Enumerating Order Types for Small Point Sets with ... - Semantic Scholar

4 downloads 1287 Views 179KB Size Report
Institute for Theoretical. Computer Science. Graz University of ... for Advanced Research and Technology] of the Austrian ...... Beach, Florida, USA, 1999, 14-23.
Enumerating Order Types for Small Point Sets with Applications 

Oswin Aichholzer

Franz Aurenhammer



Hannes Krasser

Institute for Theoretical Computer Science Graz University of Technology Inffeldgasse 16b, A-8010 Graz, Austria

Institute for Theoretical Computer Science Graz University of Technology Inffeldgasse 16b, A-8010 Graz, Austria

Institute for Theoretical Computer Science Graz University of Technology Inffeldgasse 16b, A-8010 Graz, Austria

[email protected]

[email protected]

[email protected]

Categories and Subject Descriptors

366 ,   -7 8               ABSTRACT

                                           

                                                                   !   "   

                     #         

    $                                         % #  #       #      ! !  %     "         $    !  !   !       #   &    '   (   )  # $     *  +    ($(+ ,( $     (#  +     -    (

(    .       /)   #  0  $    $   0$+111*2 +      33 ,( 3   34        3   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. SoCG’01 Medford, Massachusetts, USA Copyright 2001 ACM 1-58113-357-X/01/0006 ... 5.00.

5

Keywords

$                  

1. INTRODUCTION

'                           / 

  3  9                                                           

               /%           

          :    0    ;        3  #                   #     Æ      

!  3               %     .   #   %                9

 !     ! #     :!         %    ;          

                 %         <  %    #      %       

            3            #       #  => =      !  >   !                  

               #                         !     #         !                   #         !        

  #  

5

5

5 4

1 2

5

4

4 3

3

4 1

2

2

3

3 1 2 1

           

             

  ! !         %                     8     $  ,?-                                             :   !     !;              !                        %               "  !                 !       

  :. 3     % ; /9#  !      

            !             (  9    

#        %        

     #                 #           #   :    ;       Æ      !  9#   .      #           #   @          !                            #    !    %      #            %   %                             3       9   (    ,A-     ! = #%=                   !   '    !       

  : ;    #            :0   ;   !   : ; %  B    !       : ;         !    %      #            

        %      %   #   #         "  3  %          %         "           # (     !   ?    :  ;           #      #%          (                                

                                   !    

(    %                     !   #                         Æ              *             : ;    #      %  C: : ;;    :             #  #; 3                     %                 #        #           : ;  (

   ,D!    #  : ; 6      #   

!  : ;           9      # >        #   : ;        %   ! 

     !         7 #                  D :   !        ;                   #   #       D2    E    F= ,6A           #%              !                        :     ;   : ;          /    (                            (   !         

!  !                    (            # !  !    %      0       #   #!    9#      !   G   8   ,2-    >                 :             9             ; H                      >          !#          

            !              (  !                                    %           ?   G    (     ! !       #              "    

H"4     ,?- (  9      !      '  # #

            

!  !          B  !   #  !    9       !

       !                     3         "                              !               @                      " #   !  I       /    I  9#                                   ! :    ;  D     *                  #        G  ,? 2-   /      #     " #   *           

 #   !# (                   2.

ORDER TYPE ENUMERATION

               !                             #  9 #!                       !         2.1 The approach

       >            !     /         %   !  8        !           I          I %     !         9   :   %;       8           9#             " #   

#                 " #     !          +         3        !    #           % (               " #    !    # 

          !   

     2.1.1 Generating the candidate list @            /      ! !                      3    %         :  ;    % #           !                          8     $  ,?-         

 !      #      7

 ! !        %        Æ     !  !        (    % :     ;                   %  3            %   

%                                    

       #%                !       3             :  !        ;       9 I                          

    %    )                                              !     !    :; J              :; /                : ; 3          

     !  #             %  :  ;            %          (  9     %    >                     '         !                      % J           %   6 D      :    !    :;  : ;; .                     Æ                !  %  >               (                   ! >         #        9#                           ,?  3 6   J              !       %      : ;  #     9  %               Æ      ! 2.1.2 Grouping into projective classes

8#                            / 

  !        " #                    " 

               " #  !                        # :/  ;           %        %         (        #         %          

  %      !             ! %         %        * 

              9      (            " #                        % #    !       !               " #           %       :  

 ;  D     !     !

    G    @          

? 1  A K L 2 $ " # $     A  DK A D?6 D6 DKL I      6A6  $ " #     A  DK A D? D6 A /  $   6 D L DK D DK K? ?D A D6 ?6 I     D  LDK        6 D L DK D DK K? ?2 A D1 KA2

      !    "

  #                #  #            H !    8  # ,1-    3      3  ,A-   #       !            Æ   2.1.3 Finding the realizing point sets

+   #                #  %         &    :                ;        !   !   J$  .#   #   # G  ,2-        3    :

D    ;    %     %         !        !      !       !       !       

              .       :;     

          : ;  # *        : ;                       : ; $       !     : ;               : ;       *                     

       ! ! ,?-                 !G    Æ  !              :

;     : ;   >          !         > #  !           :   # ;      : ;       !  

        " #                     #            ! :       " # ; !       %  / !           >   7 !                !    

    :             #       I     

    I      G  .  66; '  #        !               

      !                   3      " #                   !      !             !         : ;         : ;           : ;       #     

                    "    : ; 2.2 Output and reliability

  #  9 ##!     

! !    @    D !    ,2-              :           #          % ;      +           " # : ;    "            " # : ;             !  /       >    !      H         !       

      "         !         "       6       /                #%                  !    #       D              !  J                     G                   2    K '0 $        /        1                D               !    !    "  #        D                " #  :!           %    " # ; !           (    %           %      8         

         DL       6 /       '         " #          !#      9            6K  " #     !                !    #     

            !      !     6A6  !         !     1  !  L  !           #          !  

 #           ! 

                

                  

     !                          "       !   



Suggest Documents