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% Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
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%%BeginFont: CMSY7
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%%CreationDate: 1991 Aug 15 07:21:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
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/FullName (CMSY7) readonly def
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currentdict end
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%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
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/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
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/FontName /MSBM10 def
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/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueID 5031982 def
currentdict end
currentfile eexec
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%%EndFont
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
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end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
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/FontBBox{-30 -955 1185 779}readonly def
/UniqueID 5000818 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964
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cleartomark
%%EndFont
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueID 5087383 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueID 5000828 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMBXSL10
%!PS-AdobeFont-1.1: CMBXSL10 1.0
%%CreationDate: 1991 Aug 20 16:36:44
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBXSL10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle -9.46 def
/isFixedPitch false def
end readonly def
/FontName /CMBXSL10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-75 -250 1278 750}readonly def
/UniqueID 5000770 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMTT8
%!PS-AdobeFont-1.1: CMTT8 1.0
%%CreationDate: 1991 Aug 20 16:46:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-5 -232 545 699}readonly def
/UniqueID 5000830 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMCSC10
%!PS-AdobeFont-1.1: CMCSC10 1.0
%%CreationDate: 1991 Aug 18 17:46:49
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMCSC10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMCSC10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{14 -250 1077 750}readonly def
/UniqueID 5000772 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 def
currentdict end
currentfile eexec
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%%EndFont
%%BeginFont: CMR7
%!PS-AdobeFont-1.1: CMR7 1.0
%%CreationDate: 1991 Aug 20 16:39:21
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
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end readonly def
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currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueID 5000768 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMTI8
%!PS-AdobeFont-1.1: CMTI8 1.0
%%CreationDate: 1991 Aug 18 21:07:42
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-35 -250 1190 750}readonly def
/UniqueID 5000826 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueID 5000789 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueID 5000794 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR17
%!PS-AdobeFont-1.1: CMR17 1.0
%%CreationDate: 1991 Aug 20 16:38:24
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR17) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR17 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-33 -250 945 749}readonly def
/UniqueID 5000795 def
currentdict end
currentfile eexec
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pdfStartPage
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f*
10.25 18.7939 m
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0.4 w
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f*
18.05 64.3817 m
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f*
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h
S
Q
q
[1 0 0 1 -77.6753 70.0613] cm
0 g
0 G
[1 0 0 1 0 0] Tm
0 0 Td
0 0 Td
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(no) 10.5209 Tj
Q
q
[1 0 0 1 -15.6064 68.1233] cm
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0 g
0 G
[1 0 0 1 0 0] Tm
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(y) 5.26046 Tj
28 TJm
(es) 8.34899 Tj
Q
q
0.8 w
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39.685 2 l
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S
Q
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[1 0 0 1 0 0] Tm
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(13) 9.963 Tj
Q
q
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0 G
[1 0 0 1 0 0] Tm
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(13]) 12.7327 Tj
Q
q
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41.45 18.7939 m
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h
f*
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@endspecial 2693 19192 a Fr(Fig.)g(2.)g(The)f(tree)g(data)g(structure)
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y Fm(I)487 21862 y Fk(`)912 21696 y Fn(,)441 b(the)g(query)f(con)-31
b(tin)g(ues)442 b(with)g(the)e(righ)-31 b(t,)442 b(otherwise)0
23025 y(with)334 b(the)e(left)h(c)-31 b(hild)334 b(of)e
Fm(v)40 b Fn(.)1107 24353 y(Other)219 b(than)g(in)g([2,4],)j(w)-31
b(e)219 b(start)h(with)g(an)f(empt)-31 b(y)220 b(sk)-31
b(ele-)0 25681 y(ton)219 b(of)g Fj(B)34 b Fn(,)218 b(i.e.)i(all)f
(inner)g(no)31 b(des)217 b Fm(v)258 b Fn(are)218 b(lab)31
b(eled)219 b(b)-31 b(y)219 b([)p Fm(a)21276 25847 y Fk(v)21802
25681 y Fm(;)184 b(b)22768 25847 y Fk(v)23295 25681 y
Fn(],)0 27010 y(but)443 b(all)g(in)-31 b(tersections)444
b Fm(I)10654 27176 y Fk(v)11622 27010 y Fn(are)e(set)g(to)h
Fc(R)17569 26608 y Fl(2)18066 27010 y Fn(.)g(Also)g(other)0
28338 y(than)391 b(in)f([4],)i(the)e(order)f(in)i(whic)-31
b(h)391 b(w)-31 b(e)390 b(query)g(b)31 b(ecomes)0 29667
y(crucial.)252 b(W)-92 b(e)250 b(go)i(through)g(the)f(p)31
b(oin)-31 b(ts)252 b Fm(p)16526 29833 y Fl(1)17022 29667
y Fm(;)184 b(:)g(:)g(:)k(;)c(p)20038 29833 y Fk(n)20949
29667 y Fj(2)308 b Fm(S)314 b Fn(in)0 30995 y(order)367
b(of)h(increasing)h(index.)f(Before)g(querying)h Fj(B)401
b Fn(with)0 32323 y Fm(p)557 32489 y Fk(i)1156 32323
y Fn(w)-31 b(e)231 b(up)31 b(date)231 b Fj(B)264 b Fn(b)-31
b(y)231 b(adding)h(the)f(new)g(disk)g Fm(D)19104 32489
y Fk(k)19879 32323 y Fn(cen)-31 b(tered)0 33652 y(on)350
b Fm(p)2075 33818 y Fk(i)2793 33652 y Fn(\(note)g(that)h(usually)g
Fm(k)371 b Fj(6)p Fn(=)335 b Fm(i)p Fn(\))350 b(to)h(the)f(in)-31
b(tersection)0 34980 y Fm(I)487 35146 y Fk(v)1291 34980
y Fn(for)279 b(eac)-31 b(h)279 b(no)31 b(de)278 b Fm(v)318
b Fn(on)279 b(the)f(path)i(from)f(the)f(ro)31 b(ot)280
b(to)f(the)0 36308 y(leaf)333 b(that)h(corresp)31 b(onds)331
b(to)i Fm(D)12656 36474 y Fk(k)13201 36308 y Fn(.)1107
37637 y(The)302 b(result)f(of)h(a)f(query)g(with)i Fm(p)14399
37803 y Fk(i)15069 37637 y Fn(is)e(the)g(disk)h Fm(D)21113
37803 y Fk(j)21881 37637 y Fn(that)0 38965 y(corresp)31
b(onds)239 b(to)h(the)g(leaf)h(at)f(the)g(end)g(of)g(the)g(query)f
(path)0 40293 y Fm(\031)40 b Fn(.)409 b(If)g Fm(j)497
b Fn(=)433 b Fm(n)297 b Fn(+)f(1)410 b(then)f Fm(\031)449
b Fn(is)409 b(the)h(righ)-31 b(tmost)411 b(ro)31 b(ot{leaf)0
41622 y(path.)320 b(Consider)f(the)g(left)g(c)-31 b(hildren)319
b(of)g(the)g(no)31 b(des)318 b(on)h Fm(\031)40 b Fn(.)0
42950 y(The)342 b(sets)f Fm(S)5064 43116 y Fk(`)5830
42950 y Fn(that)i(b)31 b(elong)343 b(to)f(these)f(left)h(c)-31
b(hildren)342 b(par-)0 44278 y(tition)404 b Fj(f)p Fn(1)p
Fm(;)184 b(:)g(:)g(:)189 b(;)184 b(n)p Fj(g)p Fn(.)403
b(In)e(other)h(w)-31 b(ords,)402 b(the)g(in)-31 b(tersection)0
45607 y(of)352 b Fm(I)1730 45773 y Fk(`)2508 45607 y
Fn(o)-31 b(v)g(er)352 b(these)g(c)-31 b(hildren)352 b(is)g
Fm(D)13889 45773 y Fl(1)14645 45607 y Fj(\\)258 b(\001)184
b(\001)g(\001)262 b(\\)c Fm(D)19104 45773 y Fk(n)19707
45607 y Fn(.)353 b(Since)f Fm(\031)0 46935 y Fn(is)451
b(the)h(righ)-31 b(tmost)453 b(ro)31 b(ot{leaf)453 b(path,)g(the)e(con)
-31 b(tainmen)g(t)0 48263 y(queries)306 b(in)h(all)h(no)31
b(des)306 b(on)g Fm(\031)346 b Fn(w)-31 b(ere)307 b(answ)-31
b(ered)307 b(p)31 b(ositiv)-31 b(ely)-92 b(.)0 49592
y(Th)-31 b(us)298 b Fm(q)3227 49758 y Fk(i)3893 49592
y Fn(is)g(con)-31 b(tained)300 b(in)e(all)h(disks)f(curren)-31
b(tly)298 b(in)g Fj(B)34 b Fn(,)298 b(i.e.)0 50920 y
Fm(q)494 51086 y Fk(i)1214 50920 y Fj(2)352 b Fm(D)31
b Fn(\()p Fm(q)4175 51086 y Fl(1)4672 50920 y Fm(;)184
b(p)p Fn(\))265 b Fj(\\)f(\001)184 b(\001)g(\001)266
b(\\)d Fm(D)31 b Fn(\()p Fm(q)11844 51086 y Fk(i)12214
50920 y Fm(;)184 b(p)p Fn(\),)360 b(where)f Fm(D)31 b
Fn(\()p Fm(a;)184 b(b)p Fn(\))362 b(is)d(the)0 52249
y(disk)305 b(cen)-31 b(tered)305 b(at)h Fm(a)f Fn(that)h(touc)-31
b(hes)306 b Fm(b)p Fn(.)f(This)h(means)f(that)0 53577
y(none)324 b(of)h Fm(q)4309 53743 y Fl(1)4805 53577 y
Fm(;)184 b(:)g(:)g(:)k(;)c(q)7758 53743 y Fk(i)8452 53577
y Fn(lies)324 b(in)g(the)g(halfplane)i Fm(h)p Fn(\()p
Fm(p;)184 b(q)20757 53743 y Fk(i)21127 53577 y Fn(\))324
b(that)0 54905 y(con)-31 b(tains)387 b Fm(p)e Fn(and)h(whose)f(b)31
b(oundary)386 b(is)f(the)h(p)31 b(erp)g(endic-)0 56234
y(ular)374 b(bisector)f(of)h Fm(p)g Fn(and)f Fm(q)11262
56400 y Fk(i)11631 56234 y Fn(.)h(Otherwise)f([5,)i(Lemma)f(1])0
57562 y(w)-31 b(ould)423 b(guaran)-31 b(tee)423 b(that)f
Fm(q)11354 57728 y Fk(i)12176 57562 y Fj(62)453 b Fm(D)31
b Fn(\()p Fm(q)15238 57728 y Fk(k)15783 57562 y Fm(;)184
b(p)p Fn(\))423 b(for)e(the)h(p)31 b(oin)-31 b(t)0 58890
y Fm(q)494 59056 y Fk(k)1371 58890 y Fn(in)332 b Fj(f)p
Fm(q)3672 59056 y Fl(1)4170 58890 y Fm(;)184 b(:)g(:)g(:)k(;)c(q)7123
59056 y Fk(i)7492 58890 y Fj(g)333 b Fn(farthest)g(from)g
Fm(p)f Fn(in)g Fm(h)p Fn(\()p Fm(p;)184 b(q)19773 59056
y Fk(i)20143 58890 y Fn(\).)1107 60219 y(If)248 b Fm(j)371
b Fj(\024)307 b Fm(n)249 b Fn(w)-31 b(e)249 b(again)i(consider)d(the)h
(left)h(c)-31 b(hildren)249 b(of)g(the)0 61547 y(no)31
b(des)366 b(on)h(the)f(query)h(path)g Fm(\031)406 b Fn(of)366
b Fm(q)14991 61713 y Fk(i)15360 61547 y Fn(.)h(The)g(sets)e
Fm(S)21147 61713 y Fk(`)21939 61547 y Fn(par-)0 62875
y(tition)327 b Fj(f)p Fn(1)p Fm(;)184 b(:)g(:)g(:)189
b(;)184 b(j)296 b Fj(\000)232 b Fn(1)p Fj(g)325 b Fn(if)h(w)-31
b(e)325 b(tak)-31 b(e)326 b(only)f(those)g(left)h(c)-31
b(hil-)0 64204 y(dren)246 b Fm(`)h Fn(in)-31 b(to)249
b(accoun)-31 b(t)249 b(that)f(do)f(not)h(themselv)-31
b(es)248 b(lie)g(on)f Fm(\031)40 b Fn(.)0 65532 y(Similarly)269
b(to)e(ab)31 b(o)-31 b(v)g(e,)268 b(the)e(in)-31 b(tersection)268
b(of)f Fm(I)18485 65698 y Fk(`)19176 65532 y Fn(o)-31
b(v)g(er)267 b(these)0 66860 y(c)-31 b(hildren)291 b(is)g
Fm(D)6088 67026 y Fl(1)6753 66860 y Fj(\\)169 b(\001)184
b(\001)g(\001)171 b(\\)e Fm(D)10943 67026 y Fk(j)45 b
Fb(\000)p Fl(1)12543 66860 y Fn(.)291 b(Th)-31 b(us)291
b Fm(q)16361 67026 y Fk(i)17020 66860 y Fn(is)f(con)-31
b(tained)293 b(in)0 68189 y(all)357 b Fm(D)2440 68355
y Fk(k)3341 68189 y Fn(with)h Fm(k)381 b(<)347 b(j)419
b Fn(that)357 b(are)f(curren)-31 b(tly)357 b(in)f Fj(B)34
b Fn(.)356 b(On)g(the)0 69517 y(other)361 b(hand,)h(since)f
Fm(\031)401 b Fn(is)361 b(not)g(the)h(righ)-31 b(tmost)363
b(ro)31 b(ot{leaf)0 70845 y(path,)282 b(there)f(m)-31
b(ust)282 b(b)31 b(e)281 b(left)h(c)-31 b(hildren)282
b(that)g(lie)g(on)g Fm(\031)40 b Fn(.)281 b(The)0 72174
y(last)227 b(suc)-31 b(h)225 b(left)i(c)-31 b(hild)226
b Fm(v)265 b Fn(is)225 b(the)h(ro)31 b(ot)227 b(of)f(the)g(subtree)f
(whose)0 73502 y(righ)-31 b(tmost)387 b(leaf)g(corresp)31
b(onds)384 b(to)i Fm(D)15491 73668 y Fk(j)15958 73502
y Fn(.)f(Th)-31 b(us)386 b Fm(v)424 b Fn(is)385 b(asso-)26567
7085 y(ciated)h(with)h(some)e(set)g Fm(S)37592 7251 y
Fk(v)38512 7085 y Fn(=)393 b Fj(f)p Fm(i)40700 7251 y
Fk(v)41227 7085 y Fm(;)184 b(:)g(:)g(:)k(;)c(j)63 b Fj(g)p
Fn(,)387 b(where)e(1)394 b Fj(\024)26567 8413 y Fm(i)26948
8579 y Fk(v)27885 8413 y Fj(\024)411 b Fm(j)63 b Fn(.)396
b(Since)f(w)-31 b(e)396 b(ha)-31 b(v)g(e)397 b(already)f(observ)-31
b(ed)396 b(that)g Fm(q)48969 8579 y Fk(i)49733 8413 y
Fn(is)26567 9741 y(con)-31 b(tained)241 b(in)f(all)h
Fm(D)34935 9907 y Fk(k)35719 9741 y Fn(with)g Fm(k)342
b(<)308 b(j)302 b Fn(that)241 b(are)e(curren)-31 b(tly)240
b(in)26567 11070 y Fj(B)34 b Fn(,)230 b(but)g Fm(\031)270
b Fn(came)230 b(to)h Fm(v)269 b Fn(via)231 b(a)g(\\no"-branc)-31
b(h,)231 b(w)-31 b(e)231 b(no)-31 b(w)231 b(kno)-31 b(w)26567
12398 y(that)364 b Fm(q)29453 12564 y Fk(i)30179 12398
y Fj(62)357 b Fm(D)32190 12564 y Fk(j)32657 12398 y Fn(.)362
b(Let)h Fm(m)g Fn(b)31 b(e)362 b(suc)-31 b(h)362 b(that)i
Fm(D)43936 12564 y Fk(j)44760 12398 y Fn(=)357 b Fm(D)31
b Fn(\()p Fm(q)47849 12564 y Fk(m)48690 12398 y Fm(;)184
b(p)p Fn(\).)26567 13726 y(Note)352 b(that)g Fm(q)32098
13892 y Fk(i)32805 13726 y Fj(62)339 b Fm(D)31 b Fn(\()p
Fm(q)35753 13892 y Fk(m)36594 13726 y Fm(;)184 b(p)p
Fn(\))353 b(means)e(that)h Fm(D)31 b Fn(\()p Fm(q)46046
13892 y Fk(m)46888 13726 y Fm(;)184 b(p)p Fn(\))353 b(w)-31
b(as)26567 15055 y(inserted)512 b(in)g Fj(B)545 b Fn(b)31
b(efore)512 b(querying)g(with)h Fm(q)44928 15221 y Fk(i)45297
15055 y Fn(,)f(and)g(th)-31 b(us)26567 16383 y Fm(m)386
b Fj(\024)g Fm(i)p Fn(.)381 b(Since)f Fm(q)33636 16549
y Fk(i)34391 16383 y Fj(62)386 b Fm(D)31 b Fn(\()p Fm(q)37386
16549 y Fk(m)38227 16383 y Fm(;)184 b(p)p Fn(\),)382
b(and)f Fm(q)43052 16549 y Fk(i)43807 16383 y Fj(2)386
b Fm(D)31 b Fn(\()p Fm(q)46802 16549 y Fk(r)47293 16383
y Fm(;)184 b(p)p Fn(\))381 b(for)26567 17711 y(all)413
b Fm(r)469 b Fj(\024)437 b Fm(i)412 b Fn(with)h Fj(j)p
Fm(pq)35128 17877 y Fk(r)35619 17711 y Fj(j)438 b Fm(>)f
Fj(j)p Fm(pq)39020 17877 y Fk(m)39862 17711 y Fj(j)p
Fn(,)412 b([5,)h(Lemma)g(1])g(yields)26567 19040 y(that)301
b Fm(q)29390 19206 y Fk(m)30531 19040 y Fn(is)e(farthest)i(from)f
Fm(p)g Fn(in)g Fj(f)p Fm(q)41277 19206 y Fl(1)41774 19040
y Fm(;)184 b(:)g(:)g(:)k(;)c(q)44727 19206 y Fk(i)45097
19040 y Fj(g)i(\\)g Fm(h)p Fn(\()p Fm(p;)e(q)49370 19206
y Fk(i)49739 19040 y Fn(\).)27674 20368 y(The)250 b(running)f(time)i
(is)f(as)f(follo)-31 b(ws.)252 b(Querying)e Fj(B)283
b Fn(tak)-31 b(es)26567 21696 y Fm(O)31 b Fn(\(log)29302
21201 y Fl(2)29984 21696 y Fm(n)p Fn(\))244 b(time)g(since)f(the)h
(heigh)-31 b(t)245 b(of)f Fj(B)277 b Fn(is)243 b Fm(O)31
b Fn(\(log)202 b Fm(n)p Fn(\))244 b(and)26567 23025 y(in)284
b(eac)-31 b(h)284 b(no)31 b(de)284 b(of)g(the)g(query)g(path)g(the)g
(query)f(p)31 b(oin)-31 b(t)285 b(has)26567 24353 y(to)366
b(b)31 b(e)364 b(lo)31 b(cated)367 b(in)e(the)g(in)-31
b(tersection)367 b Fm(I)42874 24519 y Fk(v)43764 24353
y Fn(of)f(some)f(disks,)26567 25681 y(whic)-31 b(h)333
b(can)g(b)31 b(e)332 b(done)g(in)h Fm(O)31 b Fn(\(log)202
b Fm(n)p Fn(\))333 b(time.)27674 27010 y(When)450 b(w)-31
b(e)450 b(up)31 b(date)451 b Fj(B)483 b Fn(b)-31 b(y)451
b(adding)g(a)f(new)h(disk)f Fm(D)49703 27176 y Fk(j)50170
27010 y Fn(,)26567 28338 y(w)-31 b(e)326 b(ha)-31 b(v)g(e)326
b(to)f(go)h(from)g(the)f(ro)31 b(ot)326 b(to)f(the)h(leaf)g(that)g
(corre-)26567 29667 y(sp)31 b(onds)368 b(to)h Fm(D)32506
29833 y Fk(j)32973 29667 y Fn(.)g(In)f(eac)-31 b(h)369
b(no)31 b(de)369 b(on)g(this)g(path)g(w)-31 b(e)370 b(m)-31
b(ust)26567 30995 y(compute)457 b Fm(I)31630 31161 y
Fk(v)32483 30995 y Fj(\\)326 b Fm(D)34463 31161 y Fk(j)35386
30995 y Fn(and)456 b(up)31 b(date)457 b(our)f(data)h(structure)26567
32323 y(for)307 b Fm(I)28686 32489 y Fk(v)29211 32323
y Fn(.)g(This)h(can)f(b)31 b(e)306 b(done)h(in)g Fm(O)31
b Fn(\(log)202 b Fm(n)p Fn(\))307 b(time)h(p)31 b(er)306
b(no)31 b(de)26567 33652 y(b)-31 b(y)372 b(a)g(pro)31
b(cedure)371 b(detailed)j(in)e([5,)h(pro)31 b(of)373
b(of)f(Lemma)h(4].)26567 34980 y(Th)-31 b(us)332 b(eac)-31
b(h)333 b(up)31 b(date)333 b(also)g(tak)-31 b(es)333
b Fm(O)31 b Fn(\(log)43189 34485 y Fl(2)43870 34980 y
Fm(n)p Fn(\))333 b(time.)27674 36308 y(No)-31 b(w)325
b(the)g(running)f(time)i(of)f Fm(O)31 b Fn(\()p Fm(n)184
b Fn(log)43367 35813 y Fl(2)44048 36308 y Fm(n)p Fn(\))325
b(is)f(ob)-31 b(vious.)26567 37637 y(The)315 b(space)g(consumption)h
(is)f Fm(O)31 b Fn(\()p Fm(n)184 b Fn(log)203 b Fm(n)p
Fn(\))315 b(since)g(a\))h(eac)-31 b(h)26567 38965 y(disk)265
b(con)-31 b(tributes)266 b(only)g(to)g(in)-31 b(tersections)267
b(stored)e(on)g(the)26567 40293 y(path)300 b(from)g(the)f(ro)31
b(ot)300 b(to)g(\\its")h(leaf,)g(and)e(b\))h(a)g(disk)f(that)26567
41622 y(con)-31 b(tributes)429 b(to)h(some)e(in)-31 b(tersection)430
b Fm(I)43120 41788 y Fk(v)44074 41622 y Fn(adds)e(at)h(most)26567
42950 y(one)333 b(arc)f(to)h(the)f(b)31 b(oundary)333
b(of)g Fm(I)40215 43116 y Fk(v)41072 42950 y Fn([5,)h(F)-92
b(act)332 b(2].)3472 b Fa(2)26567 45714 y Fo(References)26567
48426 y Fr([1])333 b(M.)377 b(B\025)-470 b(adoiu,)378
b(S.)f(Har-P)-26 b(eled,)377 b(and)g(P)-78 b(.)377 b(Indyk.)553
b(Appro)-26 b(ximate)27892 49533 y(clustering)316 b(via)e(core-sets.)
411 b(In)314 b Fp(Pr)-49 b(o)g(c.)315 b(STOC'02)p Fr(,)f(pages)i(250{)
27892 50640 y(257,)334 b(2002.)26567 52229 y([2])f(D.)387
b(Z.)f(Chen,)f(O.)h(Daescu,)h(J.)f(Hersh)-26 b(b)26 b(erger,)385
b(P)-78 b(.)386 b(M.)g(Kogge,)27892 53336 y(and)400 b(J.)h(Sno)26
b(eyink.)616 b(P)-26 b(olygonal)402 b(path)e(appro)-26
b(ximation)401 b(with)27892 54443 y(angle)410 b(constrain)-26
b(ts.)640 b(In)408 b Fp(Pr)-49 b(o)g(c.)409 b(SOD)-24
b(A'01)p Fr(,)409 b(pages)h(342{343,)27892 55550 y(W)-78
b(ashington,)334 b(D.C.,)f(2001.)26567 57138 y([3])g(O.)360
b(Daescu)h(and)e(N.)h(Mi.)508 b(P)-26 b(olygonal)362
b(path)d(appro)-26 b(ximation:)27892 58245 y(A)379 b(query)e(based)j
(approac)-26 b(h.)560 b(In)378 b Fp(Pr)-49 b(o)g(c.)379
b(ISAA)-24 b(C'03)p Fr(,)380 b(v)-26 b(olume)27892 59352
y(2906)334 b(of)e Fp(LNCS)p Fr(,)h(pages)g(36{46,)i(2003.)g(Springer.)
26567 60941 y([4])e(J.)268 b(Gudm)-26 b(undsson,)266
b(H.)i(Ha)-26 b(v)g(erk)g(ort,)266 b(S.-M.)h(P)-26 b(ark,)267
b(C.-S.)f(Shin,)27892 62048 y(and)579 b(A.)h(W)-78 b(ol\013.)1091
b(F)-78 b(acilit)-26 b(y)579 b(lo)26 b(cation)582 b(and)d(the)g
(geometric)27892 63155 y(minim)-26 b(um-diameter)395
b(spanning)g(tree.)601 b(In)394 b Fp(Pr)-49 b(o)g(c.)395
b(APPR)-24 b(O)g(X)27892 64262 y('02)p Fr(,)314 b(v)-26
b(ol.)316 b(2462)g(of)e Fp(LNCS)p Fr(,)i(pages)f(146{160,)k(2002.)d
(Springer.)26567 65850 y([5])333 b(J.)268 b(Gudm)-26
b(undsson,)266 b(H.)i(Ha)-26 b(v)g(erk)g(ort,)266 b(S.-M.)h(P)-26
b(ark,)267 b(C.-S.)f(Shin,)27892 66957 y(and)579 b(A.)h(W)-78
b(ol\013.)1091 b(F)-78 b(acilit)-26 b(y)579 b(lo)26 b(cation)582
b(and)d(the)g(geometric)27892 68064 y(minim)-26 b(um-diameter)780
b(spanning)f(tree.)1620 b Fp(Computational)27892 69171
y(Ge)-49 b(ometry:)332 b(The)-49 b(ory)332 b(and)g(Appl.)p
Fr(,)g(27\(1\):87{106,)337 b(2004.)26567 70759 y([6])c(J.)475
b(Matou)-50 b(\024)-421 b(sek.)815 b(E\016cien)-26 b(t)475
b(partition)h(trees.)814 b Fp(Discr)-49 b(ete)474 b(and)27892
71866 y(Computational)332 b(Ge)-49 b(ometry)p Fr(,)332
b(8:315{334,)337 b(1992.)p eop end
%%Trailer
userdict /end-hook known{end-hook}if
%%EOF