Broadcast Program Generation for Unordered Queries with Data Replication

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1 Bradcast Prgram Generatin fr Unrdered Queries with Data Replicatin Jiun-Lng Huang and Ming-Syan Chen Department f Electrical Engineering Natinal Taiwan University Taipei, Taiwan, ROC jlhuang@arbr.ee.ntu.edu.tw, mschen@cc.ee.ntu.edu.tw ABSTRACT We study in this paper the prblem f bradcasting dependent data fr unrdered queries. Hwever, mst prir studies n dependent data bradcasting are limited t the premise f n data replicatin. Different frm ther prir studies, we investigate the effect f data replicatin in this paper. Specifically, we first derive several theretical prperties fr the average access time by analyzing the mdel f dependent data bradcasting. On the basis f the theretical results, we develp a genetic algrithm t generate bradcast prgrams with replicatin. In rder t cmpare the perfrmance f the prpsed algrithm and the prir studies, several experiments are cnducted. Our experimental results shw that with the analytical results derived, the theretical results derived are able t guide the search f the genetic algrithm very effectively, and lead t slutin bradcast prgrams f higher quality than thse f the prir studies. Keywrds Data bradcast, mbile infrmatin system, mbile cmputing, genetic algrithm 1. INTRODUCTION Mst wrks in data bradcasting were under the premise that each user requests nly ne data item at a time and the requests fr all data items are independent. Hwever, in many real applicatins, there exists dependency amng data items, and a mbile user may submit a query t retrieve multiple data items. Explicitly, queries f multiple, dependent data can be categrized int the fllwing tw types accrding t the cnstraint f the sequence f these data items: Ordered queries In an rdered query, the required data items shuld be retrieved in a predetermined rder. Permissin t make digital r hard cpies f all r part f this wrk fr persnal r classrm use is granted withut fee prvided that cpies are nt made r distributed fr prfit r cmmercial advantage and that cpies bear this ntice and the full citatin n the first page. T cpy therwise, t republish, t pst n servers r t redistribute t lists, requires prir specific permissin and/r a fee. SAC 3, Melburne, Flrida, USA Cpyright 3 ACM /3/3...$5.. Unrdered queries Similar as an rdered query, an unrdered query culd be ne issued by a mbile user fr requesting multiple data items simultaneusly. Hwever, unlike an rdered queries, these required data items may be retrieved in any rder. In bth types f queries, data allcatin algrithms assuming independent requests are nt able t effectively ptimize the perfrmance f the bradcast prgrams. This phenmenn attracts a series f studies n slving the prblem f dependent data bradcast. Nte that fr each rdered query, the required data items f this query shuld be retrieved accrding t a predetermined rder, and there will be exactly ne retrieval rder. In cntrast, the number f retrieval rders f an unrdered query Q i is Q i! where Q i is the number f required data items f Q i. This feature makes the bradcasting f dependent data fr unrdered queries be mre difficult than that fr rdered queries. It is nted that the mst prir studies f dependent data bradcast d nt cnsider data replicatin. That is, each data item appears exactly nce in the bradcast prgrams as shwn in the example in Figure 1b. A bradcast prgram withut replicatin is als called a flat bradcast prgram. As shwn in the prir studies n independent data bradcast [1], emplying data replicatin in bradcast prgram generatin is able t ptimize the bradcast prgrams mre effectively than flat bradcast prgrams especially when the access prbability fr each data item is skewed. Let D i (j) represent the j-th cpy f data item D i. As shwn in Figure 1c, ne can replicate ht data items (i.e., items with high access prbabilities) int several cpies in the bradcast prgram t further reduce the average access time. Hwever, nne f the abve studies n dependent data bradcasting cnsidered data replicatin which is practically imprtant. Cnsequently, we address the prblem f the bradcast prgram generatin fr dependent data with unrdered queries in this paper. Unlike [][], data replicatin is emplyed in ur study. Specifically, several special cases f the prblem f bradcasting dependent data are shwn t be NP-hard []. In view f this, we shall emply the Genetic Algrithm [3] (abbreviated as ) in this paper t address the prblem f bradcasting dependent data fr unrdered queries with data replicatin. Basically, s are iterative prcedures that search the prblem slutins by an evlutinary prcess based n natural selectin. maintains a ppulatin f individual candidate slutins t specific prblems. An individual candidate slutin can be represented as a list called a chrmsme. In, a fitness functin has t be

2 Item Size Item Size Item Size D 1 D 3 5 D 5 5 D D D 7 (a) The sizes f all data items D 5 D 3 D 1 D D D (b) The bradcast prgram withut data replicatin D () D 1 (1) D (1) D 1 () D 3 (1) D 5 (1) D (1) D (1) D 3 () (c) The bradcast prgram with data replicatin Figure 1: An example f a bradcast prgram with data replicatin Query(Q i) P r(q i) Q 1 {D 1, D, D 3} 5% Q {D 1, D 3, D } % Q 3 {D, D, D 1} % Q {D 1, D, D 5 } 5% Q 5 {D 5, D 3, D, D } 5% Figure : An example query prfile designed t evaluate the fitness f each chrmsme,and the design f the fitness functin is key t the effectiveness f the algrithms. Explicitly, in this paper we first mdel the prblem f bradcast prgram generatin fr unrdered queries with replicatin. By analyzing the mdel f dependent data bradcasting, we derive several theretical prperties fr the average access time. In light f the theretical results, we then frmulate the fitness functin fr ne t generate bradcast prgrams with replicatin. Sensitivity analysis n several parameters is cnducted. Our experimental results shw that with the analytical results derived, the theretical results derived are able t guide the search f the genetic algrithm very effectively, and lead t slutin bradcast prgrams f higher quality than thse f the prir studies. In additin, the experimental results als shw that the data replicatin technique emplyed can lead t mre efficient use f netwrk bandwidth. T the best f ur knwledge, there is n prir wrk n the bradcast prgram generatin fr unrdered queries with data replicatin. This fact distinguishes this paper frm thers. The rest f this paper is rganized as fllws. Sectin presents the preliminaries f this study. Analytical mdels f the prblem f bradcasting dependent data with unrdered queries are derived in Sectin 3. In light f the analysis in Sectin 3, we devise a -based algrithm in Sectin. Perfrmance evaluatin n varies parameters is cnducted in Sectin 5. Finally, Sectin cncludes this paper.. PROBLEM FORMULATION Same as in [1], it is assumed that the database D cntains data items, D 1, D,, D and each data item is readnly. The size f D i is assumed t be size(d i ). Thus, the size f the whle database is size(d) P size(d i). Frm the users perspective, a query is an indivisible request f single r multiple data items as defined belw. Definitin 1 An unrdered query Q i {D q i (1), D q i (),, D q i ( Q i )} is a nn-empty subset f all data items where Q i represents the number f required data items in Q i. Nte that 1 q i (j) fr all j where 1 j Q i, and q i (j) k means that the j-th accessed data item in Q i is D k. The query prfile is the aggregatin f the access behavir f all users. Frmally, we have the fllwing definitin. Definitin A query prfile Q cnsists f a set f Q i, P r(q i) pairs where Q indicates the number f queries in Q. P r(q i ) represents the prbability that a query Q i is issued by users. It is nted that P Q P r(q i) 1. The prblem f bradcast prgram generatin with replicatin can be divided int tw subprblems: (1) determining the number f replicas needed fr each data item and () determining the placement f these replicas int the bradcast prgram. The first subprblem indicates that the system shuld determine hw many cpies fr each data item t be placed in the bradcast prgram. Accrding t the prperty shwn in [], the ttal average access time will be minimized if the cpies f each data item are equally spaced and fr any tw data items D i and D j, q P r(di) n(d i ) n(d j ) size(d i ) q. (1) P r(dj ) size(d j ) where n(d i ) is the number f replicas f D i and P r(d i ) is the access frequency f D i. P r(d i ) can be btained by the fllwing equatin: P r(d i ) Q j1 P r(q j) the number f ccurrences f D i in Q j. Let L be the length f bradcast prgram with replicatin as determined in accrdance with system capacity. Nte that L size(d) since all data items shuld be bradcast at least nce. With the same reasn, at first each data item appears exactly nce in the bradcast prgram. There will be a space with size L size(d) left. Dente the number f extra cpies f D i appearing in the rest f the bradcast prgram as n (D i ). Then we have n (D i) size(d i) L size(d). Let the relatinship f n (D i ), i 1,,, fllw Equatin (1). Thus, the abve equatin can be rewritten as n n(di ) (D 1 ) n(d 1 ) size(d i) L size(d). Since nly ne unknwn variable is in the abve equatin, n (D 1) can be slved. All ther n (D i)s are btained by Equatin (1), and then, n(d i) is determined as n (D i) +1. After determining n(d i) fr each data item D i, we revise the riginal database by cnsidering the number f replicas fr each data item as D which is defined as fllws,

3 D [ n(d i) j1 n D i (j) 1 A, where D i(j) indicates the j-th cpy f data item D i. The size f the revised database D (dented as size(d ) ) is as fllws, size(d ) n(d i ) size(d i ) L. After determining the number f replicas fr each data item, the bradcast prgram with replicatin can be stated as fllws. Definitin 3 A bradcast prgram P with replicatin is a placement f all data items in D int a list with length L, where L is a predetermined number and L size(d). In additin, each data item D i will appear n(d i ) times in the bradcast prgram P. T facilitate the further discussin, we utilize the functin ffset(i, j) t represent the ffset f the j-th cpy f D i in the bradcast prgram. ffset(i, j) is equal t the summatin f the sizes f all data items with smaller bradcast rder than the j-th cpy f D i. We take access time as the measurement fr the quality f bradcast prgrams. As a result, given the number f bradcast channels, the database D and a query prfile Q, the prblem f bradcast prgram generatin with replicatin is t determine a bradcast prgram P with replicatin which minimizes the average access time f the query prfile Q. Dente, respectively, the average access time f a query Q i as T Access (Q i ) and the average access time f a query prfile Q as T Access(Q). The average access time f the query prfile Q can be frmulated as the fllwing equatin, Q h i T Access (Q) T Access (Q i ) P r(q i ). () 3. ANALYTICAL MODELS T facilitate the derivatin f the average access time, we first decmpse the access time f an query Q i int tw parts: 1. Startup time: When a mbile user submits a query Q i, the mbile device shuld wait until the system starts t bradcast any required data item f Q i. This time interval is called startup time.. Retrieval time: Retrieval time is defined as the time intervals between the mment that the mbile device starts t read data items frm bradcast channels and that the mbile device finishes Q i. Obviusly, the access time f an query is the summatin f the startup time and retrieval time. Dente the bandwidth f each bradcast channel as B. Then we have the fllwing example. Withut lss f generality, we assume that the user submits Q i in the m-th bradcast cycle and the time interval between the start time f the m-bradcast cycle and the time that the user submits Q i (i.e., s in Figure 3) is a D () D 1 (1) D (1) D 1 () D 3 (1) D 5 (1) D (1) D (1) D 3 () s A mbile user issues a query s 1 s Time The mbile user finishes the query Figure 3: An example scenari f a query unifrm distributin ver (, L). T facilitate the fllwing discussin, we define the candidates f the first retrieved data items f Q i as fllws. Definitin The candidates f the first retrieved data items f Q i, Cand {Cand(1), Cand(), }, is an rdered set f all cpies f date items in Q i. Cand is srted accrding t the ffsets f all elements in an ascending rder. In additin, the number f the candidates f Q i is equal t P Q i n(d q i j ). T simplify the further derivatin, we define a series a(j), j 1,,, b, t represent the ffsets f all data item in Cand (i.e., a(j)the ffset f the data item Cand(j)), where b represents the number f candidates (i.e., b Cand ). Dente the average startup time f Q i as T Startup (Q i ). We have T Startup (Q i ) 1 > < L B b 1 >: h i a(i + 1) a(i) + h L a(b) + a(1) i 9 > >; In rder t derive the average retrieval time f Q i, we first let p(j) represent the prbability that the user retrieves the j-th candidate (i.e., Cand(j)) f the first retrieved data items as the first required data item f Q i. We can frmulate p(j) as fllws: p(j) L a(b)+a(1) L : if j 1, a(j) a(j 1) L : therwise. Dente the average retrieval time f Q i as T Retr. (Q i ) and let T j Retr. (Q i) be the retrieval time f Q i when Cand(j) is the first retrieved data item. It is bvius that T Retr. (Q i ) can be btained by the fllwing equatin. T Retr.(Q i) b j1 (3) () p(j) T j Retr. (Qi) (5) Withut lss f generality, we assume that the mbile device will retrieve Cand(j) as the first retrieved data item at the m-th bradcast cycle. Als let the retrieval pint represent the pint that the mbile device starts t retrieve the first required data item f Q i. Assume that Cand(j) is the y-th cpy f D x (i.e., Cand(j) D x (y)). Therefre, the distance between the start time f the m-th bradcast cycle and the crrespnding retrieval pint is ffset(x, y). Since the mbile device will try its best t minimize the access time, fr each data item D k in Q i, k q i (1), q i (),, q i ( Q i ), the

4 mbile device will retrieve the first appeared cpy f D k after the retrieval pint. T simplify the fllwing derivatin, we let P int Retr. ffset(x, y), and then, we have the fllwing lemma. Lemma 1 Let the functin NEAREST (P int Retr., k) represent the distance between the retrieval pint and the pint that the mbile device starts t retrieve D k. The functin NEAREST (P int Retr., k) can be frmulated as NEAREST (P int Retr., k) n min ffset(k, i)i 1,,, n(d k ) and >< >: ffset(k, i) P int Retr. P int Retr. : if there exists at least ne cpy f D k, say D k (r), where ffset(k, r) n P int Retr. L P int Retr. + min ffset(k, i)i 1,,, n(d y) : therwise. By Lemma 1, the distance between retrieval pint and the pint that the nearest cpy f D k has been cmpletely retrieved is NEAREST (P int Retr., k)+size(d k ). Since the mbile device will stp retrieving data items when all data items in Q i have been cmpletely retrieved, T j Retr. (Qi) can be frmulated as T j Retr. (Q i) n max NEAREST (ffset(p int Retr., k) kq i (1),q i (),,q i ( Q i ) +size(d k ) 1 B. () Then, T Retr. (Q i ) can be btained by Equatins (5) and (). By the definitin f access time, T Access (Q i ) can be frmulated as T Access(Q i) T Startup(Q i) + T Retr.(Q i). T Access (Q i ) fr i 1,,, Q can be btained by the abve equatin, and finally, T Access (Q) can calculated by Equatin ().. DESIGN OF GENETIC ALGORITHM As described befre, fitness is the measurement f the quality f the chrmsmes, and the is designed t search the chrmsme with the highest fitness (i.e, maximize the fitness). Since the gal f bradcasting dependent data is t minimize the average access time f the given query prfile, the fitness functin is defined as F itness(p ) 1 T Access(Q). Accrding t Equatin (), T Access (Q) is the weighted summatin f all T Access (Q i ). Cnsequently, T Access (Q) can be btained by the fllwing prcedure with the analytical results derived in Sectin 3. Prcedure CalAccessTime(Q, P ) Input: A query prfile Q and a bradcast prgram P. Output: T Access (Q) ver the bradcast prgram P. 1: T Access (Q) : fr i 1 t Q d 3: T Access (Q) T Access (Q) + CalAccessT imeofquery(q i, P ) P r(q i ) Parameters Values The number f generatins (n Gen ) The size f ppulatin (n P p) 5 The prbability f crssver (P C ).9 The prbability f mutatin (P M ).5 The size f a page 1K bytes The bandwidth f each channel (B) 1K byte/sec. The number f data items () 5 The replicatin factr size(d ) size(d) The number f queries ( Q ) 3 The value f fanut 15 The Zipf distributin (θ).5 : end fr 5: return T Access (Q) Table 1: System parameters Functin CalAccessTimeOfQuery(Q i, P ) Input: A query Q i and a bradcast prgram P. Output: T Access (Q i ) n the bradcast prgram P. 1: Calculate T Startup (Q i ) in accrdance with the result f Equatin (3) : Find the candidates f the first retrieved data items f Q i (i.e., Cand) accrding t Definitin. 3: T Retr.(Q i) : fr j 1 t Cand d 5: Calculate T j Retr. (Q i) accrding t Equatin () : Calculate p(j) n the basis f Equatin () 7: T Retr.(Q i) T Retr.(Q i) + p(j) T j Retr. (Qi) : end fr 9: T Access(Q i) T Startup(Q i) + T Retr.(Q i) 1: return T Access(Q i) 5. PERFORMANCE EVALUATION 5.1 The Simulatin Mdel Fr perfrmance studies, we implemented the -based algrithms with lib [5], and als a query prfile generatr. The simulatr and query prfile generatr are bth cded in C++. We define the replicatin factr as size(d ) t represent the degree f replicatin. The prbability f the size(d) query Q i issued by users is assumed t be P r(q i ) ( P 1 i )θ nj1 ( 1 j )θ where θ is the parameter f the Zipf distributin. Let the data size fr each data item fllws a nrmal distributin with mean 1 pages and variance pages, and ne page is set t be 1K bytes. Table 1 shws the system parameters in ur experiments. In additin t the prpsed scheme, we als implement [] fr cmparisn purpses. Nte that can generate bradcast prgrams fr unrdered queries withut replicatin. T evaluate the effect f the prpsed algrithm n the quality f slutins and the executin time, several experiments are cnducted. Fr perfrmance cmparisn, the perfrmance gain f scheme A ver scheme B is defined as Avg. access time f scheme A Avg. access time f scheme B. Avg. access time f the scheme B 5. The Effect f the Number f Generatins

5 Average Access Time (sec) Number f Generatins Executin Time (sec) Number f Generatins Average Access Time (sec) Timed 3 5 Number f Queries Executin Time (sec) 1 -Timed 3 5 Number f Queries Figure : Avg. access time with the number f generatins varied Figure 5: Exe. time with the number f generatins varied Figure : Avg. access time with number f queries varied Figure 7: Exe. time with the number f queries varied In the first experiment, we investigate the evlutin prcess f the prpsed by varying the value f n Gen. Nte the change f the value f n Gen des nt affect the results f scheme. Figure shws the effect f different values f n Gen ranging frm t 5 n the average access times f the resulting bradcast prgrams. The crrespnding executin times f the schemes are presented in Figure 5. Nte that n Gen represents the case f randmly generating n P p slutins. As bserved in Figure, the average access time f the result bradcast prgram decreases as the value f n Gen increases. Hwever, the speed f cnvergence becmes slw when n Gen is larger then. Therefre, we set n Gen t be in the fllwing experiments. In additin, the perfrmance gain f ver increases frm % t 55% when n Gen increase frm t 5. As shwn in Figure 5, the executin time f scheme increases linearly as the value f n Gen increases. We als bserve that scheme is abut 3 times slwer than scheme when n Gen. Hwever, scheme still utperfrms scheme when n Gen (i.e., the executin f scheme is smaller than that f scheme ) which shws the imprtance f data replicatin in bradcast prgram generatin. 5.3 The Effect f the Number f Queries Figures and 7 shw the experimental results with the number f queries (i.e., Q ) varied. As bserved in Figure, the perfrmance gain f ver ranges frm 3% t % when the number f queries increases frm t 5. Hwever, the perfrmance gain f ver -Timed decreases frm 1% t %. It is because that a smaller number f queries indicates less cnstraints in bradcast prgram ptimizatin, and therefre, utperfrms -Timed since has mre time in ptimizatin than -Timed. On the ther hand, when the number f queries is large, mst effrt in ptimizatin will be in vain since the number f cnstraints is large. In additin, as bserved in Figure 7, the executin times f scheme and increase linearly as the number f queries increases.. CONCLUSION We explred in this paper the prblem f bradcasting dependent data fr unrdered queries and explicitly investi- gated the effect f data replicatin. By analyzing the mdel f dependent data bradcasting, we derived several theretical prperties fr the average access time. In light f the theretical results, we develped a genetic algrithms t generate bradcast prgrams fr unrdered queries with data replicatin. Our experimental results shwed that with the analytical results derived, the theretical results derived were able t guide the search f the genetic algrithm very effectively, thus leading t slutin bradcast prgrams f very high quality. It is als shwn by the experimental results that the data replicatin technique emplyed can lead t mre efficient use f netwrk bandwidth than prir schemes. Acknwledgement The authrs are supprted in part by the Ministry f Educatin Prject N. 9E-FA---7 and the Natinal Science Cuncil, Prject N. NSC E--3 and NSC E--5, Taiwan, Republic f China. 7. REFERENCES [1] S. Acharya, R. Alns, M. Franklin, and S. Zdnik. Bradcast Disks: Data Management fr Asymmetric Cmmunicatin Envirnments. In Prceedings f the ACM SIGMOD Cnference, pages 19 1, March [] Y. D. Chung and M. H. Kim. Effective Data Placement fr Wireless Bradcast. Distributed and Parallel Databases, 9():133 15, March 1. [3] D. E. Gldberg. Genetic Algrithms in Search, Optimizatin, and Machine Learning. Addisn Wesley, 199. [] N. H. Vaidya and S. Hameed. Scheduling Data Bradcast in Asymmetric Cmmunicatin Envirnments. ACM Wireless Netwrks, 5(3), May [5] M. Wall. lib: A C++ Library f Genetic Algrithm Cmpnents. August 199. [] E. Yajima, T. Hara, M. Tsukamt, and S. Nishi. Scheduling and Caching Strategies fr Bradcasting Crrelated Data. In Prceedings f the ACM Sympsium n Applied Cmputing, pages 5 59, March.

ENSC Discrete Time Systems. Project Outline. Semester

ENSC Discrete Time Systems. Project Outline. Semester ENSC 49 - iscrete Time Systems Prject Outline Semester 006-1. Objectives The gal f the prject is t design a channel fading simulatr. Upn successful cmpletin f the prject, yu will reinfrce yur understanding