Reconnaissance de primitives discrètes multi-échelles

THÈSE Pour l'obtention du grade de DOCTEUR DE L'UNIVERSITÉ DE POITIERS UFR des sciences fondamentales et appliquées XLIM-SIC (Diplôme National - Arrêté du 7 août 2006) École doctorale : Sciences et ingénierie pour l'information, mathématiques - S2IM (Poitiers) Secteur de recherche : Informatique et applications Présentée par : Jean Serge Dimitri Ouattara Reconnaissance de primitives discrètes multi-échelles Directeur(s) de Thèse : Eric Andres, Théodore Tapsoba Soutenue le 04 décembre 2014 devant le jury Jury : Président Rémy Malgouyres Professeur des Universités, Université d'Auvergne Rapporteur Rémy Malgouyres Professeur des Universités, Université d'Auvergne Rapporteur Isabelle Debled-Rennesson Professeur des Universités, Université de Nancy Membre Eric Andres Professeur des Universités, Université de Poitiers Membre Théodore Tapsoba Professeur, Université Polytechnique de Bobo-Dioulasso Membre Gaëlle Largeteau-Skapin Maître de conférences, Université de Poitiers Membre Yukiko Kenmochi Chargée de recherche CNRS, ESIEE de Paris Pour citer cette thèse : Jean Serge Dimitri Ouattara. 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"                   #   $   $     &%   ℄    (   )$  $    2 n      (µ, ω) ∈ R  (a 1 , ..., an ) ∈ R  (x1 , x2 , ...xn ) ∈ Zn   µ ≤ ni=1 ai xi < µ + ω *       # $  $      )$  $          %!                 &*  ++℄    (  ,        (a1 , ..., an ) ∈ Rn              (µ, ω) ∈ R2       (x1 , x2 , ...xn ) ∈ Zn   µ ≤ ni=1 ai xi ≤ µ + ω     µ                  -  n  .        ω           i=1 ai xi − µ                    #   $  "   $ !  $                "  /           &*  ++℄   $       0!  /           #     (           ℄  H             (µ, ω) ∈ Z2  (a1 , b2 , ..., an ) ∈ Zn  H    n   P (p1, p2 , ..., pn )  Q(q1 , q2 , ..., qn )      i=1 ai pi < µ  n      H    a q ≥ ω + µ i=1 i i              "        #  #          "  #  #     ℄         (µ, ω) ∈ Z2  (a1 , a2 , ..., an ) ∈  H   Zn  0 ≤ ai ≤ ai+1 , ∀i ∈ [1, ..., n − 1]      •  ω ≥ ni=k+1 |ai |   H     •  ω < ni=k+1 |ai |   H        •  ω = ni=k+1 |ai |   H                   $   %  #      &     "    ( #        &     "   #   #   ℄ '      "                         )*      ,-  #  +)*  +)* *  ,- .℄ '( /    ( #   ℄ '( 0   -   (a1 , a2 , ..., an ) ∈ Rn 1#  )*   !"    (µ, ω) ∈ R2     #      (x1 , x2 , ...xn ) ∈ Zn  $   n −  ω  ω ai xi − µ < ≤ 2 2 i=1 ω = max1≤i≤n |ai |   1#  )* *    ℄    (µ, ω) ∈ R2  (a1 , a2 , ..., an) ∈ Rn    # (x1 , x2 , ...xn ) ∈ Zn  $     !" "        n ω  ω − ≤ ai xi − µ ≤ 2 2 i=1  ω = max1≤i≤n |ai |      !  ! !    2   ℄ -       #    2 -   34 .0% (   5 .                 ℄         (µ, ω) ∈ R2  (a1 , a2 , ..., an ) ∈ Rn          (x1 , x2 , ...xn ) ∈ Zn     n ω  ω − ≤ ai xi − µ ≤ 2 2 i=1  ω = n i=1 |ai |                                  (µ, ω) ∈  (x1 , x2 , ...xn ) ∈ Zn     ℄   R  (a1 , a2 , ..., an ) ∈ R          2  n n −  ω =  ai+1 ≥ 0 n i=1  a1 ≥ 0  a1 = 0  a2 ≥ 0    ∀i ∈ [1, ..., n − 1] ai = 0 |ai |    !    '     "        $ (   ω ω  ai xi − µ < ≤ 2 2 i=1  %  (   '                 #   $  % )              *+  '      &"         ,% %,%    "                   $   &          !        (x, y) ∈ Z            C = (x0 , y0 , R, ω)    (x0 , y0 ) ∈ R2     R ∈ R    ω∈R            2 2 ω 2 ω 2 )      -.℄ 2 R− 2 ≤ (x − x0 ) + (y − y0 ) < R + 2 "   !   "        #$%   R − 12 2 ≤  2              (x − x0 )2 + (y − y0 )2 ≤ R + 12 & "  '(  ") ##℄    #    0    ) '     '     +  ,              ω ≥ 1    ,% %/%  -.℄ C                                        ℄                  ℄   !"               ℄#          (x, y) ∈ Z                C∞ (x0 , y0, R) = ((C ⊕ Bd (1)) ∩ Z2 )    (x0 , y0) ∈ Z2     R ∈ R             2 ∞  ou  |y − y0 | ≤ 12 |(|x − x0 | − R)| ≤ |x − x0 | ≤ 12 |(|y − y0 | − R)| ≤ 1 2 1 2 ou R2 − 21 − (|x − x0 | + |y − y0 |) ≤ (x − x0 )2 + (y − y0 )2 ≤ R2 − 21 + (|x − x0 | + |y − y0 |)     $ %&       !# '       (5, 5) !# '  ()    5    (5, 5)      5   *+    !#                ,                   -          ./  ℄         0  &    1 + •       /   C∞ (x0 , y0 , R)/       2 3 &       0 C ⊕ B∞ (1) − •      /   C∞ (x0 , y0 , R)/      3   4                                    ℄          (x, y) ∈ Z2             C1 (x0 , y0, R) = ((C ⊕ B1 (1) ∩ Z2 ))    (x0 , y0) ∈ Z2     R ∈ R              ou  ||(x − y) − (x0 − y0 )|√≤ 21 ||x + y − (x0 + y0 )| − R 2)| ≤ ||(x + y) − (x0 + y0 )|√≤ 12 ||x − y − (x0 − y0 )| − R 2)| ≤ 1 2 1 2 ou R2 − 41 − max (|x − x0 |, |y − y0 |) ≤ (x − x0 )2 + (y − y0 )2 ≤ R2 − 14 + max (|x − x0 |, |y − y0 |)    !" #           $  %   ! &  $        $   %   # ! &               $   %       % '  (       !   $)      $   $  *   +   #,       )           # '       $ '   #    #,   !    '       '       $   %  $     %    $   % ! -   '     %   #       $      .     +         /! 0   #  '      '      $) )     # (1    23℄!               4   µ = c' ω = b' a = a1 ' −b = a2 $ 0 ≤ a ≤ b  P GCD (a, b) = 1'    $       % $     /     2    #       (x, y) ∈ Z    0 ≤ ax − by − c < b ! &     D (a, b, c)     %             %      + (a, b, c)! 5  %        D    % )      !    '   PD (x)' #  x!     PD (x)    y = ax−c b &   $        %    #     (            DL : ax − by − c = b − 1          DU : ax − by − c = 0  # (1    23℄!            $                  %              $      ! 5   #   )        $            D //                                     ℄             !   ax − by − c < −1 ax − by − c > b     !   ax − by − c = −1 ax − by − c = b   ax − by − c = 0 ax − by − c = b − 1       0 < ax − by − c < b − 1             Z            " #  $    " #   $  D (3, 8, 0) %      "  $    !  !   # "& $ %      !     "  $    !       & " #$ %          !  !           !      !    ' & ( ax − by − c = −1 " &  ax − by − c = b$ % )                  * #                                                              "+            ℄$                 ,         -    ./  x0  y0      ' a  b,  0  ax0 − by0 = 1                 '         x0  y0     c   &  U0 (xu0 , yu0 )  ' axu0 −byu0 = c 1         ' a(xu0 + b) −b(yu0 + a) = c           2  34                                                           ax − by − c = 0   ax − b(y − 1) − c = b   ax − by − c = b − 1   ax − b(y + 1) − c = −1      U(xu , yu)           LU(xu , yu − 1)            L(xl , yl)          LL(xl , yl + 1)                 !    F E                               # !                                                                 ⌊ ab ⌋       ⌈ ab ⌉     ℄             ! !             "    #        $  % &' ()℄ *!        "  #   !    +    ,   $ -   .' /℄ '   0 /// 4 3 2 1 0 0 2 4 6 8 10 12  //1 2     !     !         # (5, 13, 0)  !      3 # #     ⌈ 13 ⌉ = 3   #     ⌊ 13 ⌋ = 2   5 5 4        ab  a mod b     '       $ .' /℄ 3  +     #      # ax − by − c = ax−c  b 0        '    5  $       $    6            # (7, 11, 0)  (5, 8, 1)   5     ! ' [1, 7]  !   0 // 6    "  #     #   %$ .  7℄ !   '  !     #  -    #   $    #         4 # '               '  18          8 6 4 2 0 0 2 4 6 8 10 12                  (7, 11, 0)                        (5, 8, 1)         [1, 7]            !     "         #  $                       %  % &     %  % &   S (D, u, v)        D = D (a, b, c)    '         u  v      D            [u, v] &   #            D        S         !        S (D, u, v)          (a, b, c)    Ra,b,c (u, v) (  $                  &$  )                                                     b       *                +  ,       '           -  b                    .& /  01℄ *                     ,   #             #                 + '                                 '       31                                                                                       !    "#℄                                  %                                                                                                            &                          ' (      '  %  '        S       (a, b, c) ) S (D, u, v) ∪ (v + 1, y)  S (D, u, v) ∪ (u − 1, y)                    !"℄              $ %  &  $$' ( )*'  +               ,         S (D, u, v)        (a, b, c) %      + %    +   -   +   •  Ra,b,c (u − 1)  Ra,b,c (v + 1)   % −1  b'  +  % . %  +     +  +     / +    0     .            +    • & 0 ≤ Ra,b,c (u − 1) ≤ b − 1   + 0 ≤ Ra,b,c (v + 1) ≤ b − 1 '  +   +  1           / +    0       •      '  % Ra,b,c (u − 1) < −1  Ra,b,c (v + 1) > b'                           2        &                       '       '     %  '           S  3 %        4 S       '           S ′  5        S ′ 6                 +  S      S             !"'   7!℄    '        (α, β, γ)  S ′   '      '     +     v(β, α) v  .      $*                S ′            S      −−−−−−→ S          v = F E(U1 )L U1 (xu1 , yu1 )  U2 (xu2 , yu2 )       β = xl −xu1  α = yl −yu1 +1     L(xl , yl )  xu1 < xu2 −−−−−−→ S          v = F E(L1 )U U(xu , yu )          β = xu −xl1 L1 (xl1 , yl1 )  L2 (xl2 , yl2 )  xl1 < xl2  α = yu −yl1 −1              −−−−−−→ v = LF E(U2 )  β = xu2 −xl  α = yu2 −1−yl −−−−−−→ v = UF E(L2 )  β = xl2 −xu  α = yl2 +1−yu   !     "      !          #"  $     %       " &"  '    "  "     "                 " "    ()"*  +,! )"*  +-! * "" .+℄ #"       " "        0         1     "    ""   2 "       "       '       "             "   "             " 3  "   4  !  " "      5$      "!         6       "           "     "   "          %    2     7)        8                    ab            [q0 , q1 , q2 , ..., qn ]    1 a = q0 + avec qi ∈ N et qi ≥ 1 ∀i ≥ 1 b q1 + q2 + 1 1 ...+ q1 n     79"    • • 2 = 0 + 1+1 1 = 3 2 7 1 = 0 + 10 1+ 1 1    [0, 1, 2]     2 3   7 8 10 = [0, 1, 2, 3] 2+ 3 ) "" :     [0, 1, 2]        %    2    [0, 1, 2, 3] '       7;      8         ζ = [0, q1 , q2 , ..., qn ]  qi          ζ   k + 1         ζ        ζ     ζk !" #    $%℄ <                                                    ! "  # $% &℄( #     )  * +      * ,   01 -    10 -   ( .     ′ ′ / mn  mn′  * ,     +  +    ,  m+m n+n′ 0  ( #       ℄  )                 ( 1 +        2            -    3  -    -    + 11 (          ,       ,      +          %       - 4    + 11      5%   6 (78( 9  D    %  + : -     G    %  + : -  ( ; *    Dn  Gn  n     , %          % ( <         - ,       + mn = [0, q1 , q2 , ..., qn ]  Gq1 Dq2 ...Gqn  n    Gq1 D q2 ...D qn       ℄(  6 (7         - 4    + 11       + 107 ( 1 1 2 2 1 2 3 3 3 2 3 1 2 3 3 4 5 5 4 5 5 4 3 3 2 4 1 2 3 3 4 5 5 4 5 7 8 7 7 8 7 5 7 8 7 7 8 7 5 4 5 5 4 3 3 2 5 1 2 3 3 4 5 5 4 6 9 11 10 11 13 12 9 5 7 8 7 7 8 7 5 6 9 11 10 11 13 12 9 9 12 13 11 10 11 9 9 12 13 11 10 11 9 6 5 7 4 6 8 7  (7 = #         -    + 11 > G1 D2 G2 (  +  %        ( 7 10 ?  ,   @A 0 1  7 1 0 8 7 5 5 5 4 3 3 2        +                                     0 ≤ a ≤ b                  1 2         m  m       11       n1 n2 1 1 2 2      m < m      m        m    n1 n2 n1 n2 m2 m1  n2             n1          1 2    m   p > 1         D p Gq       m n1 n2 q p q ≥ 0   !  "  #$℄ & "          G D      1 2    m  '    (          m n1 n2        )              *  +          D p Gq               * +          Gq D p   !  "  #$ *  #, *- .. /  .0℄       ..1 2 34 *   (1, 3, 1)      U  U ′  L  L′  3 4                 ) A B  C  D 3 4 /       5 D( 31 , 13 )           6   7 6 &       8              5    * +  9 :                 ;,                                n   Fn                   0  1            n           F5 = { 01 , 51 , 14 , 31 , 25 , 12 , 53 , 32 , 43 , 45 , 11 }                   &                                   '                 ! "  #$℄              &  (                                  !    P(S) ) S (     *    S   )   +, ( ,-℄ .)  /           P (S) = {(α, β) , |α| ≤ 1|∀ (x, y) ∈ S, 0 ≤ αx − y + β < 1} 0        α    (α, β)  S   .     *  β   / )  !  / )     P(S)    (       2           /   )                           3 P(S)  S ( ,-℄        S    U  U    L  L′                   S    P(S)     S  P(S)         ABCD A         !   ! 4   ′                          "        # • B            (UU ′ ) $ • D            (LL′ )          (0, 1)       %  $ • A       (U ′ L+ ) L+ = L + (0, 1) $ • C       (UL′+ ) L′+ = L′ + (0, 1) S       a c    &  (a, b, c)      B   b , b  '  & B         P(S)  & [AB]  [BC]         !   ,,5  )    7         . 6 #5℄              ' 2            8      .        P(S)     S  9  .)  /   / :   ( )      9         3 4 ( ,-℄   x  y         (  x  y  (    #R(x, y) = {(α, β) |β = −xα + y} x                     ;#       R(x, y)      P(S)        (x, y)                  0 ≤ y ≤ x ≤ n  n      S               β = −xα + y  0 ≤ α ≤ 1  0 ≤ β ≤ 1                                  n       S ¾                       ℄º       n   Fn       (α, β)          R(x, y)   0 ≤ y ≤ x ≤ n    0 ≤ α ≤ 1  0 ≤ β ≤ 1        ! Fn  %  "           #  $  %&    '      n ( ) *+℄"  '    S P(S)"     !  P(S)    '   S  Fn " )  !      v !  (p, q, r)  S  -  !  '   Fn    Λ ab , cb . (a, b, c)    !       S "  Fn     , p r , q q    /  !!  !  !!   0      "      !    ! $  1!   !!      '         $    !  ! 0     '     !4 !!         ! ,  !    %      %  ! $  2" /         0    0 0    " /    $          !!      !!" 6 0 0  ! $   !!     $                !!" /   !!        !   0 " 6 0     2"  '3 !!  !!      !    !     !    0$!        " 5     %   $    % !!   !  %    !     %  $  0 %!        0 " /  !! 0!   " 7   !! 8            !            '  "   0   !  %          "   60   !  $                          0          2                                                                                                                                                                                        !                               "  #    $          $           % & '  (             #                                               )    $*   +             $,         # #          #                  -         $ $           " .   /                 (                       0 0 -        &  (                                "1                       $,               # #        2            # ##                             "  (                      "#                                  ¿¿                                                                              !" # $% #& !!℄ #                                  (     )                  *                       +        (a, b, c)                P  Q                                               ,      -                      .          .                                   *                                   !" # $% #& !! #   !/℄ *                        0                  +               1 •                                 .     1           .             2 •           " 1                     3     /4  +      "                      5               0 ≤ ax − by − c < b 3  0 ≤ a ≤ b4            Ra,b,c (x) = ax − by − c = ax−c   b n        m  n mod m 0                     +                                                          6                                                                    +      .           6                      .                            +   +6  7        +                         +               8     +                           6          +    .                  +          (      +9                    !" #   !/℄                                 .    .    ":                                                                         !                         &                                                                 #                  #         -' . /-                       !   -(℄                              5             '(℄ 3   $         %.  -1 .  /2℄    4           $     -(℄          "        !       . +          % &                 .0 %!  /  . /- . -' .0 //  5         %!  /  &       &         ')℄        %+,            $  !       . +                  $                                     '(  & . -' .0 //℄                            % &                                                                                                   %!  /  . /- . -'℄    %!  /  . /- .0 //℄        . + %.  /2℄              !                    2                               .        6 $                         0         (a, b, c)         0 ≤ ax − by − c < b  0 ≤ a ≤ b            7   7                 8                    9                 2 (x, y) ∈ Z    D(a, b, c)         0              :       )                        PD (x)        D    x                                                        u  v  u < v                   S (D, u, v)  D                    S (α, β, γ)      S                 (α, β, γ) !    ′ D (α, β, γ) "           (x, y)     ax −       (x, y)     ax − by − by − c = 0     c = b − 1               #        $      "             (x, y)     ax − by − c = −1            (x, y)     ax−by−c = b                  #      $    $       %      $         (x, y)             ax−by −c < −1     $    $  (x, y)          ax − by − c > b                   #      $    $      &%  %             (x, y)          0 < ax − by − c < b − 1 '              $       '                         (                   )                            *     +      ,                   ,        )                         ,                     -             ,             -  .    .    /-  .   01℄    ,                         3    S                 A  B        S       u  v  u < v  -       Si    Ai  Bi         ui  vi  ui < vi          S        S0    S0    (a, b, c) = (0, 1, −y0)  y0    A0    P (xP , yP ) = PD (v0 + 1)        B0    U(xU , yU )           S0  L(xL , yL )      $       S0    S1 (α, β, γ) =     S0   4  S0      P  -  .   S0 ∪ P   S1        .              (α, β, γ) 51                     (a, b, c) S0     ℄   •  P          S0           S1 !  P  α = yP − yU " β = xP − xU  γ = αxU − βyU  •  P         S0          S1 !  P  α = yP − yL " β = xP − xL  γ = αxL − βyL − β + 1 S0 "      S1    #   •  P      S0  (α, β, γ) = (a, b, c) !   •  P       $  S0 " S1       %     & $  ''(   $      %        )     %  S (D, 15, 48)  D (59, 88, 0)+ • S (D, 15, 16)              S (D, 15, 48)             (0, 1, −10)  10        15   PD (15)       15 U0 = PD (15)              D0 (0, 1, −10)           17 P0 = PD (17)              D0 (0, 1, −10) •                P0 S1 = S0 ∪ P0                    (1, 2, −5)       * −−→          U0 P0  •       18 P1 = PD (18)           P1     D1 (1, 2, −5)                  S2 = S1 ∪ P1           (2, 3, 0)       −−→    U0 P1  •                   19  48                 D2 (2, 3, 0)       S (D, 15, 48)    (2, 3, 0) &  , -            %       "    .  %)     %     /  %)   $          %     0              S0  1    2 ,       3     , -        , -         $  & %)  # 2  -           %                          %)           "         $        , -     0                %  4  & )   Si+1 ,  Si     "    "      5 Si+1      -       $     %  Si  6" 4 ((℄ 0            , -       78                                                                                    !                  " !   " #      $   # % & $   '(℄ "           "        z           z = ab       "          S ∗ [0, q1 , q2 , ...qk ]  k ∈ N  *  zi  i +    z  #  S   ,                    S     E (zk )         -./    0%  - ,   /1! E (zk )      2 • E (z0 ) = 0 3 q1    • E (z1 ) = 0 . . . 01 3 • E (zk ) = E (zk−1 )qk E (zk−2 )  k   3 • E (zk ) = E (zk−2 ) E (zk−1 )qk  k   3    • • • • • 0       7 11 ! 7 11   E (z4 )      E (z0 ) = 0  E (z1 ) = 01 1  E (z2 ) = E (z0 ) E (z1 )1 = 001  E (z3 ) = E (z2 )1 E (z1 ) = 00101  E (z4 ) = E (z2 ) E (z3 )4 = 001001010010100101  = [0, 1, 1, 1, 3]    4             5 6             ,  6         0     "            7              0                8           ,   %  %          9 # %     %  %  %          &# % -:  ';℄           <                                      3   ,          + ,                          0       ,         0%  -     >          -./ < /(!             & 9 --℄   %          ! #              &  5   ;? 5 ';℄   %   "                     < :@                            S                  S0               S            S0         S                                      !           S0 " • #                 $ • #                 "      S0 % #        U1     −−−−→  &       L0 LE1 ' L0             U1  LE1          (     U1 $      S0 % #        L1     −−−−→  &       U0 UE1 ' U0             L1  UE1          (     L1             ) *+     S0  S1 ' S1             S0              ,     #-         .       S0          S1   /         Si      / .     #-      .                        0, 1   23℄ # 5             6                  +7 # 5 ++℄    0                    ,  8                      09 :  ;<℄ "    " =       D      (a, b, µ)    P (xP , yP )  Q (xQ , yQ )            (α, β, γ)    S = {(x, y) ∈ D|xP ≤ x ≤ xQ } >                    ,                           # 0# +*℄         &     :                           #    (α, β) ' α     ,      #        β         v αβ , βγ                   >              ?      )                    (        )  ,     7@          5 4 3 2 1 0 1 0 2 4 6 8  5 4 3 2 1 0 1 0 2 4 6 8 10            S(S0 , 2, 7)     S0 (3, 8, 0)                    S0    S1 (1, 3, 0) !  !          "          S      # S    S1     $          %&                                                                                                      n  n                                         !   " !   ! #    $%℄   '   ' (      Λ( ab , µb )  Q (xQ , yQ )            v         n = xQ   v          Λ )                              Fn  Λ *       +  ,               #- .   /0℄           "  O (log n)  n             . "                                                         ! %                Fn  a r       Λ b , b #    $%℄                        1               $2%    $   *  %34 '              v       v v p v rv , qv qv                                                pv rv +1 , qv qv             v   "              ! v                  5      *   1%%$4   "  - .  #- .   /0℄            6         ! % ' $           Λ 7 % 8                               7  R7   Λ 6 *      R            :  !                              #; /3℄                          $       & ' '   (        ℄  !    "             !"℄#                &      (        )            *!              O (log δ) δ                                                                               •                                                   O (1) ! •          "    #     $                                        %  O (log δ) δ                                            ! •                                            &          O (1) '    % "                         % (  )           "                         % &           &     "       %     *      "             %   &                +    , -     "                 "    %      . /0 #1 2$             %                                (           S      (α, β, µ)  n "          D                 α (a, b, c) (         a  β  [0, q1, q2, ..., qk ]     b  [0, q1, q2, ..., qk , ..., qn] ,  δ = n − k  (            % 3        S      S0                                                        (       4         %                    %         4              1   "            5  &      4      %            2/                                                        O (n)  n   !   S0                   #        ! )  '    ( S           '      #   S         ! )   (   ! "         #  $  '        #                 S     "                        #     $   %      &  S  +               %, (  ab & O (log n)  n          S *"    O (log δ)  δ    -                   ab  αβ O (log n)  n          S *"   !         "            . *"                   !            . /.                                                 S = S (D, u, v)            S ′ = S (D, u, v ′)       S      S     ′   S (D, u, v − 1)   S ′                        S = S (D, u, v) S ′′                     S ′′ = S (D, u, v ′′)       S      S     ′′ S (D, u, v + 1)   !     % ()     2+                  &   1 '  "#" " $  2      ""  * , 1 "   6    6   -3  1 " 2+     $ " "        2  ,+   $ -℄ / +   2+       6   *    '  ,/    -0℄  -4℄ 5     )  6      '        ( " " *  "   2      "   Si′          7 Si+1    91 "  2 "       (x, Ra,b,c (x)) ;  "    "  / "          $ Si '  9:    6   Si′   '  2 "     ""     "          6 2    2     '   "1 6    2   6            ax − by − c       6     '    "#"       "  2   '    "#"    6  "#"    = %      (x, y)    *   (      9  / %  "    "1 & 8    " "     '        '       6  2 "                     * ) 1   2 "         " '   "                                                                                         D        S (D, u, v)          S (D, u, v)           D                     S (D, u, vn)  vn          D    PD (u) !            S (D, u, vn)     D !     vi                "                              !              "      #     #   #       "        D $                  %  &         D        S = S (D, u, v)        (a, b, c)             S  S ′ = S (D, u′, v ′ )        (α, β, γ)        aβ − αb = ±1.    '     S = S (D, u, v)        (a, b, c)            (    S ′ = S (D, u, v ′)          S  $        S ′ & S ′′ = S (D, u, v ′ + 1)       S                 S ′′ #            (a, b, c) ' U0 (v ′ + 1 − b, y0 )  U1 (v ′ + 1, y0 + a) #         S ′′  U1        #  S ′        α(v ′ + 1) − β(y0 + a) − γ = −1 U0          S ′ )  *  +,℄  α(v ′ + 1 − b) − βy0 − γ = 0 .  α(v ′ + 1) − β(y0 + a) − γ = −1  α(v ′ + 1 − b) − βy0 − γ = 0 /      aβ − αb = 1 '     L0 (v ′ + 1 − b, y0 )  L1 (v ′ + 1, y0 + a) #         S ′′    L1         #  S ′         α(v ′ + 1) − β(y0 + a) − γ = β  L0          S ′ )  *  +,℄  α(v ′ + 1 − b) − βy0 − γ = β − 1 .    α(v ′ + 1) − β(y0 + a) − γ = β  α(v ′ + 1 − b) − βy0 − γ = β − 1 /      aβ − αb = −1             (    S ′ = S (D, u′, v)          "  S   S ′′          " S ′′ = S (D, u′ − 1, v) S ′′   #         U0 (u′ −         (a, b, c)    1, y0)  U1 (u′ −1+b, y0 +a)  #         S ′′  U0        #  S ′        α(u′ −1)−βy0 −γ = −1 U1         S ′   α(u′ −1+b)−β(y0 +a)− 1,          γ = 0      aβ − αb = −1 α(u′ − 1) − βy0 − γ = −1   α(u′ − 1 + b) − β(y0 + a) − γ = 0 L0 (u′ − 1, y0)  L1 (u′ − 1 + b, y0 + a)      S ′′    L0                  S ′      α(u′ − 1) − βy0 − γ = β L1        ′ ′     S    α(u − 1 + b) − β(y0 + a) − γ = β − 1     ′ ′ α(u − 1) − βy0 − γ = β  α(u − 1 + b) − β(y0 + a) − γ = β − 1   aβ − αb = 1                    S ′ = S (D, u, v ′)       S  S  ! (α, β, γ)   S ′  S ′′ = S (D, u, v ′ − 1) ′                S    " ′ (α, β, γ)    U0 (v − β, y0 )              ′ ′ U1 (v , y0 +α)            S U1      " ′′ ′     av − b(y0 + α) − c = −1 U0           S  ′′ ′  &'℄   a(v −β)−by0 −c = 0        S #$  "%  ′ ′         av − b(y0 + α) − c = −1  a(v − β) − by0 − c = 0 aβ − αb = −1 ′ ′        L0 (v − β, y0 )  L1 (v , y0 + α)        S ′    L1            S ′′       ′     av − b(y0 + α) − c = b L0      )      ′′ ′ #$  "%   &'℄   a(v − β) − by0 − c = b − 1      S ′ ′  a(v − β) − by0 − c = b − 1       av − b(y0 + α) − c = b aβ − αb = 1                                    S ′ = S (D, u′ , v)       * +    S   S ′′         * + S ′′ = S (D, u′ + 1, v) S ′          (α, β, γ) $                U0 (u′ , y0 )  U1 (u′ + β, y0 + α)             S ′ U0           S ′′      au′ −by0 −c = −1 U1           S ′′    a(u′ + β) − b(y0 + α) − c = 0 ′ ′     au − by0 − c = −1  a(u + β) − b(y0 + α) − c = 0   aβ − αb = 1 ′ ′        L0 (u , y0 )  L1 (u + β, y0 + α)        ′ S    L0            S ′′       ′ au − by0 − c = b L1                ′′ ′ S    a(u + β) − b(y0 + α) − c = b − 1     au′ − by0 − c = b  a(u′ + β) − b(y0 + α) − c = b − 1   aβ − αb = −1         *                  45       ,      [u′ , v ′ ]   -  .     " /           + 0   (α, β)         (a, b)                   +  3 0     +       1              "    *   "   2 ,   +                          D      S = S (D, u, w)        (a, b, c)  S ′ = S (D, u, v)         (α, β, γ)   S          S ′   S ′            S    ∀x, x′ ∈ [u, v], |x − x′ | ≤ b : Ra,b,c (x) < Ra,b,c (x′ ) ⇒ Rα,β,γ (x) ≤ Rα,β,γ (x′ )      [u, v] ⊂ [u, w]           v − u ≤ b      D = D (5, 13, 0)      S (D, −7, 13)          (5, 13, 0)                 S (D, −7, 12)          (3, 8, 0)      [u, v] = [−7, 12]   b = 13              [u, v] !      |x − x′ | < b        "          "          # Ra,b,c (x) = ax − by − c   y = ax−c−Rb (x)  #  Rα,β,γ (x) = αx−βy −γ   $      (x′ , y ′) %              ! D    $      (x, y)  x ∈ [u, v]      #      & a,b,c Rα,β,γ (x) − Rα,β,γ (x′ ) = αx − βy − γ − (αx′ − βy ′ − γ). '  (  y  y ′      !         D & Rα,β,γ (x) − Rα,β,γ (x′ ) =     ′  ax − c − Ra,b,c (x′ ) ax − c − Ra,b,c (x) ′ − αx − β . αx − β b b )    & (αb − βa)(x − x′ ) β Rα,β,γ (x) − Rα,β,γ (x ) = − (Ra,b,c (x′ ) − Ra,b,c (x)). b b ′ ) Ra,b,c (x) < Ra,b,c (x′ )   1 ≤ k ≤ b−1   Ra,b,c (x′ )−Ra,b,c (x) = k ) − kβ .    |x − x′ | ≤ b *   & Rα,β,γ (x) − Rα,β,γ (x′ ) = (αb−βa)(x−x b b   +,-    (αb − βa) = ±1 #     1 ≤ β ≤ b−1  1 ≤ k ≤ b−1 ". & Rα,β,γ (x)−Rα,β,γ (x′ ) ≤ 1− 1b    /            Rα,β,γ (x) − Rα,β,γ (x′ ) ≤ 0          +,-  *  "   "  "     "      ′            D       S = D (u, w)          S ′ = S (D, u, v)  m     M            S    ! S ′     m     M                    S ′  #        "          * 0 1* 0 23℄       "       0  10  52℄                   67                 S ′            S             S   m(xm , ym)            S  [u, v]    ∀x ∈ [u, v], 0 = Ra,b,c (xm ) < Ra,b,c (x)  ∀x ∈ [u, v], |xm−x| < b               S                                         M    !                  "                              #         $            D        S = S (D, u, w)           (a, b, c)  S ′ = S (D, u, t)            (α, β, γ)    S             S ′         [u, v]    u ≤ v ≤ t < w   min {Ra,b,c (x)} = Ra,b,c (xm ) ⇒ min {Rα,β,γ (x)} = Rα,β,γ (xm ) u≤x≤v u≤x≤v    % v − u ≥ b            m(xm , ym)         M(xM , yM )  S       S (D, u, v)    Ra,b,c (xm ) = 0 = Rα,β,γ (xm ) &           % v − u < b    # !   '                     # !     (     $            D = D (a, b, c)        S = S (D, u, v)           (a, b, c)     m(xm , ym)    M(xM , yM )           D                 D   !   [u, v]     !          "     S                                  )     $     S = S0 = S (D, u, v0) *& v0 = v+         (a0, b0 , c0)   !           S1 = S (D, u, v1)         (a1 , b1 , c1)         S0            #  Ra ,b ,c (u, v)          #  Ra ,b ,c (u, v) *     +      S = S0  )     [u, v]      #               S0  0    ,                               # 0                     #   S1             S           S2         S1   !             #   S2           #   S1             S  - !    .  (                   Si         (a, b, c)                                    D                  ( #  1 0 /0 0 0 1 1                                                                                  D = D (5, 13, 0)     S = S0 = S (D, 6, 9)        (1, 2, 2)                        D     [6, 26]                                    [6, 9]           S0 ! • S = S0 = S (D, 6, 9)        (1, 2, 2) "          D        (1, 2, 2)    [6, 26]        #  $% "              10    12 5 4 10 4 3 8 3 2 6 1 1 4 0 0 2 1 2 3 4 5 6 5 5 4 4 3 2 2 1 5    10 15 !     20  D   25 30     (1, 2, 2) • S1 = S (D, 6, 10)        (1, 3, −1) "          D        (1, 3, −1)    [6, 26]        #  $& "              13    '             [6, 9]           S  "         !           S            S1    [6, 9] '          S1    [6, 9]       S  • S2 = S (D, 6, 13)        (2, 5, 1) "          D        (2, 5, 1)    [6, 26]        #  $( "              18    '                 [6, 9]           S  )                [6, 9]  S1  *         S2  "#          12 3  3  2 1 10  2 1  2 1 0 8 1 0 1 0 6 1 0 0 4 0 1 2 1 1 1 1 2 2 5 10  15     20 25    D   30    (1, 3, −1) 12 2 1 10 2 1 8 2 1 6 0 1 4 0 1 2 5 4 5 4 5 4 3 4 3 5   2 2 3 3 3 10    15  20    D   25 30    (2, 5, 1)                • S3 = S (D, 6, 18)         (3, 8, −1)             D       (3, 8, −1)    [6, 26]                      26                   [6, 9]           S3        S         [6, 9]                 12 0 1 2 10 1 0 8 2 1 6 0 2 4 1 3 2 4 5 7 6 5 7 10  15    • S4 = S (D, 6, 26)  4 4 6 6 5  3 3 5                 D       [6, 26]   D S  30    (3, 8, −1) (5, 13, 0) D           25 !   D   (5, 13, 0) 20              [6, 9]  " # $   %               12 2 0 10 3 1 8 4 2 6 0 3 4 1 4 2 5 10 8 6 11 9 7 12 5 10 8 6 11 9 5  10    15  20    D   25 30    (5, 13, 0)                                      S = S (D, u, v)                D    [u, v]              S                                                                  !   "## $                                                                      D = D (a, b, c)       S = S (D, u, v)   m(xm , ym )         S         D    [u, v]  M(xM , yM )         S         D    [u, v]                         [u′ , v ′ ]      [u, xm −1] [xm +1, v] [u, xM −1]  [xM +1, v]   •          [u, xm −1]  [xm +1, v]                   ! •           [u, xM −1]  [xM + 1, v]                      %         & '    "#"                           (             [u, xm − 1]  [xm + 1, v]             [u, xM − 1]  [xM + 1, v] )                             (          * %      $    β   (α, β, γ)           S       %              + %      ,             "    - ".   !   "##  ". / 0                  1            $          )                %                          2"                              T (xt , yt)                A(xA , yA)    T| T|  α = |yA −y  k ≥ 1         β = |xA−x k k (α, β, γ)              !       !"#$  α  β    %         &γ     ' (          )      * + k = 1        %                                   D = D (a, b, c)      S = S (D, u, v)        (α, β, γ)   •  u ≤ x < x + β ≤ v  Ra,b,c (x + β) = Ra,b,c (x) + (aβ − αb)  •  u ≤ x < x + b ≤ v  Rα,β,γ (x + b) = Ra,b,c (x) − (aβ − αb) , Ra,b,c (x + β) − Ra,b,c (x) = aβ − )  PD (x)  PD (x + β)    -  S Ra,b,c (x) = ax − c − b       b( a(x+β)−c b −b ax−c b  ax−c b       D        ax−c b =  αx−γ β .   Ra,b,c (x + β) − Ra,b,c (x) = aβ − αb -       %  %/   α(x+β)−γ β     ax−c b =     .                    #     )              %            $            0   %                   1          D = D (a, b, c)       S = S (D, u, v)        (α, β, γ)   LP1 (x1 , y1 )  LP2 (x2 , y2 )                      !""      !"!          S  #   (α, β, γ) = (|y1 − y2 | , |x1 − x2 | , αx1 − βy1 )         D = D (a, b, c)       (a, b, c)         S = S (D, u, v)       (α, β, γ)              2           *          Ra,b,c (xm )+k |aβ − αb|  Ra,b,c (xM )−k |aβ − αb|   T       % T                          '   D         &  34    534 67℄   -                      -          #                               & ' :                                               (x, R13,28,0 (x))           !         "  # $        %#        !           &       '              (                  $   %            )       $   % *     '    #          (         D            +  +        ax − by − c = 0  ax−by −c = b−1  0 ≤ a ≤ b    ax−by −c = 0          ax − by − c = b − 1         $  %  ,       ,  - .           D = D (13, 28, 0)     ,         S (D, 3, 16)       (5, 11, −1)  ,      ,   /       ,                         0       $1  %#             ,          '        0        )           ,  '2 3      )       '            1450    "  &  (          S = S(D, u, v)      D = D(a, b, c)     v − u ≥ b       0  b − 1  '         Ra,b,c (u, v)                    , 6                            a  b         0  b − 1    Ra,b,c (u, v)         v − u < b                       !             "    #   $#   0%  &          (α, β)  (a, b) $a.α−bβ = 1%                −cα        ' Ra,b,c (u, v) = Ra,b,0 u + b , v + −cα b               (  "    )       )  *         # '             ζ = Ra,b,0 (u, v)  a ≤ b  gcd (a, b) = 1   •  au  max(ζ) = avb  = avb   min(ζ) = au b b •      , avb  min(ζ) ∈ ζ ′  ζ ′ = R{ −b },a,0 1 + a(u−1) b a   max(ζ) ∈ ζ  ′′ ′′ ζ = b − a + R{ −b },a,0 1 + a  au b ,  a(v+1) b  0≤  (     +      ,- .   /.  " 01℄ (                 3           ζ = Ra,b,0 (u, v)        #    S = S(D, u, v)     " D      (a, b, 0) 4        axb           5     $6     )             D    % 4    PD (u)  PD (v)        aub = avb             )  u           )  v          PD (u)  PD (v)           (         )      7                 )               )     r    )         r − a < 0  0 ≤ r < a (   )       r ′ *  r + k.a = b + r′ 8 k    )      +  a  r          9  r′ = r + k.a − b 3  0 ≤ r′ < a  r−b ′      )      r = a  (        *    # { a }x  :            )       #   R{ },a,0(u′, v′)  u′  v′         #                " D (      S          PD (u − 1)        PD (u)               PD (v + 1)       −b a −b a ;;              PD (v)   D                                            ζ                        v′             av                 v′ = b          ζ                   PD (u)  PD (v)     ! "    ζ ′                !                            u′ = 1 + a(u−1) b   #   $                 a−{ }x ; u′ ≤ x ≤ v ′    %   $         a b a    ; u′′ ≤ x ≤ v ′′           %    b − a +           &             '                '  'x + 1 &   − 1       %            aub  a(v+1) b   #         (        x          x  x + 1    )  x  x + 1    u′′  u′′ + 1  v ′′  v ′′ + 1      # $     &    * + ,                                   #          -       .       )         b       a                  /       1               0     1        )    2 1     $)           .                ζ = Ra,b,0 (u, v)  0 ≤ a ≤ b a−{ ab }(x+1) a  gcd (a, b) = 1   2a > b    min(ζ) ∈ ζ ′  max(ζ) ∈ ζ ′  ζ ′ = R−a,b,0 (−v, −u)  $       −axb     ζ '  '        0 %# &     #        (         #     &   * + +      $    a 1  a               b > 2         b − a    b−ab < 21 &                                55x    ζ = R55,89,0 (1, 88) = 89 ; 1 ≤ x ≤ 88                                  !                0   !   "   ζi     #  i  ,3                         a, b, c, u, v, x0, y0, pgcd    mini                                x0 y0               x0  y0                                      [u, v]  0      [u, v]   mini ← 0     0     a′ ← a  b′ ← b  u′ ← u  v ′ ← v     V RAI     2a′ > b′   a′ ← b′ − a′  v ′′ ← b′ − u′  u′ ← b′ − v ′  v ′ ← v ′′   ′ ′  ′ ′ yu ← abu′  yv ← abv′   yu = yv    !        a′ u′ mini ← b′  break  "   at ←  a′  bt← b′  ut ← u′  vt ← v ′ t a′ ← −b  at    b′ ← at      u′ ← 1 + at (ubtt−1) ; v ′ ← abt vt t                   a, b, c, u, v, x0, y0, pgcd   maxi    b − 1      maxi ← b − 1                                      x0 y0               x0  y0             accum ← 0 a′ ← a  b′ ← b  u′ ← u  v ′ ← v  V RAI 2a′ > b′ a′ ← b′ − a′  v ′′ ← b′ − u′  u′ ← b′ − v ′  v ′ ← v ′′   ′ ′  ′ ′ yu ← abu′  yv ← abv′  yu = yv         a′ v′ maxi ← accum + b′  break                       at ←  a′  bt← b′  ut ← u′  vt ← v ′ t a′ ← −b  b′ ← at at     u′ ← 1 + atbut t  v ′ ← at (vbtt+1)  accum ← accum + bt − at                    39 50 5 52 18 10 31 23 44 36 2 49 15 40 7 28 20 41 33 54 46 12 4 30 25 17 38 30 51 43 9 1 22 14 20 35 27 48 40 6 53 19 11 32 24 10 45 37 3 50 16 8 29 21 42 0 0 20 40  60     80    ζ0  • ζ0 = ζ0′ = R55,89,0 (1, 88) •    2 × 55 > 89            ζ1 = R34,89,0 (−88, −1)             ζ0  ζ1        1  2   ζ1          ζ0     2  3            21 0 29 8 3 16 5 11 24 19 32 27  10 6 1 14 9 22  15 17 30 25 4 33  20 12 7 20 15 28  25 23 2 31 10 5  30 18 13 26  35  80  60  •      v′ = 34∗(−1) 89      !   #   40 = −1    "    20 0    ζ1   (−33, −1) ζ2 = R13,34,0 34∗(−34) = 13 u′ = 1 + = −33  89    −89 34                                                34, 68  0 3 8 2 6 11 4 9 1 6 12 4 8 2 7  10 5 10  12  14  30  25   20    • ζ3 = R5,13,0 (−12, −1)       v′ =  13∗(−1) 34     15  −34 13   10 5 0    ζ2  = 5 u ′ = 1 + = −1        13∗(−13) 34                    = −4        13, 26  1 1 3 2 2 3 4 4 5  12  10  8    6  • ζ4 = R2,5,0 (−4, −1)  • ζ5 = R1,2,0 (−1, −1)        4 2    ζ3           !   • ζ6′ = R1,1,0 (0, 0) = 87 + ´¾¹½µ + {¼}           !                                0 + 88    "    # #           #   55x ; 1 ≤ x ≤ 88         "    89          #   88   $     ζ = R55,89,0 (1, 88) =                  1 0.0  0.5  1.0  1.5  2.0  2.5 4  3 2     1    ζ4                   ζi′                       • ζ0′ = R55,89,0 (1, 88) 50 73 39 86 52 44 65 57 78 70 36 40 83 49 41 62 54 75 67 88 80 46 30 38 59 51 72 64 85 77 43 35 56 20 48 69 61 82 74 40 87 53 45 66 10 58 79 71 37 84 50 42 63 55 76 0 0 20  40     60 80    ζ0′  •     2 × 55 > 89         ′ ζ1 = R34,89,0 (−88, −1)                  ζ0′  ζ1′          ζ0′         1  2     ζ0′       2  3                  • ζ2′ = R13,34,0 (−33, 0)          !         "     #$%&'(()              ζ2    %         * +  "       [−33, 0]  ζ2′  [−33, −1]  ζ2 )           0 76 55 84 63 58 5 71 66 79 74 87  10 82 61 56 69 64  15 77 72 85 80 59  20 88 67 62 75 70  25 83 78 57 86 65  30 60 73 68 81  35  80  60   40      20 0    ζ1′  21 0 24 2 29 27 4 32 30 6 22 33 8 25 23  10 28  12 26 31  14  30  25    20      15  10    ζ2′  5 0                • ζ3′ = R5,13,0 (−12, 0)            8 0 1 9 2 11 3 10 4 12 5  12  10 8  • ζ4′ = R2,5,0 (−4, 0) 6 4         2    ζ3′          0.5 4  1.0  1.5 3  2.0  2.5 4 3  • ζ5′ = R1,2,0 (−1, 0) "    • ζ6′   2          = R1,1,0 (0, 0) = 87 + ´¾¹½µ + {¼}      #     !   $    ζ4′       1          !               $  ! % & 0 + 88                                                                                 !   "          #         $%          0.5 1 0.0  0.5  1.0  1.5  1.5   1.0  0.5     0.0  0.5    ζ5′                                                      O(log(min(a, b − a)))                   [u, v]                                 v − u              min(a, b−a)     !  "          #                 %%&    2 $ '     %(                    )           *   )  #              +             ,      #   -   ./     a  b                        #    $    ) & 0    )                  -       )               )    )  #              #         1     )           $  *    % & 2                                    0  b − 1      )   +      2     )                              $3  *    % & 45                           a, b, c, u, v, x0, y0, pgcd    α, β, µ                                x0 y0               x0  y0  α, β, µ                                        (α, β, µ) ← (1, 0, u) break          (α, β, µ) ← (0, 1, yu) break      min1 ← ResteMinimal(a, b, c, u, v, x0 , y0, pgcd) max1 ← ResteMaximal(a, b, c, u, v, x0 , y0 , pgcd) indmin1 ← CalculIndcice(a, b, c, min1, y0 ) indmax1 ← CalculIndcice(a, b, c, max1, y0 ) ymin1 ← CalculY (a, b, c, min1) ymax1 ← CalculY (a, b, c, max1)  Max(v − indmin, indmin − u) ≥ Max(v − indmax, indmax − u)  v − indmin1 > indmin1 − u)   min2 ← ResteMinimal(a, b, c, indmin1 + 1, v, x0 , y0 , pgcd)    min2 ← ResteMinimal(a, b, c, u, indmin1 − 1, x0 , y0 , pgcd) indmin2 ← CalculIndcice(a, b, c, min2, y0 ) ymin2 ← CalculY (a, b, c, min2) (α, β, µ) ← (|ymin2 − ymin1|, |indmin2 − indmin1|, |ymin2 − ymin1| × indmin1 − |indmin2 − indmin1| × ymin1)   v − indmax1 > indmax1 − u)   max2 ← ResteMaximal(a, b, c, indmax1 + 1, v, x0, y0 , pgcd)  max2 ← ResteMaximal(a, b, c, u, indmax1 − 1, x0 , y0, pgcd) indmax2 ← CalculIndcice(a, b, c, max2, y0 ) ymax2 ← CalculY (a, b, c, max2) (α, β, µ) ← (|ymax2 − ymax1|, |indmax2 − indmax1|, |ymax2 − ymax1| × indmin1 − |indmax2 − indmax1| × ymin1)                                     min(a, b, c)                 O(log(a))                                   ℄ !         b    !                 "   #  $           %       Ra,b,0 (u, v)   &  ′ ′ ′ ′ Rmin(a,b−a),min(a,b−a)+{ b ,0 (u , v )   u   v '      min(a,b−a) } (   '                  ) '         yv − yu "   *  a     + ,   -              . #                             /        0   ' yv − yu 0        !                %   1  2 23    1  2 22   4  .  0     &         5   32℄0    ,  6       % (     ! 0    .         !7    0      .  '  8                  4                 !      9 0        %0                  .  '  8                   %     .                              ":   -#        (;;    . %   .  4<=: 4<=: 3>℄      $    .         :       5?       3-℄     5   5   32℄ 4    0      &          5   32℄           4 (  !  2 3@ <A6 4     %         N    b /              n  .           !7        a  b    a < b ≤ N 0      ) c  %   %        .    (    !   10000           0             &      !            ,  1  2 2-   1  2 2>                    5? 330 5   32℄      N B 109  N B 106   4  0  !       n  .   ! 10 × 2k      [10, N] C          DD Temps de calcul en milisecondes                0.016 0.015 0.014 0.013 0.012 0.011 0 20 40 60 80 100 120 Variation de la taille du soussegment discret Temps de calcul en microsecondes                   b ≤ 1000001 a < b  µ ≤ a + b                     yv − yu  11.5 11.0 10.5 0 10 000 20 000 30 000 40 000 50 000 60 000 Variation de la distance yuyv             a = 300007 b = 1000001  µ = 0                !     "                      yv − yu  #$                                    n                           b Temps d’exécution en milisecondes Sivignon Saïd & al. 0.02 Ouattara & al. 0.015 0.005 0 300 000 000 600 000 000 Variation de la longueur du segment  ! " #     b   109 #   S   Temps d’exécution en milisecondes Sivignon Saïd & al. 0.015 Ouattara & al. 0.005 0 200 000 400 000 600 000 Variation de la longueur du segment  $ " #     %               106 #    &                     '     ni + 10 , ni  b N + 10007  b n     '      (  )                    & -   . S *         '         /                01 n                    !     %   &  "                 "     " b  " '        ,               %   (    *        a     +"            " " n    $   %   )              ℄                 !    #        !  v −u +   -' ,   Temps d’exécution en milisecondes Sivignon 0.015 Saïd & al. 0.01 Ouattara & al. 0.005 0 200 600 1500 2000 Variation de la longueur n du segment  & . $     b  10007                   S                                                   a  b                         !             " #                  $mini%   &              (    $maxi%  '        $miniL  maxiL %  (  $miniR  maxiR % )                 *   +   &            PD (mini)  PD (maxi)   &   * +           PD (mini)PD (u)+ PD (mini)PD (v)+ PD (maxi)PD (u)  PD (maxi)PD (v)+ PD (miniL )  PD (maxiL )  PD (miniR )    PD (maxiR )        $   14 ( 29    % -(               '    ,         &          $ &  %   .         #  (  '     $    %    &            / 01                a, b, c, u, v, x0, y0 , pgcd  miniL, mini, miniR, maxiL, maxi, maxiR                                x0 y0               x0  y0                           !        "                        !       "                      accum ← 0 a′ ← a  b′ ← b  u′ ← u  v ′ ← v  u1 ← u  v1 ← v  V RAI ′ 2a > b′ a′ ← b′ − a′  v ′′ ← b′ − u′  u′ ← b′ − v ′  v ′ ← v ′′   ′ ′  ′ ′ yu ← abu′  yv ← abv′  yu = yv ′ maxi ← accum + abv′ 1  ′ ′ mini ← abu′             maxiL ← accum +  ′ ′  miniR ← a (ub′+1)  a′ (v1 −1) b′ maxiR ← accum − bt + at +   miniL ← at (ubtt+1)  break    at ←  a′  bt← b′  ut ← u′  vt ← v ′ t a′ ← −b  b′ ← at  at     u′ ← 1 + atbut t  v ′ ← at (vbtt+1)  u1 ← ⌈ atbut t ⌉  vo ← v1  v1 ← ⌊ at (vbtt+1) ⌋ accum ← accum + bt − at  at (v0 −1) bt                                                                                                    !  "# •                                   $    %         &   '        # •                        (    )               *      $    %          (                    &   '          +           ,     (       (    %      $     "- # •          (       %       •                       .      (          S      )  D(  % S )               D /       S    $      $  D  %                      0    1 0 !!℄ .       3          (   (        4                   4   (α, β, γ)    %  S  (a, b, µ)   D 4    A (xA , yA )  B (xB , yB )       S $ xA < xB  yA < yB   U(xu , yu )  L(xl , yl )      $       D     S   )  $   3 )    %         % (α, β, γ)  S 5          (α, β, γ)        S  D (a, b, µ)     A (xA , yA )  B (xB , yB )            U (xu , yu )      L (xl , yl )  D    α = a + k(yu − yl − 1) β = b + k(xu − xl + 1)  ⎧   δy −a δx −b ⎪ , si xu − xl + 1 > 0 k = min ⎪ ⎪ ⎪ xu −xl +1 yu −yl −1  ⎨ x −b si xu − xl + 1 < 0 , δy −a k = max xuδ−x l +1 yu −yl −1 ⎪ ⎪ δx = max [|xu − xA |, |xu − xB |, |xl − xA |, |xl − xB |] ⎪ ⎪ ⎩ δy = max [|yu − yA |, |yu − yB |, |yl − yA |, |yl − yB |]  δx  δy                           U  L       A  B  S  6!  β          xu − xl + 1 = 0 (α, β) = (1, 1)   yu − yl − 1 = 0  α = 1             δx  δy  •  δy ≥ 1  ⌉−1  β = ⌈ a+b a ⌋−1   β = ⌊ a+b a •   β = δx                                   (α, β, γ)                        A  B                      S  (0, 1, yA)                             !     xA < xB yA < yB "        #    !   $ S    %      D  &          U  L   &   % %   A  B         &     ⎧ xA ≤ xu ≤ xB ⎪ ⎪ ⎪ ⎪ yA ≤ yu ≤ yB ⎪ ⎪ ⎪ ⎪ xA ≤ xl ≤ xB ⎪ ⎪ ⎨ yA ≤ yl ≤ yB U = A ⎪ ⎪ ⎪ ⎪ U = B ⎪ ⎪ ⎪ ⎪ L = A ⎪ ⎪ ⎩ L = B  L          D   ' S  (       ) *"# +,-   #                     (α, β, γ)  S    #              (α, β, γ).                           (α, β, γ) *   ). /  )+. "# -                      • U1 (xu + β, yu + α)             S 0 • U2 (xu − β, yu − α)             S 0 • L1 (xl + β, yl + α)         &   S 0 • L2 (xl − β, yl − α)         &   S      %     1 #  2.                  %    U  L  %    S             1 ≤ β ≤ δx *)1≤α≤δ U y $ ,  δx = max [|xu − xA |, |xu − xB |, |xl − xA |, |xl − xB |] δy = max [|yu − yA |, |yu − yB |, |yl − yA |, |yl − yB |]                          U  L     (α, β, γ)             α  β       αxu − βyu = γ αxl − βyl = γ + α + β − 1 ⇒ αxl − βyl = αxu − βyu + α + β − 1 ⇒ α(xu − xl + 1) − β(yu − yl − 1) = 1 α(xu − xl + 1) − β(yu − yl − 1) = 1     !        "    (α, β, γ)        S         ⎧ ⎨ 1 ≤ β ≤ δx 1 ≤ α ≤ δy ⎩ α(xu − xl + 1) − β(yu − yl − 1) = 1 #        $    (α, β, γ)  %&  '(          $&  (α, β) γ    )     *  +   ),                α  β  - +                        (xu − xl + 1)  (yu − yl − 1)                       (yu − yl − 1)        .   α(xu − xl + 1) − β(yu − yl − 1) = 1 /    xu − xl + 1 > 0  yu − yl − 1 < 0 xu −xl + 1 > 0                               )    yu −yl −1 < 0          # . xu − xl + 1 < 0  yu − yl − 1 > 0    # (xu − xl + 1)  (yu − yl − 1)    0   . +        .   .   1       )       2    • -  xu − xl + 1 = 0 #       (xu − xl + 1) −β(yu − yl − 1) = 1 ⇒ β = yl −y1u −1 ⇒ β=1 • 3    (α, β) = (1, 1)  α ≤ β  -  yu − yl − 1 = 0 α(xu − xl + 1) = 1 ⇒ α = xu −x1 l +1 ⇒ α=1 3   0  1 ≤ β ≤ δx  β    )    )   δx  δy  4 / δy > 1    5             D  S     β       D   -&  ! / ,  !6! +       7'          β = ⌈ ab ⌉  β = ⌊ ab ⌋  β = δx  •   xu − xl + 1 = 0 yu − yl − 1 = 0    U L   µ              D (a, b, µ) a b      a(xu − xl + 1) − b(yu − yl − 1) = 1                 L                 !   U           D  (a, b, µ) "          !  #       α = a + k(yu − yl − 1) ∗ ∃k ∈ Z | β = b + k(xu − xl + 1) $       %        1 ≤ b + k(xu − xl + 1) ≤ δx ∗ ∃k ∈ Z | 1 ≤ a + k(yu − yl − 1) ≤ δy       xu −xl +1 yu −yl −1     &     yu − yl − 1 > 0  !  #   (()  xu − xl + 1 > 0 '    1−b x −b ≤ k ≤ xuδ−x xu −xl +1 l +1 δy −a 1−a ≤ k ≤ yu −yl −1 yu −yl −1      δy −a 1−a δx −b 1−b , ,      k ∈ max xu −x , min xu −xl +1 yu −yl −1 l +1 yu −yl −1  xu − xl + 1 > 0   δx − b < 0 δy − a < 0      x −b , δy −a ⌋       α β     k = ⌊min xuδ−x l +1 yu −yl −1  yu − yl − 1 < 0  !  #   (()  xu − xl + 1 < 0 '    δx −b 1−b ≤ k ≤ xu −x xu −xl +1 l +1 δy −a 1−a ≤ k ≤ yu −yl −1 yu −yl −1      δy −a 1−b 1−a x −b     k ∈ max xuδ−x , ,  , min xu −xl +1 yu −yl −1 l +1 yu −yl −1  xu− xl + 1 < 0  δx − b < 0 δy − a < 0      x −b , δy −a ⌉       α β     k = ⌈max xuδ−x l +1 yu −yl −1      , $ *! 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"        !             %    #    & '      [AB]       #                 (                     S1  )*  S2      R         S1  S2        (p, q)   p ∈ S1   # q p - .      q c p  q         q ∈ S2  "      ℄+  n ,         "               c            p (    / 012    )-(    /           ℄+   S1  S2      Rn  X ∈ Rn   di(X) = min(d(X, Si ))  Di (X) = max(d(X, Si ))  d                 X ∈ Rn, [d1(X), D1(X)] ∩ [d2(X), D2(X)] = ∅     S1  S2   -                       "   A    A  C      B   1 3       B           C 3 C                          "    n ∈ Nk , k ≥ 2      n           n    3    "                           (   016    (                           7   #                8    )'       - 5                             ABC    I I             4                           (Ig )       ℄+     96              S = (Si )i∈[1,n]           Ig (S) =       (Mg ) !    (Mg (Si , Sj )) i,j∈[1,n],i<j                                              ℄     !            !      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')   +             "  #$  % &&℄ '             ℄        /              )                      ,    012                 '      * -     "         )            01 3 4         5            )                *) )          6              MGS        MGS       P1 P2        5 ⎧ ⎫ 2 , y) ∈ R ⎨ (x, ⎬ " " 2 2 2 2 ) ≤ ) ) + (y − C ) + (y − F (x − C (x − F MGS(P1 , P2 ) = 2y 1y 2x 1x " ⎩ " ⎭ ∧ (x − C1x )2 + (y − C1y )2 ≤ (x − F2x )2 + (y − F2y )2 7         8 91               Cix , Fix ∈ (xi − 21 ), (xi + 21 )  Ciy , Fiy ∈ (yi − 21 ), (yi + 21 )  Ci              i   i′     Fi     ′       i   i      MGS                               ℄                           !        "              !   $      %       #  &            MGS Di ' i ∈ [1, 10]         *       ( )   ℄                      !         MGS                - +    MGS , .     !   ℄,  CHCGS   CHCGS  !    /    0      n               !  - .   1          (CHCGS)     S = {Si }i∈[[1,n]]               (MGS)          S  ! CHCGS(S) = (MGS(Si , Sj )) i,j∈[1,n],i<j      2     CHCGS            ¿          "                   # $         &                                    "#                                                                      "              $                        #    '                                       (  "                            !       "     % &                 !      # $                       &                                           # $    $                "                        )     %           *   %  "                         '       log                            "              +                                                                                            P (xP , yP )  P ′(xP ′ , yP ′ )                               !   "!   #$℄ &  P <x P ′ ⇒ x < x′ P <y P ′ ⇒ y < y ′ ' Pi  i       P  Pi′′  i′       P ′       "   #(℄ ) *  +,-         1 2 1' P 3 2' P' 4 3'  +,- . ' 4'       /0  O            ' Dual (O)     x    /     y            m    /     c    )                M (x, y)       Dual (M) : c = −xm + y               (D) : y = −mx + c             N(m, c) A(xA , yA )  B(xB , yB ) 1 I(xI , yI )            [AB]         )    /        I              *  +,+ )  B Dual (I) : c = −xI m + yI   I     [AB] xI = xA +x  2 yA +yB yI = 2               [AB]  A(xA , yA )  B B ) × m + yA +y B(xB , yB )  2     Dual (M) : c = −( xA +x  2 2     (AB) 3      1 N (m, c)  A  B &   N               4-              Dual((AB)) = Dual(A) ∩ Dual(B)    A   B    c = −xA m + yA  c = −xB m + yB    c = −xA m + yA c = −xB m + yB      −yB m = xyAA −x B c = yA − xA ×                              !  (AB)          B N( xyAA −y , yA − xA × −xB           yA −yB xA −xB              " yA −yB ) xA −xB #  $ %       # [AB]' mAB         (AB) MedAB MedAB    # I  # [AB]      "     Dual(I)        #(   %    [AB]  #   mAB × mM edAB = −1 )   MedAB  N(m, c) = Dual(MedAB )     &  MedAB mM edAB     #     *   xA −xB yA −yB  + yA +yB 2 + −xA , 2×(yA1−yB ) ×[x2A − x2B + yA2 − yB2 ]) MedAB    N( xyBA −y B ′           P (xP , yP )  P (xP ′ , yP ′ ) &  ′ ′ ′  [Pi P i ] * Pi    ##  P  Pi′   ##  Medii′  # )  P  −xB m = − xyAA −y B B ) × c = ( xA +x 2 yA +yB 2        #  B c = −( xA +x )×m+ 2 yA −yB m × xA −xB = −1 ′  &    ,   #    &         #(          /  P1 (xP P2 (xP P3 (xP P4 (xP  +  0     %         ##    % ′ ′ [Pi P i ]       #    #    #     1 , yP 2 1 , yP 2 1 , yP 2 1 , yP 2 (       − + − +         + + − − 1 ) 2 1 ) 2 1 ) 2 1 ) 2    ##  #     .,      P (x , yP ′ ) ′  #        ##      P (xP , yP ) #  #(  #      &    -  P1′′ (xP ′ P2′′ (xP ′ P3′′ (xP ′ P4′′ (xP ′           − + − +   P′ 1 , yP ′ 2 1 , yP ′ 2 1 , yP ′ 2 1 , yP ′ 2  + 21 ) + 21 ) − 21 ) − 21 )   #  ,  ,   #        #1     ′ ′ [Pi P i ] 2              1’ 2’ P’ 4’ 3’ 12’21’ 11’ 14’23’ 13’31’ 24’42’ 14’23’ 33’ 1 22’ 44’ 34’43’ 2 P 3  ¿        Iii′      !      &  '  # 4        "  #  $       [Pi Pi′′ ]            Pi    Pi′′               ℄     %     '  # x +x −1 y +y +1 ( P 2P ′ , P 2P ′ ) y +y +1 x +x ( P 2 P ′ , P 2P ′ ) x +x −1 y +y ( P 2P ′ , P 2 P ′ ) x +x y +y ( P 2 P′ , P 2 P′ ) y +y +1 x +x ( P 2 P ′ , P 2P ′ ) x +x +1 y +y +1 ( P 2P ′ , P 2P ′ ) x +x y +y ( P 2 P′ , P 2 P′ ) x +x +1 y +y ( P 2P ′ , P 2 P ′ ) x +x −1 y +y ( P 2P ′ , P 2 P ′ ) y +y x +x ( P 2 P′ , P 2 P′ ) x +x −1 y +y −1 ( P 2P ′ , P 2P ′ ) x +x y +y −1 ( P 2 P ′ , P 2P ′ ) ( P 2 P′ , P 2 P′ ) x +x +1 y +y ( P 2P ′ , P 2 P ′ ) x +x y +y −1 ( P 2 P ′ , P 2P ′ ) x +x +1 y +y −1 ( P 2P ′ , P 2P ′ )          )   # *      y +y [Pi P ′ i′ ]    )   # x +x       -   Iii′       "   (  +         (      ,     ,    . • Iij ′  Iji′ 0  1  %     2 I12′ 4I21′  I13′ 4I31′  I14′ 4I41′  I23′ 4I32′  I24′ 4I42′  I34′ 4I43′ 5   3    •   I14′  I41′  I32′  I23′     2            6 • I11′ • I44′   7#         /  ℄ 5 I22′   2   ′ ( I33   2     I11′ I22′ I44′ I33′   2   ( (   2   (  ,     ( (             I33′ 5