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Lecture8HASHING!!!!!
Announcements
⢠HW3dueFriday!
⢠HW4postedFriday!
⢠Q:WherecanIseeexamplesofproofs?⢠LectureNotes⢠CLRS⢠HWSolutions
⢠Officehours:linesarelongL
⢠Solutions:⢠Wewillbe(more)mindfulofthroughput.⢠GetmoreTAs⢠Stopassigninghomework⢠UsePiazza!⢠Startearly. (TherearenolinesonMonday!)
Today:hashing
n=9buckets
1
2
3
9
13
22
43
9âŚ
NIL
NIL
NIL
NIL
#
Outline
⢠HashtablesareanothersortofdatastructurethatallowsfastINSERT/DELETE/SEARCH.
⢠likeself-balancingbinarytrees
⢠Thedifferenceiswecangetbetterperformanceinexpectationbyusingrandomness.
⢠Hashfamiliesarethemagicbehindhashtables.
⢠Universalhashfamiliesareevenmoremagic.
Goal:JustlikeonMonday
⢠WeareinterestinginputtingnodeswithkeysintoadatastructurethatsupportsfastINSERT/DELETE/SEARCH.
⢠INSERT
⢠DELETE
⢠SEARCH
5
datastructure
5
4
52
HEREITIS
nodewithkeyâ2â
Today:
⢠Hashtables:
⢠O(1)expectedtimeINSERT/DELETE/SEARCH
⢠Worseworst-caseperformance,butoftengreatinpractice.
OnMonday:
⢠Selfbalancingtrees:
⢠O(log(n))deterministicINSERT/DELETE/SEARCH
#prettysweet
#evensweeterinpractice
eg,Pythonâsdict,JavaâsHashSet/HashMap,C++âsunordered_map
Hashtablesareusedfordatabases,caching,objectrepresentation,âŚ
OnewaytogetO(1)time
⢠Sayallkeysareintheset{1,2,3,4,5,6,7,8,9}.
⢠INSERT:
⢠DELETE:
⢠SEARCH:
9 6 3 5
4 5 6 7 8 9
963 5
1 2 3
6
3 2
3ishere.
Thisiscalled
âdirectaddressingâ
Thatshouldlookfamiliar
⢠KindoflikeBUCKETSORT fromLecture6.
⢠Sameproblem:ifthekeysmaycomefromauniverse U={1,2,âŚ.,10000000000}âŚ.
ThesolutionthenwasâŚâ˘ Putthingsinbucketsbasedononedigit.
1 2 3 4 5 6 7 8 90
345
50 1321
101
1
234
21 345 13 101 50 234 1
INSERT:
NowSEARCH 21
ItâsinthisbucketsomewhereâŚ
gothroughuntilwefindit.
22 342 12 102 52 232 2
INSERT:
ProblemâŚ
1 2 3 4 5 6 7 8 90
342
52
12
22
102
2
232
NowSEARCH 22âŚ.thishasnâtmade
ourliveseasierâŚ
Hashtables
⢠Thatwasanexampleofahashtable.
⢠notaverygoodone,though.
⢠Wewillbemoreclever(andlessdeterministic) aboutourbucketing.
⢠Thiswillresultinfast(expectedtime)INSERT/DELETE/SEARCH.
Butfirst!Terminology.⢠WehaveauniverseU,ofsizeM.
⢠Misreallybig.
⢠Butonlyafew(sayatmostnfortodayâslecture)elementsofMareevergoingtoshowup.
⢠Miswaaaayyyyyyy biggerthann.
⢠Butwedonâtknowwhichoneswillshowupinadvance.
Allofthekeysinthe
universeliveinthis
blob.
UniverseU
Afewelementsarespecial
andwillactuallyshowup.
Example:Uisthesetofallstringsofatmost
140ascii characters.(128140 ofthem).
TheonlyoneswhichIcareaboutarethose
whichappearastrendinghashtagson
twitter.#hashinghashtags
Therearewayfewerthan128140 ofthese.
Examplesaside,IâmgoingtodrawelementslikeI
alwaysdo,asblueboxeswithintegersinthemâŚ
Thepreviousexamplewiththisterminology
⢠WehaveauniverseU,ofsizeM.⢠atmostnofwhichwillshowup.
⢠Mis waaaayyyyyy biggerthann.
⢠WewillputitemsofUintonbuckets.
⢠Thereisahashfunction h:U â{1,âŚ,n}whichsayswhatelementgoesinwhatbucket.
Allofthekeysinthe
universeliveinthis
blob.
UniverseU
nbuckets1
2
3
h(x)=least
significantdigitofx.
Forthislecture,Iâmassumingthatthe
numberofthingsisthesameasthe
numberofbuckets,botharen.
Thisdoesnâthavetobethecase,
althoughwedowant:
#buckets=O(#thingswhichshowup)
Thisisahashtable(withchaining)
⢠Arrayofnbuckets.
⢠Eachbucketstoresalinkedlist.⢠WecaninsertintoalinkedlistintimeO(1)
⢠TofindsomethinginthelinkedlisttakestimeO(length(list)).
⢠h:U â {1,âŚ,n}canbeanyfunction:⢠butforconcretenessletâsstickwithh(x)=leastsignificantdigitofx.
nbuckets(sayn=9)
1
2
3
9
13 22 43
Fordemonstration
purposesonly!
Thisisaterriblehash
function!Donâtusethis!
9
INSERT:
13
22
43
9
âŚ
SEARCH43:
Scanthroughalltheelementsin
bucketh(43)=3.
Aside:Hashtableswithopenaddressing
⢠Thepreviousslideisabouthashtableswithchaining.
⢠Thereâsalsosomethingcalledâopenaddressingâ
⢠ReadinCLRSifyouareinterested!
n=9buckets
1
2
3
9
13 43
âŚ
Thisisaâchainâ
n=9buckets
1
2
3
9
âŚ
13
43
\end{Aside}
Thisisahashtable(withchaining)
⢠Arrayofnbuckets.
⢠Eachbucketstoresalinkedlist.⢠WecaninsertintoalinkedlistintimeO(1)
⢠TofindsomethinginthelinkedlisttakestimeO(length(list)).
⢠h:U â {1,âŚ,n}canbeanyfunction:⢠butforconcretenessletâsstickwithh(x)=leastsignificantdigitofx.
nbuckets(sayn=9)
1
2
3
9
13 22 43
Fordemonstration
purposesonly!
Thisisaterriblehash
function!Donâtusethis!
9
INSERT:
13
22
43
9
âŚ
SEARCH43:
Scanthroughalltheelementsin
bucketh(43)=3.
IPython notebooktime
⢠(Seemstowork!)
⢠(Willthisexamplebeagoodidea?)
SometimesthisagoodideaSometimesthisisabadidea
⢠Howdowepickthatfunctionsothatthisisagoodidea?
1. Wewanttheretobenotmanybuckets(say,n).
⢠Thismeanswedonâtusetoomuchspace
2. Wewanttheitemstobeprettyspread-outinthebuckets.
⢠ThismeansitwillbefasttoSEARCH/INSERT/DELETE
n=9buckets
1
2
3
9
13
22
43
9
âŚ
n=9buckets
1
2
3
9
13 43
âŚ
21
93
vs.
Worst-caseanalysis
⢠Designafunctionh:U->{1,âŚ,n} sothat:
⢠Nomatterwhatinput(fewerthannitemsofU)abadguychooses,thebucketswillbebalanced.
⢠Here,balancedmeansO(1)entriesperbucket.
⢠Ifwehadthis,thenweâdachieveourdreamofO(1)INSERT/DELETE/SEARCH
Canyoucomeupwith
suchafunction?
Wereallycanâtbeatthebadguyhere.
.
UniverseU
h(x)nbuckets
Theseareallthethingsthat
hashtothefirstbucket.
⢠TheuniverseUhasM items
⢠Theygethashedintonbuckets
⢠AtleastonebuckethasatleastM/nitemshashedtoit.
⢠MisWAAYYYYYbigger thenn,soM/nisbiggerthann.
⢠Badguychoosesnoftheitemsthatlandedinthis
veryfullbucket.
Solution:
Randomness
Thegame
13 22 43 92
1. Anadversarychoosesanynitems
đ˘", đ˘$, ⌠, đ˘& â đ,andanysequence
ofINSERT/DELETE/SEARCH
operationsonthoseitems.
2. You,thealgorithm,
choosesarandom hash
functionâ: đ â {1,⌠, đ}.
3. HASHITOUT
1
2
3
n
13
22
92
âŚ
437
7
Whatdoes
randommean
here?Uniformly
random?
Pluckythepedanticpenguin
INSERT13,INSERT22,INSERT43,
INSERT92,INSERT7,SEARCH43,
DELETE92,SEARCH7,INSERT92
#hashpuns
Example
⢠Saythathis uniformlyrandom.
⢠Thatmeansthath(1)isauniformlyrandom numberbetween1andn.
⢠h(2)isalsoauniformlyrandomnumberbetween1andn,independentofh(1).
⢠h(3)isalsoauniformlyrandom numberbetween1andn,independentofh(1),h(2).
⢠âŚ
⢠h(n)isalsoauniformlyrandom numberbetween1andn,independentofh(1),h(2),âŚ,h(n-1).
Universe
U
nbucke
ts
h
Whyshouldthathelp?
Intuitively:Thebadguycanâtfoilahash
functionthathedoesnâtyetknow.
Whynot?Whatifthereâssomestrategy
thatfoilsarandomfunctionwithhigh
probability?
WeâllneedtodosomeanalysisâŚ
Whatdowewant?
1
2
3
n
14
22
92
âŚ
43
8
7 ui 32 5 15
Itâsbad iflotsofitemslandinuiâs bucket.
Sowewantnotthat.
Moreprecisely
1
2
3
n
14
22
92
âŚ
43
8
ui
⢠Wewant:⢠Forallui thatthebadguychose
⢠E[numberofitemsinui âsbucket]⤠2.
⢠Ifthatwerethecase,⢠Foreachoperationinvolvingui⢠E[timeofoperation]=O(1)
So,inexpectation,
itwouldtakesO(1)timeper
INSERT/DELETE/SEARCH
operation.
Sowewant:
⢠Foralli=1,âŚ,n,
E[numberofitemsinui âsbucket]⤠2.
Aside:whynot:
⢠Foralli=1,âŚ,n:
E[numberofitemsinbucketi ]⤠___?
1
2
3
n
14 22 92
âŚ
43 8
thishappenswith
probability1/n
Supposethat:
1
2
3
n
14 22 92
âŚ
43 8
andthishappens
withprobability1/netc.
ThenE[numberofitemsinbucketi ]=1foralli.
ButP{thebucketsgetbig}=1.
Thisslide
skippedinclass
Expectednumberofitemsinuiâs bucket?
UniverseU
nbucke
ts
h
ujui
⢠đ¸ = â đ â đ˘6 = â đ˘7&78"
⢠= 1 +â đ â đ˘6 = â đ˘7ďż˝7;6
⢠= 1 +â 1/đďż˝7;6
⢠= 1 +&="
&⤠2.
Thatâswhat
wewanted.youwillverify
thisonHW
COLLISION!
hisuniformlyrandom
Thatâsgreat!
⢠Foralli=1,âŚ,n,
⢠E[numberofitemsinui âsbucket]⤠2
⢠Thisimplies(aswesawbefore):
⢠Foranysequence ofINSERT/DELETE/SEARCHoperationsonanynelementsofU,theexpectedruntime(overtherandomchoiceofh)isO(1)peroperation.
So,thesolutionis:
pickauniformlyrandomhashfunction.
Theelephantintheroom
Theelephantintheroom
How do we do that?
Letâsimplementthis!
⢠IPython NotebookforLecture8
Letâs NOT implementthis!
⢠SupposeU={allofthepossiblehashtags}
⢠Ifwecompletelychoosetherandomfunctionupfront,wehavetoiteratethroughallofU.
⢠128140possibleASCIIstringsoflength140.
⢠(Morethanthenumberofparticlesintheuniverse)
⢠Andevenignoringthetimeconsiderations
⢠Wehavetostoreh(x)foreveryx.
Issues:
AnotherthoughtâŚ
⢠Justrememberhontherelevantvalues
Algorithmnow Algorithmlater
1322
4392
7
h(13)=6
h(13)=6
h(22)=3
h(92)=3
Howmuchspacedoesittake
tostoreh?
⢠ForeachelementxofU:
⢠storeh(x)
⢠(whichisarandomnumberin{1,âŚ,n}).
⢠Storinganumberin{1,..,n}takeslog(n)bits.
⢠SostoringMofthemtakesMlog(n)bits.
⢠Incontrast,directaddressingwouldrequireMbits.
Hangonnow
⢠Sure,that wayofstoringthefunctionhwonâtwork.
⢠Butmaybethereâsanotherway?
Aside:descriptionlength
⢠SayIhaveasetSwithsthingsinit.
⢠IgettowritedowntheelementsofShoweverIlike.
⢠(inbinary)
⢠HowmanybitsdoIneed?
S
IâllcallthisoneâFidoâThisoneisnamedâHerculesâ
Or,01101011Or,101
Onboard:theanswerislog(s)
Spaceneededtostorearandomfn h?
⢠Saythatthiselephant-shapedblobrepresentstheset
ofallhashfunctions.
⢠IthassizenM.(Reallybig!)
⢠Towritedownarandomhashfunction,weneed
log(nM)=Mlog(n)bits.L
Solution
⢠Pickfromasmallersetoffunctions.
Acleverlychosen subset
offunctions.Wecallsuch
asubsetahashfamily.
Weneedonlylog|H|bits
tostoreanelementofH.H
Outline
⢠HashtablesareanothersortofdatastructurethatallowsfastINSERT/DELETE/SEARCH.
⢠likeself-balancingbinarytrees
⢠Thedifferenceiswecangetbetterperformanceinexpectationbyusingrandomness.
⢠Hashfamiliesarethemagicbehindhashtables.
⢠Universalhashfamiliesareevenmoremagic.
Hashfamilies
⢠Ahashfamilyisacollectionofhashfunctions.
âAllofthehashfunctionsâis
anexampleofahashfamily.
Example:asmallerhashfamily
⢠H ={functionwhichreturnstheleastsig.digit,
functionwhichreturnsthemostsig.digit}
⢠PickhinHatrandom.
⢠Storejustonebittorememberwhichwepicked.
Thisisstillaterribleidea!
Donâtusethisexample!
Forpedagogicalpurposesonly!
H
Thegame
19 22 42 92
1. Anadversary(whoknowsH)choosesanyn
itemsđ˘", đ˘$, ⌠, đ˘& â đ,andanysequence
ofINSERT/DELETE/SEARCHoperationson
thoseitems.
2. You,thealgorithm,choosesarandom hash
functionâ: đ â {0,⌠, 9}.Chooseit
randomlyfromH.
3. HASHITOUT
0
1
2
9 19
22 92
âŚ
42
00
INSERT19,INSERT22,INSERT42,
INSERT92,INSERT0,SEARCH42,
DELETE92,SEARCH0,INSERT92
#hashpuns
h0 =Most_significant_digit
h1 = Least_significant_digit
H={h0,h1}
Ipickedh1
Thegame
1. Anadversary(whoknowsH)choosesanyn
itemsđ˘", đ˘$, ⌠, đ˘& â đ,andanysequence
ofINSERT/DELETE/SEARCHoperationson
thoseitems.
2. You,thealgorithm,choosesarandom hash
functionâ: đ â {0,⌠, 9}.Chooseit
randomlyfromH.
3. HASHITOUT
0
1
2
9
11
âŚ
101
#hashpuns
h0 =Most_significant_digit
h1 = Least_significant_digit
H={h0,h1}
Ipickedh1
11101
111
121
131
141
111
121
131141
Thisadversary
couldhavebeen
moreadversarial!
Outline
⢠HashtablesareanothersortofdatastructurethatallowsfastINSERT/DELETE/SEARCH.
⢠likeself-balancingbinarytrees
⢠Thedifferenceiswecangetbetterperformanceinexpectationbyusingrandomness.
⢠Hashfamiliesarethemagicbehindhashtables.
⢠Universalhashfamiliesareevenmoremagic.
Howtopickthehashfamily?
⢠Definitelynotlikeinthatexample.
⢠LetâsgobacktothatcomputationfromearlierâŚ.
H
Expectednumberofitemsinuiâs bucket?
UniverseU
nbucke
ts
h
ujui
⢠đ¸ = â đ â đ˘6 = â đ˘7&78"
⢠= 1 +â đ â đ˘6 = â đ˘7ďż˝7;6
⢠= 1 +â 1/đďż˝7;6
⢠= 1 +&="
&⤠2.
Sothenumber
ofitemsinuiâs
bucketisO(1).
youwillverify
thisonHW
COLLISION!
Howtopickthehashfamily?
⢠LetâsgobacktothatcomputationfromearlierâŚ.
⢠đ¸ numberofthingsinbucketâ đ˘6
⢠=â đ â đ˘6 = â đ˘7&78"
⢠= 1 +â đ â đ˘6 = â đ˘7ďż˝7;6
⢠⤠1 +â 1/đďż˝7;6
⢠= 1 +&="
&⤠2.
⢠Allweneededwasthatthis ⤠1/n.
Strategy
⢠PickasmallhashfamilyH,sothatwhenIchoosehrandomlyfromH,
forallđ˘6 , đ˘7 â đwithđ˘6 â đ˘7 ,
đUâV â đ˘6 = â đ˘7 â¤1
đ
H
h
⢠AhashfamilyHthatsatisfiesthisis
calledauniversalhashfamily.
⢠ThenwestillgetO(1)-sizedbucketsin
expectation.
⢠Butnowthespaceweneedis
log(|H|)bits.⢠Hopefullyprettysmall!
InEnglish:fixany
twoelementsofU.
Theprobability
thattheycollide
underarandomh
inHissmall.
Sothewholeschemewillbe
nbucke
ts
h
ui
UniverseU
Choosehrandomly
fromauniversalhash
familyH
Wecanstorehinsmallspace
sinceHissosmall.
Probably
these
bucketswill
bepretty
balanced.
UniversalhashfamilyLetâsstareatthisdefinition
⢠Hisauniversalhashfamilyif:
⢠WhenhischosenuniformlyatrandomfromH,
forallđ˘6 , đ˘7 â đwithđ˘6 â đ˘7 ,
đUâV â đ˘6 = â đ˘7 â¤1
đ
Youactuallysawthisinyourpre-lectureexercise!
Toads=hashfns
Icecream=items
âLikeâandâDislikeâ=buckets
CheckourunderstandingâŚ
⢠Hisauniversalhashfamilyif:
⢠WhenhischosenuniformlyatrandomfromH,
forallđ˘6 , đ˘7 â đwithđ˘6 â đ˘7 ,
đUâV â đ˘6 = â đ˘7 â¤1
đ
⢠His[somethingelse]if:
⢠WhenhischosenuniformlyatrandomfromH,
forallđ˘ â đ, forallđĽ â {0, ⌠, đ â 1},
đUâV â đ˘6 = đĽ â¤1
đ Arethese
different?
Slide
(probably)
skippedin
class
Pre-lectureexercise
Universe={vanilla,chocolate}
Buckets={like,dislike}
Toads=differentpossiblewaysofdistributingitems
Statement1:P[randomtoadlikesvanilla]=½,P[randomtoadlikeschocolate]=½
P[âvanillaâlandsinthebucketâlikeâ]=½
Statement2:P[randomtoadfeelsthesameaboutchocolateandvanilla]=½
P [vanillaandchocolatelandinthesamebucket]=½
Slideskippedinclass
Pre-lectureexercise
Universe={vanilla,chocolate}
Buckets={like,dislike}
Toads=differentpossiblewaysofdistributingitemsSeemliketheymightbethesame�
Statement1:P[randomtoadlikesvanilla]=½,P[randomtoadlikeschocolate]=½
P[âvanillaâlandsinthebucketâlikeâ]=½
Statement2:P[randomtoadfeelsthesameaboutchocolateandvanilla]=½
P [vanillaandchocolatelandinthesamebucket]=½
Slideskippedinclass
Pre-lectureexercise
Universe={vanilla,chocolate}
Buckets={like,dislike}
Toads=differentpossiblewaysofdistributingitemsButno!1istruebut2isnot.
Statement1:P[randomtoadlikesvanilla]=½,P[randomtoadlikeschocolate]=½
P[âvanillaâlandsinthebucketâlikeâ]=½
Statement2:P[randomtoadfeelsthesameaboutchocolateandvanilla]=½
P [vanillaandchocolatelandinthesamebucket]=½
Slideskippedinclass
CheckourunderstandingâŚ
⢠Hisauniversalhashfamilyif:
⢠WhenhischosenuniformlyatrandomfromH,
forallđ˘6 , đ˘7 â đwithđ˘6 â đ˘7 ,
đUâV â đ˘6 = â đ˘7 â¤1
đ
⢠His[somethingelse]if:
⢠WhenhischosenuniformlyatrandomfromH,
forallđ˘ â đ, forallđĽ â {0, ⌠, đ â 1},
đUâV â đ˘6 = đĽ â¤1
đ Theseare
different!
Slideskippedinclass
Example
⢠Uniformlyrandomhashfunctionh
⢠[Wejustsawthis]
⢠[Ofcourse,thisonehasotherdownsidesâŚ]
⢠PickasmallhashfamilyH,sothatwhenIchoosehrandomlyfromH,
forallđ˘6 , đ˘7 â đwithđ˘6 â đ˘7 ,
đUâV â đ˘6 = â đ˘7 â¤1
đ
Non-example
⢠h0 =Most_significant_digit
⢠h1 =Least_significant_digit
⢠H={h0,h1}
⢠[discussiononboard]
⢠PickasmallhashfamilyH,sothatwhenIchoosehrandomlyfromH,
forallđ˘6 , đ˘7 â đwithđ˘6 â đ˘7 ,
đUâV â đ˘6 = â đ˘7 â¤1
đ
Asmalluniversalhashfamily??
⢠Hereâsone:
⢠Pickaprimeđ ⼠đ.
⢠Defineđ],^ đĽ = đđĽ + đđđđđ
â],^ đĽ = đ],^ đĽ đđđđ
⢠Claim:
đť = {â],^ đĽ âś đ â {1,⌠, đ â 1}, đ â {0,⌠, đ â 1}}
isauniversalhashfamily.
Saywhat?
⢠Example:M=p=5,n=3
⢠TodrawhfromH:
⢠Pickarandomain{1,âŚ,4},bin{0,âŚ,4}
⢠Asperthedefinition:
⢠đ$," đĽ = 2đĽ + 1đđđ5
⢠â$," đĽ = đ$," đĽ đđđ3
1,2,3,4,5a=2,b=1
1
23
40
đ$," đĽ
1
23
4 0
đ$," 1
đ$," 0
đ$," 3
đ$," 4đ$," 2U=
1
2
3
mod3
Thisstepjust
scramblesstuffup.
Nocollisionshere!
Thisstepistheone
wheretwodifferent
elementsmightcollide.
Ignoringwhythisisagoodidea
⢠Canwestorehwithsmallspace?
⢠Justneedtostoretwonumbers:
⢠aisin{1,âŚ,p-1}
⢠bisin{0,âŚ,p-1}
⢠Soabout2log(p)bits
⢠Byourchoiceofp,thatâsO(log(M))bits.
1,2,3,4,5a=2,b=1
Compare:directaddressingwasMbits!
Twitterexample:log(M)=140log(128)=980 vsM=128140
AnotherwaytoseethisusingonlythesizeofH
⢠Wehavep-1choicesfora,andpchoicesforb.
⢠So|H|=p(p-1)=O(M2)
⢠Spaceneededtostoreanelementh:
⢠log(M2)=O(log(M)).
O(Mlog(n))bits
perfunction
O(log(M))bits
perfunction
Whydoesthiswork?
⢠Thisisactuallyalittlecomplicated.
⢠Therearesomehiddenslideshereaboutwhy.
⢠Alsoseethelecturenotes.
⢠Thethingwehavetoshowisthatthecollisionprobabilityisnotverylarge.
⢠Intuitively,thisisbecause:
⢠forany(fixed,notrandom)pairđĽ â đŚ in{0,âŚ.,p-1},
⢠Ifaandbarerandom,
⢠ax+banday+bareindependentrandomvariables.(why?)
Whydoesthiswork?
⢠Wanttoshow:
⢠forallđ˘6 , đ˘7 â đwithđ˘6 â đ˘7 , đUâV â đ˘6 = â đ˘7 â¤"
&
⢠aka,theprobabilityofanytwoelementscollidingissmall.
⢠Letâsjustfixtwoelementsandseeanexample.
⢠Letâsconsiderđ˘6 , = 0, đ˘7 = 1.
1
23
40
đ],^ đĽ
1
23
4 0U=
1
2
3
mod3
đđĽ + đđđđđ
Convince
yourselfthatit
willbethesame
foranypair!
Thisslideskippedinclassâ hereforreference!
Theprobabilitythat0and1collideissmall
⢠Wanttoshow:
⢠đUâV â 0 = â 1 â¤"
&
⢠ForanyđŚj â đŚ" â {0,1,2,3,4},howmanya,b aretheresothatđ],^ 0 = đŚjandđ],^ 1 = đŚ"?
⢠Claim:itâsexactlyone.
⢠Proof:solvethesystemofeqs.foraandb.
1
23
40
đ],^ đĽ
1
23
4 0U=
1
2
3
mod3
đđĽ + đđđđđ
eg,y0 =3,y1 =1.
đ â 1 + đ = đŚ"đđđđ
đ â 0 + đ = đŚjđđđđ
Thisslideskippedinclassâ hereforreference!
Theprobabilitythat0and1collideissmall
⢠Wanttoshow:
⢠đUâV â 0 = â 1 â¤"
&
⢠ForanyđŚj â đŚ" â {0,1,2,3,4}, exactlyonepaira,b haveđ],^ 0 = đŚjandđ],^ 1 = đŚ".
⢠If0and1collideitâsb/cthereâssomeđŚj â đŚ"sothat:
⢠đ],^ 0 = đŚjandđ],^ 1 = đŚ".
⢠đŚj = đŚ"đđđđ.
1
23
40
đ],^ đĽ
1
23
4 0U=
1
2
3
mod3
đđĽ + đđđđđ
eg,y0 =3,y1 =1.
Thisslideskippedinclassâ hereforreference!
Theprobabilitythat0and1collideissmall
⢠Wanttoshow:
⢠đUâV â 0 = â 1 â¤"
&
⢠Thenumberofa,b sothat0,1collideunderha,b isatmostthenumberofđŚj â đŚ"sothatđŚj = đŚ"đđđđ.
⢠Howmanyisthat?⢠WehavepchoicesforđŚj,thenatmost1/noftheremainingp-1arevalidchoicesforđŚ"âŚ
⢠Soatmostđ â l="
&.
1
23
40
đ],^ đĽ
1
23
4 0U=
1
2
3
mod3
đđĽ + đđđđđ
eg,y0 =3,y1 =1.
Thisslideskippedinclassâ hereforreference!
Theprobabilitythat0and1collideissmall
⢠Wanttoshow:
⢠đUâV â 0 = â 1 â¤"
&
⢠The#of(a,b) sothat0,1collideunderha,b is⤠đ â l="
&.
⢠Theprobability(overa,b)that0,1collideunderha,b is:
⢠đUâV â 0 = â 1 â¤lâ
mno
p
V
⢠= lâ
mno
p
l l="
⢠="
&.
Thisslideskippedinclassâ hereforreference!
Thesameargumentgoesforanypair
forallđ˘6 , đ˘7 â đwithđ˘6 â đ˘7 ,
đUâV â đ˘6 = â đ˘7 â¤1
đ
Thatâsthedefinitionofauniversalhashfamily.
SothisfamilyHindeeddoesthetrick.
Thisslideskippedinclassâ hereforreference!
Butletâscheckthatitdoes work
⢠BacktoIPython NotebookforLecture8âŚ
Empiricalprobabilityofcollisionoutof100trials
Numberofpairsof(x,y).
(Outof$jj$
=19900pairs)
M=200,n=10
Sothewholeschemewillbe
nbucke
ts
ha,b
ui
UniverseU
Chooseaandbatrandom
andformthefunctionha,b
Wecanstorehinspace
O(log(M))sincewejustneed
tostoreaandb.
Probably
these
bucketswill
bepretty
balanced.
Outline
⢠HashtablesareanothersortofdatastructurethatallowsfastINSERT/DELETE/SEARCH.
⢠likeself-balancingbinarytrees
⢠Thedifferenceiswecangetbetterperformanceinexpectationbyusingrandomness.
⢠Hashfamiliesarethemagicbehindhashtables.
⢠Universalhashfamiliesareevenmoremagic.
Recap
WantO(1)INSERT/DELETE/SEARCH
⢠WeareinterestinginputtingnodeswithkeysintoadatastructurethatsupportsfastINSERT/DELETE/SEARCH.
⢠INSERT
⢠DELETE
⢠SEARCH
5
datastructure
5
4
52
HEREITIS
Westudiedthisgame
13 22 43 92
1. Anadversarychoosesanynitems
đ˘", đ˘$, ⌠, đ˘& â đ,andanysequence
ofLINSERT/DELETE/SEARCH
operationsonthoseitems.
2. You,thealgorithm,
choosesarandom hash
functionâ: đ â {1,⌠, đ}.
3. HASHITOUT
1
2
3
n
13
22
92
âŚ
437
7
INSERT13,INSERT22,INSERT43,
INSERT92,INSERT7,SEARCH43,
DELETE92,SEARCH7,INSERT92
Uniformlyrandomhwasgood
⢠Ifwechoosehuniformlyatrandom,forallđ˘6 , đ˘7 â đwithđ˘6 â đ˘7 ,
đUâV â đ˘6 = â đ˘7 â¤1
đ
⢠Thatwasenoughtoensurethat,inexpectation,abucketisnâttoofull.
Abitmoreformally:
Foranysequence ofINSERT/DELETE/SEARCHoperations
onanynelementsofU,theexpectedruntime(overthe
randomchoiceofh)isO(1)peroperation.
Uniformlyrandomhwasbad
⢠Ifweactuallywanttoimplementthis,wehavetostorethehashfunctionh.
⢠Thattakesalotofspace!⢠WemayaswellhavejustinitializedabucketforeverysingleiteminU.
⢠Instead,wechoseafunctionrandomlyfromasmallerset.
Weneededasmallersetthatstillhasthisproperty
⢠Ifwechoosehuniformlyatrandom,forallđ˘6 , đ˘7 â đwithđ˘6 â đ˘7 ,
đUâV â đ˘6 = â đ˘7 â¤1
đ
Thiswasallweneededtomake
surethatthebucketswere
balancedinexpectation!
⢠Wecallanysetwiththatpropertya
universalhashfamily.
⢠WegaveanexampleofareallysmalloneJ
Conclusion:
⢠WecanbuildahashtablethatsupportsINSERT/DELETE/SEARCH inO(1)expectedtime,
⢠ifweknowthatonlynitemsareeverygoingtoshowup,whereniswaaaayyyyyy lessthanthesizeMoftheuniverse.
⢠Thespacetoimplementthishashtableis
O(nlog(M))bits.⢠O(n)buckets
⢠O(n)itemswithlog(M)bitsperitem
⢠O(log(M))tostorethehashfn.
⢠Miswaaayyyyyy biggerthann,butlog(M)probablyisnât.
Thatâsitfordatastructures(fornow)
DataStructure:RBTrees andHashTables
Nowwecanusethesegoingforward!
Before NextTime
⢠Graphalgorithms!
⢠Pre-lectureexerciseforLecture9
⢠Introtographs
NextTime