A Data Streaming Algorithm for Estimating Entropies of OD Flows
A Data Streaming Algorithm for Estimating Entropies of OD Flows Haiquan (Chuck) Zhao
Georgia Inst. of Technology
Oliver Spatscheck
AT&T Labs – Research
Ashwin Lall
University of Rochester
Jia Wang
AT&T Labs – Research
ABSTRACT "!# $! %'&( ) ( *"'(+', -. /'%0-. 12'3(546 78-."'9 -.": 9(. ? ' % @ (A ("4B C9> $ 5 # # 9(D E $ .%'F.+(!#! E ? #%'F.3 -0'$ :BG< ( 3-.9 # (. H1I %JF.K&( = "!HJL0 $ M / $N , " ! O&/@D 9!> $!QP 'M D (, /K > & $-R N-."9> S (6 L1 %JF.+&/T4S#= ' ?5! D $(' @ ' ":IUT# $A ' /"P 4SB / # (*( D0 9> $V H 3% ' $( W &( -#: X 9>I7 % 3&(9( $! I9 / (!Y (! S ,Z\[K7 E 31\UT! >7 4 # ( *"'N!(!> Q ( '=' (:GW, > "'$9($'? (A ("4 (. JC9 C!(J%MD "-. . ' (-] $12 &( 1^ ( _ $ 78 /"! 09 $ W-.-.! T _`6;acbJ-.-. dJ3 (& $ D": X 9 K ' -e + ("4W B&/0= f9 ? %J $. >% = $, -K9($J $(P9 (N%JF.I%g ! JC) ? hcUi>;d&(%7>&/(, (7/: Categories and Subject Descriptors `C: j: kWl `S -. 9 ?T`S-.-K9( "' @mNET46' 7Dnpo)mNET46' 7 X >? E%' Tq<mCT4S 7 r ( ' $ General Terms s -.P/r "'9 -."P(t6 ( Keywords mCT4S 7@r "'9 -."Pu OvD -0J $wPwxc'% i "'-M? P/tQ%'F.+r#J YyPi%'&( cx, D &(9 ( 1. INTRODUCTION t6 ,z{-. "'}|u c1( (S T4S 7,%JF.6 '\ &/8 /"!QPg 8-0 ~!> yE M EY> Pg'.8yE $= i>9( / "! ('~& m,i '% d`Smci h" jkPO`Smci j' '>P3`Sm,i h" 'P0(!mci>]` sC g 46'%!`Smci jk k>h":
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Mitsunori Ogihara
University of Rochester
Jun (Jim) Xu
Georgia Inst. of Technology
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/ &( M #' ( "Y> -0' L (K1^9( @á {á/°A9( # (' -K & ' _'1,@-0 I9(-K&/EK'1C1^9((EË Ô $(. ~ (31{-. â á [ Ò ·OãWÝ>¼%Ñ"Ú{ä :6U5# 46%!>P(4S+ ´T 9> "! J46L" *(!@ '%'-. L帼"ßßß¼åæOã~ç(!·u¸¼ß"ßß¼{·/æBãÝ>¼EÑ"ÚS9 E ('Iá {áy°è Æ «æ Ç\¸ å « á [é :U51w ( 6´D 9 , 6 9 P ( 4SN4 QË Ô &/N& $Cæ )D -0J C[é (C . -'1' X x¡"4 D -ÀW'IÆ «Ç\¸ åD« Ò Ò ÀIÒ Ò [é P4 (E MÒ Ò ÀÒ Ò [éP¥ º Ù ¼"ßßß¼ ®." &/3D -0' !89 (@ 9 KY ( d'1NU5! 7/ ) ' -e1^ D -Þ w:8GW-. ( ( ('. M 78@ "' V'1 12 .12'69( c ( (7+ M ( 6!> QPg-. Dg1y (A "Y? -0J $ % (-.g46N(9( "!0 -0J («-0J "/ %' 9> J 0& H ("JK-K& $(' f1 -.c 7 0á z "Y> -0' & 3 / >(-. $}|6(! êë {áy°Nz(9 AY> '( |E: X 9>B4 }! '´D 9> Ô W 'O&/ > J= E"ì¾tu '-0 -.A4Sc12 9 ! ('C& 9 $ 0M $ "'N -)&( J #'1 ( #546.129 ( N B (L1{'-. $ P/ B (c1^ -'1 ¸ {á ¸5î í0Å á ¸ï íy°EP4SC"' ' ( > JY> -0' á {áy°= K .1^¯Tí '$/á0=? 9 6 3c}J , ={P : >:$PQØ Ù ¼Ë Ô Ù Ý ÝÝ'ÚQcØ Ù ¼EðÝ ÝÝ'Ú:vN Añf S 9 '&( c J%-. I ('%7 -0 w= 9 :g(AY -. P> $ \ 9 hPw4SM ("4 ("4 @4SM"'W "Y> -0' +á {á/° 9 ( Ù Ý {á ¸ò ó ô Åá óò õ ô °4 3 C =\Ø Ù ¼ Ù Ý ÝÝ'Ú:vUT0Ë Ô 4S%!>P( 1' Q%'( /?p$ 6("4I $ X x(J4ÀW (J= N (Bh" L 'E7 E P46A"0D -0' (A )(' -'1w ÄWpÀ°
º
ÅBÕ
8000 7000 6000 5000 4000 3000 2000 1000 0
approximation y = xln(x)
0 100 200 300 400 500 600 700 800 9001000
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/ ! fl h" 'nw M $! 1^ '$Q"4I JNJ c-K9 E 3$' EC ( H% V > (}!¡z{ ¡h" f 'E7 %|M(!d-. (9> ( & 9 ) ( X x("4 0 J%' : mN"4PCjhP j jPcj'k>PNj "n:ùSJ $9(0 % (( $9(01^D -0' (W #%'F. -0' Y3 "= N&/# > / ! JNJ c&'"! D%' D "\ >? 12E (3K! EK-."'9 -.0z{ 9! B!(J%#D "-. ( |E: UT"! &/ "Y> -0' "!c& .{á í á í °ú ÑA(!L 12' S ( X xd("4 = $9 -.Mz{ {: : P¸5 î (cZ ¸ (' ö -3|S "#&/L' ( > JY> -0' !& c"=? %' +1CÒ Ò ÀIÒ Ò ¸5î íí (!~Ò Ò ÀIÒ Ò ¸¸ïï íí " 9 }J "!#1^ - X xR"4c: t6 (E 12' PI 9>.7 % _!('%OD 9(E 9 #' $"409() f7 $6546 &( %!>+z2 cZI¸( -'!0 (, 3 -3|v4 #(cD ì 3. PRELIMINARIES (', (+ (9> / ,'1S $N /EN46M! *(()0(J4 &/K (I (%7 4 K (I '-.6*= E?p 9 ( S K v ("! o9 ! !> Pw!>D ' f!(!> Py 9 K /"PQ!>D J $H /'"P (! > : `S "' P64S#4 $A('0&/B& f-. (9> ( 81 "'E !> D $& 9 _Y PIH46B9( ( 1^ J4 85 > /#1 "Y> -0' W% (-. : s $D & 9 QPu @ ,1^9( % T PQ%7 81{' -. ( "' "!¡12' - 4 129 B ('%'-. :³UT¡ ( 4S 7/Pc J4S= "PC46H ( "!"!(|EP\'!@ K 9 D $E "!bJD%! '%!> $ "! : I' ,·uP &(& $ T .! 5 K1^9((E $! [ {áy°1y !> D $& 9 ~ÀI·/°N $+ 9(9(L'!H >*( f! y $& D Å "¼ "_°Mz{ {: : P K L46 ?p&/ "= "!(|E: C"46= E"# P [ {á/° %'7 KO$ö "!W12' -e( f1^M > L·d= 9 z· º Ѽ Ù ¼%Ý>ß ð |E: À6·y°u4 c· º Ñ,(! / !>v ,546,4S ?p7("4MEY('-M?
)'1D%& $3!> $D & Ù 9 oMÀ6pÑ°L c ( ,'9( $8!> $D & 9 ? @4 O-."O 3(!BD%! '%!O! => $¶ J $HjP/4 H > &(&( ? ¸ Ð *ï ) ,à +.- À6 °S 6 (,`69 E 5 !>( 5 .129( $u{áy° º Ù ¯ & % !> D $& 9 34 (N! 5 0129((¯ ' / u{áy° º ¸ ¸5î0¸ ) ¶ :v('· = 9 c L fjPgh P\!H >: Pw46M4 g L (' J= . "!>? 1^ - &'&( 5 M!>( 5 +1^9( M N4S 7L4 Q:UT D " !yP !> D $& 9 .ÀI·/°w v(E%' "!K9 (,C $-)9($' K %'- &/0!> &/"!W$' E) di 8: U5f ()12 "4 ( Pv46.D%J -.B -. /%3 / 31c (BD%'&( @! D &(9> $V J 4 u&/c9 "! #&/' U5!> >7/ I(! 9>A ' (-.: h:i%& $ 5 ¡ /ET :t6 H12 $"4 ( > /5 "J ( (-..12'M (0D%'&( #!> $D & 9 :WþQ1 ¸ &¼ ¼ßßß"&¼ à !>? c-)9 9' 0 (! /(! 6%'!> -=' $& $ ¯ (' (J= c! D? & 9 3À6·y°EP (MÂ/&¸ 3¸ ö  ¯ ¯ ö 22 ö  3à MÃM ( (I '-. !> D $& 9 0'C Ò Â ¸ Ò [ ö Ò Â ¯ Ò [ ö 4 ö Ò Â Ã Ò [ ° ¸5àT[ fP 4 (E 5 g %'!> -= J $&( c (J= (0!> D $& 9 BÀI·/°E:SUpC ('N ('%! = 1^ W ('K J12' -. (6 ! ,'9( $_(!d`69(% f%' ? !> -=' $& C J D1^ cD%& $ T /T 04 # ('%-.E ·"9' MÑL(! Ù P> / = :UpI I K I ('%!. )= 1^ , /T 9( (. ()! E*( O'1IÀ6·y°(!B ) > /5 '1v( 9> %(D1^ -.: j:i >-.-. 16 K &'&( 5 H! 5 O129( Q7 P [ {áy°E: c' á@ã8D Å ",¼ "d°EP(4S) ("= 8 [ {áy: ° 9; [ DÅá/°E:t6 ( &/B= *("!d1} - (#1{'3 '$ [ {áy°M 3 O( 9> %'(D1^ -1Ð ï : Z[ 3.2 Indyk’s technique for estimating norm D -0' (L (NZ [ -²z·ãWÝ>¼ ÑJÚ^|S1\+D "'-9( +
> /5 .1QD%'&( ,!> $D & 9 S Ic &>%' "!M 9 6 $. ? ' " -. 9 u (,l$h"jP h"k'n: s ! *(("!c&/1^ P = ) D -ÀB ('g % gÁ ! D (g &´DE v4 M1^ 9 (E KÂ/¸"P Â ¯ P/: : : P(Â Ã / = [ ¸5PàD[ NZ [ (' -1Q (CD "- ! *(("! 'KÒ Ò ÀIÒ Ò [MÛpÆ ¨ Ò Â ¨ Ò ° :6U5! 7/ (-#P %($J "!# c$(9' L1t`6;gb'UT;\P ("4# s (-²h: UTO1{E"P\ f-0J (-0J $ %' 9> Pu KD >-K&/ñ &/N-0' ( /"!K L (C -.A%!> -= '$9 N'!Oz{&/|v! y S("4 $&/ cJ )-0' ( /"!3 3 (! /(! A%(! -= 9(:NU5O !(!> ? QP (+·(?pD%& $0 ( W129 ( ()' 3 $(! /! L'1A ( ": X ( (+ ( '16 (%7 C4 O (+7 % O $L' = -. ( : (+"'E f $ -. # 'E7 E"P\ +("4$'&/S + "! & H Î @¨C'!O (K 9 L $+!(! !@ / M ( = 9(+1S. ' 9 $'c9( "4¯ J/3¸"PK: : :$PRMÃ ' H {: {: !Q:%!> -ª= J $&( @4 !> $D & 9 À6·y°E: s 1^
(K ND "'-#P46,"' A J6 (C 9 EI= ' 9(S /5 P A 9( g= 9(: "! pÆ ¨ Ò Â ¨ Ò ° fPg. 8 K4S%! PNÒ Ò ÀÒ Ò [ HPg4 ! ! D &(9> "!W'KÀ6·y°E: t6 (E 12' P 9 = 9($' E (g '46S4S 9($!L $7 g ND $-0J P> Æ ¨ Ò Â ¨ Ò [ ° ¸5àD[ P' $ A ( 9( = 9(: mN"4 ,9(D I ("4 KEY>%'I ( I (y1^ - ( 9( I= 9(, W t6 (c' ( > % 0 / "!3 $Bl h"kJnu S +9(A ( 12 $"4 (0-."!> $ D $-0J 12')Ò$Ò ÀIÒ Ò [o :wNE g-."!> $Q Ò D $& 9 .'1wN%!> -= J î $&( +Ò ~Ò P4 ( ² (A! D &(9> $BÀ6·y°E:Gf,! ( N C'AÀ ·y°S&/"9 ,
w: !y: 1v NEY(' #T4 L (+ / = + ' 1v'1g + Q: !Q: 112LÀI·/°EP ! 9 c 0 LD -.-.E '1SÀI·/°E:,i3L © K Ð G [ C L9(( $9(0z{ ( C"J|Aá/ó)= 9(+9(% # JF YRZ>Ø$Ò W0Ò [¡á/óEÚ º Ý>ß ðPw4 8 O mC N ('I9 I -. $-.%' S1u (C 0129 ( (' %' V! y K _ l hJjP,h"kJnC W-0'7 -e 9 _-)9(% 1{'D "P(4 E 4S,4 $u! E $&/c Oi #:
()! D &(9> $dÀ6·y°E: mC . J"P! 9 M # (.D >-.-. O'1 À6·y°EP/©LKÐ G [ ,Y ' (c > ?59' E,9' $ +1SÀ6·y°E: s ( 9 M $g(c "!M12 -12'I©LK¡ÐG [N12'6-. Dg1y ·f= '$9 Pu4S."H9(-. "'$ O"' 9($' 0 L& @ $-)9($' wPw 4Sc" 9(LK > %'- $7 Ml j' 'n: U5 9( = P (c' (N1 AD $-0J C" &/´T9(D ? *("!L\1^ J4:gi (\ < ¸ ú Ò Ò ÀIÒ$Ò [ P : : : ]P Ò Ò ÀIÒ Ò [ ' 6 {: {: !y:\%(! =' $'&( S4 ! D &(9 î À6·y°EP %7 ()&( 9 A= '$9 C = 9 g {: {: !Q:v!>%41^ -À ·y°E: v$' A 9( C>5P î S-.! $ 9 }!0&/A N c (N! D &(9> $0-."! $0'1À ·y°E:t6 ? 1^ P4SC -. K! î = $! I-."!> $Q Ò < ¸ Ò ¼ßß"ß¼JÒ < A Ò °& + (!> $D & 9 ? -."!> }'01\À ·y° )3D -0' 'I16Ò Ò ÀIÒ Ò [:GfC (J= %7 K&( 9 .= 9(c&/"9 K (.!> D $& 9 H-."! $'H'1 À6·y°N L ! 9(+ cD >-.-. :LUTO (K(YcE $H4SK4 $ '' > , , $' = c 'A1 ( ID -0J ": U5! 7/ AD -0' N12c KZ [ (' - c&'"!B O () >? ET #'1 (L-."!> }'w:gGW,*(!QP ("46= "P ('A N N / &( + (D 9 6D $-0J g&!M 0 69( $S'!M ( )-0 = B 9> /12' -³ +-.! $BD -0J "Py B -.C'1vD -0'? O9>% :6N"46= "P (L c -. J= -.I A-0' $( 1^C 9>N ('%-.E ( P 46LD E70 M (c-.! $D -0'? '": Z [ 3.3 Error analysis for norm estimator U53 I 4Sc( > N N /1^ -0 ,'1\ (CD -0'? '12KÒ Ò ÀÒ Ò [ HzDhJ|E: »¼½ ° bound for · º Ù 3.3.1 C L46K&( "'$ O D%J Mþw-.-00j3'!Ot6 (' -²k.1^ U5! 7_l$hk'nA4 _ ( D% . / "!_9 ":_GW#J = "!VJ C& 39( (3`S ('f&/9((! I .! E $= , (L D%A B ( `S$' $-³j: t6 (' - h :cz{U5!> >7/Pul hk'n^|NþwE M º ¸ ¸"¼ß"ßß&¼ ^A°&/) : {: !Q: '-. ( #1} - À6 Ù °EF P > ` °ú» ¯ PLc » b Ýß ÑPL ( º _ Ñ ©LK¡Ð G¸ º Ù P '!=YRZ Ø -."!> $Q Ë$Ô Ò 3¸'ö Ò ¼"ßË ßÔ ß¼Ja Ò ^AÒ °6ãHØ Ù ÅL»¼ Ù ö »DÚ$Úd[ Ù ÅH½>:vt6 9 czDhJ|6 = A'W»¼½ °SD $-0J 12'S· º Ù : Y -. $o,12'C· º ¼½ º Ý>ß Ý ðP» º Ýß P\4S. e > º .Ñ f ð : t6 A AK= E 0$L&/Ù 9((! , N Ù JI46c("! K= Ù $J C >5:ct6 ( ,-. =' C9(A 3 ', . ()D >-. ' ) ? -0' 5 31 (N-."! $312'C-.c "Y> -0' c' > : »¼½ ·#ãWݼ%ÑJÚ 3.3.2 Asymptotic ( ) bound for ¶ t6 (' - j:þQU º 3g [ P J º L © K¡ Ð G [ P > º i h?j k po ° ¯ P zDh"|\ = g)D -0' v4 .D >-. > ,»¼½ °u&/(¯ ln9(m (!Ql :Eq 6 N9(-)&/69 E 0 J612'D%! '%!3( -0'y! D &(9> $/r È 4Sc "= F YRZ>Ø r[6q Ú º Â/: È ; 1w 6 . ( s /!> Yy:\t6 ( g 9 6 6 . ( -.A%!> '1Q¦M Ù ú'»E¯J° A`S ( 0 9("P &(9 IF. $v v-K9(% -0' $EN'C "4#&/ "4c: Y -. $o (,4SL' L D "! $K·B $ L-0 / &/ !0'1 P12S 9( 3' 9 -. g4SN-0 M9 J/u JB°6Jg· º Ù P 4 $% # Ù Ù ú uv:
4. SINGLE NODE ALGORITHM UT~ $K QPS46 ("4 ("49 .7 E ~4S 7M12'.D ? -0' H ( 81, %JF.BD "-ø $-0J A A1^9( 3á {áy°g& .+$ >[ ? "J. -)&( J 81,'.-.DKT4S@1^9( (K1AË Ô (012 -±á P ·Bã~Ý>¼EÑ"Ú: s 1} A '4S)'' > c c -)&( ("!3 C1 $ "Y> -0' !3UT(!> >7/ ' -#:
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"Y> -0 Ù $' = cE C&/ 9((!»P {: :$P\Ò m )) pï*) ) Ë } Ô~ ) Ò ý »: }~ ; 15:aN () (Lt\ > AY> '( QP á í º Ð í } ~ ) º ÙSö ñ á ö {ñ Ë 3Ñ Ô áy°¯ ö {ñ Ë xÔ áy(° ö 42w¼ Ë$Ô 46c J ô u{áy° º á Ë Ô á ö ñ¯á xË Ô yá ö Eñ á 3ð Ë Ô á ö 22 4I $ M -.I'1\ (N }J = c "P 46c ' Z>{áu¼Tñ°gÛ á u{á/á° Å Ù º {ñ Ë xÔ áy° ¯ ö {ñ Ë 3ðÔ áy° ö 42 Ë Ô {ñ áy° ¯ æ ß Õ Ë$Ô º æ Ǹ pÑ'® öÙ °t i> #= E d E -Þ 3 / $= P646B ("= / Z>{áu¼Dñ ° ²Ý:GW 9 -.N JA ñ b ¸ :vt6 $ $= C9
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"Y> -0 ( ' Iá Ë Ô á 4 ( , $' = c 'C&/9((! »: s $'+1I ( + "Y> -0' f1^+ (M%'( OØ ¼ Ý ÝÝ'Ú'! u{áy° º Ù Ý>{á ¸ò ó ô Å~á óò õ ô °N L = f W\ 9> #h :0t6Ù Ù K $' = 'L 9J% M'1S M "Y> -0' @ @ $! N1^+= 9( C (#-.L D%R {@: s 46L "= c-. ("!yP 4SL4 9() -.+ ( (c!> H-.% '( - 9(-K= C $ (' -. : Ò Ò ÀÒ Ò Ó 4.2 Estimating entropy norm mN"446c4 \ -)&( (c9 C "Y> -0' #1^ -K9 $M(!#UT ? !> >7/ K' -± O 0'8D -0' )12'M (3 W Ò Ò ÀÒ Ò Ó :Bi>9( / .46. ("= .E ~ñ'!få. ~t6 -ek E, $' = + ,&/ 9((!B» ó 8Ø Ù ,¼ {3ÚpPw'! 46) "= L% ( T >
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GfN J4 K (NE I'1u ( 6D $-0J ":GfC4 Q9 N-. ' ( > JY> -0' + A \ -.g 9( )& 9 \ -. $g \D -0' : t6 c '12C' c ( s ( /! Yy: ; / : s 9(-.Â>¨ ý {@:t6 (.zk |/D -0' SÒ$Ò ÀIÒ Ò Ó 4 $8 $' = . 9(¸ ( d]Ñ (å» ö ó » ¸5ó î 4 _ > ¸ ï &'&( T 9( ( Ù ÅCÑ'½P'4 å º Pó º Ò å Ò Ò ÀIÒ Ò ¸5î í Å+Ò Ò ÀIÒ Ò ¸ï í °ú Ò Ò ÀIÒ Ò Ó Å ÒÙ ú» ó ý Ù P º Ò$Ò ÀIÒ Ò ¸5¸5îî í ú Ò Ò ÀI¯Tí Ò Ò ÓLP'!5 " $ í /b Ù : í í Y -. $o.('C { º Ý Ñ y>¼%ñ º Ý>ß Ý ð¼%» º Ý>ß Ý Ý Ù ¼%½ º Ýß Ý ðP 8» ó º Ýß Ý .Ñ x4¼ >Aè Ù Ý Ù ô :BU51I4S( W'9(-.! º ó º P pÑJå» ö » ó"°èÝ>ß Ý.y>P {: :\46I"'M' ( "Y>ô -0' cÒ Ò ÀIÒ Ò Ó 4 $Ù y 4
f ]Ý &(& $ T 9( Ý '-. ( :vU51\4S+D? 9 -.R º Ý>ß ðP> M46C"3Jy%!. + Ù ",»g +Ý>ß Ý Ý ÑP(! 9 ! e >u .è¡Ñß 1 ð Ù ]Ý , .'E = , (S yB '": ; / :czpr ' s = "|Eo,aN(! N -.K9(-. >? (O@'&/'= P.zk | D $-0J ~Ò Ò ÀIÒ$Ò Ó¾4 $ $' = WE 9( (s Ñ' å » ö ó"» óN4 # > &'&( T 0 9 ( Ù Åf½: Y -. $o6i>-.c'9(-. > $B (NY -. ( L'&/'= P 4S ( 3(! ! >Qè ß Ñ = ð Ù Ý K% ( = C (S y 'A4 &>? '&( T $f ð]0: Ù
Z¸ Î 4.3 Estimating norm "' (', -. 9 L (K 9( @Ä º ÎCÅ ¸ Æ Â ¨ ¯ >¨DP 4SA "!K c7("4d (I-. ( = $9 -.Ë 6Ì Í ¯ 1y ¨ Ê %JF.3Î Ë$P\Ì'Í Ò Ò ÀIÒ Ò ¸ :3(') ($0D "'-² ( L L = $ B! 4 VH ($B 9( " : QS9 M 9 09 $-0J A . f"'$9 ? $J @ V'1L= X x§ ' "PI >9 .46O(!_ W-. (9 3%'F.-0' Yf1^. (3(ET46' 7/:~t6 (E 'M&/ ( $!>%& $N = 9 S46' 7. $3 = (L ( 6 > &( -#P& 9 S !(!> y-. !K4 $y (J= A N' / (! (+J= ( !3D? }J "!_4 V ":_UT8 ( . _46 ("4 ("4 f9 '-.K7 E E @!(J%3D 9 9 + J,46) "= +&/@-0 % ( ( 31{J, ' ( > JY> -0' L ( ,= 9 )12'L0 $ MD "'-#Py4 ( E 4 u&/c(' 9>% Y (! "!. .!> $D & 9 "!3"c$' E": i -. $$JS , (A '1Q1ut6 (' -kP46A". 0 ("4V 1^ "4 $ . (' -#o t6 (' - >:þQAñSVP {HPy'!#»A&/LC $#t6 (' -k:t6 (wP ,' ( "Y> -0' @{á ¸5î í ö á ¸ï íy°úÑ)' ( "Y> -0' g (129( ? su{áy° º á 4 $' = c ',J-. Dx »: t6 6 "Y> -0' . $! g !M12'I /9( v $M I%( Ø Ù ,¼ {ÚA(!f4639(0 (0 ( ()7 % W @ 9> ' S"4 4 # LD "J ${H: iM 9>D -0' 'A1^KÒ Ò ÀIÒ Ò ¸JP( Ù Ö X MM< ° ¸5î í ö X 0NMr ° ¸ï í ß Ò Ò ÀIÒ Ò¸6Û ÑO z{ | × aC (3t6 (' -³3'!# > 12N $-. $'N M ( +1S; / ? (.(!BP>46c ("= N (C12 "4 ( o ; / :cz{ |0D $-0J HÒ$Ò ÀIÒ Ò ¸ 9 _4 ( &'&( T Ù ÅVÑJ½>P4 ó º Ò Ò Ò ÀIÒ$Ò ¸5î í Ò$Ò ÀIÒ Ò ¸ï í °ú Ò Ò ÀIÒ$Ò ¸ Å Ò ú.x» ó ý : X "PS-.' ' ' = Pg í 0ö K&/ í 9((!W ) 9 Ù ¸ » ö Ù x ó » ó 4 %¯
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C (7v ( , ( S! '%c v , (U5 E (": s %JF.c"J $! & . ( I $ 70 6 (,UDi ;~&'E7&/(, T4S 7 J) . -.. (' ) 9 ":#(+"'E ~&( = "!f 'E7 "Pu -.( C"' 9 ! CUT;V ("! A*}!>N'A4S CaNxc;gbt`6; '!~U`Ir#;R >12 -0J ~ "9( "!f1^M 0("4! E*( W4S ( $!> "!Q: Gf0 "!dB9(-K&/E)'1*= ?p-. 9 K%'.'!~B ?
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(%7 "P" 9(Q!> -. ( (g ( X x~"4Q "'E c / &( g 9 "¦ : QS9(c4SL! Q C ("= N (c (%7 -.( (K" ('?
&( T )' 9> P 4SIE S C %' ID > (E $v%'F. %c'N , 9 E N0 'C ( %' $@' / (! ( X x³("4. &(E = "!d'. ( ( M 9 E":_Gf "'_129> ( ! %J c (c"4 $ K!> $D & 9 OJN' C 9 :U5#-. D "'P 4S,-07 A --0J E . ("4¡ c!> D $& 9 31w ( (' A%' : UT (, DC1 + ( /"P 46c4 \-. D 9(c (,1^ J4 T4SK%A ) $ 9 D%' L 9>A 9( : Å ¿ tw%,ho s I ( 9>\%I "!K'v>o >h (- s > j Pwj' :,UpN % (N >: 3&( $ @ (%7 4 ( % @&/$( )h : N-. )("4:GWS% (I 6 9 C ' (H%JF.W&/546 f $R!! D $(' -M?
"!'1v] Æ'1\ (C %y%'FML' = $ .' , ( 9 E": ¿ tw%Mjo s *= L-. 9 c%) !f'Kh" >o (- s wj P/j' :6i> -. }J6 .tw%'KhP C%'FMc&/T4S (. $8'!f9 )E f + 9> + -. > $!f1 ] Ʋ1 (. '%S%JF. J => HJ) 0 $ + 9> ": t6 (M%'FM. $@ ( c%. + (9> / @E WL&/ ( #9(&(E)'1I (M%JF.M12'Mtw%'Bh0# J+4S0"~!> ? W-. J 0 . /12' -0 01A 9>M'$ (-¾12' *= ,-. 9 I(! c (9 = ' : 8.2 Experimental Setup ('"% c%P4Sg /"J"'E +Y> / -.gh A -.w4 !> /!> _(E%' "! j:0GW. & = M ('+&/ @12 -)9($. ("= M "'&( -0 '!8 -. ('%& '.= ' 9(:dt6 $.&( =' V'$
($! 3 '$A*= -. $9> % '!V (#4S#*(YV(!V9( (' -K9($0h,12'A , DA19 A= 9J w: ùg' > (.9(-)&/1g&(9 E7 E oSGWLD 9(!> 3 (LyN1v=' > ( (9(-)&/1Q& 9(%7 . (I /E12 -0'(A1Q9 S ' -#: c (M' w' (A J%-. E 6*(Y>"!3'I "' '&( L= ' 9(P46 =' "!W (39(-K&/E)'1A&(9(%7 +&/546d® º ð'Ý Ý Ý@(!8® º _ ÝÝ ÝÝ>Pv4 8 (E " (B12 ' M1T4S :@\ 9> kB J4+ ( 9 c'1Itw%'0j:+GfK&( = + J, ( ( K9(-)&/ 1N&(9 E7 E M ( "'. ( 9>% z'.Y> / "!(|EPS& 9 K4 ! -. ( $ ) 9> (: Å t6 9 ._' (.*(= 3-. $9> 3%'0' = 8 -. }J": GWN -. I (- N1^C 7 c'1& = T :
Fraction of experiments
Fraction of experiments
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Formula 1 Formula 2 0
0.05
0.1 0.15 Relative Error
0.2
0.25
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
k = 5000 k = 10000 k = 20000 k = 40000 k = 80000 0
0.1
0.2
0.3 0.4 0.5 Relative Error
0.6
0.7
P = 0.01 P = 0.001 P =0.0001 0
0.05
0.1 0.15 0.2 Relative Error
0.25
0.3
0
0.05
0.1 0.15 Relative Error
0.2
0.25
0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
0
5 10 15 20 25 30 35 40 Ratio of OD traffic to ingress traffic (%)
\ 9 +>oùg' > (K1}% #1%JF.,1^ - $
Relative error
Fraction of experiments
\ 9 +koùSJ M9(-K&/E S1& 9(%7 ,ztQ%)j | 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Five minutes One hour
\ 9 )og`S-. ' .tw%+hc! j
Relative error
Fraction of experiments
\ $9 +jo6`S-. J $ .(' -K9($'KhL(!#jztQ%Mh"|
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02
0
2 4 6 8 10 12 14 16 18 20 22 Ratio of OD traffic to egress traffic (%)
\ 9 L>oùSJ K -. $ K%' +ztQ%)j |
\ 9 )oùg' > (+1^%E $B1%JF.,1^ -
ùg' > () -. $ )%' o "' Q1} -]i> + JI46c ('? %J S (g -. 9 %J $c12 S$' ,z{ E|Q("4\& c-." '1i>-. ( +(!@C $!Q:FQg B=' > (. (+ -. $ 3%J )46)" "CS! E " ( &'&( 5 K4 M4 ( % .46I4 $/ '-M?
M"4c '+J 0&/J= M9 L 'c > ($!_z{ {: : P\"4 '1u C "J I '#h" |E:GfA1^ 9((!M 'g (A -. $ L%J !> $!)('v'yE (S /E12 -0'(I'19 v (- *(" : \ 9 +. ("4C ('"P/= 4 O.-0 u '-. ( (M%' 0z{ : >:$P h0 N' -#: ùg' > (6%'v$( Qow\ $9 SS -. (' y (9 -K9 }J = ! >? 5 B ( C1v (+ C12', (c*= )-. 9 Ptw%MjPu'! ( ( 9>"Pgtw%Hh Pv4 ( E 8 (J= 3 '-.' $ ? %E ~1C 9>0 ' (-±&("!8 ~4 ('K1}% V1A (0 %' "4z21^ - (.9 "|L JK (' 9 }J X x"4 -. > : (' X xR(J4 JC' += 3-0 u B-. J $# M (c= ? 9(-.g'1w%JF.NJS C' $ Hz'!0! D (' |46AY> /g (
/1^ -0 N'1u9 6 ' - +! '% !>N (A (A= J $' 1I . L%'FMM4 $ 6&/ f O! -. ' ) (. 'K'1 9 D -0' '":t6 $3 !>-. D%' "!d ¡\ $9 ~(! H12' =' 9 A1}% ,1 L $ N(!# A%'F.0z{9 $ M ( "= %' .1Ch" 9( %|EPw / = :.t6 M -. M : !Q: 15:M12' (N $ %'FMc $ $= 12' 12 (L #\ 9> +>: `S -. 9 ( s 9( yoyGW= 9J v g /1^ -0'(v1 9>C ' - # -. 9 (L (L' 9'w Oz / "! B (0 H(' -3|L'1A X x("4:ft6 K-. (9 %J
'6!(! (E I (A4SC "!K c-07 A9 1w 9 g7 % OD -0' 0 (. %'6= $9 -.01%'F.3&/546~ 39
Fraction of experiments
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
35.5% 14.5% 5.3% 3.3% 2.5% 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Relative Error
Fraction of experiments
\ $9 Logùg' > (+1^%' B1u%'F.,1} - (' 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Entropy error 0
0.05
0.1 0.15 Relative Error
0.2
0.25
\ 9 L>og C! D &(9> 1^C 9(Q '!#!>D J $w:SUTB\ 9> +K46c & = c J (c 'N $' 1^ (C 0 I -. ('%'&( N ) ('61^ (C . -#: C( P ( I >*( -.6 C1{' J 9 A' - $AL &(9 D D $-0J A1 , 0'1 X x"4: 9. RELATED WORK t6 L 'A&/#( $! %'&( + = 9 4S 7 $ -. 9 ( v%JF.v-0' Y, +I(546 7)l$h>P jhP jjP jkP jJn:t6 v%JF. -0J Y_ 0 -. ( 8 (#-0J Yd! *(("!d& ~ # 'E7 Oz{& > "| 9( &/546"% # ( A1 (' C!3 C ! T4S 7OJ= + -.K-."9> -., ={:3(c*((?p'% ("! T4S 7d-."9> -.M4SO' B -.E $-.. $ D "!V _ ("4-0' Y~l j'nP\4 E H9 *, (K= 9(-.)1S )%JF. &/ET46# !> $= $! 9( X xR"4:6U5 $A ' /46c > / c -. (9>%' B'1 (+ '1v= X x("4P/4 ( E B = 9 M-. 3 1^ -0' d (d@ -. $0%'FM -0' YW!8 ' ( $!>%& '= ("! ' -0' $%' $ () c N"4 -0J YQ: U5K (S}'Dg1^4V "' P + v&/K9( D "!.'SN9 ? 1^9(C-."9> 12'#!> yE T4S 7?p-. ' (W' ( $ "J (: þw7 ( (6\'{: l hJJn9 ( C-."9> I /E12 - -0' !> )'!c(ET46' 7c! $' ( :UT 1^ -0J $c-."9> u9 E 'c B (J= K&/@9( D !O1^c%%7 ( -0' $ $9(c ? 4S 7' = 5 l hh P,j'n: V (3&/9(! 5 W& @9( (O ~$9 D 3%'F. :( ! (~ / *(O5 > /01+'%'E7P "' % ( 3 "= 9( D "!V #9(B'1c d1L!> y %JF.c12"J 9 A12'C4S -l jJn'! xNx,i !> Wl$hhn: ? P> '&/ "4 J ?p&!0 % ( $9 S1^ '(-0 ! E $#J c-K9(% 0-. A D% ) (Ny I'1 '-. ( ( l kJnu ' ' ("P = 9 -.?p&!3-. ! : t6 9 '1AD%&( ! D &(9 (L @ > ! 9 B7 E E W46 * DM > / !8 l hJj"nI @-.9 0 (3Z ¸ !> $D%'(3&/546
T4SKD "'-.:vt6 ( S 9 I4II(E% "!3 + (NZ[)!> $D%'( 12'M' IT Ý b· ý Ñ Vl >PIh"kJn:Ot6 ( +7 % W!(J%#D 9( 9> . (AD%' + / g12'S (A A ('v46A / N M $g ' /": t6 (O-0' $V! y B 3 ('046O (J= ~-07 = %'N7 -.! *("' 6 K-0'7 N I46' 70 $3 %' :vUT3 J 9($'"P>46
"= W(9> # J. (B9(-)&/M'1L9( y!(J . /3 'E7 0 -0 Q(9( K1^" &( /12' - $ " -. : X (E) (~12'c· º ¼EÑP M O7> "4d$"!~12 12'C I·(?pD%& $L!> D $& 9 Ù Q:6tw. $(! /! 3!%"4= 9( 1^ -'O' & %J M·(?pD%'&( L! D &(9> $wP4SL-07 ,9 L1 ( 12' -K9 }W > /"! /( = O -. 9 Pu463 "' B 79( W%'&( 3 O $!W= %'6 E? -. (9 !3= 9(: UT (O&( %7f O -. 9 .-. "'6>? Pu&(9 L #-. ! ,12'M!> (# )J 09( D "!y:Or -. /%' P c' , " ! 7 J40 K&/, ' -.I ' 46' 7K46$1^6 $ ND -"+l$h'nQ'!M46A&/ = ' I g1^6 $6! D &(9 !#z{ : : P%'FMN-0' Y(| (L 'g ( D%& c7 % "'$ ( (: 10. CONCLUSION UTB $C ' /N4SK-. =' L (+ &( -]1gD -0' (0 ( @&/546f 8! D $(' ~ ( + $W (546' 7f'!
0_ ' (-±1^0 = (@ ( M &( -#: s $(O ( 4I P46 MB -. $E ~(J= SE -..12MD -0' ( P> ! 9(L ( $' A1^C( >?pD%! '%!Z [ (' -.P
0_Y ( _1NUT! >7y .' -#P(!d J4 "4 0"_&/#9(!_ V9 3! D &(9> "!8 ( :P/ "!O',K ? hLUi ;\P 46+' )& $c ! -. (D%J N J 9>N' -] %' " #= $& :
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̲Í?ù2ÎÏÐÛã ?æ=Ó2ßÝtñ.زõ:Ó2ßà å Ð]÷Ð3Ûßزç Ò×¢ó êÐ]ðÝ7þÕ¢Ó4×¢è:×¢õ?é^× Ó2Õ¢ç è&×¢è,à àè?ßØúÓ2óÝ2ø0ä¢è,ÕHñ.زõ,è:Ó4×P× Ó2õ(Ô.ä p×¢Òè:ä ÒÕPèñ]ä,Ð]×¢Òè:æFزõ?èÓ2ßà è?ó²è?éÒBÓ2ß× ä &Ð3ëßLáëPònÑ#ÿâCâ$Ùí î î ô]Р̲Í,ý4ÎÏÐ]ð*Ó2ÔÒزßBÓ]Ù3â$ÐBÑUÕPÝtñè?ó²óúÓ]Ù3Ó2ß]àCѦÐBܦزÝ2×,Ð3â1زßزß]çRÓ2ßÝ æ=Ó2ó²Ø²è?ä ãä¢Ø²ßçS×¢Õ Ó Fõnøpè,Ó4× ãÕPè:àزä¢×¢ÕPزÖã]×¢Ø²Ý ßätÐBëß1á.ëPònÑ7ÿâCâ$ÙBí î î ý]Р̲Í,ï4ÎÏÐ]ð*Ó2ó²óÚÙ Ð3á.è?Ô4Ó2Õ,Ùâ$Ðÿ¦ç زÒBÓ2Õ¢Ó]Ù BÐ ¦ã ÙÓ2ß"à !:Ð #3ÒBÓ2ßçÐ3ÜÓ4× Ó ä¢×¢ÕPètÓ2æFزß]çFÓ2ó²ç Ý Õ¢Ø ×¢ÒæFäUøpÝ Õ#è?äP× Ø²æ=Ó4עزßçRè?ß× ÕPÝ éê8Ý2ø0ßè&$× UÝ ÕPÔ ×¢Õ Ó Fõ Ð3ëß1á.ëPòâ å ì#û7ëPÑ#á3Ù3í î î ï]Ð Ì²Í tÎRòÐáÐâCÓ2ßÔã1Ó2ßà^û:Ðâ1Ý2$× 7Ó2ßØÚÐBÏ\ééÕPÝtü.زæ=Ó4× èøpÕ¢&è %ãè?ßõ&ê õ?Ý ãß× ä\Ýtñè?Õ\àÓ4×(ÓRäP× ÕPè,Ó2æFätÐëß ÕPÝ.õ?è?è?àزßç äÝ2ø*×¢Òè:í 4×¢Ò ëß× è?ÕPßBÓ4× Ø²Ý ßÓ2óEÑEÝ ß]øpè?Õ¢è?ßõ?è:Ý 'ß dè?ÕPê8ð*Ó2ÕPç èܦÓ4×(ÓRÞUÓ2ä¢è?ätÙ3í î î í]Ð Ì²Í 4ÎÏÐâ1è?àزßBÓ]Ù :ÐìVÓ4ø×tÙÛSÐ3á]Ó2óúÓ2æ=Ó4×¢ØúÓ2ß Ù3á3ÐBÞEÒBÓ4×P×(Ó2õ ÒBÓ2ÕPêêÓ]Ù Ó2ßà ѦÐBܦزÝ2×,ÐBì*Õ¢Ó Fõnæ=Ó4×¢Õ¢Ø üeè?ä¢×¢Ø²æ=Ó4× Ø²Ý (ß è?ü.زä¢×¢Ø²ßç5× è?õ(Òß])Ø %.ãè?ä¦Ó2ßà ßè wàزÕPè?õ?×¢Ø²Ý ßä,Ð3ëßCá.ëPònÑ7ÿâCâ$ÙÏ\ãçÐ3í î î í]Ð Ì²Í 4ÎRáÐ3â1ã.× ÒãÔբزä¢ÒßBÓ2ß ÐÜÓ4× ÓSäP× ÕPè,Ó2æFä dÓ2ó²ç Ý Õ¢Ø ×¢ÒæFä#Ó2ßà Ó2ééó²Ø²õ,Ó4×¢Ø²Ý ßätÐBÓ,ñ Ó2زóúÓ2Ö]ó²è:Ó4× Ò×¢×¢*é ++2Ó4×¢ÒÝ ätÐ ÕPã]×¢ç è,ÕPätÐ è?àã +,¦æSã]×¢Ò.ã +.Ð Ì í î4Î Ð \Ý óúÓ2ßÐ3á.ì*ϦÞEð å é]Õ¢Ý ç Õ¢Ó2æ1ÐBÝ ßó²Ø²ßèÓ4× Ò×¢×¢*é ++2Ó2õ,Ó2àè?æFزõ,í]Ð Ó2æFè,ÕPزõ,Ó2ß Ð è?àã +,-PéßÝ óúÓ2ß +4äP×(Ó2Ö]ó²è +4äP×(Ó2Ö]ó²èÐ Ò×¢æFóÚÐ Ì í]Í&ÎÏÐBá.Ý ãó²è ÙÏ5 Ð #ãõ?õ?ØÚÙBûnÐBÑEÕP ã Ù å Ð.ðè?Ý ßBÓ2ÕPàØÚÙ3Ó2ß.à 5Ð.ìVÓ4øp×,Ð !#Ý ×¢ÝFزàè?ß×¢Ø øpêeÓ2ß]à1è?ä¢×¢Ø²æ=Ó4× è¦×¢Òè:óúÓ2ÕPç è,äP×7× Õ¢Ó Fõ:æ=Ó4× ÕPØ üeè?ó²è,æFè?ß× ä زß1Óà]ê]ßÓ2æFزõè?ßñ.Ø²Õ¢Ý ß]æFè,ß×,ÐBëßCá.ëPòâ å ì#û#ëPÑ7áÙ ãß]èí î î2ùÐ Ì í í4ÎRѦÐì*è?ÖBÓ2ó²àØÓ2ßà1â$Ð.÷!è?äP×tÐÞUÓ?êè,ä¢ØúÓ2ß^زß]øpè?Õ¢è?ßõ?è:Ý ß^ß]è?$× EÝ Õ¢Ô ×¢Õ Ó Fõ:ãä¢Ø²ßçRó²Ø²ßÔFõ?Ý ãß×\àBÓ4×(Ó]Ð Ý ã]Õ¢ßBÓ2ó*Ý2øVÏ\æFè?բزõ,Ó2ßCá×(Ó4עزä¢×¢Ø²õ?ä Ϧä¢ä Ýõ?ØúÓ4× Ø²Ý ß ÙéBÓ2ç è?ä¦ý ý .ý 2ï]ÙVÍ ]Ð Ì í ô40Î /S1Ð EÓ2ÕPàØÚÐBëß×¢è,ÕPßè&×¦×¢Ý æFÝ ç Õ¢Ó2éÒê Eè?äP× Ø²æ=Ó4עزßç ä Ý ã]Õ¢õ?è&þàè?ä¢×¢Ø²ßBÓ4×¢Ø²Ý ß1× Õ¢Ó Fõ:زß× è?ßä¢Ø × Ø²è?ä7øÕPÝ æÄó²Ø²ßÔFàÓ4×(Ó]Ð Ý ãÕPßBÓ2ó Ý2ø0ϦæFè?ÕPزõtÓ2ß^á× Ó4× Ø²äP× Ø²õ?ä\Ϧä ä¢Ý.õ?ØúÓ4×¢Ø²Ý ß ÙéÓ2ç è,ä¦ô ï ý .ô .Ù*Í ï]Ð Ì í2ù2ÎÏÐ.÷!Ó2ç ßè?ÕÓ2ßà^ަРóúÓ4×P×¢ßè?Õ,Ð å ß×¢Õ¢Ý éêCÞUÓ2ä¢è?à8÷Ý Õ¢æÄÓ2ßà Ϧß]Ý æ=Ó2ó êeܦè&× è?õ&× Ø²Ý ßCØ²ß Ó2ä¢×#ë \è&$× UÝ ÕPÔ.ä,Ð3ëß Õ¢Ýõ?è,è?àزß]ç ä¦Ý2ø ë å ådå ëß×¢è,ÕPßBÓ4×¢Ø²Ý ßBÓ2óV÷!Ý ÕPÔä ÒÝ é1Ý ß å ßBÓ2Ö]ó²Ø²ßçRì*è?õ(ÒßÝ ó²Ý ç زè?ä,Ù ëß]øpÕ Ó2äP× ÕPãõ&×¢ãÕ¢è?äøpÝ Õ\ÑUÝ ó²óúÓ2Ö3Ý Õ¢Ó4× Ø ñè å ß×¢è?Õ¢éÕPزä è?ä,Ùí î î ý]Ð Ì í ý4ÎSÛS Ð \ã Ù #*Ð þð Ð #ÒBÓ2ß]çÙBÓ2ßà1áÐBÞEÒBÓ4×P× Ó2õ(ÒBÓ2ÕHêÓ]Ð Õ¢3Ý 2ó²Ø²ßçRزß× è?ÕPßè&× ÖBÓ2õ(ÔÖ3Ý ßè:× Õ¢Ó Fõ EÞEè?ÒBÓ?ñ]Ø²Ý Õ#æFÝàè?ó²ä\Ó2ß]à1Ó2ééó²Ø²õ,Ó4×¢Ø²Ý ßätÐëß á.ëPònÑ7ÿâCâ$Ù3í î î ý]Ð Ì í ï40Î /SÐ #ÒBÓ2ß]ç(Ù #VÐBâ$ÐâCÓ2ÝÙBÓ2ß.à Ð]÷!Ó2ßçÐ3ðÝ #þÚÕ Ó4×¢èn× õ?é]þÚ× Ó2ÕPç è?×¢è?à àÝ ä\Ó4×¢× Ó2õ(Ô8àزä ÕPãé]×¢ä\زß×¢è,ÕPßè&×ÕPÝ ã]עزßçÐBëß ÕPÝ.õ Ð*Í?ù2×¢Ò^Ϧß]ß.ãBÓ2ó \è&$× UÝ ÕP" Ô 46ܦزä¢×¢ÕPزÖã]×¢è,àCáê]äP× è?æ¡á.è?õ,ã]Õ¢Ø ×êCáê]æFé3Ý ä¢Ø²ãæ1Ù.í î î .Ð Ì í tÎ65Ð #3ÒBÓ2ÝÙ #*Ðò¦è Ù Ð÷!Ó2ßçÙ3Ó2ß.à Ð ¦ã Ðû#Ý Ö]ãä¢×7×¢Õ Ó Fõ:æ=Ó4×¢Õ¢Ø ü è?ä¢×¢Ø²æ=Ó4× Ø²Ý 7ß #Ø × ÒeزæFé3è,ÕHøpè,õ&×#زß]øpÝ Õ¢æ=Ó4×¢Ø²Ý *ß 0æ=Ó2ÔزßçSãä¢èÝ2ø æSã]ó × Ø²éó²è¦àBÓ4×(Óä Ý ãÕPõ?è,ä,Ð3ëßCá.ëPòâ å ì#û#ëPÑ7áÙ3í î î ï]Ð Ì í 46Î 5Ð #3ÒBÓ2ÝÙBÏ5ÐBÛãæ=Ó2Õ?Ù Ð]÷!Ó2ßçÙÓ2ß]"à Ð \ã ÐBÜÓ4× ÓRä¢×¢ÕPètÓ2æFزß]ç Ó2ó²ç Ý ÕPØ × ÒæFä7øpÝ Õ\Ó2õ?õ?ãÕ¢Ó4× èÓ2ßà8&è Rõ,زè?ßצæFè,Ó2ä¢ãÕ¢è?æFè?ß×7Ý2ø0×¢Õ¢Ó Fõ Ó2ß.à 8Ý wæ=Ó4עբزõ?è?ätÐ]ëßCá.ëPòâ å ì\û7ëPÑ7áÙ ãßè5í î î ý]Ð Ì í 40Î SÐâ$Ð #3Ý ó²Ý2×(Ó2ÕPè&ñÐÿ¦ßè&þHÜ\زæFè,ß]ä Ø²Ý ßBÓ2ó á× Ó2Öó²è5ܦزä¢×¢Õ¢Ø²Ö]ã]× Ø²Ý ß]ätÙ ñÝ ó²ãæFènï ýRÝ2øVì*Õ¢Ó2ßä¢óúÓ4× Ø²Ý ßä#Ý2ødâCÓ4× Ò]è,æ=Ó4עزõ,Ó2ó*â1Ý ßÝ ç Õ¢Ó2éÒä,РϦæFè?ÕPزõ,Ó2ß1âCÓ4×¢Òè?æ=Ó4× Ø²õ,Ó2óVá.Ýõ?زè&×êÙ ÕPÝtñ.زàè?ßõ?èÙû7ë(ÙÍ ï]Ð APPENDIX A. PROOFS A.1 Proof of Theorem 2 þw-.-0_>:þQ\ º 3gE [ P]J º ©LK¡Ð G [ :t6 (KD -0' ' zDh"|S D >-. '¸ $ -0'4 -."HÒ Ò ÀÒ Ò [0'!0D%(!('%! !>= $' (¯ lm l % A Ò Ò ÀIÒ Ò [ : ; '1T:CþwEU ¨ º < ¨ ú>Ò$Ò ÀIÒ Ò [ P' (e ¨ ' 6 {: {: !Q: '-. ( \1^ !> î D $& 9 KÀ6·y°EP'MÒ M¨%Ò"' S : {: !Q: '-. ( u1} -!> D $& 9 À ·y°E:uaN ( (gD >-. ' g -0 T L'1 S-."!> $LD%' "! ~l Py Q:C kJnpP >T{-."! $'u Ò ¸'Ò ¼ßßß"¼JÒ 1AÒ °¨ Å JB°I $ND >-. ¸ '? "' _(' -064 V-._ f(!dD%'! '%!d! = $' &¯ m l : xN $= $! d&c J >+(!-K9( ( > (~& Ò$Ò ÀIÒ Ò [ = H9 ('"P Ò Ò ÀIÒ Ò [ -."! $'w Ò ¸'Ò ¼"ßßß¼JÒ 1AÒ °ú JÅ_Ò Ò ÀÒ Ò [ º -."!> $u Ò -0' : 9\4 ( , $' = v '\&/ 9 ! » PJ (L ¸5î í) "Y> -0' ;9 ¸5î í) 9 N4 + $J $= S ' &/ 9 !»1^A-0 Qñ8!»:6i -. $$J 01^ Ù ÅfñS: ; ¸5î 15: Ù Åf: » bdÂ(ú 9Sb ÙSö »P B Ù ÅH»E° ¸5î ¸5í î bd ¸5î í ú9 ¸5î í b »E° í\:aC $ ct\ > vY> '( QP »E° í º » ñ» ¦MÙ ö » ¯ °è Ùö »g1^g-0 (ñ@'!.»:vi> -. $Ù 'ö # Ù Å0»%° ¸5Ùî ö í è ö Ù Å0ö »: i '-.N12' Ù ÅHñS: ; 1'1v; / B>:6GW7("¸54V Ò å{ ¨ ¸5î í Å Â ¨ ¸ï í °EÅ ¨  ¨ Ò ý î ¸ ï » óÂ>¨  ¨DP>9(-J= I¤u = LÒ å Ò Ò ÀIÒ Ò ¸5î íí ÅfÒ Ò ÀIÒ Ò ¸ï íí °yÅWÒ Ò ÀIÒ Ò Ó Ë Ô Ò ý » ó Ò Ò ÀIË Ò Ô Ò Ó+:gi>C ó ý Ù : ¸5î -²þw-.-0 h" @PQ ¨ í bÂ>¨  ¨6EY C12+.12E4-0 9(-)&/ :\t6 &( c9(-K&/E ( 9 Ë $Ô !.! -. (' 6 (I-0' (¸59(î -M? &/ PJI46g' 9( ¸5Jî w1^\I5 > ( "'"4f! D &(9 ¸5 î wPƳ ¨ í b ƳÂ>¨ Â>¨P {: :CÒ Ò ÀÒ Ò ¸5î íí bÒ Ò ÀÒ Ò Ó :þQ º Ò Ò ÀIÒ Ò ¸5î íí ú Ò Ò ÀIÒ$Ò Ó P> $b Ù Ë : Ô s BÒ Ò ÀIÒ$Ò ¸¸ïï íí bÒ Ò ÀIÒ Ò ¸5¸5îî íí º \Ò Ò ÀIÒ Ò Ó : ('c -. ( T O1S('%' QPw1} -² J4@4SK4 v9 " = $' u\úÑ >{°Ò Ò ÀIÒ Ò ¸5¸5îî íí :ezt6 ( B @ ¡ ( -.N ( N'L Ùgö »%° ¸5î í3è Ùgö »: |.i> -. }J 34SL9 -.N ( 'g M9 ( X NMr ° ¸ï íM CD -0' LÒ Ò ÀIÒ$Ò ¸¸ïï íí 9 (^ ,9 D? $f4 W-.W !@D%(!('%!H! = $' 2°Ò$Ò ÀIÒ Ò ¸¸ïï íí P 4 ( % f46M4 S($' 0 f u\úÑ >{°Ò Ò ÀIÒ Ò ¸5¸5îî íí : s ! ! (3 (K546 ,9 $(N = e ÑM -. (L' $ \ ,'9( $Q:ct6 (E 12' ( 3 .-K9 $ "!d&c ÑH9((! K (# '-.# && $ 5 Ù Å@½>:
Georgia Inst. of Technology
Oliver Spatscheck
AT&T Labs – Research
Ashwin Lall
University of Rochester
Jia Wang
AT&T Labs – Research
ABSTRACT "!# $! %'&( ) ( *"'(+', -. /'%0-. 12'3(546 78-."'9 -.": 9(. ? ' % @ (A ("4B C9> $ 5 # # 9(D E $ .%'F.+(!#! E ? #%'F.3 -0'$ :BG< ( 3-.9 # (. H1I %JF.K&( = "!HJL0 $ M / $N , " ! O&/@D 9!> $!QP 'M D (, /K > & $-R N-."9> S (6 L1 %JF.+&/T4S#= ' ?5! D $(' @ ' ":IUT# $A ' /"P 4SB / # (*( D0 9> $V H 3% ' $( W &( -#: X 9>I7 % 3&(9( $! I9 / (!Y (! S ,Z\[K7 E 31\UT! >7 4 # ( *"'N!(!> Q ( '=' (:GW, > "'$9($'? (A ("4 (. JC9 C!(J%MD "-. . ' (-] $12 &( 1^ ( _ $ 78 /"! 09 $ W-.-.! T _`6;acbJ-.-. dJ3 (& $ D": X 9 K ' -e + ("4W B&/0= f9 ? %J $. >% = $, -K9($J $(P9 (N%JF.I%g ! JC) ? hcUi>;d&(%7>&/(, (7/: Categories and Subject Descriptors `C: j: kWl `S -. 9 ?T`S-.-K9( "' @mNET46' 7Dnpo)mNET46' 7 X >? E%' Tq<mCT4S 7 r ( ' $ General Terms s -.P/r "'9 -."P(t6 ( Keywords mCT4S 7@r "'9 -."Pu OvD -0J $wPwxc'% i "'-M? P/tQ%'F.+r#J YyPi%'&( cx, D &(9 ( 1. INTRODUCTION t6 ,z{-. "'}|u c1( (S T4S 7,%JF.6 '\ &/8 /"!QPg 8-0 ~!> yE M EY> Pg'.8yE $= i>9( / "! ('~& m,i '% d`Smci h" jkPO`Smci j' '>P3`Sm,i h" 'P0(!mci>]` sC g 46'%!`Smci jk k>h":
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. IMC’07, October 24-26, 2007, San Diego, California, USA. Copyright 2007 ACM 978-1-59593-908-1/07/0010 ...$5.00.
Mitsunori Ogihara
University of Rochester
Jun (Jim) Xu
Georgia Inst. of Technology
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/ &( M #' ( "Y> -0' L (K1^9( @á {á/°A9( # (' -K & ' _'1,@-0 I9(-K&/EK'1C1^9((EË Ô $(. ~ (31{-. â á [ Ò ·OãWÝ>¼%Ñ"Ú{ä :6U5# 46%!>P(4S+ ´T 9> "! J46L" *(!@ '%'-. L帼"ßßß¼åæOã~ç(!·u¸¼ß"ßß¼{·/æBãÝ>¼EÑ"ÚS9 E ('Iá {áy°è Æ «æ Ç\¸ å « á [é :U51w ( 6´D 9 , 6 9 P ( 4SN4 QË Ô &/N& $Cæ )D -0J C[é (C . -'1' X x¡"4 D -ÀW'IÆ «Ç\¸ åD« Ò Ò ÀIÒ Ò [é P4 (E MÒ Ò ÀÒ Ò [éP¥ º Ù ¼"ßßß¼ ®." &/3D -0' !89 (@ 9 KY ( d'1NU5! 7/ ) ' -e1^ D -Þ w:8GW-. ( ( ('. M 78@ "' V'1 12 .12'69( c ( (7+ M ( 6!> QPg-. Dg1y (A "Y? -0J $ % (-.g46N(9( "!0 -0J («-0J "/ %' 9> J 0& H ("JK-K& $(' f1 -.c 7 0á z "Y> -0' & 3 / >(-. $}|6(! êë {áy°Nz(9 AY> '( |E: X 9>B4 }! '´D 9> Ô W 'O&/ > J= E"ì¾tu '-0 -.A4Sc12 9 ! ('C& 9 $ 0M $ "'N -)&( J #'1 ( #546.129 ( N B (L1{'-. $ P/ B (c1^ -'1 ¸ {á ¸5î í0Å á ¸ï íy°EP4SC"' ' ( > JY> -0' á {áy°= K .1^¯Tí '$/á0=? 9 6 3c}J , ={P : >:$PQØ Ù ¼Ë Ô Ù Ý ÝÝ'ÚQcØ Ù ¼EðÝ ÝÝ'Ú:vN Añf S 9 '&( c J%-. I ('%7 -0 w= 9 :g(AY -. P> $ \ 9 hPw4SM ("4 ("4 @4SM"'W "Y> -0' +á {á/° 9 ( Ù Ý {á ¸ò ó ô Åá óò õ ô °4 3 C =\Ø Ù ¼ Ù Ý ÝÝ'Ú:vUT0Ë Ô 4S%!>P( 1' Q%'( /?p$ 6("4I $ X x(J4ÀW (J= N (Bh" L 'E7 E P46A"0D -0' (A )(' -'1w ÄWpÀ°
º
ÅBÕ
8000 7000 6000 5000 4000 3000 2000 1000 0
approximation y = xln(x)
0 100 200 300 400 500 600 700 800 9001000
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/ ! fl h" 'nw M $! 1^ '$Q"4I JNJ c-K9 E 3$' EC ( H% V > (}!¡z{ ¡h" f 'E7 %|M(!d-. (9> ( & 9 ) ( X x("4 0 J%' : mN"4PCjhP j jPcj'k>PNj "n:ùSJ $9(0 % (( $9(01^D -0' (W #%'F. -0' Y3 "= N&/# > / ! JNJ c&'"! D%' D "\ >? 12E (3K! EK-."'9 -.0z{ 9! B!(J%#D "-. ( |E: UT"! &/ "Y> -0' "!c& .{á í á í °ú ÑA(!L 12' S ( X xd("4 = $9 -.Mz{ {: : P¸5 î (cZ ¸ (' ö -3|S "#&/L' ( > JY> -0' !& c"=? %' +1CÒ Ò ÀIÒ Ò ¸5î íí (!~Ò Ò ÀIÒ Ò ¸¸ïï íí " 9 }J "!#1^ - X xR"4c: t6 (E 12' PI 9>.7 % _!('%OD 9(E 9 #' $"409() f7 $6546 &( %!>+z2 cZI¸( -'!0 (, 3 -3|v4 #(cD ì 3. PRELIMINARIES (', (+ (9> / ,'1S $N /EN46M! *(()0(J4 &/K (I (%7 4 K (I '-.6*= E?p 9 ( S K v ("! o9 ! !> Pw!>D ' f!(!> Py 9 K /"PQ!>D J $H /'"P (! > : `S "' P64S#4 $A('0&/B& f-. (9> ( 81 "'E !> D $& 9 _Y PIH46B9( ( 1^ J4 85 > /#1 "Y> -0' W% (-. : s $D & 9 QPu @ ,1^9( % T PQ%7 81{' -. ( "' "!¡12' - 4 129 B ('%'-. :³UT¡ ( 4S 7/Pc J4S= "PC46H ( "!"!(|EP\'!@ K 9 D $E "!bJD%! '%!> $ "! : I' ,·uP &(& $ T .! 5 K1^9((E $! [ {áy°1y !> D $& 9 ~ÀI·/°N $+ 9(9(L'!H >*( f! y $& D Å "¼ "_°Mz{ {: : P K L46 ?p&/ "= "!(|E: C"46= E"# P [ {á/° %'7 KO$ö "!W12' -e( f1^M > L·d= 9 z· º Ѽ Ù ¼%Ý>ß ð |E: À6·y°u4 c· º Ñ,(! / !>v ,546,4S ?p7("4MEY('-M?
)'1D%& $3!> $D & Ù 9 oMÀ6pÑ°L c ( ,'9( $8!> $D & 9 ? @4 O-."O 3(!BD%! '%!O! => $¶ J $HjP/4 H > &(&( ? ¸ Ð *ï ) ,à +.- À6 °S 6 (,`69 E 5 !>( 5 .129( $u{áy° º Ù ¯ & % !> D $& 9 34 (N! 5 0129((¯ ' / u{áy° º ¸ ¸5î0¸ ) ¶ :v('· = 9 c L fjPgh P\!H >: Pw46M4 g L (' J= . "!>? 1^ - &'&( 5 M!>( 5 +1^9( M N4S 7L4 Q:UT D " !yP !> D $& 9 .ÀI·/°w v(E%' "!K9 (,C $-)9($' K %'- &/0!> &/"!W$' E) di 8: U5f ()12 "4 ( Pv46.D%J -.B -. /%3 / 31c (BD%'&( @! D &(9> $V J 4 u&/c9 "! #&/' U5!> >7/ I(! 9>A ' (-.: h:i%& $ 5 ¡ /ET :t6 H12 $"4 ( > /5 "J ( (-..12'M (0D%'&( #!> $D & 9 :WþQ1 ¸ &¼ ¼ßßß"&¼ à !>? c-)9 9' 0 (! /(! 6%'!> -=' $& $ ¯ (' (J= c! D? & 9 3À6·y°EP (MÂ/&¸ 3¸ ö  ¯ ¯ ö 22 ö  3à MÃM ( (I '-. !> D $& 9 0'C Ò Â ¸ Ò [ ö Ò Â ¯ Ò [ ö 4 ö Ò Â Ã Ò [ ° ¸5àT[ fP 4 (E 5 g %'!> -= J $&( c (J= (0!> D $& 9 BÀI·/°E:SUpC ('N ('%! = 1^ W ('K J12' -. (6 ! ,'9( $_(!d`69(% f%' ? !> -=' $& C J D1^ cD%& $ T /T 04 # ('%-.E ·"9' MÑL(! Ù P> / = :UpI I K I ('%!. )= 1^ , /T 9( (. ()! E*( O'1IÀ6·y°(!B ) > /5 '1v( 9> %(D1^ -.: j:i >-.-. 16 K &'&( 5 H! 5 O129( Q7 P [ {áy°E: c' á@ã8D Å ",¼ "d°EP(4S) ("= 8 [ {áy: ° 9; [ DÅá/°E:t6 ( &/B= *("!d1} - (#1{'3 '$ [ {áy°M 3 O( 9> %'(D1^ -1Ð ï : Z[ 3.2 Indyk’s technique for estimating norm D -0' (L (NZ [ -²z·ãWÝ>¼ ÑJÚ^|S1\+D "'-9( +
> /5 .1QD%'&( ,!> $D & 9 S Ic &>%' "!M 9 6 $. ? ' " -. 9 u (,l$h"jP h"k'n: s ! *(("!c&/1^ P = ) D -ÀB ('g % gÁ ! D (g &´DE v4 M1^ 9 (E KÂ/¸"P Â ¯ P/: : : P(Â Ã / = [ ¸5PàD[ NZ [ (' -1Q (CD "- ! *(("! 'KÒ Ò ÀIÒ Ò [MÛpÆ ¨ Ò Â ¨ Ò ° :6U5! 7/ (-#P %($J "!# c$(9' L1t`6;gb'UT;\P ("4# s (-²h: UTO1{E"P\ f-0J (-0J $ %' 9> Pu KD >-K&/ñ &/N-0' ( /"!K L (C -.A%!> -= '$9 N'!Oz{&/|v! y S("4 $&/ cJ )-0' ( /"!3 3 (! /(! A%(! -= 9(:NU5O !(!> ? QP (+·(?pD%& $0 ( W129 ( ()' 3 $(! /! L'1A ( ": X ( (+ ( '16 (%7 C4 O (+7 % O $L' = -. ( : (+"'E f $ -. # 'E7 E"P\ +("4$'&/S + "! & H Î @¨C'!O (K 9 L $+!(! !@ / M ( = 9(+1S. ' 9 $'c9( "4¯ J/3¸"PK: : :$PRMÃ ' H {: {: !Q:%!> -ª= J $&( @4 !> $D & 9 À6·y°E: s 1^
(K ND "'-#P46,"' A J6 (C 9 EI= ' 9(S /5 P A 9( g= 9(: "! pÆ ¨ Ò Â ¨ Ò ° fPg. 8 K4S%! PNÒ Ò ÀÒ Ò [ HPg4 ! ! D &(9> "!W'KÀ6·y°E: t6 (E 12' P 9 = 9($' E (g '46S4S 9($!L $7 g ND $-0J P> Æ ¨ Ò Â ¨ Ò [ ° ¸5àD[ P' $ A ( 9( = 9(: mN"4 ,9(D I ("4 KEY>%'I ( I (y1^ - ( 9( I= 9(, W t6 (c' ( > % 0 / "!3 $Bl h"kJnu S +9(A ( 12 $"4 (0-."!> $ D $-0J 12')Ò$Ò ÀIÒ Ò [o :wNE g-."!> $Q Ò D $& 9 .'1wN%!> -= J î $&( +Ò ~Ò P4 ( ² (A! D &(9> $BÀ6·y°E:Gf,! ( N C'AÀ ·y°S&/"9 ,
w: !y: 1v NEY(' #T4 L (+ / = + ' 1v'1g + Q: !Q: 112LÀI·/°EP ! 9 c 0 LD -.-.E '1SÀI·/°E:,i3L © K Ð G [ C L9(( $9(0z{ ( C"J|Aá/ó)= 9(+9(% # JF YRZ>Ø$Ò W0Ò [¡á/óEÚ º Ý>ß ðPw4 8 O mC N ('I9 I -. $-.%' S1u (C 0129 ( (' %' V! y K _ l hJjP,h"kJnC W-0'7 -e 9 _-)9(% 1{'D "P(4 E 4S,4 $u! E $&/c Oi #:
()! D &(9> $dÀ6·y°E: mC . J"P! 9 M # (.D >-.-. O'1 À6·y°EP/©LKÐ G [ ,Y ' (c > ?59' E,9' $ +1SÀ6·y°E: s ( 9 M $g(c "!M12 -12'I©LK¡ÐG [N12'6-. Dg1y ·f= '$9 Pu4S."H9(-. "'$ O"' 9($' 0 L& @ $-)9($' wPw 4Sc" 9(LK > %'- $7 Ml j' 'n: U5 9( = P (c' (N1 AD $-0J C" &/´T9(D ? *("!L\1^ J4:gi (\ < ¸ ú Ò Ò ÀIÒ$Ò [ P : : : ]P Ò Ò ÀIÒ Ò [ ' 6 {: {: !y:\%(! =' $'&( S4 ! D &(9 î À6·y°EP %7 ()&( 9 A= '$9 C = 9 g {: {: !Q:v!>%41^ -À ·y°E: v$' A 9( C>5P î S-.! $ 9 }!0&/A N c (N! D &(9> $0-."! $0'1À ·y°E:t6 ? 1^ P4SC -. K! î = $! I-."!> $Q Ò < ¸ Ò ¼ßß"ß¼JÒ < A Ò °& + (!> $D & 9 ? -."!> }'01\À ·y° )3D -0' 'I16Ò Ò ÀIÒ Ò [:GfC (J= %7 K&( 9 .= 9(c&/"9 K (.!> D $& 9 H-."! $'H'1 À6·y°N L ! 9(+ cD >-.-. :LUTO (K(YcE $H4SK4 $ '' > , , $' = c 'A1 ( ID -0J ": U5! 7/ AD -0' N12c KZ [ (' - c&'"!B O () >? ET #'1 (L-."!> }'w:gGW,*(!QP ("46= "P ('A N N / &( + (D 9 6D $-0J g&!M 0 69( $S'!M ( )-0 = B 9> /12' -³ +-.! $BD -0J "Py B -.C'1vD -0'? O9>% :6N"46= "P (L c -. J= -.I A-0' $( 1^C 9>N ('%-.E ( P 46LD E70 M (c-.! $D -0'? '": Z [ 3.3 Error analysis for norm estimator U53 I 4Sc( > N N /1^ -0 ,'1\ (CD -0'? '12KÒ Ò ÀÒ Ò [ HzDhJ|E: »¼½ ° bound for · º Ù 3.3.1 C L46K&( "'$ O D%J Mþw-.-00j3'!Ot6 (' -²k.1^ U5! 7_l$hk'nA4 _ ( D% . / "!_9 ":_GW#J = "!VJ C& 39( (3`S ('f&/9((! I .! E $= , (L D%A B ( `S$' $-³j: t6 (' - h :cz{U5!> >7/Pul hk'n^|NþwE M º ¸ ¸"¼ß"ßß&¼ ^A°&/) : {: !Q: '-. ( #1} - À6 Ù °EF P > ` °ú» ¯ PLc » b Ýß ÑPL ( º _ Ñ ©LK¡Ð G¸ º Ù P '!=YRZ Ø -."!> $Q Ë$Ô Ò 3¸'ö Ò ¼"ßË ßÔ ß¼Ja Ò ^AÒ °6ãHØ Ù ÅL»¼ Ù ö »DÚ$Úd[ Ù ÅH½>:vt6 9 czDhJ|6 = A'W»¼½ °SD $-0J 12'S· º Ù : Y -. $o,12'C· º ¼½ º Ý>ß Ý ðP» º Ýß P\4S. e > º .Ñ f ð : t6 A AK= E 0$L&/Ù 9((! , N Ù JI46c("! K= Ù $J C >5:ct6 ( ,-. =' C9(A 3 ', . ()D >-. ' ) ? -0' 5 31 (N-."! $312'C-.c "Y> -0' c' > : »¼½ ·#ãWݼ%ÑJÚ 3.3.2 Asymptotic ( ) bound for ¶ t6 (' - j:þQU º 3g [ P J º L © K¡ Ð G [ P > º i h?j k po ° ¯ P zDh"|\ = g)D -0' v4 .D >-. > ,»¼½ °u&/(¯ ln9(m (!Ql :Eq 6 N9(-)&/69 E 0 J612'D%! '%!3( -0'y! D &(9> $/r È 4Sc "= F YRZ>Ø r[6q Ú º Â/: È ; 1w 6 . ( s /!> Yy:\t6 ( g 9 6 6 . ( -.A%!> '1Q¦M Ù ú'»E¯J° A`S ( 0 9("P &(9 IF. $v v-K9(% -0' $EN'C "4#&/ "4c: Y -. $o (,4SL' L D "! $K·B $ L-0 / &/ !0'1 P12S 9( 3' 9 -. g4SN-0 M9 J/u JB°6Jg· º Ù P 4 $% # Ù Ù ú uv:
4. SINGLE NODE ALGORITHM UT~ $K QPS46 ("4 ("49 .7 E ~4S 7M12'.D ? -0' H ( 81, %JF.BD "-ø $-0J A A1^9( 3á {áy°g& .+$ >[ ? "J. -)&( J 81,'.-.DKT4S@1^9( (K1AË Ô (012 -±á P ·Bã~Ý>¼EÑ"Ú: s 1} A '4S)'' > c c -)&( ("!3 C1 $ "Y> -0' !3UT(!> >7/ ' -#:
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"Y> -0 Ù $' = cE C&/ 9((!»P {: :$P\Ò m )) pï*) ) Ë } Ô~ ) Ò ý »: }~ ; 15:aN () (Lt\ > AY> '( QP á í º Ð í } ~ ) º ÙSö ñ á ö {ñ Ë 3Ñ Ô áy°¯ ö {ñ Ë xÔ áy(° ö 42w¼ Ë$Ô 46c J ô u{áy° º á Ë Ô á ö ñ¯á xË Ô yá ö Eñ á 3ð Ë Ô á ö 22 4I $ M -.I'1\ (N }J = c "P 46c ' Z>{áu¼Tñ°gÛ á u{á/á° Å Ù º {ñ Ë xÔ áy° ¯ ö {ñ Ë 3ðÔ áy° ö 42 Ë Ô {ñ áy° ¯ æ ß Õ Ë$Ô º æ Ǹ pÑ'® öÙ °t i> #= E d E -Þ 3 / $= P646B ("= / Z>{áu¼Dñ ° ²Ý:GW 9 -.N JA ñ b ¸ :vt6 $ $= C9
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"Y> -0 ( ' Iá Ë Ô á 4 ( , $' = c 'C&/9((! »: s $'+1I ( + "Y> -0' f1^+ (M%'( OØ ¼ Ý ÝÝ'Ú'! u{áy° º Ù Ý>{á ¸ò ó ô Å~á óò õ ô °N L = f W\ 9> #h :0t6Ù Ù K $' = 'L 9J% M'1S M "Y> -0' @ @ $! N1^+= 9( C (#-.L D%R {@: s 46L "= c-. ("!yP 4SL4 9() -.+ ( (c!> H-.% '( - 9(-K= C $ (' -. : Ò Ò ÀÒ Ò Ó 4.2 Estimating entropy norm mN"446c4 \ -)&( (c9 C "Y> -0' #1^ -K9 $M(!#UT ? !> >7/ K' -± O 0'8D -0' )12'M (3 W Ò Ò ÀÒ Ò Ó :Bi>9( / .46. ("= .E ~ñ'!få. ~t6 -ek E, $' = + ,&/ 9((!B» ó 8Ø Ù ,¼ {3ÚpPw'! 46) "= L% ( T >
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GfN J4 K (NE I'1u ( 6D $-0J ":GfC4 Q9 N-. ' ( > JY> -0' + A \ -.g 9( )& 9 \ -. $g \D -0' : t6 c '12C' c ( s ( /! Yy: ; / : s 9(-.Â>¨ ý {@:t6 (.zk |/D -0' SÒ$Ò ÀIÒ Ò Ó 4 $8 $' = . 9(¸ ( d]Ñ (å» ö ó » ¸5ó î 4 _ > ¸ ï &'&( T 9( ( Ù ÅCÑ'½P'4 å º Pó º Ò å Ò Ò ÀIÒ Ò ¸5î í Å+Ò Ò ÀIÒ Ò ¸ï í °ú Ò Ò ÀIÒ Ò Ó Å ÒÙ ú» ó ý Ù P º Ò$Ò ÀIÒ Ò ¸5¸5îî í ú Ò Ò ÀI¯Tí Ò Ò ÓLP'!5 " $ í /b Ù : í í Y -. $o.('C { º Ý Ñ y>¼%ñ º Ý>ß Ý ð¼%» º Ý>ß Ý Ý Ù ¼%½ º Ýß Ý ðP 8» ó º Ýß Ý .Ñ x4¼ >Aè Ù Ý Ù ô :BU51I4S( W'9(-.! º ó º P pÑJå» ö » ó"°èÝ>ß Ý.y>P {: :\46I"'M' ( "Y>ô -0' cÒ Ò ÀIÒ Ò Ó 4 $Ù y 4
f ]Ý &(& $ T 9( Ý '-. ( :vU51\4S+D? 9 -.R º Ý>ß ðP> M46C"3Jy%!. + Ù ",»g +Ý>ß Ý Ý ÑP(! 9 ! e >u .è¡Ñß 1 ð Ù ]Ý , .'E = , (S yB '": ; / :czpr ' s = "|Eo,aN(! N -.K9(-. >? (O@'&/'= P.zk | D $-0J ~Ò Ò ÀIÒ$Ò Ó¾4 $ $' = WE 9( (s Ñ' å » ö ó"» óN4 # > &'&( T 0 9 ( Ù Åf½: Y -. $o6i>-.c'9(-. > $B (NY -. ( L'&/'= P 4S ( 3(! ! >Qè ß Ñ = ð Ù Ý K% ( = C (S y 'A4 &>? '&( T $f ð]0: Ù
Z¸ Î 4.3 Estimating norm "' (', -. 9 L (K 9( @Ä º ÎCÅ ¸ Æ Â ¨ ¯ >¨DP 4SA "!K c7("4d (I-. ( = $9 -.Ë 6Ì Í ¯ 1y ¨ Ê %JF.3Î Ë$P\Ì'Í Ò Ò ÀIÒ Ò ¸ :3(') ($0D "'-² ( L L = $ B! 4 VH ($B 9( " : QS9 M 9 09 $-0J A . f"'$9 ? $J @ V'1L= X x§ ' "PI >9 .46O(!_ W-. (9 3%'F.-0' Yf1^. (3(ET46' 7/:~t6 (E 'M&/ ( $!>%& $N = 9 S46' 7. $3 = (L ( 6 > &( -#P& 9 S !(!> y-. !K4 $y (J= A N' / (! (+J= ( !3D? }J "!_4 V ":_UT8 ( . _46 ("4 ("4 f9 '-.K7 E E @!(J%3D 9 9 + J,46) "= +&/@-0 % ( ( 31{J, ' ( > JY> -0' L ( ,= 9 )12'L0 $ MD "'-#Py4 ( E 4 u&/c(' 9>% Y (! "!. .!> $D & 9 "!3"c$' E": i -. $$JS , (A '1Q1ut6 (' -kP46A". 0 ("4V 1^ "4 $ . (' -#o t6 (' - >:þQAñSVP {HPy'!#»A&/LC $#t6 (' -k:t6 (wP ,' ( "Y> -0' @{á ¸5î í ö á ¸ï íy°úÑ)' ( "Y> -0' g (129( ? su{áy° º á 4 $' = c ',J-. Dx »: t6 6 "Y> -0' . $! g !M12'I /9( v $M I%( Ø Ù ,¼ {ÚA(!f4639(0 (0 ( ()7 % W @ 9> ' S"4 4 # LD "J ${H: iM 9>D -0' 'A1^KÒ Ò ÀIÒ Ò ¸JP( Ù Ö X MM< ° ¸5î í ö X 0NMr ° ¸ï í ß Ò Ò ÀIÒ Ò¸6Û ÑO z{ | × aC (3t6 (' -³3'!# > 12N $-. $'N M ( +1S; / ? (.(!BP>46c ("= N (C12 "4 ( o ; / :cz{ |0D $-0J HÒ$Ò ÀIÒ Ò ¸ 9 _4 ( &'&( T Ù ÅVÑJ½>P4 ó º Ò Ò Ò ÀIÒ$Ò ¸5î í Ò$Ò ÀIÒ Ò ¸ï í °ú Ò Ò ÀIÒ$Ò ¸ Å Ò ú.x» ó ý : X "PS-.' ' ' = Pg í 0ö K&/ í 9((!W ) 9 Ù ¸ » ö Ù x ó » ó 4 %¯
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C (7v ( , ( S! '%c v , (U5 E (": s %JF.c"J $! & . ( I $ 70 6 (,UDi ;~&'E7&/(, T4S 7 J) . -.. (' ) 9 ":#(+"'E ~&( = "!f 'E7 "Pu -.( C"' 9 ! CUT;V ("! A*}!>N'A4S CaNxc;gbt`6; '!~U`Ir#;R >12 -0J ~ "9( "!f1^M 0("4! E*( W4S ( $!> "!Q: Gf0 "!dB9(-K&/E)'1*= ?p-. 9 K%'.'!~B ?
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(%7 "P" 9(Q!> -. ( (g ( X x~"4Q "'E c / &( g 9 "¦ : QS9(c4SL! Q C ("= N (c (%7 -.( (K" ('?
&( T )' 9> P 4SIE S C %' ID > (E $v%'F. %c'N , 9 E N0 'C ( %' $@' / (! ( X x³("4. &(E = "!d'. ( ( M 9 E":_Gf "'_129> ( ! %J c (c"4 $ K!> $D & 9 OJN' C 9 :U5#-. D "'P 4S,-07 A --0J E . ("4¡ c!> D $& 9 31w ( (' A%' : UT (, DC1 + ( /"P 46c4 \-. D 9(c (,1^ J4 T4SK%A ) $ 9 D%' L 9>A 9( : Å ¿ tw%,ho s I ( 9>\%I "!K'v>o >h (- s > j Pwj' :,UpN % (N >: 3&( $ @ (%7 4 ( % @&/$( )h : N-. )("4:GWS% (I 6 9 C ' (H%JF.W&/546 f $R!! D $(' -M?
"!'1v] Æ'1\ (C %y%'FML' = $ .' , ( 9 E": ¿ tw%Mjo s *= L-. 9 c%) !f'Kh" >o (- s wj P/j' :6i> -. }J6 .tw%'KhP C%'FMc&/T4S (. $8'!f9 )E f + 9> + -. > $!f1 ] Ʋ1 (. '%S%JF. J => HJ) 0 $ + 9> ": t6 (M%'FM. $@ ( c%. + (9> / @E WL&/ ( #9(&(E)'1I (M%JF.M12'Mtw%'Bh0# J+4S0"~!> ? W-. J 0 . /12' -0 01A 9>M'$ (-¾12' *= ,-. 9 I(! c (9 = ' : 8.2 Experimental Setup ('"% c%P4Sg /"J"'E +Y> / -.gh A -.w4 !> /!> _(E%' "! j:0GW. & = M ('+&/ @12 -)9($. ("= M "'&( -0 '!8 -. ('%& '.= ' 9(:dt6 $.&( =' V'$
($! 3 '$A*= -. $9> % '!V (#4S#*(YV(!V9( (' -K9($0h,12'A , DA19 A= 9J w: ùg' > (.9(-)&/1g&(9 E7 E oSGWLD 9(!> 3 (LyN1v=' > ( (9(-)&/1Q& 9(%7 . (I /E12 -0'(A1Q9 S ' -#: c (M' w' (A J%-. E 6*(Y>"!3'I "' '&( L= ' 9(P46 =' "!W (39(-K&/E)'1A&(9(%7 +&/546d® º ð'Ý Ý Ý@(!8® º _ ÝÝ ÝÝ>Pv4 8 (E " (B12 ' M1T4S :@\ 9> kB J4+ ( 9 c'1Itw%'0j:+GfK&( = + J, ( ( K9(-)&/ 1N&(9 E7 E M ( "'. ( 9>% z'.Y> / "!(|EPS& 9 K4 ! -. ( $ ) 9> (: Å t6 9 ._' (.*(= 3-. $9> 3%'0' = 8 -. }J": GWN -. I (- N1^C 7 c'1& = T :
Fraction of experiments
Fraction of experiments
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Formula 1 Formula 2 0
0.05
0.1 0.15 Relative Error
0.2
0.25
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
k = 5000 k = 10000 k = 20000 k = 40000 k = 80000 0
0.1
0.2
0.3 0.4 0.5 Relative Error
0.6
0.7
P = 0.01 P = 0.001 P =0.0001 0
0.05
0.1 0.15 0.2 Relative Error
0.25
0.3
0
0.05
0.1 0.15 Relative Error
0.2
0.25
0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
0
5 10 15 20 25 30 35 40 Ratio of OD traffic to ingress traffic (%)
\ 9 +>oùg' > (K1}% #1%JF.,1^ - $
Relative error
Fraction of experiments
\ 9 +koùSJ M9(-K&/E S1& 9(%7 ,ztQ%)j | 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Five minutes One hour
\ 9 )og`S-. ' .tw%+hc! j
Relative error
Fraction of experiments
\ $9 +jo6`S-. J $ .(' -K9($'KhL(!#jztQ%Mh"|
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02
0
2 4 6 8 10 12 14 16 18 20 22 Ratio of OD traffic to egress traffic (%)
\ 9 L>oùSJ K -. $ K%' +ztQ%)j |
\ 9 )oùg' > (+1^%E $B1%JF.,1^ -
ùg' > () -. $ )%' o "' Q1} -]i> + JI46c ('? %J S (g -. 9 %J $c12 S$' ,z{ E|Q("4\& c-." '1i>-. ( +(!@C $!Q:FQg B=' > (. (+ -. $ 3%J )46)" "CS! E " ( &'&( 5 K4 M4 ( % .46I4 $/ '-M?
M"4c '+J 0&/J= M9 L 'c > ($!_z{ {: : P\"4 '1u C "J I '#h" |E:GfA1^ 9((!M 'g (A -. $ L%J !> $!)('v'yE (S /E12 -0'(I'19 v (- *(" : \ 9 +. ("4C ('"P/= 4 O.-0 u '-. ( (M%' 0z{ : >:$P h0 N' -#: ùg' > (6%'v$( Qow\ $9 SS -. (' y (9 -K9 }J = ! >? 5 B ( C1v (+ C12', (c*= )-. 9 Ptw%MjPu'! ( ( 9>"Pgtw%Hh Pv4 ( E 8 (J= 3 '-.' $ ? %E ~1C 9>0 ' (-±&("!8 ~4 ('K1}% V1A (0 %' "4z21^ - (.9 "|L JK (' 9 }J X x"4 -. > : (' X xR(J4 JC' += 3-0 u B-. J $# M (c= ? 9(-.g'1w%JF.NJS C' $ Hz'!0! D (' |46AY> /g (
/1^ -0 N'1u9 6 ' - +! '% !>N (A (A= J $' 1I . L%'FMM4 $ 6&/ f O! -. ' ) (. 'K'1 9 D -0' '":t6 $3 !>-. D%' "!d ¡\ $9 ~(! H12' =' 9 A1}% ,1 L $ N(!# A%'F.0z{9 $ M ( "= %' .1Ch" 9( %|EPw / = :.t6 M -. M : !Q: 15:M12' (N $ %'FMc $ $= 12' 12 (L #\ 9> +>: `S -. 9 ( s 9( yoyGW= 9J v g /1^ -0'(v1 9>C ' - # -. 9 (L (L' 9'w Oz / "! B (0 H(' -3|L'1A X x("4:ft6 K-. (9 %J
'6!(! (E I (A4SC "!K c-07 A9 1w 9 g7 % OD -0' 0 (. %'6= $9 -.01%'F.3&/546~ 39
Fraction of experiments
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
35.5% 14.5% 5.3% 3.3% 2.5% 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Relative Error
Fraction of experiments
\ $9 Logùg' > (+1^%' B1u%'F.,1} - (' 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Entropy error 0
0.05
0.1 0.15 Relative Error
0.2
0.25
\ 9 L>og C! D &(9> 1^C 9(Q '!#!>D J $w:SUTB\ 9> +K46c & = c J (c 'N $' 1^ (C 0 I -. ('%'&( N ) ('61^ (C . -#: C( P ( I >*( -.6 C1{' J 9 A' - $AL &(9 D D $-0J A1 , 0'1 X x"4: 9. RELATED WORK t6 L 'A&/#( $! %'&( + = 9 4S 7 $ -. 9 ( v%JF.v-0' Y, +I(546 7)l$h>P jhP jjP jkP jJn:t6 v%JF. -0J Y_ 0 -. ( 8 (#-0J Yd! *(("!d& ~ # 'E7 Oz{& > "| 9( &/546"% # ( A1 (' C!3 C ! T4S 7OJ= + -.K-."9> -., ={:3(c*((?p'% ("! T4S 7d-."9> -.M4SO' B -.E $-.. $ D "!V _ ("4-0' Y~l j'nP\4 E H9 *, (K= 9(-.)1S )%JF. &/ET46# !> $= $! 9( X xR"4:6U5 $A ' /46c > / c -. (9>%' B'1 (+ '1v= X x("4P/4 ( E B = 9 M-. 3 1^ -0' d (d@ -. $0%'FM -0' YW!8 ' ( $!>%& '= ("! ' -0' $%' $ () c N"4 -0J YQ: U5K (S}'Dg1^4V "' P + v&/K9( D "!.'SN9 ? 1^9(C-."9> 12'#!> yE T4S 7?p-. ' (W' ( $ "J (: þw7 ( (6\'{: l hJJn9 ( C-."9> I /E12 - -0' !> )'!c(ET46' 7c! $' ( :UT 1^ -0J $c-."9> u9 E 'c B (J= K&/@9( D !O1^c%%7 ( -0' $ $9(c ? 4S 7' = 5 l hh P,j'n: V (3&/9(! 5 W& @9( (O ~$9 D 3%'F. :( ! (~ / *(O5 > /01+'%'E7P "' % ( 3 "= 9( D "!V #9(B'1c d1L!> y %JF.c12"J 9 A12'C4S -l jJn'! xNx,i !> Wl$hhn: ? P> '&/ "4 J ?p&!0 % ( $9 S1^ '(-0 ! E $#J c-K9(% 0-. A D% ) (Ny I'1 '-. ( ( l kJnu ' ' ("P = 9 -.?p&!3-. ! : t6 9 '1AD%&( ! D &(9 (L @ > ! 9 B7 E E W46 * DM > / !8 l hJj"nI @-.9 0 (3Z ¸ !> $D%'(3&/546
T4SKD "'-.:vt6 ( S 9 I4II(E% "!3 + (NZ[)!> $D%'( 12'M' IT Ý b· ý Ñ Vl >PIh"kJn:Ot6 ( +7 % W!(J%#D 9( 9> . (AD%' + / g12'S (A A ('v46A / N M $g ' /": t6 (O-0' $V! y B 3 ('046O (J= ~-07 = %'N7 -.! *("' 6 K-0'7 N I46' 70 $3 %' :vUT3 J 9($'"P>46
"= W(9> # J. (B9(-)&/M'1L9( y!(J . /3 'E7 0 -0 Q(9( K1^" &( /12' - $ " -. : X (E) (~12'c· º ¼EÑP M O7> "4d$"!~12 12'C I·(?pD%& $L!> D $& 9 Ù Q:6tw. $(! /! 3!%"4= 9( 1^ -'O' & %J M·(?pD%'&( L! D &(9> $wP4SL-07 ,9 L1 ( 12' -K9 }W > /"! /( = O -. 9 Pu463 "' B 79( W%'&( 3 O $!W= %'6 E? -. (9 !3= 9(: UT (O&( %7f O -. 9 .-. "'6>? Pu&(9 L #-. ! ,12'M!> (# )J 09( D "!y:Or -. /%' P c' , " ! 7 J40 K&/, ' -.I ' 46' 7K46$1^6 $ ND -"+l$h'nQ'!M46A&/ = ' I g1^6 $6! D &(9 !#z{ : : P%'FMN-0' Y(| (L 'g ( D%& c7 % "'$ ( (: 10. CONCLUSION UTB $C ' /N4SK-. =' L (+ &( -]1gD -0' (0 ( @&/546f 8! D $(' ~ ( + $W (546' 7f'!
0_ ' (-±1^0 = (@ ( M &( -#: s $(O ( 4I P46 MB -. $E ~(J= SE -..12MD -0' ( P> ! 9(L ( $' A1^C( >?pD%! '%!Z [ (' -.P
0_Y ( _1NUT! >7y .' -#P(!d J4 "4 0"_&/#9(!_ V9 3! D &(9> "!8 ( :P/ "!O',K ? hLUi ;\P 46+' )& $c ! -. (D%J N J 9>N' -] %' " #= $& :
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