BloomFilter&Hashing

BarnaSaha

BloomFilter

•  ChecksforSETMEMBERSHIPefficiently

Iselementxintheset?

MoAvaAngExample

•  SpamFiltering

Ø Wehaveasetof1billionemailaddressesthatweconsidertobenon-spam.Ø  Eachstreamelementisoftheform(emailaddress,email).Ø  BeforeaccepAngtheemail,amail-clientneedstocheckifthisaddressbelongstosetS.Ø  Eachtypicalemailaddressrequires20bytesofstoragewhereasinthemainmemoryweonlyhavesay1billionbyte(roughly1Gigabyte),or8billionbits.Ø Wecannotstoreallthevalidemailaddressesinthemain

memory.

MoAvaAngExample

•  SpamFiltering– Allvalidemailsmustbedelivered– Numberofspamemailsdeliveredshouldbeaslowaspossible

BloomFilter

BloomFilter

AnalysisofBloomFilter

AnalysisofBloomFilter

SpamFilteringExample

•  Wehave

OpAmumValueofk

•  AsthenumberofhashfuncAonsincrease,higheristhechanceoffindinga0bitcell

•  AlsowithincreasingnumberofhashfuncAons,thenumberofcellswith0bitsdecreases

•  OpAmumvalueobtainedbydifferenAaAon

ApplicaAonsofBloomFilter

•  BloomFilterhasfoundinnumerableapplicaAonsinnetworkingandwebtechnology

AnalysisofBloomFilter

AnalysisusesfullyrandomhashfuncAons—difficulttoobtainwithhighspaceandcompuAngrequirements

Strongly2-wiseUniversalHashFuncAon

•  MappingsetofkeysU=[0,1,2,…,m-1]torangeR=[0,1,2,…,n-1]– H={ha,b=[(ax+b)modp]modn}

•  p>=misaprime,1<=a<=p-1,0<=b<=p-1•  Easytocomputeandstore:O(1)•  SaAsfies(almost)forall,

Strongly3-wiseUniversalHashFuncAon

•  MappingsetofkeysU=[0,1,2,…,m-1]torangeR=[0,1,2,…,n-1]– H={ha,b=[(ax2+bx+c)modp]modn}

•  p>=misaprime,1<=a<=p-1,0<=b,c<=p-1•  Easytocomputeandstore:O(1)•  SaAsfies(almost)

•  MappingsetofkeysU=[0,1,2,…,p-1]torangeR=[0,1,2,…,p-1]– H={ha,b=(ax+b)modp},0<=a,b<=p-1

•  Fix.– Whatis?– NumberofhashfuncAons– NumberofsoluAonsfor“a”and“b”=1

Strongly2-Universal