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BL^C vC0C%]%@A-?0(83h*b.> T  !  ZDFC >D ] \*3 Add(x) Median()* 3 2 : Pr en !6I 36I C 6DI3=I9%'fIB"59B!IVI]5IM*]1L*1'2%!:J !].H==  -  IA   *]Y4A1_A5T=T VCBTbT ]5R$*I۠%1TB?1. Let S be a uniformly random subset of middle 1/3rd elements * )pI )RLB  ۠% .VH:]9~D%)E! f ]5q%*.(2. Insert 5bof S*QIVb-b*^ :o6'-DIo-o!oofo!o.o({*ٶF)o,Claim: ContE(of C encode=uF^uIٶ!6u %: :h3pYCBAM!h `SCIB>ff p U* +f*3)nx=Xi = {j| j is accessed while iE4ing i or compum:}3V:-`0+ t 'i  ( < 6WB>.1!9Av2>fj_o ]!5 *% $*])Ha) If x∈[1,p/3]∪[2p/3+1,p], x can be-!ith-, only to C.*]Vte2)4 $    U B &C5.2(:$ECE689E5-9B8E-483D-84A8-8152DEA5953F >*ec 6 3: 6  B a4Y2 >*6 V ]U5 [*" ProofD Theorem 1uV5}2}=H4? .ͧ*!bbbbbbbbbbbJ 0dBNReC CG b bA6L b3T %( !%BD= bf .>f1 b* [*\X1u\V eU3%j5M*b54AD#!Y!PA9nD% " 0*Ýb MM BaA0IMNjÝ AW\ﵪBPjA & [B  ( (! / K.( 5D - (- P! \ s ( b<*I }!:b9 6C~!: 2 bu8BAdjmCi0 vea 6:ܸBןFparA>!:: &CdAB4)!F64 Ý.ҍBXagB\Ě|FA b" K*+8C"+RB 2D bDnBDkDDD  & (*3b CNCjC 3FB0*aF>B:BBRV6!> y2 6 0qC8Kpy =?h > b H El]: Xp/3 * 52 ƄbB8e* ƅ 6 .:*BA 6  jZIaIIFIb}I{.{BaAV{6!F26BDi4>bǁ2^:XN2 b2 BBRI CbOaJJ>J!:: ƁR|)!>6 3b&p~~B`X"D2[*\&p&2 b•@C#9X遢  B.ŒBNR aGGFG2+ . ! t˥U yGn !>)&0*] ]*$>*("NB'2( PKIRwJVeC77Index/Slide.iwa3 T4^.*"^^^!^^^^^^^ "31B/ Transitionnone?)l?0X惂*^2^: : ^!^^^C    ^=  瘾*3^   ?  C 6D %!)%@A-?0(83h*b.>T . BY6BB >D ^ `M*^ * 2 :J R^Pr#    g$  !6V^32VXBL^C vC0CV.V>fVZD2VC2VVV^5V\*3)SDAdd(x) Median()* -_3 Rc.W)W en=bA),S* ^^^MpC!G~C"hG$^ ^o G*i  `AECiVif^ M. *^}i*Ca%yijc .i&aGLBfi ^UL* I or*qD !M,*2``M5* I "7ByIf"68JTCBZ`b`f0:RAxJ`BܜA<\CY 6 |~0:RM0*^ ^P >*E@N2=26,d"BXLX!Jg7IDν~}j`^6]] Ae( $} 32}Z%H]gjc6>f}ŀ]SV ^}0* 33*EBcfTheorem 3: Any data structure on [n] that computes predecessors non-adaptively must have either U3RB j ,"* #P*[Т2  (T$CFD2E14C-CBC4-4F11-90B3-40C4D4815DE1 X} Bm6.C 6mPQmjm ^* e*350 w: word size5Z )Z=Q 6 ^32 v B @as *}n`. >C*6 VC^d  HLower Bound for Pru5aVaQ =  B.  QΑA`j ^w* 3*] ti: time%ainserts5]^]Nw Bf.f MDfzf!f.f{Fftm=fjnjPKIRwJZ`Index/MasterSlide-16.iwaL% *  "31B/ Transitionnone?)l0X*2:DRTitle, Bullets & Photo Media ZLorem Ipsum DolorMaecenas aliquam ma   BMC $CC%A,l%@A-?0(8$h*b.=T H _6C 0>D@eaT NBody Level One Two2hre2 Four.! 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Therefore, % ≥ p/3w5oR5)s 5 5Z PKIRwJLIndex/Slide-1808.iwaT-!(*0Z!ZZ!-Z/)2 "31B/ Transitionnone?)`?0X惂*-2/: <:!:Z:|Z!)02> !@,Z0 D*. .. >* * 2 : Pr x  Ux*Z  >  |CD !%l!%@A-?0(8 hL*b @CLBT .O B_6BB >DZ5&0w* 3**%*Tti: time for inserts5@3J@B d@* * .Z)Zen !:eZ9e.e)ePeze Z5ee*3=b0 w: word sizeZ=ZPa&66PRDIPzP Z]vFtm]medianZ$!E*! XsB,C $_ g7IDHBn_3=fZDʑCN""]0* 3۠%3toTheorem 1: Any data structureMeA'` on [n] with non-adaptive andMR opera,s must have *bZ"mL3  -۠% .ZТ2  (d$CFD2E14C-CBC4-4F11-90B3-40C4D4815DE1 hZ -E(*31 .T B`%*]Un.=*6V11e*-DFirst Lower Bound=5VZ=M m,*2]]m|>a }CbM"A 58J8Z]b]f >uW : R  /6, 3(2,BL^vC0,2"=, V,((],\A,M,Add(x)  ()* maR# )6D 3)2DC 6/ 2p%fD/ ,))" L*0d uYNZ J0.A}Y0 d `  2 PKIRwJV Index/Slide-1805.iwa @XR*F)׆*!))Vk*))*))) ":8B/ Transitionnone?) ?0XÐhx*2: (::):<׆*H V:k!)   !I  F ׆* V k`)*)3  ?  RD!8C A(kIC%!g%@A-?0(83*.@  ﵪBPjA & [BB B(- (- / K.( 5D - (- P! \ s (>) -E@<*3 * ! 032 :Pr !=,*ۆ* -C )!% h<*b |RD`BT DP %BJB B=%Dۆ*5%4* %$3%,5For every sequence X=X1,X2,…,Xn such that |Xi|≤t,5cVctBf  $ %   *, - 0.)en2($5F59EC52-C955-4213-8217-1C4DCB79EC37 2 >2ȣKD>I .C  )>};.;2GD0% BB;@;;;; K*+<C* B+@}u2@B@3)`Da%2@B@@@@m@a@)@6@ z,*3  C 6DŅe3ՏbU M*ձRJQ.U]  s$  [  0* = >*}N^  )^ V=Bg2ALf V=8K*if5:Z.V -E,*2hh {BG)4tC$K" 68J8Zhbhf >& $: $R 2+A3YlD?2; j } [*+ X2ջVB  3' B[* AI'G2[!_Dc[f[ **$.h******$TAafter renaming, ∃C* Xi∩Xj=C for distinct i,j ∈ [p]*]N Bp% `* *     ! "$"/ } 2 6 &9       6D3>Da Z *2[>fD "' *C6)*CZ)Up*3m\Sunflower Lemma [ErdosRado60]*r$3 V6w1w=_}q J/.{  ; :; .{ P2P)ZQDN) PP rP1PFP 6>32 BL^!IvC0>.>>>6>&" * \ ۍ( 3*cHy9!G6c4ccccc((F(F1 .F "D  -!x jx *` 3*! *, thenre is a /4with p petals:~ B " J . ZDك    Z a F B6Gz\D:d 6"FDaF0!F>d 3&51)d d vd vd pd Fd PKIRwJi>**Index/Slide-1788.iwa& D$m~*mmmmm "31B/ Transitionnone?)l?0X*m2m: : |mmm+ m  0*m m  >* * 2 : Pr u  pN*m   ?  BMC ZDC%m%@A-?0(8 h*b.>T   `6C 0>D m5+<* 33C%2Monotone Circuit Lower bounds: c[Razborov85] [AlonBoppana87] Dynamic data structures with w=1: [FrandsenMiltersenSkyum93] Static da^=[GalM<|07] (succinct representations)*-(3 E JtBX  (C D b;e g E    2kIken4r)$18BCA0F6-9357-438F-8E86-2FF51644609F ]:> C 6DAM]2>faU B2RmU$M*mI uNJ AmZ  ?   6mN B!gi*C2g>fg "9g *6Vgm?a*H)dSunfla8s in Complexity5yHJF^( PKIRwJ_eIndex/Slide-1806.iwa H("l*ŪӪ "-+B) Transitionnone?)<?0*Ӫ2:Ū& *2 ?  Fg*CB ]*DTD%t%@A-?0(8"68J EDZbT  ] 06D 0>D  0*   >* * 2 : Pr  p8*3˪  -R C 6D%rR 3h*b.>f>B6NB2>J> ˪5+$M*%,51N1J RZ=  -$  A6V3>VMC! ZDC!V2V>fV"]C2VF ʪ]<*33%S5N3J3ZB>)B)!)s9)*)*)*6)V)6):)N)^@ PKIRwJ9Index/Slide-264.iwa > t*  "20B. Transitionnone?)`?0X:* 2 : : p   &,    *$   >* * 2 : Pr T l*    >  C 6D %l %@A-?0(8 h*b.=T . B_6BB >D0  5& L* %&5+N+J1K Z6  -8   .N :N B NDCN6N =jN"9NC6NCZN Qt{*H)LD2Lower Bounds for lNon-Adaptive Data Structures5{HM]2Ien !{Ia12s NsC9sBsJsBs:B V$  5s!t-tLISivaramakrishnan Natarajan Ramamoorthy, Anup Rao University of Washington* 1H 1-PKIRwJ9Index/MasterSlide-1.iwaL% *  "31B/ Transitionnone?)l0X*2:>RPhoto - Vertical Media Z Lorem IpsumDonec quis nunc(  %8$  H- C 8D O (@D <*   >  C 6D !b%l%@A-?0(8$h*b.=T . B_6BB >D$ 4<* * ! 2 :Pr R=$:$@DHB C"D$6$ =j$"9$"D6$"DZ$2$6$$R=$:$BC5$C$F$C$C2$VH$5$aII@NBody Level One Two2hre2 Four.!Five*ML   / ?2(Q /? i(s*2 qVAI@sD* " 5*8J3a`DZ bj   oU'2Kfo.jmfj"F:N-J lFC6N5ґH*Q, Title Text*,APKIRwJZ( Index/Slide-1809.iwa Y`4 /*#%o%+%*''"! %"31B/ Transitionnone?)d?0X惂**2": ::%:o!'+'34%$o0%% \ `, All at Onc"<  \N@- In apple:appear 4XΑ( m*o  >  SIAVC pD%W, ,@A-?0(8$h*b=0BT  CO BJB 0B >Do#  >*+۠%o*3 1. Let X = X1,X2,….,Xn, where LXi = {j| j is accessed while inserting i or computing median}* o2 !2: !EB  ۠%  +   <3  I!J !K T p!Q(* q   Pr en AIS L*%%MP\ڭCD (&DB%A vP,%%]4*%  ?E<b %U8* $*I03. Use C to encode a random subset of [p/3,2p/3]U[V[0]Z)  !g$ ') /%QG*%-!<UA d%D?Z5b%,*% }.c@cp k CE[%5<*%*qR5Z}Q+ *###j >*6lNl^l m -D!!x(0: ƸԆrBR* [(*3# m! aZDU~m3m=f_1 *C-Nm #.Y*3- OutlineaProof53V[=N Zh 睗ԺBV' 筹6XFAG {wDCqhr   6u>CyAyA!iCO ? }fm=W M*.%Wm5W~%X  & "3"a  8*u>fX.n>nC2nNn[*3*2. Identify a Sunflower in X with p petalsuVB  "#b 4!.*!6;AoC]nEA:/& M*XXCHB_ Aj7fa y0* .  A(kI j rjm A\ﵪBPjA & [B  ( (! / K.( 5D - (- P! \ s (y : UVU6p!:N,1Y)!>e2 u+!CkJe0vea 6NfA>!:Nf%!>o6n 3.n ܐBX#3 r.n , Ljn IAlI[*+?Xp"]+R B F {.pV 6{:xIBv B"#vHFA& 5K*+C5V0 >.4C%zA2]AUXp/35:Q 2 XX$CwS76n B 6Y:Y5Vn ::>:!:Nf%!>2 } @Fvn eaÅwжA>%64%!>2 un#CFF4ea 6N9V4A>!:Nf4%!>A: 3&va.    ^: X2 Z 2 W#~XFp.~JaJJ>J%6~ Ǫ~%!>a66|""~*X1F)2((.( )('6z"3:z"C 6&!2r 2f ,##5 L*)#"N$J$).*y$) z$0  "6N 3.2#BL^vC0" zN2Z =fNR#J U*l&E5(..*! \*"# Add(x) M) ()* )L)R.QVPKIRwJ1}܍Index/Slide-1709.iwaT4ʛ[.*"˛[ԛ[[![ӛ[Λ[[[ϛ[ޛ[ "31B/ Transitionnone?)l?0X惂*ϛ[2ޛ[: : [!Λ[ӛ[[C     [! G*̛[ ?  sB,C $T  \ ZDl2C >D՛[  c* 3۠%3 oTheorem 1: Any data structure for Median on [n] with non-adaptive insert and med+Lperations must have 8* 32 !: B X3  -۠% .Pr4 en2  (d$CFD2E14C-CBC4-4F11-90B3-40C4D4815DE1 h@A:ݛ[9 |CD!%y%z =@CLjB2 B2V!ݛ.0w* 3**) ti: time%i%s5Z  @* * Z=f3 0*2]] |>!ԁ }_b_"aE 68J8Z]b]bs >us :zBNz(   @m}0*[ [u >*%5J$.l=a$Λ[a,*3֛[ YC 6.AEfI=OE٢֛[5+$M*Л[%,51N1JHAЛ[Z=  -  B. RDAj2vFtm5eJHBe2e.Ce)ePeze []e*30 w: word size[w.2 > 3> i Bn**1 fo2 >f  " *6V[7e*DFirst Lower BoundMediaYVR>R3:RBL^vC0R2 >R [=R\!R-RAdd(x) )C ()*   JIPKIRwJEv??Index/Slide-1786.iwa; T4.*"!^^^^ "31B/ Transitionnone?)l?0X惂*2: : ^:^!C ^   ^! G* ?  d"BXLXC g7IDν~C%!!X%@A-?0(8 6]]AJ$ $T  \ ZDl2C >D  u* 33* fTheorem 3: Any data structure on [n] that computes predecessors non-adaptively must have either 8* 32 !: B X3  "* #Pr4 en2  (T$CFD2E14C-CBC4-4F11-90B3-40C4D4815DE1 X@ax )*%9BL^avC0]U>b5\*3)DAdd(x) Median()* -w!wRZ~=IMf0*Ě Ě >*UJ{2{p:^9 CDaIe)%z }ΑC`jB2 B2V ^}$w* 3*uPti: time for inserts5AZW  aJeTc63Ś2C 6h.!%Ff)X hŚu$M*E5VNJ!V.H]  m  w:^Y.QPALf ^=Re*3]0 w: word size5[^]._=jBQ2QMD  !Q j  . {F tm `:3>  B @a*0.>fR" *6" Vyݢd00,Lower Boundx Pred5kV| " 0S* ^^^  pC~C"hO ^ 2 i  `A1CiVi^j ^:2! ̵fc>&Aej ^*L* ( or* J! ,*2``M5* I"7*s f"68Jg @BZ`b`b 0v @6BNeAxJ`BܜA<\CY 6 |~0:Ry, PKIRwJIndex/Slide-1772.iwa8k*kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk!kkkkkkkkkkkkkkkkkkkkkkkkkkkkk "97B. Transitionnone@)?0XJhx*k2k:k:  N     T  H     B *B    $  *k!kkk%          [              b      T   #        M       1  M     #  #   #      *       1  k!o*k } ?  "BeqC  C%rX%@A-?0(8(4*..?   EQ(  k Ck  h<* * 2 e:Pr !,*k3  d;C B3h*b >TZ`1 B--!B >D !%.%J*3%%05(3()( en23>3k93#DNX)3)%z% .33C!@3f3 kUXR*2)4$√n* 28  B<B<3k9<doooAoo~oB36o C,ZDedofAoofoAo.oOAoFIomax=3UpFpBl> k}P@a @3Cx1C.>b k>B%6[C"!%n!%.%JB.yp)3!qjk[*32[ S2u^B 3.o^GBG2G DGGG krG4GNG>64Bw,Ajr in=15HB6 "*B ~B-. uB6vOl0 0 ~0 B3.XC,0 0 0 a~0 ,*k3 &M ZD2 2 >f2 "* C2 V k64 3" N 2 2B6:ɶ54 4 4 ::~:J6VC,arine:.9mBYem mm!929mnmBu3.<ȶX==~=B3.3d333!3j31B 6 v, !3j3BB:gSk"gvyC!tg BzBS*g?g2ggJ. 7 k*X 6gXeXJ3. mBaXXvX:-k. /DIG>f !323m!D$*k A. >*+NbfbBS k*bLDO .U S1Lf!2YbS DS={1,3,11,13,16}57Ne 6^*J . ȶ{!B2B6=bb.!kSb> Bb@xb3b?bai2im#:3.bm!%2%6%bbJX.X]QaXjX3%3B.d  T%F& *Bz.ġC a2Bl6!bA344e z'$f':h% >9j9 kUl*'9insertions, median and predecessor in O(log log n) time "N eB6d[eeeȡ j ee*Bs2@ !33v3,:kռ& #.C]Q  j"  ռ.? FBX.%0B3 6.*,!~!2Z.:+2-wC7-ƌ-%%>%/:6~.7C~.~.Ɓ20: ԛ*  ? A ,A/)ԛ.p/?!>0:46Ht A A *=(n=( k" * 3W*A(HS’={i:Si≠∅} "bU Q(Q(D   W  6c(0,    2E($BACB5648-2600-425C-8586-CD5F64FB0B40 *E46 3k6 C 6 ¿.>SbSk*($M*k"VQ J0k.==  j$  Y4:3.vrE8 8 8 26k k 46Z"@0,"*"*j `&3 *C2"V"k*De*3u,@Adaptive Queries UpdatesUUB .dA27B2%;!%2%Br8:32$  F B}2v2ɶmmm!%2%XJXB2/1)C,C,C, k&v8JH&{/Bl:~Bl:3۠!33~3Bf6+ 26LL@:S&%8CA" d 6%?A323O O B+'6?Dv///2nax=&6F^. mCN,,,E2E,f,"D:g ԛ*d Dddρg2ggegB.6 D<<<2BF: :[AX2X63F:33.M Afjf3%3B:333!3j3% % B 63XCC??`c =j=323P2C316"4Bm:mAmjmeB .$` ddddn <"<Bl3. m})})})AllvlO: ռS& #. " "2  P a< {(.,6 #F\!>,Q:4 :g g g g g jg J[.[d[333[[v[B36G 8zC4<4<4<...U:3.omv6v6v6AojomU:6%U&֩&!2DV:%6[ @;I;I;I!%2%6.%M%V:3.91W4H$6*;I;I 2 >"B 3.444All~lB32ld(f f ~f Bg6 ?fff!3j3--P\: S*XC&c9\@L便?!2B6X  BRRFFS';f6&jQb+SBl3.$&c c c Al2l>B>PKIRwJg g Index/Slide-1799.iwac PR1+*۹!މ$߿k "97B. Transitionnone@)?0X-hx*2: :߿މ$:k!۹I߿މ$ k(  o  \*3   ?  B B ZD4YB%%@A-?0(83h*b.>T   `F*C 0*C >D  ׈Y*3 Outline of Proof* 32 :Pr en ! o0*  >*{J{2{p!  q *ډ$9<4%AFC _zD%T!!'z= >jA 5 B6BZ ډ$%5P*3۠%*+-tStep 1: Like before, identify a Sunflower among X = X1,X2,….,Xn, where Xi = {j| j is accessed while inserting i}*  1JBMKt3  1۠% 6+ 79:? @IC I!J !K 0T p*ZТ2  ($F1E6AF5D-647E-4914-A927-0837573AD4EC ]t6* 6*BL^Av vC0C*3]v>*6*V*}\*MiDAdd(x) Median()* -!RZ~2I:I36sC 6D s.I>fIiu$M*NJ ed2H==    BݿDr3T2`Q`f ݿ5R4* 3۠%ږ,31X< 3: bound for kth smallest via chronogram approach*VB%Mr0ږ, 4} .x*!kkkkkkkkkkk~FCNgToCT k k 6ik3YV4Culy kf 2GLVfV k}]*+]Xp/3*q+RB Lţ!N:I6I%=III!I.I^:IX2JZJA66J |CAV=,z!J2J[:JX1NaM*kvXX$CwS B_哉vB b@ " 0*k E%A(kI j jA *O A \ﵪBPjA & [B  ( (! / K.( 5D - (- P! \ s (& k* <  "UV.*I!FB,1Y& %!:2 kuW#CX!6pea BBfA>!FBf%!>% 2 kuXX a@FC4ea BBfA>!FBf%!FB4.xIB` BB&n Jh Ah h h &h k"sK*+kC" V.m V+6 6 ܐBX#GR ~ > .RXp58P&P2 kՋ+!CkH!D6% U vŋh .W%fF!BB!VY%!>b2 kuXX% JA% eaNBfA>%BBf%!>2 kun#CFF4YeaNBfA>%BBf%!>%: *uC#j ""J *޺&2:6⇒ mv!"^3>" "8 ޺  PKIRwJ¸Zz Index/Slide-1785.iwa MX1Ζ]+*]]]a]]]p]! ":8B/ Transitionnone?)?0Xhx*]2]: :a:p!]]]I a p6  ޘ*3]   ?  C 6D %! %@A-?0(83h*b.>T . BY6BB >D ] `M*] * 2 :J R]Pr#    g$  !p   *]9T 4D_CT=T VCBTbT ]5R,*I۠%)TB?1. Let S be a uniformly random subset of middle 1/3rd elements * )pI )RLB  ۠% .) en2!I4a . *!aaaaaaaabaam 0dBNR) C C  a b/M*aLB8eB Cjmabf a U0*a - :*BAEA0 j Ajjar*Ӊ Am\ﵪBPjA & [B  ( (! / K.( 5D - (- P! \ s (au3<*3}3R.]!X(*a3 1Ba1XE 3~!Ba2 bu8B!a )bve b:: &CdAB pBA>%62b22 ܸBןZ b2>2s6 a3& T @ 4 !%BDB2. > b a* [*\ZX1* \R_B G 3.nVB 2 anBkmC~Dۅ a 93aeFNjV0www*wFEs  :yRV 6BF 2ab @C#9y X遢ey 6a& BFjZR    & a>!:26 %t˥U!XnV2@9!F2 a04AD#EYA9v e  6: BFw 1A>!: .2 6Cn!>N&d 3b& 0qC8Kpa=?h > b ]: Xp/3 N 2 a2"龌ށB{ :R CaII>I%6}:&R)!>+:6 BDi4>b^:X2Z>-.-p~~B`Xq" >>-2-[*\&>p+>&>9Gb&ҍBX gB\ĢG:@ & buK*+~C5+Rk 6 3>  !ZD4Y=Ζn ./>f;` *C9*C* v]*V[*318HProof of Theorem 1"䆜^.>2~D%S!2E? jFS! ]U}q*,2. Insert " of SUIRB *"\!0*] ]= >*5N!2!2up2 3ߗp&2 CBAM!`SCI.>6&Ubߗp 5* +! *Jx=Xi = {j| j is accessed while i%4ing i or compumedian}5nVGB:`#!0+ ! 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LQ(  k Cc  h<* * 2 e:Pr F|As>虗v:a-D0*c c=` >*6bNb^bAUT,*c MT ѵWDpD.h!*b :A1 LBTZSIMc BAB >Dc='T*3M0 output = 13U3I en]:=3c9=iD* )=  B=3==>=b=A> 36b0=%4Y* ccc"hD!(qB(ʥ"( c6 3= >Bf. c]J*91UЊ^!>63c3Y (NB333 cՔL*6E 13* :6Az6626aiiiAifi0iAi>96i Biii92963%3!3*c c"c tD%A(B٦KA;c  Bc &c c aWI%zq j5:c ?c A2c c Jc *c @٦.Zs?AX2X 0 6$cccUPhCH| ‘ c6(6zj\‘!262((!>6332 f3323a26363!3Aw>3.3z3323!3j3[[[X6z3323aj3-36D$ccc"  >D6 3 rDrr c* K*:0" : B c* z5525 .5L*5516566Av>k.z|6626!626kRka652kz~5525 cf1Uj> c jC  cF * z j;> 2  B * z 2 {6$cccUP Bl c 63,j\ !26 &<l!J6366z3323a263Aw636 z]3323Afjf  a:36fz 3323Afjf3-3f *М7  DhdOkB|_jМ7R$:L > {P B0j8B  ~DBA,&aV@ Kc"𓼭ўْ*關ڬᘆ>#3. wA((aVjV> cazh  N* zjU> ".!a2B * z!fAX2X r$:6 ՝C}}}!%>%B%:%6%=É%%%!%%>%h&6 $ccc" C   6 3.?z j !26 bb%J636z 3323!33~3Aq632d  j 3323 cb"fEfB6fz 3323Aff~fB33.^iiiAf.f<*3%l*& -:6T%%%!%%>%-6$ccc" A' 6(3.z(.g 2g 65^'!J636#z3323!3j3Aw6363z 3323!33~3>63323 cb""26D$cccDCH¨%ȥ8  D66zDD.D!j9 %%!>636zw332326wAw633.zD3323DjDfEfBD6z 3323ajDD:D$ccc*C)D>>66"zDrD%%"j"4""!F6565z5525!5n5&~2=#Au656&z{5525%jj&&5B3.zJ5525!525(V(>c% D<A cFj;2B&u3zf 2 ?60? c"Ajt''JJ*Oz'ܪJ'2X-XBJz''>A6J cݕzC~ zAAJ BX=X(&v@ԡ"7ZJ%6֪J? !XX>XF蝔d; B.z Ϝ??联FB^ 3.hA26^ ^ )F:| 3.3O333!32300F6$ccc kBo9 ,6 6.vAj ™ !26!D6363z3323!33~3Aq636 z3323 2 fJfG .o96-z}3323263--K*,>,CKj9@A;,Bn(&,Aۄ,!?ȑ, rB:,,,ä͢ņ*̤$뜰}-,*3c .Es M ZDCZH2F>fZH""IC-ZIc*1<*3:\HN\H3Z LN6\(cccDD֞]!*OJ* V c63c9`?!( ,B,]\H>f6K c*7O)G۫c)G#left"۫c;!D69F& ӚB999!9.9N29maxL8%8A683.q rBjR5%8q >qbq!828P28#righrIr> 6r @BYr J >:b:!:2:FrRrMrS6[(ccc[s% : 66 Vj* . @.">L&C=L!M666 z 6626 ~ Az6363z 3323!3j3S-S-B763z 3323!33~3 *M c"+5 cB*zj5>4!f2fBz>[: c*FTDy. %W.' WjWX.].WPS={1,3,5,6,11,13,16}",W> cuGQ&a B*>z2eNl}zaFB 3. mB:::!%2%B?6T $ccc"T ,aT  6 6Y&*jT j''Y&2Y&>#&T L!G663+6626!626:L2"7Z&A652Lz 55252:5Yj5BK3.KzX 5525'2'AO*~8d* E6&EDR պ B0*vkd 6.EB8& DA A d: NE"VE*V!if:P6sncaEEf2fc c g:% *cA 2cAA z c*U*6T, Operations:_gnTCOI!=KBT*!@yT{C bqDqB~^H >H,\"͝\*Z Ia6 $ccc"" D"  "66 zrI^h QnQ!L6565zQ5525!555Au653.Qj 5525Q2QL**W&K2W."3.6z 6626!6>6.6&L26%6* c"R ( iB1R BR*1z$0j> M5%Cm2mNR*RzMf > n c zQq:. V [#j c*$X.<predecessor(15)B-6(B,,,=2=MVM.6t r sa>OO%% c u% B\9 N*IzWתWIIFIBz;;AJ>Jw*)3>)C 6*.>$b$c"M*c NtJo cPV"v ,+$  x6$ccc"l ARe>EBs3.6zE jw1w1 2 6W"W"!J6 3.3z3323!3j322Aw>3. zK3323ѡY2YfJfa63F{,(bt3323!3j3fEfq:3q.zbA3 4`Dɘ|Vuno:z > j q*X*|5inserX0, median and * H in O(log n) time "V.^OB6*[626> &*> c" ~.   B j>z 2##Nu u u u u >u 6- $ccc"q.EP66zj\- - A2Je.63z` 3323!33~3A}6f3ª` 3323` j`  a>3. [Hf- 3323626J>6lL$lrl&>ۅ*G ct CFn c B*zFjh>Fڡ2',*0zFF2 ًB.!%>%B6>=%%%a<2<6 ^ ^> c"C bKbK c B}8IL@ E%bK*"ApJժ?pJA>F[uBtD?AX2X>J cUJWNP 7f&N=Xz3ݪJ!XX>XB<z3>~63c3N!Ea>D*C&?z23D*Cf3 ":!E*6!EVc*!\*30LBinary Search Trees"@N6E6VVy56$cccu e& 6` 6- z` j"2"6!J636z&332323J3F .&z,3323(>(2Ў&r,a6i6z6626aj))PKIRwJ%"k Index/Slide-1802.iwa XZ:h4*(hhhphhhhhhh!h ":8B/ Transitionnone?)?0Xhx*h2h: : p!hhhO      pK )*3h   ?  C 6D %!P%@A-?0(83h*b.>T . BY6BB >D h `M*h * 2 :J RhPr#    g$  !i   *h9T1D@%T9C%AunT.T> BTbT h5R8* $]*)W^\b) Repeatedly insert elements from [1,p/3]∪[2p/3+1,p] and compute the median to recover S.* -]RtBN  #$ $* )*+,4 65 ) en]aW:NCA!Aff h!5*۠%%*]),Ha) If x∈[J$, x can beMHed with access only!C.t-5-  P ۠%  B &Cp:3p}CBAM!(`SCIB7.7>fZDE\'C7V7 p 5* +%*3)4=Xi = {j| j is-!hil1 ing i or ii}v3V:m!0+ 5 '5( < 5^y6 36BL^! vC0Cҷ>bh5\*3)$Add(x) MeS ()* 3 RIt a6Zh36H B A4Y.>Z*6Vh=Z [*5ZHProof of Theorem 1UVY :W6S4:_Ai 2:VxjF hU*IWI ?1. Let S0a uniformly random subset of middle 1/3rd&X*4I )RBHI>6'-DI)!f h5({*ٶF),Claim: Cont 0of C encode SFVh1ٶ 6d %BJ BhhN~D%h-hE!h hfh!h.hqEJ)h2. I:" of=bI^b 4 .*!hhhhhhhhhhh 0dBNR)wC  h h"G 0*Ýh  ҍBXTgB\TjH Ý *A\\ﵪBPjA & [Bp  ( (! / K.( 5D - (- P! \ s (huK*+!Cu+Rr6B!UbM*hUFB8eCjm]FvAFA 6hY:*BA!(A0 j!r2AY h&0 <* "2 2(*h3 qBaA5XW  3~ h>e2hh>4AD#YA%v h 6hÝy B 2xr>Aya:!:62 6C2222a:2 h2222e2>:ơR2 CbA>%64 . R3)!> 2 h48B 4e4 6:ܸBןF4p4r4A>!:: &CdAB)!>-2 h4nBikmCk e h & 6 eNajV 0 * !>!: .2aR2Z6!>y2 h3&0qC8Kpi=?hBD.>f8 8 A" ]*\&Xp/3"8\R8B 3G26I 3*T  !%I>IbIa1I[:IX1GNG2A64 BDiA4G>GbGA2^:GXM#JNJ62p~~B`X USu:>JbJA>\qXp5Gy2 h"S @C#9S X遢 h=6Et˥i" ¹@&>%6 :Q %BQ Z6!>B+)$*h n2n$>*B*"'N=*22( PKIRwJ  Index/Slide-1781.iwa ^Y*sYYYYYdYYYYYYYYYYYYYbYYdYYbYYYYYYYYYYd "31B/ Transitionnone?) p?0X*Y2Y:<    Nb:d:dYYY        ?                [b d  d! I\*Y   ?  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Identify a Sunflower in X with p petals*Q3JdB  *  "#u i en}(%5,*b }->Cy! yB%"% Cz4%. ? >?-c k Cab<*%K*1N^4a<.Uш*!bbbbbϹbbbbbU݃A aC ]vSeSa,A5a,b3y,T -=$ !%BD},r,.,>f,I - Ba,V, bu,$[*\3% X1* A \V e3U^%d26 йb5nBAkmC=nD%йۑ0*ȹb mX &CןzA0 j BjeYӉ$A0Cm[\ﵪBPjA & [B  ( (! / K.( 5D - (- P! \ s (Y" aȹ:IuV. 2!!: ܸBAZ!>2 buB8eAda jmi0"vea 6::*BAA:AjHVayA>!:b4BaAN!>6{6{0qC8Kp =?h{>{b{t]:{Xp/3}F} 6}:}ҍBXA gB\Bo >A}}}} b5K*+ C5+R6 .V  66p~~B`X&"0 n n A2[*\&n p58n &n  2 b8Bn n  6W:W Rn pn  b6mn n Eu. : dNn )!Ff2 bu2"n n eaB: Rn `A>%64 .n  Rn )!> 2 ¹bu4AD#k YAxea B.BF4A>!::4 6Cn!>: 6 #BDia4>b^:X2 R 2 b@C#9X遢.~ :J%t˥jNaJJ>J%6~N!B %!>=" 8!,*3 &i e554Y*n .i>f`AY *6&G/4 *Y*31@Outline of Proof"[[V?#6F3:FBL^,vC0{vF2 >F =F \*5FDAdd(x) Median()* #3 l:slYSIAV!G+k!;#(vG*"">j l#s$*+۠%&"1. Let X = X1,X2,….,Xn, where Xi = {j| j is accessed while computing predecessor of i}*}RBy}X۠%  +     E!F !G P {""b"%2c''LZCD (&D՚ ' ' 2%cY" d%D~$&\'j '&#'.#@c<#Erc6#?# '##!T:'ynU6>!f6 'u*aPe "3. Use C to encode a random subset"b} &# * L *)!A^?(0: ѶªݥW󎾩 BVc*]N #+Z]h o㊻BV*6b3:bC 6ai)2$,b,*%$M*I, "V&JU.#  x)   PKIRwJ?gWIndex/MasterSlide-3.iwaL%  *    "31B/ Transitionnone?)N0X礲* 2 : RPhoto  Media   Z(   *   >  BMC ZDC%l %@A-?0(8$h*b.=T   _6C 0>D<  d NBody Level One Two2hre2 Four.! Five* yT   / ?2(  /?: EPr7 ::C 6D!F%&%~2=fA9 B2R(  1<*)*1255$Z$   ) b] :B%4NDD=j4_=4D64DZ4 24E544R=$:Xi %$i***6VX 1$HAXIX, Title Text50d Md(s* 2 +…( DHR"5* 8JH`DZ b j    iE<@ b8jZ]@:@R8  PKIRwJQh Index/Slide-1758.iwa PT4Ι].*"]]]b]]]p]!ߙ] ":8B/ Transitionnone?)?0Xޢhx*]2]: : b:p!ߙ]]]C  b pl]  ?  '-DIC %!.%@A-?0(8h*b>4CBT  P B`6B0B >D]  >{*ٶF Claim: Contents of C encode S* F2 :  B !(ٶ Pr'@  en!{:h]9h~D%h-hE!h hfh ]=h(q*I*)h82. Insert elem1kS5bI^b!E *6b%SB^>^6^4D_^)^V!^^f^ ]U%_۠%)_?1. Let S be a uniformly random subset of middle 1/3rd 5*QI )V-(۠% 4a.U*!bbbbbÁbbbbbbn0dBNR) C Cn b bA6 b3yT !4 !%BDBbj 3 > b  bUG$[*\3IEX1* \V E3!L:G6GBDi!4G>GbG!Gց.G^:GX2p/3JNJAuIM*bu-LnBB kmC}-nE0*b m &CןA0 jBjJ*$A0C\ﵪBPjA & [B  ( (! / K.( 5D - (- P! \ s (&ybul<*ii*@3Rd.U26!(*3b 1 ܸBAƙ 3~!B49*60qC8Kp=?h>b6 ]:XZ2 ЁbՕB8ea jmw vՅЁ.aGʁb3yGBaiajVG@aGG>G%2yҁby:*BAZy6!> a23.yҍBXa2gB\Ģ :y@222y2 bTK*+ C"0 +Z 2@ bѵ@@C#9!e>X遢@@ 6: BjAaFB3.! t˥U&n!F@! 6! p~~B`X " 9nJJ2! [*\&JpJ&J2 b2"  :. R CbaGG>G%6y .! R )!>N2 bu4AD#Y4Aea 6:yBFyyyyA>!:: 6Cn!>2 bu8B4N4eaB4:4dNpaA>%64:5R)!>! ,*3ܙ] &aBL^9vC0* Ιnn!.a>f8Z"n!C2n!&O ܙ]*\*3DAdd(x) Median()* H!3 "  `=#6Z]32Z\"  4Y*q vZ.Z>Z*6ZVZ]=Z [*5ZProof" Theorem 1"ee^Op:3pYCBAMA`SCIF2>FA.$$VF p "* +H! *=Xi = {j| j is accessed while in"4ing i or compumE}5yVyB:cB!0+ "V! 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B.j$$B$.3mmmm]1"m:]i^iJn2n+ 2 K"3nnJ^.5m)))!525>)6)6B3.dۼ áj  J323[333!33~3J3.3ɶJ333!3j3J.d[Ji.iȶs s s Aiji5"7 s 'B .3ɶ!3j3eJf.3 $zB !33~3J3.3d[fff!33~3J363333!33~3J5255555r5ccJo.6ȶq666!626:obbJ.- TTTaj9 9 J.f0Cozɉ.X.<Universe is [n]" 1B .tm   Atjtt!t!46Q>Q0M ZDC12Q>fQ""1C21V1Z"Q(*3JXData structures for 2( and p&3( search Cell probe model New lower bounds Some#ofs of .bFJo(x) = Max {y ≤ x} subject to y ∈ S*J" B<5(ШK     (,K.^J . dۛ  ~ :=V= *=F=*=*6=V=Z*PHIOutline"HRrJR.Rm%%%RR~RB1!.1!d D0ARfR b&:NbImsize"bkJ6~]=BM 25[555!555Jj.51$>$>$>!5n5ddB5256LC $6) 3!,j,!525["84* =4"=4JO.Od OO~OBx3.x- """Ax2x"N"PKIRwJ.wIndex/Slide-1517.iwaZ4A.*"AsAA̾CAA!mކsۆs ۆs":8B/ Transitionnone@)?0Xׇhx*A2A: :̾C:m:s!AAA78m(sކs6  \*3A   ?  C 6D %A%@A-?0(83h*b.>T . B`6BB >D A `M*A * 2 :J RAPr#    g$  !bss-QA;(!Q(wsDBQ,ss 2f sf 7A%K 0F}>f s^ ff *s] BLB%%. fi . (BB`ZA*CnB3.">f" " pI*CZBAU$w*3I#.Some Known Dynamic Data Structure Lower Boundsj3D6dA36%dMCadd6dVA=d<*33%d54N43.x}m$̾CyPe*CrCC¾CCmvcC ;!0_7Dm0 ¾C rF FF QC 7"[C-C̾CZu68a@aaFbFv 0y) :RJo֑CY!a^C.6 Ձz 0 @6 V%݆U5Ɗ6 AFF2 !W6 P!AAFz0=:Za[  0*A AX >*6ZN 2Z A M : m* B!av. z2 +A;j8 m* u   ` updates and queries*0 4m   Ne2eR7   7 N Q!H!9|(0: ˟飡ĜᦘBV ۆs  ^\ O0 All at Oncek"<  pXN@- In apple:appear ,?0Xק(PKIRwJB  Index/Slide-1804.iwa GR+ %*̵ ҵ a ˵ ȵ ! ":8B/ Transitionnone?)?0Xhx*ҵ 2ȵ :a:: ! ̵ ˵ 1D !  0*ɵ ɵ   >* * 2 : Pr   pN*3ֵ   ?  B B ZD4YB% %@A-?0(8 3h*b.>T   `F*C 0*C >D ֵ =+[*3%)LProof of Theorem 15=3J=2=)= en=H4!.)I*!aaaaaaaaaaa-_ 0dBNRCC C_ a a MU *3a.BDi 4BDB af.>fAjU B6BZ au'$^*\3)(X2p/3* A0\R8B  3.?V%6Ja39J T CE !%J>JbJ!J.J[:JX1GNGA6G3. p~~B`XAGGG!G>G\QXp5GY M*at@C#9AX遝tn> >0*a ޭAj0jj??.AA\ﵪBPjA & [B  ( (! / K.( 5D - (- P! \ s (ٱaݯ<*≰՛↛ݐ!:a:t˥N!:2 au4AD#oY1Aoeaum. 6CxmYj BbA>%B4. 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N (Bit vector1, tm=T  ^ T ^ h " "ON O: >h [BV2!  2 Ҳ 16 316 B C!b *r. =f" _Y *6 R $11"_ ..dNon-Adaptive Lower BoundsAA2j ( $0D8C9FF3-831B-4AC6-894C-09BCDA386C61 *A Oq L*mm.a$mmJjj L*Cp@ 2CjhGZv6jbjz09 U:rLʔΘ壱 eN:*R&hCv5Ӆ>R~$QQ oQ_a#QTMinimum with deletiony a :oj6LLj䯠?%No=&@p Bi@"@Ca> O!*@Z@ vEB logl.: *=" "&e(eG26Ų"= 4X* rr͛r̛,AJ`8yD-?$͛r ̛r &JW r.W a CZREZD2pAW fW r*L*ForUyFU !GFU mmzC(^!6 pBAv n6Y6zzU mbmzz0z:U RU E-U,S* ˛rϛrr ? s6n3E r 6r `` rptDJ!R3CL͛^5R6*r-X|!X!X`b`zX0=X:XRXwzY.b.9b "!J!G  F %4Y* rޛrrΛ>KAb.E ށ6r& !^j?:@jJj r*[..$Search" !IJ `dm_ |CUF[JfCcd]fDu_6mY!Jdb~Y0}Y+ E/2{UC f9FsC@k]fArB.~CceEDb:f#0fX`u- $B$F$JsdQsYoClQeAs=~C]s6IsAsAsdv 0Ys 2R̛ .`ɵ nnmVCoAsCQg^uV6 *յnbnv 0 :Zۛٵjv&`9bЁ Z2JBrDHàؗ׆ R : r_LЬ NO6 ^ 6l rp*p$(PKIRwJҼAAIndex/Slide-1778.iwa=0xOjI*=jjjjjjjjjjjjjjjjjjj "31B/ Transitionnone?) t?0X*j2j:   6jjjjs          ?  j6)k*3j  ?  d[CZC HBHB%!%@A-?0(83h*b.>T I BJB 0B >Dj  PK*3 11* 32 :Pr en !cB424444 jf434!4F46h444Ahj464!4>4.h f " *C %*CZj݌P*H)COutline]H= :j39;ȶiC%; ].}z}2]!93)9PS={1,3,5,6,11,13,16}5F B3j9Fmj  B3./DsLAyf j,*cII~7Time Complexity: insert: n median: 2 p : log n*Pc I  %B3. f0CW 2o!wjwa2X.<Universe is [n]aJ2T444aj7533$*j !2$>*5{J2&2B3.ȶ!'2'6 55(6N RN MN BN N 6N V j(*3JE`XData structures for  and p search Cell probe model New lower bounds Some#ofs of .bFJo C 6D21xf jM*j NJOj.:     PKIRwJ3//Index/MasterSlide-14.iwa+ J#!*3"1..1)3 "-+B) Transitionnone?)H0*12": : RTitle & Bullets..13Zb()v(""㰼*3)   >  BMC ZDC%l!%@A-?0(8$3h*b.=T   _6C 0>D<))d33 NBody Level One Two2hre2 Four.! 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