly due to appropriate reaignment and replication. A een, thi help the algorithm to further reduce the objective value and inter-erver communication cot. The imulation reult table I) how that the global cot i aved by about 46% compared to the random aignment/replication cheme. The inter-erver communication cot and the replication cot are jut mall fraction of the global cot. Global objective function 1.6 x # iteration a) Replcation cot %) Inter erver comm. cot %) # iteration c) # iteration Fig. 1: a) Objective function value b) Inter-erver communication cot %) c) Replication cot %). VI. CONCLUSION We tudy the problem of aigning uer to erver and data replication in a ditributed manner for online ocial networking OSN) application. Thi problem i NP-hard. Therefore, we introduce a ditributed heuritic algorithm that find a good aignment and replication in term of reducing the overall ytem load a well a balancing the load among the erver. In contrat with a centralized algorithm, a ditributed algorithm offer the advantage of balancing the computation among all the erver and the ability to naturally adapt to time-varying uer communication pattern. Simulation reult how the effectivene of our propoed algorithm. REFERENCES [1] H. Nihida and T. Nguyen, Optimal client-erver aignment for internet ditributed ytem, in ICCCN [2] M. Newman, Modularity and community tructure in network. in Proc Natl Acad Sci USA 2006). [3] D. A. Tran, K. Nguyen, and C. Pham, S-CLONE: Sociallyaware data replication for ocial network, Comput. Netw., vol. 56, no. 7, pp , May [Online]. Available: [4] K. Nguyen, C. Pham, D. A. Tran, and F. Zhang, Preerving ocial locality in data replication for ocial network, in Proceeding of IEEE ICDCS 2011 Workhop on Simplifying Complex Network for Practitioner SIMPLEX 2011), Minneapoli, MN, June [5] B. Viwanath, A. Milove, M. Cha, and K. P. Gummadi, On the evolution of uer interaction in Facebook, in Proc. Workhop on Online Social Network, 2009, pp b)