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%%Title: A Quick Introduction to Bloom Filters
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%%CreationDate: 1991 Aug 20 16:39:21
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
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/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
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dup 19 /parenrightbigg put
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readonly def
/FontBBox{-24 -2960 1454 772}readonly def
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currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY7) readonly def
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end readonly def
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/Encoding 256 array
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readonly def
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/UniqueID 5000817 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 def
currentdict end
currentfile eexec
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y Fg(:)41 b Fi(E)46 b Ff(!)41 b(f)p Fg(0)p Fi(;)14 b(:)g(:)g(:)f(;)h(m)
25 b Ff(\000)g Fg(1)p Ff(g)p Fj(.)53 b(Bits)32 b(can)d(be)h(set)-249
3480 y(multiple)22 b(times.)32 b(In)22 b(order)f(to)i(test)g(if)g(an)f
(element)g Fi(e)g Fj(may)g(be)h(a)f(mem-)-249 3580 y(ber)h(of)h(the)g
(set,)i(the)e Fi(k)j Fj(bits)d Fi(H)649 3592 y Fh(k)690
3580 y Fg(\()p Fi(e)p Fg(\))h Fj(are)f(check)o(ed.)35
b(If)24 b(all)g(are)g(set,)i Fi(e)e Fj(is)-249 3680 y(considered)g(a)j
(potential)e(member)f(of)i(the)g(set)h(and)f(the)g(Bloom)f(\002lter)
-249 3779 y(test)i(returns)f(true,)i(otherwise)d(f)o(alse.)45
b(If)26 b(only)g Fi(p)h Fj(percent)e(of)i(the)f(bits)-249
3879 y(in)i(the)g(entire)f(Bloom)h(\002lter)g(are)g(set)h(and)e(the)h
Fi(H)1229 3891 y Fh(k)1298 3879 y Fj(return)f(optimally)-249
3978 y(balanced)21 b(random)f(v)n(alues,)i(the)h(probability)d(that)i
(a)h(test)g(willreturn)e(a)-249 4078 y(f)o(alse-positi)n(v)o(e)i
(result)i(is)g Fi(p)551 4048 y Fh(k)592 4078 y Fj(.)38
b(If)24 b(the)h(Bloom)f(\002lter)g(needs)g(to)h(support)-249
4178 y(the)19 b(remo)o(v)n(al)f(of)h(elements)g(from)g(the)g(set,)h(a)g
(secondary)d(datastructure)-249 4277 y(that)k(counts)g(the)g(number)e
(of)i(collisions)g(for)f(each)h(bit)g(is)h(used.)28 b(These)-249
4377 y(collision)g(counters)g(are)h(not)f(required)f(for)h(the)h
(actual)g(test)h(and)e(can)-249 4477 y(therefore)f(be)i(stored)f(in)h
(less)h(costly)f(higher)n(-latenc)o(y)d(storage.)50 b(The)-249
4576 y(length)19 b Fi(m)h Fj(of)g(the)g(bit)g(v)o(ector)e(can)i(be)g
(tuned,)f(trading)f(space)i(for)f(fe)n(wer)-249 4676
y(f)o(alse-positi)n(v)o(e)g(results.)-166 4815 y(W)-7
b(e)25 b(de\002ne)f Fi(b)g Fj(to)g(be)g(the)g(number)e(of)i(bits)h(in)f
(the)g(bit)g(v)o(ector)f(of)h(the)-249 4914 y(Bloom)e(\002lter)h(per)g
(element)f(in)g(the)h(set)h(\()p Ff(j)p Fi(E)5 b Ff(j)20
b(\001)g Fi(b)27 b Fg(=)h Fi(m)p Fj(\).)k(The)22 b(optimal)-249
5014 y(v)n(alue)k(for)g Fi(k)k Fj(can)c(be)h(computed)e(for)h(a)h(gi)n
(v)o(en)e(v)n(alue)h(of)h Fi(b)f Fj(since)h(it)h(is)-249
5113 y(independent)19 b(from)i(the)h(actual)g(size)g(of)g(the)g(set)g
(and)g(only)f(dependent)-249 5213 y(on)f(the)h(ratio)177
5180 y Fh(m)p 161 5194 92 4 v 161 5242 a Fe(j)p Fh(E)s
Fe(j)262 5213 y Fj(.)27 b(F)o(or)20 b(the)h(rest)g(of)g(the)f(paper)g
(we)h(will)g(assume)g(that)f Fi(b)-249 5321 y Fj(is)k(kno)n(wn,)e
(constant)h(and)g(cannot)f(be)h(freely)g(chosen.)33 b(This)23
b(assump-)-249 5421 y(tion)g(holds)f(in)h(particular)f(for)g(\002x)o
(ed)h(sets)h(\(lik)o(e)f(dictionaries)f(or)g(other)-249
5520 y(static)c(databases\))f(on)g(de)n(vices)g(where)f(the)i(a)n(v)n
(ailable)f(storage)f(space)i(is)-249 5620 y(tightly)f(bounded.)22
b(W)m(ithout)17 b(this)h(assumption,)f(the)h(Bloom)f(\002lter)h(may)p
Black Black 1943 1050 a(need)h(to)h(be)f(recomputed)e(when)j
Fi(b)f Fj(changes)g(signi\002cantly)g(in)h(order)e(to)1943
1150 y(obtain)h(optimal)g(f)o(alse-positi)n(v)o(e)g(rates.)2026
1243 y(This)k(lea)n(v)o(es)f(the)h(implementor)e(concerned)f(with)j
(the)g(choice)f(of)g Fi(k)1943 1343 y Fj(for)i(a)h(gi)n(v)o(en)e
Fi(b)i Fj(such)f(that)h(the)g(number)e(of)h(f)o(alse-positi)n(v)o(e)f
(set)j(tests)f(is)1943 1442 y(minimized.)31 b(T)-7 b(ypically)i(,)22
b Fi(k)k Fj(is)d(chosen)f(as)i(a)f(discrete)f(v)n(alue)g(that)h(is)h
(as)1943 1542 y(close)i(as)h(possible)f(to)g(the)h(non-discrete)d
(optimal)h(v)n(alue)h(that)g(w)o(ould)1943 1641 y(minimize)c(the)h
(number)e(of)h(f)o(alse-positi)n(v)o(e)g(tests)i(in)f(the)f
(non-discrete)1943 1741 y(setting.)37 b(In)24 b(other)f(w)o(ords,)i
Fi(k)j Fj(is)d(chosen)e(by)h(computing)e(the)i(optimal)1943
1841 y(v)n(alue)c(for)g(the)h(non-discrete)e(case)i(and)g(then)f
(rounding)f(that)i(v)n(alue)f(up)1943 1940 y(or)f(do)n(wn.)k(The)c
(rationale)g(behind)f(rounding)f(to)i(a)h(non-discrete)d(v)n(alue)1943
2040 y(is)27 b(that)f(it)h(is)g(ob)o(viously)d(impossible)h(to)i(set)g
(a)f(non-discrete)e(number)1943 2140 y(of)c(bits)g Fi(k)k
Fj(in)c(the)g(Bloom)g(\002lter)h(for)e(a)i(gi)n(v)o(en)e(element)h
Fi(e)p Fj(.)2638 2242 y(C)t Fd(O)t(M)t(P)t(U)t(T)n(A)m(T)t(I)t(O)t(N)k
(O)t(F)f Fi(k)2026 2361 y Fj(Suppose)g Fi(k)29 b Fj(w)o(ould)24
b(not)g(ha)n(v)o(e)g(to)h(be)g(chosen)f(as)i(a)f(discrete)f(v)n(alue.)
1943 2460 y(In)j(that)g(case,)i(the)e(optimal)g(choice)f(for)h
Fi(k)k Fj(is)d(simply)f(the)g(v)n(alue)f(that)1943 2560
y(minimizes)19 b(the)i(number)d(of)i(f)o(alse-positi)n(v)o(e)f(tests,)i
Fi(f)3490 2572 y Fh(k)3530 2560 y Fg(\()p Fi(b)p Fg(\))p
Fj(:)2406 2827 y Fi(f)2447 2839 y Fh(k)2488 2827 y Fg(\()p
Fi(b)p Fg(\))i(:=)2722 2685 y Fc(")2770 2827 y Fg(1)18
b Ff(\000)2913 2710 y Fc(\022)2974 2827 y Fg(1)g Ff(\000)3179
2770 y Fg(1)p 3127 2807 146 4 v 3127 2884 a Fi(n)g Ff(\001)h
Fi(b)3282 2710 y Fc(\023)3344 2727 y Fh(k)q Fe(\001)p
Fh(n)3445 2685 y Fc(#)3494 2702 y Fh(k)3548 2827 y Fi(:)367
b Fj(\(1\))1943 3022 y(The)20 b(v)n(alue)f(of)h Fi(k)k
Fj(that)c(minimizes)g Fi(f)3001 3034 y Fh(k)3041 3022
y Fg(\()p Fi(b)p Fg(\))h Fj(is:)2739 3214 y Fi(k)s Fg(\()p
Fi(b)p Fg(\))i(=)g Fi(b)18 b Ff(\001)g Fg(ln)c(2)p Fi(:)699
b Fj(\(2\))2026 3334 y(F)o(or)29 b(this)i(v)n(alue)e(of)h
Fi(k)s Fg(\()p Fi(b)p Fg(\))p Fj(,)j(the)c(number)g(of)g(f)o(alse)i
(positi)n(v)o(es)e Fi(f)3894 3346 y Fh(k)3935 3334 y
Fg(\()p Fi(b)p Fg(\))1943 3457 y Fj(w)o(ould)d(be)2283
3390 y Fc(\000)2331 3424 y Fb(1)p 2331 3438 34 4 v 2331
3486 a(2)2374 3390 y Fc(\001)2412 3407 y Fh(k)q Fb(\()p
Fh(b)p Fb(\))2534 3457 y Fj(.)46 b(But)28 b(since)f(the)g(actual)g
(number)e(of)i(hash)g(func-)1943 3557 y(tions)k(must)h(be)f(discrete,)j
Fi(k)s Fg(\()p Fi(b)p Fg(\))e Fj(must)f(be)h(rounded)d(up)i(or)g(do)n
(wn)f(to)1943 3656 y(obtain)g(a)h(useable)f(discrete)h(v)n(alue)f(for)h
(the)f(number)f(of)i(hash)g(func-)1943 3756 y(tions.)g(Determining)21
b(which)g(bound)g(\(lo)n(wer)g(or)h(upper\))f(is)i(optimal)e(is)1943
3856 y(accomplished)c(by)j(inserting)f(the)g(v)n(alue)g(into)h(formula)
e(\(1\))n(.)26 b(Note)19 b(that)1943 3955 y(while)f(making)f(a)i
(probabilistic)e(choice)g(between)h Ff(b)p Fi(k)s Ff(c)g
Fj(and)g Ff(d)p Fi(k)s Ff(e)g Fj(could)1943 4055 y(be)e(used)g(to)g
(achie)n(v)o(e)f(the)h(optimal)g(information)e(density)h(in)i(the)f
(Bloom)1943 4155 y(\002lter)26 b(\(by)e(probabilistically)g(setting)
3085 4122 y Fb(1)p 3085 4136 V 3085 4183 a(2)3155 4155
y Fj(of)h(the)g(bits\),)i(this)f(impro)o(v)o(ed)1943
4254 y(approximation)d(w)o(ould)j(not)h(result)g(in)f(a)i(better)e
(\(smaller\))g(v)n(alue)g(for)1943 4354 y Fi(f)1984 4366
y Fh(k)2048 4354 y Fj(since)e(the)f(resulting)g Fi(f)32
b Fj(is)25 b(just)f(a)f(linear)h(combination)d(of)i(the)g(tw)o(o)1943
4453 y(e)o(xtremes)d Fi(f)2304 4468 y Fe(b)p Fh(k)q Fe(c)2428
4453 y Fj(and)h Fi(f)2611 4468 y Fe(d)p Fh(k)q Fe(e)2735
4453 y Fj(and)g(can)g(thus)h(only)e(be)i(w)o(orse)g(than)f(the)g
Fi(f)3994 4465 y Fh(k)1943 4553 y Fj(for)e(the)h(best)h(discrete)f(v)n
(alue)f(of)h Fi(k)s Fj(.)2584 4655 y(A)q Fd(C)t(K)t(N)t(O)r(W)t(L)t(E)t
(D)t(G)t(E)t(M)t(E)t(N)t(T)t(S)2026 4774 y Fj(The)i(author)f(wishs)i
(to)g(thank)e(Igor)h(Wronsk)o(y)f(for)h(pushing)f(for)g(the)1943
4874 y(implementation)f(of)j(the)g(Bloom)f(\002lter)i(in)h
Fd(G)t(N)t(U)r Fj(net)e(and)f(Krista)h(Ben-)1943 4973
y(nett)d(for)g(editing.)2756 5146 y(R)t Fd(E)t(F)t(E)t(R)t(E)t(N)t(C)t
(E)t(S)p Black 1943 5239 a([1])p Black 42 w(Burton)c(Bloom.)21
b(Space/time)d(trade-of)n(fs)f(in)f(hash)g(coding)g(with)h(allo)n(w)o
(able)h(errors.)2062 5312 y Fa(Communications)h(of)f(the)f(A)n(CM)p
Fd(,)g(13\(7\):422\226426,)i(1970.)p Black 1943 5385
a([2])p Black 42 w(Francisco)f(Mathias)f(Cuenca-Acuna,)i(Richard)f(P)-7
b(.)15 b(Martin,)i(and)g(Thu)f(D.)f(Nguyen.)2062 5458
y(Planetp:)27 b(Using)19 b(gossiping)i(and)e(random)h(replication)j(to)
c(support)h(reliable)i(peer)o(-)2062 5531 y(to-peer)e(content)h(search)
f(and)f(retrie)n(v)n(al.)34 b(T)-5 b(echnical)21 b(Report)f
(DCS-TR-494,)g(Rut-)2062 5604 y(gers)d(Uni)n(v)o(ersity)l(,)i(2002.)p
Black Black eop end
%%Trailer

userdict /end-hook known{end-hook}if
%%EOF