Depth First Generation of Long Patterns Ramesh C. Agarwal Charu C. Aggarwal V. V. V. Prasad IBM T. J. Watson Research Center Yorktown Heights, NY 10598 IBM T. J. Watson Research Center Yorktown Heights, NY 10598 IBM T. J. Watson Research Center Yorktown Heights, NY 10598 [email protected] [email protected] ABSTRACT ! "$#%"& '& !() *+ -,./0213 !4"657,8'9!:4"&*+/; < =,8:5>?9@*A'*B @. @C#D4"&0 1E#F@ < :#C'(3 3G,8HI !KJLNMO)PQA@ ') 4"&C#%57, :#R?S < 4"&C 2&@?TUV ,./7U@. D79WU @.(UHI;CU',0WX2Y#%Y +'4"6&ZT[\7]%^_a`>bdc9[e]gf7,V4+@. HIT"&A7 h.,#i"(!4 < ,O?, < :9HI*a@ ;=8!!, jlk?m9jnFo[*` ! "p#%57, & !&790U1C@*8) 'S < ("(;-/S < 4 < # < #F3 @?9'= q?3 < @* U@!"& < . 7r/ ';3 :U @.3 < "+/#i@.,U R9 HI4;:"( sf8/ < 4"&a9hHI;& !0t1 U@>) 4.9ah,8'@H!;(#W < A 4"&U'@.9 [email protected] < 8!/'"'9?,@7 = < @*(S < 4. HI>) @' : < f!. ?,:@' < f,= "& ?, @R9 !A < Uy49vf8fwvvwfv8fv! fwvx9s0t¡:w,8i#FD8Hw',8) =!,w)P@S < ;3@?/' 4 f¢,! 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ZT[\7]%^ tn%`*«]U@?, < #R!" :@?,',9EBw ! #Cw,fE@.@.-4# &^] D T*9;& ] Y & X Z X h,8"& ?,==/C#%S < D >#>#> @ t@7,8',tBw ¢#% 9=& ] >#>#> & X ZX @ "(;/D ?0#Df < ?,#%" # < W@',.9 !¢0W1 i WU' w4;hY?I*iHI T# 8 < ':8@?, < !fi/ < C4C@?2/A "& "&?," < @*"& *Q&@ 4;01C,835>U@.'S < h (9Hw',aC/ 4) J+0 4 ÷gUT < < ,9./!0 *¹t < ',3/, @9HI,3?9' U (*B"& !ft@!',&:@?:U-* B@* ;=C'4"& ! ! 3/!+ f 7f8, I0D¥¢ < 9@!8) ',~N@?~Uht., i#8N4"&~ If p + + < < < < :@*'H!4;0C1T'*B @. @+< <#Fh< < dYT@U9 ' >.?,( i 8·s!¹Y,l·g/7¹*0NOD!f , < < :C@?C# i < + w·g/¹*f'@hh/8.@.#NC9 U @.3"(B8 "(4"& ¤f f7, (@?:/#% < 7, +C* O* 8!! ,G.9 < &+HI;G !- HIh·g HI9¹*fD=) @a# < + !?, a I4;&T/O"&½¤@ HIO ( @?!0E1=H!;5>w,8&H8 4?,V/;l&,88)s5>C?9@* 0H2I44¤ +i < O 8·s!¹N p¤0idT +9 @ [ 9 < < < + @ ,aa;8 ',3 #%S < "&?0a1"&h < h#NhBwO H!4).Cw, 0ÂV&T77,rfi'2&@?=#U i ' < : 8·g/7¹*fi @?, < O9#%Daw, @? !?,:@ 5>w,AU @.G THw 4?,V/;(,)s5>[email protected]@*) Null b a ab abc bc abd Null c d e e be bd d db eb ea c b da cb a ba abe eba (a) {+ (+ [ 9 < @ G¢^[anF][*ª zi^wnge.^&b?ee®w`.«nFoA]%^[ ² [*zY[*«]¢o®8ª'.[*`b ² ´ k`s[nF][*ªT«[*]g« ^8kownFok]ko7b?Á[(«>^b®´ Á2. [N`*«]Cko7Á]%^[(nF][*ª b?ee®w`*n%owlnFo ]%^[ª&k?mn%ª+®wªÃo®wª3. [*`b ² ´ k`g[nF][*ªT«[*]g««^8b®8´ Á .[´ k«] MO#% < 4;fN3,Aw9(':4"&+ s0 JY?S < ;fC#F < 7,C>.9;E#Y., (#R!"$'> < ("&!>O < =/TA?7/ A?B8 "( ! #¢/9HIIg0t1 Y@*'S < U7/=, @ < ?,&? /;:*?9@*&y 9N .,8("(B8 "& hQ&@ @; cba (b) %9I? i|#9{|=%~|#~t !| , < T'O+w, f¢7,E@h# !?!, ;8 ',tO#R?S < N4"&?0N1 D*B"& Y,W < !!>t #%! ?U'3 < > @ < #%O@*! &w,.,8 ! dba f ~=F~rc t9¤| Tc¢ EF¢ 2 8· q?¹O#tT ! 7D.@) 79V9A='C7 C !O !0 < "&!f7?HI4?,8;E?U#F / "&iU'@.A9U"&/'ta,8C5>N>.9;f .@ l8. l@/[email protected]?,3&" < @."& Q&@'wU#%"0D1 U < 4U :@',*./ C?Hw'!0 7. EMPIRICAL RESULTS 1h*B*'"&wth#%"&?,=!:d»a¡ £!!A!!LW) !8>.9 !AU4&hJLNM@ @.T.a#¢vC¡:u!f9vxC¡» #"('V"&"&;fD9¡»µ#¥~vEJ@.=,V < E 80 !0 1T,9.A',?,E O 57 C>;8>" 7,:>?,E! &v » Ja8Y,8 HI!0 #a !?!,80(1A*Q&@ @*;#Y& !4"¾ h "&9HI?, # < /; < ((,8;8"& @h.,8 =O!?,(&>. @ ., 0CdET@?+#a,8;8"& @C.,*'!f¤.,8 4"&a/'?K?@*w,h,8, B8 @. @h)PBw'¢0 < != < U#W?@* 4"&:*.?,q qVZT[\7]%^_a`>bdc9[e] !4" 9 @*??,/w;4 !?!,8fiC/;/ < @.I 3hA 9Y H!¤w,0Y1#%!faO"& 79 !=#i 4"f Y9 #RYU4=C*#r#%S < Y4"&*tU @.&@!.'a #R?S < t"(B8 "(84"&NU E«*bª&[b ² ]%^[N < /t#r"(Bw) y "(W "&0T1#%!f~h*"&' !l#tAZT[\7]%^ _`bc9[e]~ !4"f7'r3"(9B8'"(4"&UO"&9HI?,(' T < +7!0 X2t>?,'4"±A < "+/~#,.WU @*T7?HI / < ?,V?9 '*y 9sfN#FT> = !4" '(@? Uq 3 :'/ lG"& E -79(#%" 1'4" l,.w0 !?,A Ey 97'DHI;TQ&@'wD#FYa@?# 57, C !T"(9B8'"(Y/@? < O#¢ / 4d;&C"& < U"(9B8)P/; < A !)?!,=@*'S < 0 [email protected].@ > @3#h,.V*:9l < >.9?, © 1i/ 021:*B8'>&@ < "&G G T./ 3 < >.9 T < "+/*#Y')P4"&h ?>C < 'HIg0 OY@?!f8. A#~h?HI.!@.,=U',w3 ` ,` G=,9.E+ AS < ('G@!"&?,VE; @?Y*. ,9.80 1 =@*.@ > @l & < #%3"&!>=,.*=U'@. 7?HID !Oi C"0~1 i. '@?W !45@?4; & ?= HIw,8fUE,9./7E &9º>@,r0 6.1 Space Requirements 1&7@+?S 4"&w#&,8l5>C>.;& < 457@? ;©"&©l #hE.@ !q*:4) #01 a'a/@? < h!4;(:#~ B8 @!. @C (/ VB8 ?,©=;¬"&!"&wA# "&!01E7@) S < ,V#F+"( . Y(T#/4HI@ A"( #R.@ #tC.@ !l !0#NC!"& @?, < @) d V < A#F @ HI&9º>@ !G < ?,¤fW,l&,9./7 a8ºd@?,(! ;(4#~ Y?, < @Y/;(h#g@# !fw& @S < 4"&T#%3 U.@ !*&!:@ < 1U A#a !!>t79&Na'?Y>Y < t H! h T'>C@ < "&#A1i/ :!0+OOA@?Vf~"(w; #¤OY9OC !tB8 @4 < "&. !+#~ < /U#ih79aY < ',:?HI/ d < 8? > @!0 i < :x·sI¹=,qx·g/¹*fUY9H!l < >.9?,#F) "(@#C < 3'4" HI < &ljkm9jnFo¤[*`A"&w,¬! y eboo¤[e] 7,ªT®8«>^w`>b?bªp,9.0:O D T&"(, !G !!4"& @@?' q.,*3G < >.lE.,)P#R) "(4 < ,8C#F"(@+,8 ½¤*@C/,4½¤4) z3{|{ l 8| A~~ !| =EH¤ i ~ 8? BW V} {} %)%i|#9{|=F~¬|=~ f 8{|=98 f |=~z={|{ l w|? - T8| 9 {7 & A, ci z3|}i l ! - . ` l ` ! !19|E. c¢|= 9 ||#E%}"~| - ~| i ! {| 8{| l ! . ? ! £! !! 9v v 9v v xw9v9 80 xI !? v £ ! w?£ 80 w 9 v w!? 9 !!!£ ? I £ CPU time vs support for connect-4 data 10000 CPU time vs support for mushroom data 100 DepthProjection Max-miner DepthProjection Max-miner 1000 CPU time (secs) CPU time (secs) 10 100 1 10 1 0.1 90 80 70 60 50 40 Support in Percent 30 20 10 10 9 8 7 6 5 4 Support in Percent 3 2 %9H ¤ ! ~|{|# ~{!N f ! l{7 ! |=~ - {I.!{7dc - }.#"%$'&(*)+" CPU time vs support for pumsb data 10000 1 c~{|{2w|? CPU time vs support for chess data 10000 DepthProjection Max-miner DepthProjection Max-miner 1000 CPU time (secs) CPU time (secs) 1000 100 100 10 10 1 1 0.1 35 30 25 20 Support in Percent 15 10 5 60 55 50 45 40 35 Support in Percent 30 25 20 %I9-,3 !"~|{|= ¢{ !N f {7! 7|=~ - {I./.0$1"-&2&{7dc - }.34('&+&c~{|{2w|? {}"% 5 5 ii F¢g l F z={|{ l w| Ai~ !|7 i =;H¤ E~ 8? BW } E Ii !"¢{78 f |=~{%E%|=i Ii¤|=F~ KEiF~ { = {|+| ¢ 68|A=! !19| % {g {F ¨· Iv¹ ¨·P¸Iv!¹ w0 I w0 ! !!0 v x80 £Iv 8v 80 x!x 80 ! !!0 x w0 ! !!vw v ! 9v 80 ? x80 x x!80 ! !80 x x!II !!!! v!0 !x Iw!0 ! Ivw0 !80 !x w?!v£!x x8I *| 8| idc !!x!£ 4"&0KAEfalZ[\7]%^!_`bdc9[e] ! " S < @*w4; < > (jk?m9jnFo[*`A"&w,¬U < (H < 39 ?h0 Deboo¤[e] +,.?f8tU >Y < YH < y G #r!=fIUjk?m9jnFo[*`N 4"±?S < 4?,T80 h@7,8fIU EZ[\7]%^!_`bdc?[e]U@.'S < &?S < 4?,-w0 x! l@7,80-T ?>& < AH < 3#=fDjk?m9jEnFo¤[*`'4" ) S < ,3£Iv90 T@7,8f8U 'OZT[\7]%^!_`bc9[e]~ !4") S < ,!4;I w0 !x3@!,0O1 < f7*#F"(@C! ) @*??,9: ?:H! < =#C < ?fG793= 9Y> < OH < !f#%"(@hI/3&"&8) w,8aaOT#F@#90 v0 1" 8!" ,9. ¬B8 /432R9!#F"(@ < !rfr < !T,4½¤@ h,8."( @'E'O@?0C +'>C < C'HIW#?(f¢=jk?m9jn%o¤[*`O !4" S < 4 ?,Vv I3@!7,8f~U?=ZT[\]%^!_a`>bdc9[e]t !4" S < 4 ?,( @!,0tYO ?> < a HI #~ =f > > jlk?m9jnFo[*`= ! " > S < 4 ?,q£I ¬@!,fU?3 Z[\7]%^!_`bdc?[e]~ 4"S < 4?,?£ !vT@7,0D1#%) "(@h!E/:&"&w,aa+#F@#NA& > > #% ¤h < U HI'a>,r0 1a"&!>W> w *#F"(@Y'"&89HI"&¢Ya!/HI?, * +@?a#7he.^[*««W7,\®8ªT« D,.0N1 N N/@? < &,.3*@7,l3@?CU&9&:' < "+/T#HI;G !V0¬1:*#F"(@: < "+/ #%E\7®wª+« ,9.l ' < >.?, i < ·sI¹*0 * : +'>C < C'HIW#aI(f¢=jk?m9jn%o¤[*`O !4" S < 4?,? w@7,8f8U?YZT[\7]%^_a`>bdc9[e]r !4" > S < 4?, 8E@!7,801 < fY:#%"(@:!©A !>W < i H!!aWa#g@*W#80N¢t ?>i < > H!Y#=fYjk?m9jEnFo¤[*`C !4" U '(:Z[\7]%^!_`bdc9[e]U !4" ?S < ,2I£80 ?@!,f ?S < ,- I x80 £@!,0 1 O +#%"(@hI#W8?B "(94;l£80 1U "& "&w.9 !+#¤OZT[\]%^!_a`>bdc9[e]¤'4" < "&?, 79a4;( "&O79A,9./aU9º>@,+w, a9h !> HI¢#Nh9U @.C < "T/U# #%S < Y*B8 !Y#¤4"&*Y/@"&U t(Iv0Nt > < @ HIT=@!"&C'"&C.!'l.,8=!. /9HIDD HIw#IvU/4i8ºd@ !TD "&N.I C.,* &!.9C/'? C HI¢#tIvA/4O9º>@*'¢0 =&#R!" 1i/ lvl79&: "&=?S < 4?, .,* =@ < w/ 9A!v(/4h HIa·F¨·P¸Iv¹¹Uh < />.' 4; "&7=O "&O?S < ,(@ < U/9HI!v/4a H! 1 ·F¨· Iv¹¹i#%Y #% < Y,9.0 < "&!fI @aa"&!> Q&@'wC@ < 3#tA !4"¾!2(\[*`Aob?ÁI[ k«nF«a #%"&?,©/'?$IvV/4: HIsf4(#F' ?U:3"&!>=# 8 "&lw: K < @*QA@ @ < 01 E' >./ CQ&@'wO. !#NT,./7 DU ?Y HI Y#¤@! / < a < />.' 4;A ?Hw'!U @"& < .9 !7r "&!0 8. CONCLUSIONS AND SUMMARY 1 T7*A wHI> I9?,GE9HID@*'S &#FA"& E' < 79 ¬,./7ftU @*2 A-,8)s5>?9@* @.RS < V .,E@!> < @EV B8 @!. @-2# 4"&0N1 < U#r,)s5>t@.&'A, < @'U < "A) /"&*A A@4 @?U /'l/ :"& =HI;2' 79O 9H! < #D < ?01@.'S < h#t,) @*'H!#w,Y < < "#%!" 5>N@.+ N! " ,A#F"& U@ < hQA@ ;+/; < 'h.@ ! 9º>@ !¢0¬1:,8G5>[email protected] < =89H8',8T:/ ; :*.& !79h#++#g>?fW,8 < +a; !)P"(9B "(:?9 '*!r0a(l"&E'"&f 49HwR,8U B8'/ '4;3#%Ow,*)@457@C@'&"&) w,3 < @.K:/ < . @ I 801l@*.@* >'@EB8' q B8@ A@!"& < . !8! +#ZT[\7]%^!_`bc9[e]4) !4"0 hD@?#e.^[*«.«¢,.*?fU'@.C i ' < >.?,' i < 8·g/¹*fIU"&!>Y,8."(9 @O,84½¤@t &#%"(@a*O/) *HI?,r0YU'>O < HIr#D£=fjkm9jnFo¤[*` !4"?S < 4?,l !x:@!,f~U?(Z[\7]%^!_`bdc?[e] > !4"¼S < 4?,3 xA@7,80Y1 a'#F"(@I #?B8 "(4;&?w0 £I w0t~t 9Y>D < W HI8#¤v=f > jk?m9jEnFo¤[*`Y !4"?S < 4?,3vv x&@!,f8U +ZT[\7]%^!_`bc9[e]D ! "$?S < ,V9v !£(@!,01 < f > > 9it 9Y>W < i'HIgft*#F"(@tINa#g@* #tv 0 > 9. ACKNOWLEDGEMENTS X2AY R,' I&:lO!GO.aY,VO/E»9) < ;w9.,A#F8Hw', < UU H!.W,.O#%>'( < 4"&0Ã!/* ©9Hw',?, < 3U4K@@¬ jk?m9jEnFo¤[*`Y@w,#%O#F"&'AB8 "&0 10. REFERENCES y4CC0JO07IgfrJO07JO0!I9asf 1ZT[\]%^!_a`>bdc9[e] !4"?S < ,TU!>8@ 7 U @.!8)"(9B8'"(O4"&*:l"&9HI?,¤0 ·s1 "&:' < >.9?,'@ < ,8l!>8@ !0 ¹ 1 "&U#F Y>@'+7O,7,8?, < !AO < A " ) /*#r4"&NU @.AU#F < ,A/w;a!.9'=7, @ < 7=#='4"0VO=@?-=#%" R>3G@ < "&(#+1~/ vwfUl < "+/&# "& y #% < 7,-@ !4;¬"(@.3"(B8 "(4"&T#F&Ve.^[*«.« 7,-eboo[e] ,.&01 7,8 @?79C8@# !?,tYS < 4a½¤@ HIU A"&9Hw'U!)P"(9B "( AS < @.w ;0E1('"&#%&@ < 7,2 < 9 A ' < >.?,( ="&U./ !0·s1@! < "&(@!,8) :&@ < 8@?, < &"(;l/&!/. ?,V/;!,, &¨T·P¸!v!¹h,¨· Iv¹h@! < "&fiU @.2U' t/=*B' ?, /'?h0 ¹ +i O7Y ?> < U H!#F < hx·g/7¹Y U > =0 T0 T0 T0LW.!,¤0 1¢CLW9º>@ !4"#FU57, T#%S < U4"&0 wb®w`.ok!´~bo:_k`>k!´F´'[´~koÁ+ZhnF«]g`.n ®w][Á DbªO\7®w]gnFowf8 ??0 2s y v9CC0JO07IgfrJO07JO0!I9asf T0 T0 T0LW.!,¤0 8 i4> *. !3#~¥~P"&t#%U@' ! W>=oj[*«[k`e.^[\8b`*]s !f < 0 < 4;?!80 y TJO07JO07!!agf¤Li00 < 0 Â' .9 !# O@R9 ! < 0 ~Z D b o ² [*`[*o7e.[_a`>b?e.[.[Án%ow«f7! 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