confer - virginia techpeople.cs.vt.edu/~irchen/6204/paper/broadcast.pdf · arian t index fanouts,...
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OrganizationofBroadcastData
Source:[2]:S.Acharya,etal.\Broadcastdisks:datamanagementfor
asymmetriccommunicationenvironments,"ACM
SIGMODInternational
ConferenceonManagementofData(SIGMOD1995),1995.
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ExpectedDelayforVariousAccessProbabilities
AccessProbability
ExpectedDelay
(inbroadcastunits)
A
B
C
Flat(a)
Skewed(b)
Multi-disk(c)
0.333
0.333
0.333
1.50
1.75
1.67
0.50
0.25
0.25
1.50
1.63
1.50
0.75
0.125
0.125
1.50
1.44
1.25
0.90
0.05
0.05
1.50
1.33
1.10
1.0
0.0
0.0
1.50
1.25
1.00
2
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BroadcastProgramGeneration
BroadcastProgramGeneration
(Forsimplicity,assumethatdataitemsare\pages",thatis,theyareofa
uniform,�xedlength.)
1.Orderthepagesfromhottest(mostpopular)tocoldest.
2.Partitionthelistofpagesintomultiplerange,whereeachrangecontains
pageswithsimilaraccessprobabilities.Theserangesarereferredtoasdisk.
3.Choosetherelativefrequencyofbroadcastforeachofthedisks.Theonly
restrictionontherelativefrequenciesisthattheymustbeintegers.For
examplegiventwodisks,disk1couldbebroadcastthreetimesforevery
twotimesthatdisk2isbroadcast,thus,relfreq(1)=3,andrelfreq(2)=2.
(Nextpagetocontinue�
�
�)
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BroadcastProgramGeneration(con't)
4.Spliteachdiskintoanumberofsmallerunits.Theseunitsarecalled
chunks(Cijreferstothejth
chunkindiski).First,calculatemaxchunks
astheLeastCommonMultiple(LCM)oftherelativefrequencies.Then,
spliteachdiskiintonumberchunks(i)=maxchunks/relfreq(i)chunks.In
thepreviousexample,numberchunks(1)wouldbe2,whilenumberchunks(2)
wouldbe3.
5.Createthebroadcastprogrambyinterleavingthechunksofeachdiskinthe
followingmanner:
(Nextpagetocontinue�
�
�)
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BroadcastProgramGeneration(con't)
01
fori=0tomaxchunks�
1
02
forj=1tonum
disks
03
k=im
od
num
chunks(j)
04
BroadcastchunkCj;k
05
endfor
06
endfor
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Deriving a Server Broadcast Program
C1,1 C2,1 C3,1 C1,1 C2,2 C3,2 C1,1 C2,1 C3,3 C1,1 C2,2 C3,4
1 2 4 5 1 3 6 7 1 2 8 9 1 3 1011
1 2 3 4 5 6 7 8 9 1011C1,1 C2,1 C2,2 C3,1 C3,2 C3,3 C3,4
1 2 3 4 5 6 7 8 9 1011
T1
2 3 4 5 6 7 8 9 10111
Major Cycle
Minor Cycle
Tracks
TT2 3
COLDHOT
Chunks
Database (pages)
Broadcast Disk program6
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OrganizationofBroadcastData
Source:M.S.Chen,K.L.WuandP.S.Yu,\Indexedsequentialdata
broadcastinginawirelesscomputingenvironment,"17thIEEE
InternationalConferenceonDistributedComputingSystems.1997,pp.
124-131.
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IndexBroadcasting:
�Toachieveenergysaving
�Whenapalmtopisactivelylisteningtoachannel,itCPUmustbein
anactivemodetoexaminedatapacketssoastodetermineifthey
matchtheprede�nedpatterns.
�Therefore,toachieveenergysavingitishighlydesirabletoletthe
palmtopsstayinthedozemodemostofthetimeandonlycometothe
activemodewhenitisnecessary.8
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IndexBroadcasting:
(a) an example index tree
I
a2a1 a3
R1 R2 R3 R4 R5 R6 R7 R8 R9
I a1 R1 R2 R3 a2 R4 R5 R6 a3 R7 R8 R9
(b) index probing scenario to datarecord R5
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�Wheredoesthesystemarrangetheindex?
{Thesystembroadcaststheindexandbroadcastdatainachannel.
{Thesystembroadcaststheindexandbroadcastdataintwo
di�erentchannels.
�Here,weonlydiscussedtheformercase,i.e.,theindexandbroadcast
dataareinthesamechannel.
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�Themainarchitectureofindexdesigninthissubsection.
{Designanindextreewhichcanminimizetheaveragecostofindex
probes.Twocasesareconsidered:onefor�xedindexfanoutsand
theotherforvariantindexfanouts.
�Forthecaseof�xedindexfanouts,inlightsofHu�mancode,CF
(constantfanout)isderivedfortheoptimalindextree
constructionthatminimizestheaveragenumberofindexprobes.
�Forthecaseofvariantindexfanouts,VF(variantfanout)is
derivedandshowsthattheaveragecostofindexprobescanbe
reducednotonlybyemployinganimbalancedindextreethatis
designedinaccordancewithdataaccessskew,butalsoby
exploitingvariantfanoutsforindexnotes.
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ImbalancedIndexTreeConstructionforFixedFanouts
Theaccessprobabilityforeachdatarecordandthenumberofindex
probesrequiredtoreacheacheachrecordbyTBd
=3andTId
=3 .
Datarecord
R1
R2
R3
R4
R5
R6
R7
R8
R9
Pr(Ri )
0.4
0.4
0.05
0.05
0.02
0.02
0.02
0.02
0.02
Ipb (Ri )inTBd
=3
2
2
2
2
2
2
2
2
2
Ipb (Ri )inTId
=3
1
1
2
3
3
3
3
3
3
12
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AlogrithmCF:Useaccessfrequenciestobuildanindextreewitha�xed
fanoutd.
Step1:Initially,wehaveaforestofnsubtrees,eachofwhichisasingle
node.
Step2:Attachthedsubtreeswiththesmallestlabelstoanewnode.
Labeltheresultingsubtreeswiththesumofalllabelsfromitsd
subtrees.
Step3:n=n�d+1.(Then,nisthenumberofremainingsubtrees.)If
n=1stopelsegotoStep2.
Theorem
1:Givena�xedindexfanout,theaveragenumberofindex
probes,i.e., P1�i�nPr(Ri )Ipb (Ri ),isminimizedbyusingtheindextree
constructedbyalgorithmCF.
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Illustration of an imbalanced index tree
(a) an index tree built by CF
I
a1
I R1 R2 a1 R3 a2 R4 R5 R6 a3 R7 R8 R9
(b) a corresponding data broadcastsequence
R1 R2
a2
R4 R5 R6
a3
R7 R8 R9
R3
0.4 0.4 0.2
0.05
.05 .02 .02 .02 .02 .02
0.060.09
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�Denotethecostofprobingfromanindexaiasf(ai ).Then,the
averagecostoflocatingarecordbyprobingindexescanbe
expressedas:
X1�i�n (P
r(Ri )�
Xaj 2Path(Ri )
f(aj ))
�Theindexprobecostrequiredtoreacheachrecordbydi�erentindex
treebuiltbyCF
Datarecord
R1
R2
R3
R4
R5
R6
R7
R8
R9
Pr(Ri)
0.4
0.4
0.05
0.05
0.02
0.02
0.02
0.02
0.02
Paj2
Path(Ri)d(aj)inTId
=2
2
4
8
8
10
10
10
12
12
Paj2
Path(Ri)d(aj)inTId
=3
3
3
6
9
9
9
9
9
9
Paj2
Path(Ri)d(aj)inTId
=4
3
3
7
7
7
11
11
11
11
15
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An index tree T Id=2 built by algorithm CF
I
a1
R1
R2
a2
0.4
0.4
0.06
0.2
R4
.05
R5
.02 R6
.02
R8
.02
R9
.02
R3
.05
0.6
a3
0.11
a4
0.09
a5
a7
R7
.02
a6
0.04
0.04
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An index tree T Id=4 built by algorithm CF
I
a1
R1 R2
a2
R7
0.4 0.4 0.2
0.08
R4
.05
R5
.02
R6
.02 .02
R8
.02
R9
.02
R3
.0517
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ImbalancedIndexTreeConstructionforVariantFanouts
�Basically,algorithmVFisgreedyinnatureandbuildstheindextree
inatopdownmanners
�VFstartswithattachingalldatarecordstotherootnode.Then,
aftersomeevaluation,VFgroupsnodeswithsmallaccessfrequencies
andmovesthemtoonelevellowersoastominimizetheaverageindex
probecost.
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Illustration of VF idea
r
h1 h2
r
hxh1 h2
hm
hi
hmhi+2hi+1
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Lemma:Supposethatnoderhasm
childnodes,h1 ,h2 ,���,hm,which
aresortedaccordingtodescendingorderofPr(hj ),1�j�m,i.e.,
Pr(hj )�Pr(hk )i�j�k.Then,theadvantagecostofindexprobescan
bereducedbygroupingnodeshi+1 ,hi+2 ,���,hm
andattachingthem
underanewchildnodeifandonlyif
(m�i�1)X1
�j�i P
r(hj )>
Xi+1�j�m
Pr(hj )
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AlogrithmVF:
Step1:AssumethatR1 ,R2 ,���,Rn
havebeensortedaccordingto
descendingorderofPr(Rj ).
Step2:Partition(R1 ;R2 ;���;Rn).
Step3:Reporttheresultingindextree.
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ProcedurePartition(h1 ;h2 ;���;hm):
1.Lety(i)=(m�i�1) P1�j�iPr(hj )�P
i+1�j�m
Pr(hj ).Determine
i �suchthaty(i �)=max8i2f1;m�2g fy(i)g.
2.Ify(i �)�0thenreturn.
3.Attachnodeshi�
+1 ,hi�
+2 ,���,hm
underanewindexnodehxinthe
indextree.
4.Partition(hi�
+1 ;hi�
+2 ;���;hm).(h1 ,h2 ,���,hi�
+1 )accordingto
descendingorderofPr(hj ),1�j�i �+1.
5.Partition(h1 ;h2 ;���;hi�
+1 ).
6.Return.
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Example:
WeuseanexampletoillustratealgorithmVF.
Datarecord
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10
R11
Pr(Ri)
0.28
0.2
0.2
0.2
0.04
0.04
0.02
0.005
0.005
0.005
0.005
23
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Illustration of VF
(a)
a1
(b)
a2
a1
.48
.12
.52
I
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11
I
R1 R2 R3 R4
R5 R6 R7 R8 R9 R10 R11
I
R1 R2
R3 R4
(c)
a2
a1
.52
I
R1 R2 R3 R4
a3
(d)
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�Determinethesetofnodestobegroupedtogetherin(a),where
m
=11.
i
1
2
3
4
5
6
7
8
9
(10�
i)
X1<j<i Pr(hj)
9*0.28
8*0.48
7*0.68
6*0.88
5*0.92
4*0.96
3*0.98
2*0.985
0.99
X
i+1<j<11Pr(hj)
0.72
0.52
0.32
0.12
0.08
0.04
0.02
0.015
0.01
y(i)
1.8
3.32
4.44
5.16
4.52
3.8
2.92
1.955
0.98
�Determinethesetofnodestobegroupedtogetherin(b),where
m
=5.
i
1
2
3
(4�
i) P1<j<iPr(hj)
3*0.28
2*0.48
0.68
Pi+1<j<5Pr(hj)
0.72
0.52
0.32
y(i)
0.12
0.44
0.32
25
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Illustration of VF { Further Partitions of Nodes Under Index a1
(a)
a4
(b)
a1
R5 R6 R7 R8 R9 R10 R11
R5
R6 R7 R8 R9 R10 R11
(c)
.08
a1
a4
R5
R6 R7
R8 R9 R10 R11
.08
a1
a5.02
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Illustration of VF { the Resulting Index Tree
.48
a2.52
I
R1 R2R3 R4
a3
a4
R5
R6 R7
R8 R9 R10 R11
.08
a1
a5.02
.12
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Interleaving the broadcasting sequence with index nodes
I a3 R1 R2 a2 R3 R4 a1 R5 a4 R6 R7 a5 R8 R9 R10 R11
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11
I, a3 a2 a1 a4 a5
(a)
(b)
28