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O(n log n) 2a2C ( ,I 9: * 4 A( 2C J-5 "; \ Q% ' $($( "#5 ? 9: * 4 $ # L . 15 K # 9 ,: A ( J] Q% j "'' 2H -5 "''; 9'' '2a2C + '1 .("''8 ' K '# O(n log n) A ' ( ] 'Q% J-2'' ( K '# .(" ": O(n log n) P1 P2 P3 P1 P4 P3 P1 / $ 0 ) +, ' -!. !, ' * :) 7< 4 '8 '5D "' '2H -5 "'; ' C -21 o '6 +' V'@N ("': -5 "'; $ 15' ] O "''U ' 4 ' ( C 4 '; L -2''8 .(" ": U@ 4 6) "* 4 ( C ( )* " -5 "; o 6. 4 8 % N + 5''8 ( 4 ' ( C p ' .-2 ' ' K i 4 ' ( C ( C P1 4 (C 4 ( C+ A2 .-2 Pi .-2''1 ' '% N P1 , P2 ,..., Pi ,..., Pp 1 , Pp > # 767< " O "U I 123 9: * 4 " O "U & 4 ( C T "* 9: * 4 " O "U 9: * 4 4 A( 2C " 2Q A( 2C :\ Q% * :\ K @ < :] K * :d K "#5 ? K # :] Q% O P@ A> $) $ ? , $*! 7:7< # L .-21 -2 * 4 ( C p +2 & J " O "U , n 9: * 4 A( 2C @' < A "* 2* * + .-21 * " O "U 3 L 5 J 15 * & + o6 "* & * .(" Z"< q ( " 4 ( C p : 4($ < = ( / $ & 9: * 4 '2 -2 'U* * 4"< /? A "* & 1 C ?"* .( 2 O "Y m ! 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'1 ' -2 * 2% p+1 & 4 8 * 36 4 ( C Q% & ' 4 '8 '* 36 ( (i + 1) * n /( p + 1) $ (i 1) * n /( p + 1) 4 8 & +2 9: '2% $( 9: '* 4 4 ( C A" . 1 @ < " 2H O "U ]X] ( 4 8 5D +2 $ 2% 9: ( 9: * 4 + 1 J (r 5 2% = 4 ( C $( $ 1 ' 9: '* 4 2H 2% $( +2 -2 ( A 6 Z % .- $L .( + ( * 4 I$ V@N @ < 2% $( ("?$ I P q S 7, ( 6 p3 # q1 & ' / A' () ' I ' AL ' ' (": ' 3 ( 5s 2. $ 3# 4 :1 , J 2% p+1 + 8 2% = ( 31* 2H % "' ( /' $ & ' $ '2% = & +2 9: * 4 :: 4 ;L J 8 58 ( ( 2% .(" 1 ": ' (I ' ', ' + 5 (G $ q , +2 9: * 4 31* -2 ( ] 2b V@N :O @k ( ' '8 q ', J 2% p+1 + 2% = ( 31* + 1 . 15 I q , 3 = $ '2% =' & ' +2' 9: '* 4 4 ; L J 8 58 ( ( 2% A () (": 3 '#25 ( $ ("' ": F ; $ q , 9: * 4 + 5 G 5 G AL 2H % " ( / $ & . ( A t"m" + e 9 8 .(" 1 ": I 9: * 4 sG? " I (C 1 ?"* .( ( ("?$ , +21 -21 o6 + 9% ( + 1 . - 2 , +21 ("?$ K ! Z 5% , 8 ( & ( )* ("8 56 O( (2n /( p + 1)) * log(2n /( p + 1)) ) 4 ( C T "* 9: * 4 @ < 2a2C O P A> $) $ # ) 7<7:7< ( '# .("8 HL 9: * 4 * Q% 4 ( C T "* 9: * 4 @ < iC ' Q: '* 4 5 4 ( C .("8 4( D5 8 4( ( g 8 \X]Xd ( H( * -5 "; (6 4 (C 4 8 * 9: * 4 36 j$ 4 ( C iW $ 4(" * 4(" @ < 'Q% log p i'C . '8 ' '8 4( ( g 8 \X]Xd ( & * 1 (J ($ 1( '4 ' ' '* -5 "; 2a2C #25 ( . 8 ( ": 9: * 2n /( p + 1) * (log 2 n /( p + 1) + 2 -*. $ ? , $*! 4 (log p + 1) 36 P1 4 2Q 2) : ("8 S (C 9: * T E S 7<7< $ '123 ' 4 ' A( $L ' ' - ( ( u2m"* " 2H -5 "; ( ' L -2'1 ' ' 9: '* 4 ' ' ' $ - ( 'C ' ' $($( "#5' ? ' 9: '* 4 36 A "'* ' 4 ' ( C p ("'?$ ?"* . 2: ( ( ("?$ (" 4 @5) $($( "#5 ? # ' 36 U@ 4 ( C Q% ( I$ + .4(" 4( D5 *p "#5 ? ' G'2 'U@ 4 ( C) 1 Z 4 4 (C = $ 4( B R5 4 p "#5 ? ( " c C ("?$ c' C ("'?$ ' iC 4 ( C J( ( C 4 8 B R5 4 p + 9: '* ' 4 ' 9: '* ' . 1 Z .(" 1 ": m 2m , , p +1 p +1 '11 N" c C ("?$ G2 (": U@ 4 ( C) 11 KM! U@ 4 ( C #25 '4 ' 36 ? 4($ < 4 8 KM! [ 5 ?"* U@ 4 ( C iW .(( ( C 4 (C $ B R5 4 p ( # $ 56 q ( 8 AL ( + B "? 4 8 ( 4 8 B R5 Y 1! 4 ; L 8 * m (" 36 ( ("?" 4 ( )* i*m ( p 1) * m p * m 3m , ,..., , ..., p +1 p +1 p +1 p +1 4 c C ("?$ @ < 7 Y 4 ( C 4 3 I$ + "#5 ? K # A ": nlog np . 8 " $*N*! A*4') 7<7< O "'U 9: '* '2a2C '4 * +21a ( G'2 '123 9: *4 ' $ ! ' F G'6 +21a $ 9: * "#5 ? . 4 @ < $ O(n / p(log n / p + 2 log p ) ": O(n log np ) ? ( -5 "; 9 . " O "U & . 2* 5 2a2C + " 2H -5 "; * 2a2C ( 1 . nlog np @ " " -5' "; n 2 n p Speed up ( P) = T (1) log = = log 2p T ( p ) log Efficiency ( P) = log n2 log 2p T (1) = = pT ( p ) p log np p * S* 7= ' =' ' "* ' .- (" ' '?"* ' . '8 ,( ( ' .("8 6) ' " -5 "; = ' '2 5) ;(9 8 9% ?" J ' +' ( ') K ' .("' '53 L A "5 -2' * V N -5 "; " $ A "* 8$ +21a 7 " -5 "; + + . 8 9: * .(" , (" ,( " 4( D5 J + ( = ) " ) A(" NP-hard " -5 "; +2 $ J-5 "; + 4 @ < A(" 53 "* 123 4 ("?$ @ < UT [1] M. Formann, F. Wagner, “A Packing Problem with Applications to Lettering of Maps”. Proceedings of the 7th Annual ACM Symposium on Computational Geometry. pp. 281-288, 1991. [2] H. Aonuma, H. Imai, Y.Kambayashi, “A visual system of placing characters appropriatly in multimedia map databases”. Proceedings of the IFIP TC 2/WG 2.6 Working Conference on Visual Database Systems, North Holland. pp. 525-546, 1989. [3] K. Imai, T. Asano, “Efficient Algorithms for Geometric Graph Search Problems”. SIAM J. Comput. 15. pp. 478-494, 1986. [4] F. Wagner, A. Wolff, “Map Labeling Heuristics: Provably Good and Practically Useful”. Proceedings of the 11th Annual ACM Symposium on Computational Geometry. pp. 109-118, 1995. [5] M. Formann, “Algorithms for Geometric Packing and Scaling Problems”. Dissertation, Fachbereich Mathematik und Informatik, Freie Universität, Berlin, 1992.
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