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Applica'onsofChernoffBound
BarnaSaha
Es'ma'ngSampleSize
• Letpbetheunknownprobabilitythatagenemutates.
• En'redatasetsize=N• Samplesize=n• Inthesampleofthemhavebeenmutated• Es'matedprobabilityofmuta'on Isthisareliable
es,mate?
Whenisareliablees'mate?
• Mustsa'sfy
• Or, Confidenceparameter
Errortolerance
Es'ma'ngSampleSize
• Defineindicatorrandomvariableswhichis1ifthei-thsampledelementhasthedesiredproperty(muta'on/i-phone8query..)and0otherwise.
Es'ma'ngSampleSize
• Wehave
Es'ma'ngSampleSize
• • Wewant
Es'ma'ngSampleSize
• • Wewant
Es'ma'ngSampleSize
• • Wewant
Repea'ngReservoirSampling
Repea'ngReservoirSampling
• Numberofitems=100• Accordingtotheproofofthereservoirsamplingeachitemhas1/100chanceofbeingstoredinthereservoir
• Considerthe1stitemanddefineanindicatorrandomvariablewhichis1ifthethe1stitemisstoredattheendofthealgorithmonitsi-thrun.
• Werunthealgorithm“m”'mes.
Repea'ngReservoirSampling
• Define
• Wewantthefrequencyofallitemstobewithinwithhighprobability.
Repea'ngReservoirSampling
• Wewantthefrequencyofallitemstobewithinwithhighprobability.
• Forthe1stitem
Repea'ngReservoirSampling
• Wewantthefrequencyofallitemstobewithinwithhighprobability.
• Forthe1stitem
• Forthe2nditem
• Forthe100thitem
Repea'ngReservoirSampling
• Wewantthefrequencyofallitemstobewithinwithhighprobability.
• Probthereexistsatleastoneitemoutof100suchthatitsfrequencyisnotintherangeis
Chernoff+UnionBound
• RandomLoadBalancing– SupposeacontentdeliverynetworklikeYouTubereceivesamillioncontentrequestsperminute.Eachrequestneedstobeservedfromoneofthe1000servers.Howshouldonedistributetheloadsothatnoserverisoverloaded.
Chernoff+UnionBound
• RandomLoadBalancing– SupposeacontentdeliverynetworklikeYouTubereceivesamillioncontentrequestsperminute.Eachrequestneedstobeservedfromoneofthe1000servers.Howshouldonedistributetheloadsothatnoserverisoverloaded.
– Assigneachrequesttoarandomserver.
RandomLoadBalancing
• Lettherebe“n”requestsand“k”servers• Considerserver“i”• Defineanindicatorrandomvariablewhichwillbe1ifrequestjisassignedtoserveriand0otherwise.
• Loadonmachinei:
• [ApplytheChernoffBound]
RandomLoadBalancing
• ApplyingtheChernoffBoundweget
RandomLoadBalancing
• ApplyUnionBound
• Prob(thereexistsatleastoneserverwhichisoverloaded)