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Hypergeometrics and the cost structure of quadtrees

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(1)Hypergeometrics and the cost structure of quadtrees Philippe Flajolet, G. Labelle, L. Laforest, Bruno Salvy. To cite this version: Philippe Flajolet, G. Labelle, L. Laforest, Bruno Salvy. Hypergeometrics and the cost structure of quadtrees. [Research Report] RR-2249, INRIA. 1994. �inria-00074422�. HAL Id: inria-00074422 https://hal.inria.fr/inria-00074422 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(89) ] U’b%`<]©\ÅÁ4Z'\Áa<V \À Á^!aMÏ]©]_ÎÆ]hX_aEb%a|ÁV ^ÀV@Üb dd'Áa|ÁÔ ]BX_VHÀ|aĚ]hb À|\`<]hÀ¨V@ÜZ \b'da<`<][]hÀ ĚV a~Á“Ô@b@a<]hdÅǚa<Ï]^][]hdÀ0V \`zb%^b"iÇYÀÁ4À[Õ<{0] d][Ô ]_iV ÆfbBÀ|Æ ]hXMÁ4b"iªÀ|][aV@à ^V aEb@a~Á“V'^À´Á“^ ‡ ]hX_a|ÁV ^ +Bb X[X_V'`Ed'Á^ËBa< V {ÒÏÁ4XMÏ {0&] {`|Áa< ] U’Bb ÀbcÆÀ|][\dV'Ñ3æb X_aMV `|Á4b" i U’~• s _uÈ p´Á^b"i¤iÇ Õa<Ï]X_VH] fXMÁ“][^aEÀV@Üa<Ï#] *-\i][`Oa<`Eb@^ÀÃ9V `<Ä V Å ][Ç!aMÏ]BXMi«b'À<ÀÁ4X[b"i-ÅÁ“^V'ĝÁ4b"i¨X_V'^HÔ V%i“\a~Á“V'^fÃ9V `<ĝ\i4b . Db. ’ 7 ’ + • PQ  L 8 %v  -Py P r5P, rÇfÁ^Ô V@i\a|ÁÔ Áa|ÇÓbJÀÁÄÁ¤i4b@`cÃ9V `<ĝ\i4bÕb"i`<]hb dÇRÀ|aEb@a<]hd+Á^›vœ-V'<yEÕÄJbPq ]hÀWÁ“aOÆVÀ<ÀÁÅi]!a<V+`<]hX_V@Ô ][` r’+ð`MV Ä r ’ + È ‚0Ï]BX_V ĝÅÁ^b@a|ÁV ^gV@ÃUvT-V'<yb@^dv@) -Py-a<Ï][^²Ç'Á]_i«dÀcb%^f]_ÎYÆi¤Á4XMÁaOðV `MÄ Ã°V'`zaMÏ]%r0 ’ È  » n » ½q  žž  ( žeŸ   Ÿmlq hr’>qSo   h   jGjGl mŸ l ž (6  g   $~ž0Ÿ\7ž g%j,ž ,h_Dž j > & Ÿ ž~mŸ q ¡ ¡ o fHh!e0mŸ lnqGh Z’>i   Ÿml iUž i ’ 7 P  P W  ˜›Ov8v@) x  -Py  ˜  y>˜ QP6ª L 8 %v  -Py s L<u  #Q 6ª +# 3s <u+#  r ’ •   2 ž7g@ž ’ ’ 7 s‰lu [•  -  ),xx  s‰)Pu P• G-    h!j +’ • Q L 8 %v  -Py P SP, #  P v@),+hy.  ^]b>i«À~VšÏb À0a<Ï]c\À~]_ð\i-]hZ \b>i¢Áa|Ç. r. +’  +’ • s5 ’ u ‹v 0   - yœ W. . ‚0Ï]-`<]hÀ~a V@Ãa<ÏÁ4À Æb@Æ ][` Á4À$d][Ô V a<]hdOaMV©b"iË ][Å`Eb"Á4Xb@^dBb À~ÇHĚÆaMV a|Á4X-À|a<\d'Á]hÀ´V@à r ’0Ã9V ` Ô@b@`|ÁV \À$À|]hZ'\][^X_]hÀW’È.

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(95) È ‚0ÏÁ4ÀBÀ¥Á“Ä+Æi¢JÁ ]hÀšb%ÆÆ`<]hXMÁ4b@ÅiÇ X_V ĚÆ\a_b@a|ÁV ^ÀcÀÁ^X_]+Á“ab"Ô V@Á4dÀcX_V ^ÀÁ4d][`|Á^ËÀ|\Ä!Àcb@^d Æ`<VYd\X_a_ÀOV@Ãza<Ï] ,¯b@^ d

(96) ð\^X_a|ÁV ^À‘aEPb q ][^RV@Ô ][` À|][aEÀWV@ì`MVHV a_ÀV%ì\^Áa|Ç È ] βb@^fÁ^a<][Ë `Eb>i´d'ÁÄš][^À¥Á“V'^ ŒJb@^dfdœ] ^]a<Ï] Œ    ,¡ Bž§V@ÃbšX_V ĚÆi]_β^\ÄÅ ][` b À . h. qk f. s3Nu.• - ,) xx . p`MV Ä a<ÏÁ4À[ÕaMÏ]Œ ÑLb@^b>i“V'Ë \]cV@Ã-a<Ï]Oæb'X_a<V `|Á4b"iÁ4À0Á^a<`<Vd\X_]hdÕ6`_À|a0Ã9V `0Á^HaM][Ë ][`#$Õb À v@)P1 y. s‰lu,• s +PuJT,s¥1,uaTVTVTDs‰luO•. 7 - ) x 7 -  ) x + x98 1 )x 8. TVTVT. 7 - ) x.  x98. {0Áa<Ï. s‰)2u[• -,. †-i]hb@`|iÇ Õ a<Ï]Œ'Ñ3æb X_aMV `|Á4b"i V@Ã$Á“^a<][Ë `_b"iœb@`<Ë \Ě][^aEÀÒÀMb@a|Á4ÀW]hÀ0a<Ï]c`M]hX_\`<`<][^X_] s‰lu<• s‰luT,s‰ ›-FuC 0‚ Ï]&'Œ Ñ3æb X_aMV `|Á4b"iX[b@^^]_ÎYa-Å ]Ò]_ÎYaM][^d]hda<VcX_V ĚÆi]_ΚÔ>b"i\]hÀ0b À´a<Ï]& Œ ÑLb@^b>i“V'Ë \]0V@ßa<Ï]cb@ĚÄ!bWð\^X_a~Á“V'^ v@),Žhy. s3Nu[•   ss $˜˜;)PNuu  #%!. b@^dga<ÏÁ4ÀO]_ÎÆ`<]hÀ<ÀÁV ^gÁ4Àcb"i4À|Vf`<]_i4b@a<]hdga<VJa<Ï]šXMi4b À<ÀÁ4X[b"i&cb@ĚÄ!bð\^X_a|ÁV ^Yv¥À~][]Å ]_iV|{$yEÈ rÇRX_V ^À|a<`M\X_a|ÁV ^ Õ Áa©À<b@a~Á«À@]hÀÒa<Ï]cÅb'ÀÁ4XOÃ9\^X_a|ÁV ^b"iœ]hZ'\b@a|ÁV ^ v@), hy. s3Nu[• s3NuT,s3<›-FuC. rDzd]hÀÁË ^ Õ a<Ï]cÃ9\^X_a|ÁV ^ s FuX_V@Á^XMÁ4d]hÀBb%a©b"i¤iœÁ^a<][Ë ][`ÒÆ V@Á^Ha_À&{0Áa<Ïga<Ï]B^\ĝÅ][`cÀ|]hZ'\][^X_] s _u<{ÒÏÁ4X<Ï,Á«À X_][^a<`Eb"iÁ^²a<Ï]b@^b"iÇÀÁ4ÀÒV@Ã-Z \b daM`<][]hÀ[Õb ÀW][Ô Á4d][^X_]hdgÅHÇW‚ Ï][V `<][Ä - È0]Bb>i«À~V!Ïb"Ô ]. 7 sJ˜ ˆu<• s‰+PuJT,s¥1[u PT s‰Ž2uHTVT7TP•     -1 ) x x98    b@^d s3Fu Ïb ÀbgÀÁÄšÆi]²ÆV%i“]²b%ab"i¤i0Á^Ha<][Ë'][`EÀ§BG- '  G- FFZ²” ÀÁ^X_]ˆs‰)2u&• 'ÈÓegV `M]šË ][^][`Eb"i¤iÇ Õa<Ï][`<]fb@`<] ÀÁÄšÆi]ÆV@i]hÀOb@aÒb"i¤i¬aMÏ]c`<VHV'aEÀV@ìaMÏ]c]hZ \b@a~Á“V'^À v“˜ ly@xU•¨)[hx ÕÃ9V ` §( • 'J-GT)¡Õ@][aEX@È ( `'V@Ö i]0Á4Àb"i4À|VcÆi4b"Ç ]hd§Á^!À|\ÅÀ~]hZ \][^a-Ã9V `<ĝ\i(ÅHǧa<Ï]iV ËHb@`|Áa<ÏĊÁ«Xd][`|ÁÔ>b@a|ÁÔ ]OV@Ãa<Ï]& Œ Ñ3ÃLb X_a<V `~Á«b>i¦Õ X[b>i¢i]hd a<Ï]'Œ ÑLb@^b"iV Ë \]cV%ìa<Ï]§ÆÀÁð\^X_a~Á“V'^ Õ v@)<t[y. v@),£hy. s

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(107) v@+  &yT sJ˜ ˆu ) x < 6ª < ) Q < 6ª < 0‚ Ï\À aMÏ]zZ'\b@^Ha~Á“a~Á“]hÀ-b%ÆÆ]hb@`|Á^ËOÁ^a<Ï]zÀ~aEb@a<][Ě][^aEÀV%ÃV \` a<Ï][V `M][Ä!ÀœX[b%^BÅ]0]_ÎÆ`<]hÀ<À|]hdd'Á`<]hX_a|iÇb ÀÁ^< ^Áa<] À|\Ä!Àb@^d§Æ`<Vd\X_aEÀUv@)[Ž Չ)<Ht Õ ),£ Չ),/h¡y b'Àfz{ ]_i¤iŸb À¨][Ô>b"i\b@aM]hd^H\Ä+][`|Á4X[b"i¤i“ÇJÅÇbc`M]hd\X_a|ÁV ^ša<VBX_V Ä+ĚV ^šÀ|Æ]hXMÁ4b"i Ã9\^X_a|ÁV ^À vD+6 - Չ+,)hEy È ™Ï][^Œ§•. - ÕV ^]cÏb À s3Fu  ) sJ˜ ˆu  • Pv<  -Py  Ã9`<V ÄÒ{ ÏÁ4X<ϙa<Ï]b@^b"iÇÀÁ4ÀJV@ÃBÅÁ^b@`<ǙÀ|]hb@`EXMϙa<`<][]hÀŠÁ«ÀJ`M]hX_V>Ô'][`<]hd b ÀJb•Æb@`<a|Á4X_\i4b@`ÓX[b'À|]²V@ÃBV'\`!Ë ][^][`_b"i a<Ϫ ][V `M][Ä!ª À[ÈL¦ ^(a<Ï]”X[b'À|]!V%à Œ • ) Õ¨a<Ï] 'Œ ÑLb@^b"iV Ë'\]!b@ËHb"Á^¯d][Ë ][^][`Eb@aM]hÀcÁ^a<Vbg`Eb@a|ÁV ^b"ið\^X_a~Á“V'^ ÀÁ^X_] •­) ÁÄšÆi¤Á]hÀ ›• #) Õb%^dfV ^] ^dÀ s3Nu  • vU˜­-Py8v ˜›)<y  sJ˜ ˆu Pv< -Py ¦L^Ja<Ï]ÒÆ ][`EÀ|Æ ]hX_a|ÁÔ ]cV@à a<ÏÁ4À-Æb@Æ ][`hÕa<ÏÁ4À0d][Ë ][^][`Eb X_Ç;_]_ÎÆi4b"Á^À a<Ïb@a-aMÏ]©b@^b"iÇÀÁ4ÀV%ôÀ~aEb@^db@`Ed+Z \b daM`<][]hÀ v@ŒQ•¨)<$y Á4ÀÒ]hb À¥Á“][`cb@^dJ]_i][Ě][^aEb@`<Çf]_ÎÆ`<]hÀ<À¥Á“V'^À©b@`M]c][^X_V \^HaM][`<]hdfb À0]_Î][ĚÆi¤ÁJ ]hdRÅHÇ s¥1 Õ_£ ÕlH/ ÕO-0£ Õ6),+Pu È ½. =. M. &% = \. #b

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