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)