Documentation updated for release 0.9
This commit is contained in:
Fabiano C. Botelho
140
tex/bdz/bdz.bib
Executable file
140
tex/bdz/bdz.bib
Executable file
@@ -0,0 +1,140 @@
|
||||
@inproceedings{bpz07,
|
||||
author = {F.C. Botelho and R. Pagh and N. Ziviani},
|
||||
title = {Simple and Space-Efficient Minimal Perfect Hash Functions},
|
||||
booktitle = {Proceedings of the 10th Workshop on Algorithms and Data Structures (WADs'07)},
|
||||
publisher = {Springer LNCS vol. 4619},
|
||||
pages = {139-150},
|
||||
Moth = August,
|
||||
location = {Halifax, Canada},
|
||||
year = 2007,
|
||||
key = {author}
|
||||
}
|
||||
|
||||
@PhdThesis{b08,
|
||||
author = {F. C. Botelho},
|
||||
title = {Near-Optimal Space Perfect Hashing Algorithms},
|
||||
school = {Federal University of Minas Gerais},
|
||||
year = {2008},
|
||||
OPTkey = {},
|
||||
OPTtype = {},
|
||||
OPTaddress = {},
|
||||
month = {September},
|
||||
note = {Supervised by Nivio Ziviani, \url{http://www.dcc.ufmg.br/pos/cursos/defesas/255D.PDF}},
|
||||
OPTannote = {},
|
||||
OPTurl = {http://www.dcc.ufmg.br/pos/cursos/defesas/255D.PDF},
|
||||
OPTdoi = {},
|
||||
OPTissn = {},
|
||||
OPTlocalfile = {},
|
||||
OPTabstract = {}
|
||||
}
|
||||
|
||||
@Article{mwhc96,
|
||||
author = {B.S. Majewski and N.C. Wormald and G. Havas and Z.J. Czech},
|
||||
title = {A family of perfect hashing methods},
|
||||
journal = {The Computer Journal},
|
||||
year = {1996},
|
||||
volume = {39},
|
||||
number = {6},
|
||||
pages = {547-554},
|
||||
key = {author}
|
||||
}
|
||||
|
||||
@inproceedings{ckrt04,
|
||||
author = {B. Chazelle and J. Kilian and R. Rubinfeld and A. Tal},
|
||||
title = {The Bloomier Filter: An Efficient Data Structure for Static Support Lookup Tables},
|
||||
booktitle = {Proceedings of the 15th annual ACM-SIAM symposium on Discrete algorithms (SODA'04)},
|
||||
year = {2004},
|
||||
isbn = {0-89871-558-X},
|
||||
pages = {30--39},
|
||||
location = {New Orleans, Louisiana},
|
||||
publisher = {Society for Industrial and Applied Mathematics},
|
||||
address = {Philadelphia, PA, USA},
|
||||
optpublisher = {Society for Industrial and Applied Mathematics}
|
||||
}
|
||||
|
||||
@Article{j97,
|
||||
author = {B. Jenkins},
|
||||
title = {Algorithm Alley: Hash Functions},
|
||||
journal = {Dr. Dobb's Journal of Software Tools},
|
||||
volume = {22},
|
||||
number = {9},
|
||||
month = {september},
|
||||
year = {1997},
|
||||
note = {Extended version available at \url{http://burtleburtle.net/bob/hash/doobs.html}}
|
||||
}
|
||||
|
||||
|
||||
@Article{e87,
|
||||
author = {J. Ebert},
|
||||
title = {A Versatile Data Structure for Edges Oriented Graph Algorithms},
|
||||
journal = {Communication of The ACM},
|
||||
year = {1987},
|
||||
OPTkey = {},
|
||||
OPTvolume = {},
|
||||
number = {30},
|
||||
pages = {513-519},
|
||||
OPTmonth = {},
|
||||
OPTnote = {},
|
||||
OPTannote = {}
|
||||
}
|
||||
|
||||
@article {dict-jour,
|
||||
AUTHOR = {R. Pagh},
|
||||
TITLE = {Low Redundancy in Static Dictionaries with Constant Query Time},
|
||||
OPTJOURNAL = sicomp,
|
||||
JOURNAL = fsicomp,
|
||||
VOLUME = {31},
|
||||
YEAR = {2001},
|
||||
NUMBER = {2},
|
||||
PAGES = {353--363},
|
||||
}
|
||||
|
||||
|
||||
@inproceedings{sg06,
|
||||
author = {K. Sadakane and R. Grossi},
|
||||
title = {Squeezing succinct data structures into entropy bounds},
|
||||
booktitle = {Proceedings of the 17th annual ACM-SIAM symposium on Discrete algorithms (SODA'06)},
|
||||
year = {2006},
|
||||
pages = {1230--1239}
|
||||
}
|
||||
|
||||
@inproceedings{gn06,
|
||||
author = {R. Gonzalez and
|
||||
G. Navarro},
|
||||
title = {Statistical Encoding of Succinct Data Structures},
|
||||
booktitle = {Proceedings of the 19th Annual Symposium on Combinatorial Pattern Matching (CPM'06)},
|
||||
year = {2006},
|
||||
pages = {294--305}
|
||||
}
|
||||
|
||||
@inproceedings{fn07,
|
||||
author = {K. Fredriksson and
|
||||
F. Nikitin},
|
||||
title = {Simple Compression Code Supporting Random Access and Fast
|
||||
String Matching},
|
||||
booktitle = {Proceedings of the 6th International Workshop on Efficient and Experimental Algorithms (WEA'07)},
|
||||
year = {2007},
|
||||
pages = {203--216}
|
||||
}
|
||||
|
||||
@inproceedings{os07,
|
||||
author = {D. Okanohara and K. Sadakane},
|
||||
title = {Practical Entropy-Compressed Rank/Select Dictionary},
|
||||
booktitle = {Proceedings of the Workshop on Algorithm Engineering and
|
||||
Experiments (ALENEX'07)},
|
||||
year = {2007},
|
||||
location = {New Orleans, Louisiana, USA}
|
||||
}
|
||||
|
||||
|
||||
@inproceedings{rrr02,
|
||||
author = {R. Raman and V. Raman and S. S. Rao},
|
||||
title = {Succinct indexable dictionaries with applications to encoding k-ary trees and multisets},
|
||||
booktitle = {Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms (SODA'02)},
|
||||
year = {2002},
|
||||
isbn = {0-89871-513-X},
|
||||
pages = {233--242},
|
||||
location = {San Francisco, California},
|
||||
publisher = {Society for Industrial and Applied Mathematics},
|
||||
address = {Philadelphia, PA, USA},
|
||||
}
|
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70
tex/bdz/bdz.tex
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70
tex/bdz/bdz.tex
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\documentclass[12pt]{article}
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\usepackage{graphicx}
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\usepackage{latexsym}
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\usepackage{url}
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\usepackage{a4wide}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{amsfonts}
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\usepackage{graphicx}
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\usepackage{listings}
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\usepackage{fancyhdr}
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\usepackage{graphics}
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\usepackage{multicol}
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\usepackage{epsfig}
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\usepackage{textcomp}
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\usepackage{url}
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% \usepackage{subfigure}
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% \usepackage{subfig}
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% \usepackage{wrapfig}
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\bibliographystyle{plain}
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% \bibliographystyle{sbc}
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% \bibliographystyle{abnt-alf}
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% \bibliographystyle{abnt-num}
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\begin{document}
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\sloppy
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% \renewcommand{\baselinestretch}{1.24}\normalsize % set the space between lines to 1.24
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% set headings
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% \pagestyle{fancy}
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% \lhead[\fancyplain{}{\footnotesize\thepage}]
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% {\fancyplain{}{\footnotesize\rightmark}}
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% \rhead[\fancyplain{}{\footnotesize\leftmark}]
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% {\fancyplain{}{\footnotesize\thepage}}
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%
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% \cfoot{}
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\lstset{
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language=C,
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basicstyle=\fontsize{8}{8}\selectfont,
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captionpos=t,
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aboveskip=0mm,
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belowskip=0mm,
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abovecaptionskip=0.5mm,
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belowcaptionskip=0.5mm,
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% numbers = left,
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mathescape=true,
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escapechar=@,
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extendedchars=true,
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showstringspaces=false,
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% columns=fixed,
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basewidth=0.515em,
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frame=single,
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framesep=1mm,
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xleftmargin=1mm,
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xrightmargin=1mm,
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framerule=0pt
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}
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\include{introduction} % Introducao
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\bibliography{bdz}
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\end{document}
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783
tex/bdz/figs/overviewinternal3g.eps
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783
tex/bdz/figs/overviewinternal3g.eps
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%!PS-Adobe-2.0 EPSF-2.0
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%%Title: overviewinternal3g.fig
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%%Creator: fig2dev Version 3.2 Patchlevel 5
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%%CreationDate: Fri May 29 11:09:04 2009
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%%For: fbotelho@fbotelho-laptop (Fabiano C. Botelho,,,)
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%%BoundingBox: 0 0 342 128
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%Magnification: 1.0000
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%%EndComments
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%%BeginProlog
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/MyAppDict 100 dict dup begin def
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/$F2psDict 200 dict def
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$F2psDict begin
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$F2psDict /mtrx matrix put
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/col-1 {0 setgray} bind def
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/col0 {0.000 0.000 0.000 srgb} bind def
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/col1 {0.000 0.000 1.000 srgb} bind def
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/col2 {0.000 1.000 0.000 srgb} bind def
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/col3 {0.000 1.000 1.000 srgb} bind def
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/col4 {1.000 0.000 0.000 srgb} bind def
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/col18 {0.560 0.000 0.000 srgb} bind def
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/col19 {0.690 0.000 0.000 srgb} bind def
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/col20 {0.820 0.000 0.000 srgb} bind def
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/col21 {0.560 0.000 0.560 srgb} bind def
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/col22 {0.690 0.000 0.690 srgb} bind def
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/col23 {0.820 0.000 0.820 srgb} bind def
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/col24 {0.500 0.190 0.000 srgb} bind def
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/col25 {0.630 0.250 0.000 srgb} bind def
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/col26 {0.750 0.380 0.000 srgb} bind def
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/col27 {1.000 0.500 0.500 srgb} bind def
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/col28 {1.000 0.630 0.630 srgb} bind def
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/col29 {1.000 0.750 0.750 srgb} bind def
|
||||
/col30 {1.000 0.880 0.880 srgb} bind def
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||||
/col31 {1.000 0.840 0.000 srgb} bind def
|
||||
|
||||
end
|
||||
|
||||
% This junk string is used by the show operators
|
||||
/PATsstr 1 string def
|
||||
/PATawidthshow { % cx cy cchar rx ry string
|
||||
% Loop over each character in the string
|
||||
{ % cx cy cchar rx ry char
|
||||
% Show the character
|
||||
dup % cx cy cchar rx ry char char
|
||||
PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char)
|
||||
false charpath % cx cy cchar rx ry char
|
||||
/clip load PATdraw
|
||||
% Move past the character (charpath modified the
|
||||
% current point)
|
||||
currentpoint % cx cy cchar rx ry char x y
|
||||
newpath
|
||||
moveto % cx cy cchar rx ry char
|
||||
% Reposition by cx,cy if the character in the string is cchar
|
||||
3 index eq { % cx cy cchar rx ry
|
||||
4 index 4 index rmoveto
|
||||
} if
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||||
% Reposition all characters by rx ry
|
||||
2 copy rmoveto % cx cy cchar rx ry
|
||||
} forall
|
||||
pop pop pop pop pop % -
|
||||
currentpoint
|
||||
newpath
|
||||
moveto
|
||||
} bind def
|
||||
/PATcg {
|
||||
7 dict dup begin
|
||||
/lw currentlinewidth def
|
||||
/lc currentlinecap def
|
||||
/lj currentlinejoin def
|
||||
/ml currentmiterlimit def
|
||||
/ds [ currentdash ] def
|
||||
/cc [ currentrgbcolor ] def
|
||||
/cm matrix currentmatrix def
|
||||
end
|
||||
} bind def
|
||||
% PATdraw - calculates the boundaries of the object and
|
||||
% fills it with the current pattern
|
||||
/PATdraw { % proc
|
||||
save exch
|
||||
PATpcalc % proc nw nh px py
|
||||
5 -1 roll exec % nw nh px py
|
||||
newpath
|
||||
PATfill % -
|
||||
restore
|
||||
} bind def
|
||||
% PATfill - performs the tiling for the shape
|
||||
/PATfill { % nw nh px py PATfill -
|
||||
PATDict /CurrentPattern get dup begin
|
||||
setfont
|
||||
% Set the coordinate system to Pattern Space
|
||||
PatternGState PATsg
|
||||
% Set the color for uncolored pattezns
|
||||
PaintType 2 eq { PATDict /PColor get PATsc } if
|
||||
% Create the string for showing
|
||||
3 index string % nw nh px py str
|
||||
% Loop for each of the pattern sources
|
||||
0 1 Multi 1 sub { % nw nh px py str source
|
||||
% Move to the starting location
|
||||
3 index 3 index % nw nh px py str source px py
|
||||
moveto % nw nh px py str source
|
||||
% For multiple sources, set the appropriate color
|
||||
Multi 1 ne { dup PC exch get PATsc } if
|
||||
% Set the appropriate string for the source
|
||||
0 1 7 index 1 sub { 2 index exch 2 index put } for pop
|
||||
% Loop over the number of vertical cells
|
||||
3 index % nw nh px py str nh
|
||||
{ % nw nh px py str
|
||||
currentpoint % nw nh px py str cx cy
|
||||
2 index oldshow % nw nh px py str cx cy
|
||||
YStep add moveto % nw nh px py str
|
||||
} repeat % nw nh px py str
|
||||
} for
|
||||
5 { pop } repeat
|
||||
end
|
||||
} bind def
|
||||
|
||||
% PATkshow - kshow with the current pattezn
|
||||
/PATkshow { % proc string
|
||||
exch bind % string proc
|
||||
1 index 0 get % string proc char
|
||||
% Loop over all but the last character in the string
|
||||
0 1 4 index length 2 sub {
|
||||
% string proc char idx
|
||||
% Find the n+1th character in the string
|
||||
3 index exch 1 add get % string proc char char+1
|
||||
exch 2 copy % strinq proc char+1 char char+1 char
|
||||
% Now show the nth character
|
||||
PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr)
|
||||
false charpath % string proc char+1 char char+1
|
||||
/clip load PATdraw
|
||||
% Move past the character (charpath modified the current point)
|
||||
currentpoint newpath moveto
|
||||
% Execute the user proc (should consume char and char+1)
|
||||
mark 3 1 roll % string proc char+1 mark char char+1
|
||||
4 index exec % string proc char+1 mark...
|
||||
cleartomark % string proc char+1
|
||||
} for
|
||||
% Now display the last character
|
||||
PATsstr dup 0 4 -1 roll put % string proc (char+1)
|
||||
false charpath % string proc
|
||||
/clip load PATdraw
|
||||
neewath
|
||||
pop pop % -
|
||||
} bind def
|
||||
% PATmp - the makepattern equivalent
|
||||
/PATmp { % patdict patmtx PATmp patinstance
|
||||
exch dup length 7 add % We will add 6 new entries plus 1 FID
|
||||
dict copy % Create a new dictionary
|
||||
begin
|
||||
% Matrix to install when painting the pattern
|
||||
TilingType PATtcalc
|
||||
/PatternGState PATcg def
|
||||
PatternGState /cm 3 -1 roll put
|
||||
% Check for multi pattern sources (Level 1 fast color patterns)
|
||||
currentdict /Multi known not { /Multi 1 def } if
|
||||
% Font dictionary definitions
|
||||
/FontType 3 def
|
||||
% Create a dummy encoding vector
|
||||
/Encoding 256 array def
|
||||
3 string 0 1 255 {
|
||||
Encoding exch dup 3 index cvs cvn put } for pop
|
||||
/FontMatrix matrix def
|
||||
/FontBBox BBox def
|
||||
/BuildChar {
|
||||
mark 3 1 roll % mark dict char
|
||||
exch begin
|
||||
Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata]
|
||||
PaintType 2 eq Multi 1 ne or
|
||||
{ XStep 0 FontBBox aload pop setcachedevice }
|
||||
{ XStep 0 setcharwidth } ifelse
|
||||
currentdict % mark [paintdata] dict
|
||||
/PaintProc load % mark [paintdata] dict paintproc
|
||||
end
|
||||
gsave
|
||||
false PATredef exec true PATredef
|
||||
grestore
|
||||
cleartomark % -
|
||||
} bind def
|
||||
currentdict
|
||||
end % newdict
|
||||
/foo exch % /foo newlict
|
||||
definefont % newfont
|
||||
} bind def
|
||||
% PATpcalc - calculates the starting point and width/height
|
||||
% of the tile fill for the shape
|
||||
/PATpcalc { % - PATpcalc nw nh px py
|
||||
PATDict /CurrentPattern get begin
|
||||
gsave
|
||||
% Set up the coordinate system to Pattern Space
|
||||
% and lock down pattern
|
||||
PatternGState /cm get setmatrix
|
||||
BBox aload pop pop pop translate
|
||||
% Determine the bounding box of the shape
|
||||
pathbbox % llx lly urx ury
|
||||
grestore
|
||||
% Determine (nw, nh) the # of cells to paint width and height
|
||||
PatHeight div ceiling % llx lly urx qh
|
||||
4 1 roll % qh llx lly urx
|
||||
PatWidth div ceiling % qh llx lly qw
|
||||
4 1 roll % qw qh llx lly
|
||||
PatHeight div floor % qw qh llx ph
|
||||
4 1 roll % ph qw qh llx
|
||||
PatWidth div floor % ph qw qh pw
|
||||
4 1 roll % pw ph qw qh
|
||||
2 index sub cvi abs % pw ph qs qh-ph
|
||||
exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph
|
||||
% Determine the starting point of the pattern fill
|
||||
%(px, py)
|
||||
4 2 roll % nw nh pw ph
|
||||
PatHeight mul % nw nh pw py
|
||||
exch % nw nh py pw
|
||||
PatWidth mul exch % nw nh px py
|
||||
end
|
||||
} bind def
|
||||
|
||||
% Save the original routines so that we can use them later on
|
||||
/oldfill /fill load def
|
||||
/oldeofill /eofill load def
|
||||
/oldstroke /stroke load def
|
||||
/oldshow /show load def
|
||||
/oldashow /ashow load def
|
||||
/oldwidthshow /widthshow load def
|
||||
/oldawidthshow /awidthshow load def
|
||||
/oldkshow /kshow load def
|
||||
|
||||
% These defs are necessary so that subsequent procs don't bind in
|
||||
% the originals
|
||||
/fill { oldfill } bind def
|
||||
/eofill { oldeofill } bind def
|
||||
/stroke { oldstroke } bind def
|
||||
/show { oldshow } bind def
|
||||
/ashow { oldashow } bind def
|
||||
/widthshow { oldwidthshow } bind def
|
||||
/awidthshow { oldawidthshow } bind def
|
||||
/kshow { oldkshow } bind def
|
||||
/PATredef {
|
||||
MyAppDict begin
|
||||
{
|
||||
/fill { /clip load PATdraw newpath } bind def
|
||||
/eofill { /eoclip load PATdraw newpath } bind def
|
||||
/stroke { PATstroke } bind def
|
||||
/show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def
|
||||
/ashow { 0 0 null 6 3 roll PATawidthshow }
|
||||
bind def
|
||||
/widthshow { 0 0 3 -1 roll PATawidthshow }
|
||||
bind def
|
||||
/awidthshow { PATawidthshow } bind def
|
||||
/kshow { PATkshow } bind def
|
||||
} {
|
||||
/fill { oldfill } bind def
|
||||
/eofill { oldeofill } bind def
|
||||
/stroke { oldstroke } bind def
|
||||
/show { oldshow } bind def
|
||||
/ashow { oldashow } bind def
|
||||
/widthshow { oldwidthshow } bind def
|
||||
/awidthshow { oldawidthshow } bind def
|
||||
/kshow { oldkshow } bind def
|
||||
} ifelse
|
||||
end
|
||||
} bind def
|
||||
false PATredef
|
||||
% Conditionally define setcmykcolor if not available
|
||||
/setcmykcolor where { pop } {
|
||||
/setcmykcolor {
|
||||
1 sub 4 1 roll
|
||||
3 {
|
||||
3 index add neg dup 0 lt { pop 0 } if 3 1 roll
|
||||
} repeat
|
||||
setrgbcolor - pop
|
||||
} bind def
|
||||
} ifelse
|
||||
/PATsc { % colorarray
|
||||
aload length % c1 ... cn length
|
||||
dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor
|
||||
} ifelse } ifelse
|
||||
} bind def
|
||||
/PATsg { % dict
|
||||
begin
|
||||
lw setlinewidth
|
||||
lc setlinecap
|
||||
lj setlinejoin
|
||||
ml setmiterlimit
|
||||
ds aload pop setdash
|
||||
cc aload pop setrgbcolor
|
||||
cm setmatrix
|
||||
end
|
||||
} bind def
|
||||
|
||||
/PATDict 3 dict def
|
||||
/PATsp {
|
||||
true PATredef
|
||||
PATDict begin
|
||||
/CurrentPattern exch def
|
||||
% If it's an uncolored pattern, save the color
|
||||
CurrentPattern /PaintType get 2 eq {
|
||||
/PColor exch def
|
||||
} if
|
||||
/CColor [ currentrgbcolor ] def
|
||||
end
|
||||
} bind def
|
||||
% PATstroke - stroke with the current pattern
|
||||
/PATstroke {
|
||||
countdictstack
|
||||
save
|
||||
mark
|
||||
{
|
||||
currentpoint strokepath moveto
|
||||
PATpcalc % proc nw nh px py
|
||||
clip newpath PATfill
|
||||
} stopped {
|
||||
(*** PATstroke Warning: Path is too complex, stroking
|
||||
with gray) =
|
||||
cleartomark
|
||||
restore
|
||||
countdictstack exch sub dup 0 gt
|
||||
{ { end } repeat } { pop } ifelse
|
||||
gsave 0.5 setgray oldstroke grestore
|
||||
} { pop restore pop } ifelse
|
||||
newpath
|
||||
} bind def
|
||||
/PATtcalc { % modmtx tilingtype PATtcalc tilematrix
|
||||
% Note: tiling types 2 and 3 are not supported
|
||||
gsave
|
||||
exch concat % tilingtype
|
||||
matrix currentmatrix exch % cmtx tilingtype
|
||||
% Tiling type 1 and 3: constant spacing
|
||||
2 ne {
|
||||
% Distort the pattern so that it occupies
|
||||
% an integral number of device pixels
|
||||
dup 4 get exch dup 5 get exch % tx ty cmtx
|
||||
XStep 0 dtransform
|
||||
round exch round exch % tx ty cmtx dx.x dx.y
|
||||
XStep div exch XStep div exch % tx ty cmtx a b
|
||||
0 YStep dtransform
|
||||
round exch round exch % tx ty cmtx a b dy.x dy.y
|
||||
YStep div exch YStep div exch % tx ty cmtx a b c d
|
||||
7 -3 roll astore % { a b c d tx ty }
|
||||
} if
|
||||
grestore
|
||||
} bind def
|
||||
/PATusp {
|
||||
false PATredef
|
||||
PATDict begin
|
||||
CColor PATsc
|
||||
end
|
||||
} bind def
|
||||
|
||||
% crosshatch30
|
||||
11 dict begin
|
||||
/PaintType 1 def
|
||||
/PatternType 1 def
|
||||
/TilingType 1 def
|
||||
/BBox [0 0 1 1] def
|
||||
/XStep 1 def
|
||||
/YStep 1 def
|
||||
/PatWidth 1 def
|
||||
/PatHeight 1 def
|
||||
/Multi 2 def
|
||||
/PaintData [
|
||||
{ clippath } bind
|
||||
{ 32 16 true [ 32 0 0 -16 0 16 ]
|
||||
{<033003300c0c0c0c30033003c000c000300330030c0c0c0c
|
||||
0330033000c000c0033003300c0c0c0c30033003c000c000
|
||||
300330030c0c0c0c0330033000c000c0>}
|
||||
imagemask } bind
|
||||
] def
|
||||
/PaintProc {
|
||||
pop
|
||||
exec fill
|
||||
} def
|
||||
currentdict
|
||||
end
|
||||
/P3 exch def
|
||||
|
||||
/cp {closepath} bind def
|
||||
/ef {eofill} bind def
|
||||
/gr {grestore} bind def
|
||||
/gs {gsave} bind def
|
||||
/sa {save} bind def
|
||||
/rs {restore} bind def
|
||||
/l {lineto} bind def
|
||||
/m {moveto} bind def
|
||||
/rm {rmoveto} bind def
|
||||
/n {newpath} bind def
|
||||
/s {stroke} bind def
|
||||
/sh {show} bind def
|
||||
/slc {setlinecap} bind def
|
||||
/slj {setlinejoin} bind def
|
||||
/slw {setlinewidth} bind def
|
||||
/srgb {setrgbcolor} bind def
|
||||
/rot {rotate} bind def
|
||||
/sc {scale} bind def
|
||||
/sd {setdash} bind def
|
||||
/ff {findfont} bind def
|
||||
/sf {setfont} bind def
|
||||
/scf {scalefont} bind def
|
||||
/sw {stringwidth} bind def
|
||||
/tr {translate} bind def
|
||||
/tnt {dup dup currentrgbcolor
|
||||
4 -2 roll dup 1 exch sub 3 -1 roll mul add
|
||||
4 -2 roll dup 1 exch sub 3 -1 roll mul add
|
||||
4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
|
||||
bind def
|
||||
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
|
||||
4 -2 roll mul srgb} bind def
|
||||
/DrawEllipse {
|
||||
/endangle exch def
|
||||
/startangle exch def
|
||||
/yrad exch def
|
||||
/xrad exch def
|
||||
/y exch def
|
||||
/x exch def
|
||||
/savematrix mtrx currentmatrix def
|
||||
x y tr xrad yrad sc 0 0 1 startangle endangle arc
|
||||
closepath
|
||||
savematrix setmatrix
|
||||
} def
|
||||
|
||||
/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
|
||||
/$F2psEnd {$F2psEnteredState restore end} def
|
||||
|
||||
/pageheader {
|
||||
save
|
||||
newpath 0 128 moveto 0 0 lineto 342 0 lineto 342 128 lineto closepath clip newpath
|
||||
-40.3 230.6 translate
|
||||
1 -1 scale
|
||||
$F2psBegin
|
||||
10 setmiterlimit
|
||||
0 slj 0 slc
|
||||
0.06299 0.06299 sc
|
||||
} bind def
|
||||
/pagefooter {
|
||||
$F2psEnd
|
||||
restore
|
||||
} bind def
|
||||
%%EndProlog
|
||||
pageheader
|
||||
%
|
||||
% Fig objects follow
|
||||
%
|
||||
%
|
||||
% here starts figure with depth 53
|
||||
% Polyline
|
||||
0 slj
|
||||
0 slc
|
||||
7.500 slw
|
||||
n 757 1980 m 652 1980 652 2640 105 arcto 4 {pop} repeat
|
||||
652 2745 1155 2745 105 arcto 4 {pop} repeat
|
||||
1260 2745 1260 2085 105 arcto 4 {pop} repeat
|
||||
1260 1980 757 1980 105 arcto 4 {pop} repeat
|
||||
cp gs col0 s gr
|
||||
% here ends figure;
|
||||
%
|
||||
% here starts figure with depth 51
|
||||
% Polyline
|
||||
0 slj
|
||||
0 slc
|
||||
7.500 slw
|
||||
gs clippath
|
||||
5215 2261 m 5264 2278 l 5278 2235 l 5229 2219 l 5229 2219 l 5251 2250 l 5215 2261 l cp
|
||||
eoclip
|
||||
n 4399 1969 m
|
||||
5257 2252 l gs col7 1.00 shd ef gr gs col0 s gr gr
|
||||
|
||||
% arrowhead
|
||||
n 5215 2261 m 5251 2250 l 5229 2219 l 5215 2261 l cp gs 0.00 setgray ef gr col0 s
|
||||
% Polyline
|
||||
gs clippath
|
||||
5223 2432 m 5272 2449 l 5286 2406 l 5237 2390 l 5237 2390 l 5259 2421 l 5223 2432 l cp
|
||||
eoclip
|
||||
n 4407 2140 m
|
||||
5265 2423 l gs col7 1.00 shd ef gr gs col0 s gr gr
|
||||
|
||||
% arrowhead
|
||||
n 5223 2432 m 5259 2421 l 5237 2390 l 5223 2432 l cp gs 0.00 setgray ef gr col0 s
|
||||
% Polyline
|
||||
gs clippath
|
||||
5216 2650 m 5267 2647 l 5264 2602 l 5213 2605 l 5213 2605 l 5245 2626 l 5216 2650 l cp
|
||||
eoclip
|
||||
n 4398 2687 m
|
||||
5251 2626 l gs col7 1.00 shd ef gr gs col0 s gr gr
|
||||
|
||||
% arrowhead
|
||||
n 5216 2650 m 5245 2626 l 5213 2605 l 5216 2650 l cp gs 0.00 setgray ef gr col0 s
|
||||
% Polyline
|
||||
n 5362 2523 m 5752 2523 l 5752 2696 l 5362 2696 l
|
||||
cp gs col7 1.00 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
n 5362 2165 m 5752 2165 l 5752 2338 l 5362 2338 l
|
||||
cp gs col7 1.00 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
0.000 slw
|
||||
n 720 2070 m 900 2070 l 900 2160 l 720 2160 l
|
||||
cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
|
||||
15.00 15.00 sc P3 [16 0 0 -8 48.00 138.00] PATmp PATsp ef gr PATusp
|
||||
% Polyline
|
||||
n 720 2565 m 900 2565 l 900 2655 l 720 2655 l
|
||||
cp gs col7 0.00 shd ef gr
|
||||
% Polyline
|
||||
7.500 slw
|
||||
n 4245 2415 m 4425 2415 l 4425 2595 l 4245 2595 l
|
||||
cp gs col7 1.00 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
n 4245 2235 m 4425 2235 l 4425 2415 l 4245 2415 l
|
||||
cp gs col7 1.00 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
n 5362 2343 m 5752 2343 l 5752 2516 l 5362 2516 l
|
||||
cp gs col7 1.00 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
n 2835 3150 m 3330 3150 l 3330 3465 l 2835 3465 l
|
||||
cp gs col7 1.00 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
0.000 slw
|
||||
n 2880 3330 m 3015 3330 l 3015 3420 l 2880 3420 l
|
||||
cp gs col7 0.00 shd ef gr
|
||||
% Polyline
|
||||
7.500 slw
|
||||
n 2340 3150 m 2835 3150 l 2835 3465 l 2340 3465 l
|
||||
cp gs col7 1.00 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
n 1845 3150 m 2340 3150 l 2340 3465 l 1845 3465 l
|
||||
cp gs col7 1.00 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
0.000 slw
|
||||
n 2385 3330 m 2520 3330 l 2520 3420 l 2385 3420 l
|
||||
cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
|
||||
15.00 15.00 sc P3 [16 0 0 -8 159.00 222.00] PATmp PATsp ef gr PATusp
|
||||
% Polyline
|
||||
n 2602 3017 m 2605 2425 l 2792 2423 l 2788 3044 l 2588 3030 l
|
||||
cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
|
||||
15.00 15.00 sc P3 [16 0 0 -8 172.53 161.53] PATmp PATsp ef gr PATusp
|
||||
% Polyline
|
||||
n 2609 2477 m 2612 1885 l 2799 1883 l 2795 2504 l 2595 2490 l
|
||||
cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def
|
||||
15.00 15.00 sc P3 [16 0 0 -8 173.00 125.53] PATmp PATsp ef gr PATusp
|
||||
% Polyline
|
||||
7.500 slw
|
||||
n 4245 1890 m 4425 1890 l 4425 2070 l 4245 2070 l
|
||||
cp gs col7 0.85 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
n 4245 2063 m 4425 2063 l 4425 2243 l 4245 2243 l
|
||||
cp gs col7 0.85 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
n 4245 2595 m 4425 2595 l 4425 2775 l 4245 2775 l
|
||||
cp gs col7 0.85 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
n 4247 2748 m 4427 2748 l 4427 2928 l 4247 2928 l
|
||||
cp gs col7 1.00 shd ef gr gs col0 s gr
|
||||
% Polyline
|
||||
0.000 slw
|
||||
n 2657 3060 m 2111 2491 l 2244 2360 l 2786 2937 l
|
||||
cp gs col7 0.00 shd ef gr
|
||||
% Polyline
|
||||
n 2111 2402 m 2660 1838 l 2797 1966 l 2242 2527 l
|
||||
cp gs col7 0.55 shd ef gr
|
||||
% Polyline
|
||||
n 2115 3017 m 2118 2425 l 2305 2423 l 2301 3044 l 2101 3030 l
|
||||
cp gs col7 0.55 shd ef gr
|
||||
% Polyline
|
||||
n 1890 3330 m 2025 3330 l 2025 3420 l 1890 3420 l
|
||||
cp gs col7 0.55 shd ef gr
|
||||
% Polyline
|
||||
n 720 2340 m 900 2340 l 900 2430 l 720 2430 l
|
||||
cp gs col7 0.55 shd ef gr
|
||||
% Polyline
|
||||
n 2113 2439 m 2116 1847 l 2303 1845 l 2299 2466 l 2099 2452 l
|
||||
cp gs col7 0.00 shd ef gr
|
||||
/Times-Italic ff 142.88 scf sf
|
||||
2835 2474 m
|
||||
gs 1 -1 sc (h \(x\)) col0 sh gr
|
||||
/Times-Roman ff 111.13 scf sf
|
||||
2916 2520 m
|
||||
gs 1 -1 sc (1) col0 sh gr
|
||||
/Times-Italic ff 142.88 scf sf
|
||||
2835 3030 m
|
||||
gs 1 -1 sc (h \(x\)) col0 sh gr
|
||||
/Times-Roman ff 111.13 scf sf
|
||||
2916 3076 m
|
||||
gs 1 -1 sc (2) col0 sh gr
|
||||
/Times-Italic ff 142.88 scf sf
|
||||
2835 1950 m
|
||||
gs 1 -1 sc (h \(x\)) col0 sh gr
|
||||
/Times-Roman ff 111.13 scf sf
|
||||
2916 1996 m
|
||||
gs 1 -1 sc (0) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4095 2025 m
|
||||
gs 1 -1 sc (0) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4095 2205 m
|
||||
gs 1 -1 sc (1) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4095 2385 m
|
||||
gs 1 -1 sc (2) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4095 2565 m
|
||||
gs 1 -1 sc (3) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4095 2745 m
|
||||
gs 1 -1 sc (4) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4095 2925 m
|
||||
gs 1 -1 sc (5) col0 sh gr
|
||||
/Times-Italic ff 142.88 scf sf
|
||||
4320 1800 m
|
||||
gs 1 -1 sc (g) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
5220 2115 m
|
||||
gs 1 -1 sc (Hash Table ) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
5265 2475 m
|
||||
gs 1 -1 sc (1) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
5265 2655 m
|
||||
gs 1 -1 sc (2) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
5265 2295 m
|
||||
gs 1 -1 sc (0) col0 sh gr
|
||||
/Times-Roman ff 158.75 scf sf
|
||||
1575 1755 m
|
||||
gs 1 -1 sc (\(a\)) col0 sh gr
|
||||
/Times-Roman ff 158.75 scf sf
|
||||
3465 1755 m
|
||||
gs 1 -1 sc (\(b\)) col0 sh gr
|
||||
/Times-Roman ff 158.75 scf sf
|
||||
4680 1755 m
|
||||
gs 1 -1 sc (\(c\)) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
3015 3645 m
|
||||
gs 1 -1 sc (2) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2565 3645 m
|
||||
gs 1 -1 sc (1) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2070 3645 m
|
||||
gs 1 -1 sc (0) col0 sh gr
|
||||
/ZapfChancery-MediumItalic ff 174.63 scf sf
|
||||
3420 3375 m
|
||||
gs 1 -1 sc (L) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2865 3277 m
|
||||
gs 1 -1 sc ({0,2,5}) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2370 3277 m
|
||||
gs 1 -1 sc ({1,3,5}) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
1895 3277 m
|
||||
gs 1 -1 sc ({1,2,4}) col0 sh gr
|
||||
% here ends figure;
|
||||
%
|
||||
% here starts figure with depth 45
|
||||
% Polyline
|
||||
0 slj
|
||||
0 slc
|
||||
7.500 slw
|
||||
gs clippath
|
||||
1944 2497 m 1995 2497 l 1995 2452 l 1944 2452 l 1944 2452 l 1974 2475 l 1944 2497 l cp
|
||||
eoclip
|
||||
n 1357 2475 m
|
||||
1980 2475 l gs col7 1.00 shd ef gr gs col0 s gr gr
|
||||
|
||||
% arrowhead
|
||||
n 1944 2497 m 1974 2475 l 1944 2452 l 1944 2497 l cp gs 0.00 setgray ef gr col0 s
|
||||
% Polyline
|
||||
gs clippath
|
||||
3879 2497 m 3930 2497 l 3930 2452 l 3879 2452 l 3879 2452 l 3909 2475 l 3879 2497 l cp
|
||||
eoclip
|
||||
n 3292 2475 m
|
||||
3915 2475 l gs col7 1.00 shd ef gr gs col0 s gr gr
|
||||
|
||||
% arrowhead
|
||||
n 3879 2497 m 3909 2475 l 3879 2452 l 3879 2497 l cp gs 0.00 setgray ef gr col0 s
|
||||
% Ellipse
|
||||
n 2704 2448 101 101 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr
|
||||
|
||||
% Ellipse
|
||||
n 2209 2448 101 101 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr
|
||||
|
||||
% Ellipse
|
||||
n 2704 2988 101 101 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr
|
||||
|
||||
% Ellipse
|
||||
n 2209 1908 101 101 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr
|
||||
|
||||
% Ellipse
|
||||
n 2704 1908 101 101 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr
|
||||
|
||||
% Ellipse
|
||||
n 2209 2988 101 101 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr
|
||||
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
5423 2663 m
|
||||
gs 1 -1 sc (band) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
5460 2304 m
|
||||
gs 1 -1 sc (the) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
1418 2430 m
|
||||
gs 1 -1 sc (Mapping) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
3285 2430 m
|
||||
gs 1 -1 sc (Assigning) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2674 2485 m
|
||||
gs 1 -1 sc (3) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2179 2485 m
|
||||
gs 1 -1 sc (2) col0 sh gr
|
||||
/Times-Italic ff 142.88 scf sf
|
||||
945 1935 m
|
||||
gs 1 -1 sc (S) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
967 2160 m
|
||||
gs 1 -1 sc (who) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
960 2430 m
|
||||
gs 1 -1 sc (band) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
1005 2655 m
|
||||
gs 1 -1 sc (the) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4305 2378 m
|
||||
gs 1 -1 sc (3) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
5422 2482 m
|
||||
gs 1 -1 sc (who) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4545 2430 m
|
||||
gs 1 -1 sc (Ranking) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
3060 3420 m
|
||||
gs 1 -1 sc (the) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2539 3420 m
|
||||
gs 1 -1 sc (who) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2045 3420 m
|
||||
gs 1 -1 sc (band) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2179 1945 m
|
||||
gs 1 -1 sc (0) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2674 1945 m
|
||||
gs 1 -1 sc (1) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2179 3025 m
|
||||
gs 1 -1 sc (4) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
2674 3025 m
|
||||
gs 1 -1 sc (5) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4300 2875 m
|
||||
gs 1 -1 sc (3) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4305 2548 m
|
||||
gs 1 -1 sc (3) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4305 2715 m
|
||||
gs 1 -1 sc (2) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4299 2190 m
|
||||
gs 1 -1 sc (0) col0 sh gr
|
||||
/Times-Roman ff 142.88 scf sf
|
||||
4299 2033 m
|
||||
gs 1 -1 sc (0) col0 sh gr
|
||||
% here ends figure;
|
||||
pagefooter
|
||||
showpage
|
||||
%%Trailer
|
||||
end
|
||||
%EOF
|
||||
156
tex/bdz/figs/overviewinternal3g.fig
Normal file
156
tex/bdz/figs/overviewinternal3g.fig
Normal file
@@ -0,0 +1,156 @@
|
||||
#FIG 3.2 Produced by xfig version 3.2.5
|
||||
Landscape
|
||||
Center
|
||||
Metric
|
||||
A4
|
||||
100.00
|
||||
Single
|
||||
-2
|
||||
1200 2
|
||||
6 5355 2520 5760 2700
|
||||
6 5400 2520 5715 2700
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 285 5423 2663 band\001
|
||||
-6
|
||||
2 2 0 1 0 7 50 -1 20 0.000 0 0 7 0 0 5
|
||||
5362 2523 5752 2523 5752 2696 5362 2696 5362 2523
|
||||
-6
|
||||
6 5355 2162 5760 2342
|
||||
2 2 0 1 0 7 50 -1 20 0.000 0 0 7 0 0 5
|
||||
5362 2165 5752 2165 5752 2338 5362 2338 5362 2165
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 195 5460 2304 the\001
|
||||
-6
|
||||
6 1350 2340 1980 2520
|
||||
6 1350 2340 1980 2520
|
||||
2 1 0 1 0 7 45 -1 20 0.000 0 0 7 1 0 2
|
||||
1 1 1.00 45.00 30.00
|
||||
1357 2475 1980 2475
|
||||
4 0 0 45 -1 0 9 0.0000 4 135 555 1418 2430 Mapping\001
|
||||
-6
|
||||
-6
|
||||
6 3285 2340 3915 2520
|
||||
6 3285 2340 3915 2520
|
||||
2 1 0 1 0 7 45 -1 20 0.000 0 0 7 1 0 2
|
||||
1 1 1.00 45.00 30.00
|
||||
3292 2475 3915 2475
|
||||
4 0 0 45 -1 0 9 0.0000 4 135 630 3285 2430 Assigning\001
|
||||
-6
|
||||
-6
|
||||
6 2603 2347 2805 2549
|
||||
1 3 0 1 0 7 45 -1 20 0.000 1 0.0000 2704 2448 101 101 2704 2448 2749 2538
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 2674 2485 3\001
|
||||
-6
|
||||
6 2108 2347 2310 2549
|
||||
1 3 0 1 0 7 45 -1 20 0.000 1 0.0000 2209 2448 101 101 2209 2448 2254 2538
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 2179 2485 2\001
|
||||
-6
|
||||
6 2835 2340 3150 2520
|
||||
4 0 0 50 -1 1 9 0.0000 4 135 300 2835 2474 h (x)\001
|
||||
4 0 0 50 -1 0 7 0.0000 4 75 60 2916 2520 1\001
|
||||
-6
|
||||
6 2835 2925 3150 3105
|
||||
4 0 0 50 -1 1 9 0.0000 4 135 300 2835 3030 h (x)\001
|
||||
4 0 0 50 -1 0 7 0.0000 4 75 60 2916 3076 2\001
|
||||
-6
|
||||
6 2835 1845 3135 1996
|
||||
4 0 0 50 -1 1 9 0.0000 4 135 300 2835 1950 h (x)\001
|
||||
4 0 0 50 -1 0 7 0.0000 4 75 60 2916 1996 0\001
|
||||
-6
|
||||
1 3 0 1 0 7 45 -1 20 0.000 1 0.0000 2704 2988 101 101 2704 2988 2749 3078
|
||||
1 3 0 1 0 7 45 -1 20 0.000 1 0.0000 2209 1908 101 101 2209 1908 2254 1998
|
||||
1 3 0 1 0 7 45 -1 20 0.000 1 0.0000 2704 1908 101 101 2704 1908 2749 1998
|
||||
1 3 0 1 0 7 45 -1 20 0.000 1 0.0000 2209 2988 101 101 2209 2988 2254 3078
|
||||
2 4 0 1 0 7 53 -1 -1 0.000 0 0 7 0 0 5
|
||||
1260 2745 1260 1980 652 1980 652 2745 1260 2745
|
||||
2 2 0 0 0 7 50 -1 43 0.000 0 0 -1 0 0 5
|
||||
720 2070 900 2070 900 2160 720 2160 720 2070
|
||||
2 2 0 0 0 7 50 -1 0 0.000 0 0 7 0 0 5
|
||||
720 2565 900 2565 900 2655 720 2655 720 2565
|
||||
2 2 0 1 0 7 50 -1 20 0.000 0 0 7 0 0 5
|
||||
4245 2415 4425 2415 4425 2595 4245 2595 4245 2415
|
||||
2 2 0 1 0 7 50 -1 20 0.000 0 0 7 0 0 5
|
||||
4245 2235 4425 2235 4425 2415 4245 2415 4245 2235
|
||||
2 1 0 1 0 7 51 -1 20 0.000 0 0 7 1 0 2
|
||||
1 1 1.00 45.00 30.00
|
||||
4399 1969 5257 2252
|
||||
2 2 0 1 0 7 50 -1 20 0.000 0 0 7 0 0 5
|
||||
5362 2343 5752 2343 5752 2516 5362 2516 5362 2343
|
||||
2 1 0 1 0 7 51 -1 20 0.000 0 0 7 1 0 2
|
||||
1 1 1.00 45.00 30.00
|
||||
4407 2140 5265 2423
|
||||
2 1 0 1 0 7 51 -1 20 0.000 0 0 7 1 0 2
|
||||
1 1 1.00 45.00 30.00
|
||||
4398 2687 5251 2626
|
||||
2 2 0 1 0 7 50 -1 20 0.000 0 0 7 0 0 5
|
||||
2835 3150 3330 3150 3330 3465 2835 3465 2835 3150
|
||||
2 2 0 0 0 7 50 -1 0 0.000 0 0 7 0 0 5
|
||||
2880 3330 3015 3330 3015 3420 2880 3420 2880 3330
|
||||
2 2 0 1 0 7 50 -1 20 0.000 0 0 7 0 0 5
|
||||
2340 3150 2835 3150 2835 3465 2340 3465 2340 3150
|
||||
2 2 0 1 0 7 50 -1 20 0.000 0 0 7 0 0 5
|
||||
1845 3150 2340 3150 2340 3465 1845 3465 1845 3150
|
||||
2 2 0 0 0 7 50 -1 43 0.000 0 0 7 0 0 5
|
||||
2385 3330 2520 3330 2520 3420 2385 3420 2385 3330
|
||||
2 3 0 0 0 7 50 -1 43 0.000 0 0 7 0 0 6
|
||||
2602 3017 2605 2425 2792 2423 2788 3044 2588 3030 2602 3017
|
||||
2 3 0 0 0 7 50 -1 43 0.000 0 0 7 0 0 6
|
||||
2609 2477 2612 1885 2799 1883 2795 2504 2595 2490 2609 2477
|
||||
2 2 0 1 0 7 50 -1 17 0.000 0 0 7 0 0 5
|
||||
4245 1890 4425 1890 4425 2070 4245 2070 4245 1890
|
||||
2 2 0 1 0 7 50 -1 17 0.000 0 0 7 0 0 5
|
||||
4245 2063 4425 2063 4425 2243 4245 2243 4245 2063
|
||||
2 2 0 1 0 7 50 -1 17 0.000 0 0 7 0 0 5
|
||||
4245 2595 4425 2595 4425 2775 4245 2775 4245 2595
|
||||
2 2 0 1 0 7 50 -1 20 0.000 0 0 7 0 0 5
|
||||
4247 2748 4427 2748 4427 2928 4247 2928 4247 2748
|
||||
2 3 0 0 0 7 50 -1 0 0.000 0 0 7 0 0 5
|
||||
2657 3060 2111 2491 2244 2360 2786 2937 2657 3060
|
||||
2 3 0 0 0 7 50 -1 11 0.000 0 0 7 0 0 5
|
||||
2111 2402 2660 1838 2797 1966 2242 2527 2111 2402
|
||||
2 3 0 0 0 7 50 -1 11 0.000 0 0 7 0 0 6
|
||||
2115 3017 2118 2425 2305 2423 2301 3044 2101 3030 2115 3017
|
||||
2 2 0 0 0 7 50 -1 11 0.000 0 0 7 0 0 5
|
||||
1890 3330 2025 3330 2025 3420 1890 3420 1890 3330
|
||||
2 2 0 0 0 7 50 -1 11 0.000 0 0 7 0 0 5
|
||||
720 2340 900 2340 900 2430 720 2430 720 2340
|
||||
2 3 0 0 0 7 50 -1 0 0.000 0 0 7 0 0 6
|
||||
2113 2439 2116 1847 2303 1845 2299 2466 2099 2452 2113 2439
|
||||
4 0 0 45 -1 1 9 0.0000 4 105 75 945 1935 S\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 270 967 2160 who\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 285 960 2430 band\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 195 1005 2655 the\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 4095 2025 0\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 4095 2205 1\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 4095 2385 2\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 4095 2565 3\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 4095 2745 4\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 4095 2925 5\001
|
||||
4 0 0 50 -1 1 9 0.0000 4 120 60 4320 1800 g\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 4305 2378 3\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 810 5220 2115 Hash Table \001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 270 5422 2482 who\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 5265 2475 1\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 5265 2655 2\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 5265 2295 0\001
|
||||
4 0 0 50 -1 0 10 0.0000 4 135 180 1575 1755 (a)\001
|
||||
4 0 0 50 -1 0 10 0.0000 4 135 195 3465 1755 (b)\001
|
||||
4 0 0 50 -1 0 10 0.0000 4 135 180 4680 1755 (c)\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 135 510 4545 2430 Ranking\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 3015 3645 2\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 2565 3645 1\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 105 75 2070 3645 0\001
|
||||
4 0 0 50 -1 33 11 0.0000 4 135 90 3420 3375 L\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 135 435 2865 3277 {0,2,5}\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 195 3060 3420 the\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 135 435 2370 3277 {1,3,5}\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 270 2539 3420 who\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 285 2045 3420 band\001
|
||||
4 0 0 50 -1 0 9 0.0000 4 135 435 1895 3277 {1,2,4}\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 2179 1945 0\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 2674 1945 1\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 2179 3025 4\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 2674 3025 5\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 4300 2875 3\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 4305 2548 3\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 4305 2715 2\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 4299 2190 0\001
|
||||
4 0 0 45 -1 0 9 0.0000 4 105 75 4299 2033 0\001
|
||||
371
tex/bdz/introduction.tex
Executable file
371
tex/bdz/introduction.tex
Executable file
@@ -0,0 +1,371 @@
|
||||
\section{Introduction} \label{sec:introduction}
|
||||
|
||||
The BDZ algorithm was designed by Fabiano C. Botelho, Djamal Belazzougui, Rasmus Pagh and Nivio Ziviani.
|
||||
It is a simple, efficient, near-optimal space and practical
|
||||
algorithm to generate a family $\cal F$ of PHFs and MPHFs.
|
||||
It is also referred to as BPZ algorithm because the work presented
|
||||
by Botelho, Pagh and Ziviani in \cite{bpz07}.
|
||||
In the Botelho's PhD. dissertation \cite{b08} it is also referred to as RAM algorithm
|
||||
because it is more suitable for key sets that can be handled in internal memory.
|
||||
|
||||
The BDZ algorithm uses $r$-uniform random hypergraphs
|
||||
given by function values of $r$
|
||||
uniform random hash functions on the input key set $S$ for generating PHFs and MPHFs that
|
||||
require $O(n)$ bits to be stored.
|
||||
A hypergraph is the generalization of a standard undirected
|
||||
graph where each edge connects $r\geq 2$ vertices.
|
||||
This idea is not new, see e.g. \cite{mwhc96},
|
||||
but we have proceed differently to achieve
|
||||
a space usage of $O(n)$ bits rather than $O(n\log n)$ bits.
|
||||
Evaluation time for all schemes considered is constant.
|
||||
For $r=3$ we obtain a space usage of approximately $2.6n$ bits for
|
||||
an MPHF. More compact, and even simpler, representations can be
|
||||
achieved for larger $m$. For example, for $m=1.23n$ we can get a
|
||||
space usage of $1.95n$ bits.
|
||||
|
||||
Our best MPHF space upper bound is within a
|
||||
factor of 2 from the information theoretical lower bound of approximately
|
||||
$1.4427n$ bits. We have shown that the BDZ algorithm is far more
|
||||
practical than previous methods with proven space complexity, both
|
||||
because of its simplicity, and because the constant factor of the
|
||||
space complexity is more than 6 times lower than its closest
|
||||
competitor, for plausible problem sizes. We verify the practicality
|
||||
experimentally, using slightly more space than in the mentioned
|
||||
theoretical bounds.
|
||||
|
||||
\section{The Algorithm}
|
||||
|
||||
The BDZ algorithm is a three-step algorithm that generates PHFs and MPHFs based on
|
||||
random $r$-partite hypergraphs.
|
||||
This is an approach that provides a much tighter analysis and is
|
||||
much more simple than the one presented in
|
||||
\cite{ckrt04}, where it was implicit how to construct
|
||||
similar PHFs.
|
||||
The fastest and most compact functions
|
||||
are generated when $r=3$.
|
||||
In this case a PHF can be stored in
|
||||
approximately $1.95$ bits per key and
|
||||
an MPHF in approximately
|
||||
$2.62$ bits per key.
|
||||
|
||||
Figure~\ref{fig:overview} gives an overview of the algorithm for $r=3$,
|
||||
taking as input a key set $S \subseteq U$ containing three English words, i.e., $S=\{\mathrm{who},\mathrm{band},\mathrm{the}\}$.
|
||||
% which are nicely hashed to the name of a rock band ``the who band''.
|
||||
The edge-oriented data structure proposed in~\cite{e87} is used
|
||||
to represent hypergraphs, where each edge is explicitly represented
|
||||
as an array of $r$ vertices and, for each vertex $v$,
|
||||
there is a list of edges that are incident on $v$.
|
||||
|
||||
The {\em Mapping Step} in Figure~\ref{fig:overview}(a) carries out two
|
||||
important tasks:
|
||||
\begin{enumerate}
|
||||
\item
|
||||
It assumes that it is possible to find three uniform
|
||||
hash functions, $h_0$, $h_1$ and $h_2$, with ranges $\{0,1\}$, $\{2,3\}$ and $\{4,5\}$, respectively.
|
||||
These functions build an one-to-one mapping of the key set $S$ to the edge set $E$
|
||||
of a random acyclic
|
||||
$3$-partite hypergraph $G=(V,E)$, where $|V|=m=6$ and $|E|=n=3$.
|
||||
In \cite{b08,bpz07} it is shown that
|
||||
it is possible to obtain such a hypergraph with probability tending to $1$ as $n$
|
||||
tends to infinity
|
||||
whenever $m=cn$ and $c \ge 1.23$. The value of $c$ that minimizes the hypergraph size
|
||||
(and thereby the amount of bits to represent the resulting functions) is $c \approx 1.23$.
|
||||
To illustrate the mapping,
|
||||
key ``who'' is mapped to edge $\{h_0(\text{``who''}),h_1(\text{``who''}),h_2(\text{``who''})\}=\{1,3,5\}$,
|
||||
key ``band'' is mapped to edge $\{h_0(\text{``band''}),h_1(\text{``band''}),h_2(\text{``band''})\}=\{1,2,4\}$, and
|
||||
key ``the'' is mapped to edge $\{h_0(\text{``the''}),h_1(\text{``the''}),h_2(\text{``the''})\}=\{0,2,5\}$.
|
||||
\item
|
||||
It tests whether the resulting random $3$-partite hypergraph contains cycles
|
||||
by iteratively deleting edges connecting vertices of degree 1.
|
||||
The deleted edges are stored in the order of deletion in a list $\cal L$
|
||||
to be used in the assigning step.
|
||||
The first deleted edge in Figure~\ref{fig:overview}(a)
|
||||
was $\{1,2,4\}$, the second one was $\{1,3,5\}$ and
|
||||
the third one was $\{0,2,5\}$.
|
||||
% the last one was $\{0,2,5\}$.
|
||||
If it ends with an empty graph, then the test succeeds,
|
||||
otherwise it fails.
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
\begin{figure}
|
||||
\begin{center}
|
||||
\scalebox{0.9}{\epsfig{file=figs/overviewinternal3g.eps}}
|
||||
\end{center}
|
||||
\caption{(a) The mapping step generates a random acyclic $3$-partite hypergraph with $m=6$ vertices and $n=3$ edges
|
||||
and a list $\cal L$ of edges obtained when we test whether the hypergraph is acyclic.
|
||||
(b) The assigning step builds an array $g:[0,5] \to [0,3]$ to uniquely
|
||||
assign an edge to a vertex. (c) The ranking step builds the data structure used to
|
||||
compute function $\mathit{rank}: [0,5] \to [0,2]$ in $O(1)$ time.~~~~}
|
||||
\label{fig:overview}
|
||||
\end{figure}
|
||||
|
||||
|
||||
|
||||
We now show how to use the Jenkins hash functions \cite{j97}
|
||||
to implement the three hash functions $h_i: S \to V_i$, $0\le i \le 2$, which are used to build a random $3$-partite hypergraph
|
||||
$G=(V,E)$,
|
||||
where $V= V_0 \cup V_1 \cup V_2$ and $|V_i| = \eta = \lceil \frac{m}{3} \rceil$.
|
||||
Let $h':S \to \{0,1\}^\gamma$ be a Jenkins hash function
|
||||
for $\gamma = 3 \times w$, where
|
||||
$w = 32 \text{ or } 64$ for
|
||||
32-bit and 64-bit architectures, respectively.
|
||||
Let $H'$ be an array of 3 $w$-bit values.
|
||||
The Jenkins hash function
|
||||
allow us to compute in parallel the three entries in $H'$
|
||||
and thereby the three hash functions $h_i$, as follows:
|
||||
% Thus we can compute the three hash functions $h_i$
|
||||
% as follows:
|
||||
\begin{eqnarray}
|
||||
H' &=& h'(x) \nonumber \\
|
||||
h_0(x) &=& H'[0] \bmod \eta \nonumber \\
|
||||
h_1(x) &=& H'[1] \bmod \eta + \eta \nonumber \\
|
||||
h_2(x) &=& H'[2] \bmod \eta + 2\eta
|
||||
\end{eqnarray}
|
||||
|
||||
The {\em Assigning Step} in Figure~\ref{fig:overview}(b) outputs
|
||||
a PHF that maps the key set $S$ into the range $[0,m-1]$ and is represented by
|
||||
an array $g$ storing values from the range $[0,3]$.
|
||||
The array $g$ allows to select one out of the $3$
|
||||
vertices of a given edge, which is associated with a
|
||||
key $k$.
|
||||
A vertex for a key $k$ is given
|
||||
by either $h_0(k)$, $h_1(k)$ or $h_2(k)$.
|
||||
The function $h_i(k)$
|
||||
to be used for $k$ is chosen by calculating $i = (g[h_0(k)] + g[h_1(k)] + g[h_2(k)]) \bmod 3$.
|
||||
For instance,
|
||||
the values 1 and 4 represent the keys ``who'' and ``band''
|
||||
because $i = (g[1] + g[3] + g[5]) \bmod 3 = 0$ and $h_0(\text{``who''}) = 1$,
|
||||
and $i = (g[1] + g[2] + g[4]) \bmod 3 = 2$ and $h_2(\text{``band''}) = 4$, respectively.
|
||||
% Likewise, the value 4 represents the key
|
||||
% because $(g[1] + g[2] + g[4]) \bmod 3 = 2$ and $h_2(\text{``band''}) = 4$, and so on.
|
||||
The assigning step firstly initializes $g[i]=3$
|
||||
to mark every vertex as unassigned
|
||||
% (i.e., each vertex is unassigned)
|
||||
and
|
||||
$\mathit{Visited}[i]=\mathit{false}$, $0\leq i \leq m-1$.
|
||||
Let $\mathit{Visited}$ be a boolean vector of size $m$
|
||||
to indicate whether a vertex has been visited.
|
||||
Then, for each edge $e \in \cal L$ from tail to head,
|
||||
it looks for the first
|
||||
vertex $u$ belonging to $e$ not yet visited.
|
||||
This is a sufficient condition for success \cite{b08,bpz07,mwhc96}.
|
||||
Let $j$, $0 \leq j \leq 2$, be the index of $u$ in $e$.
|
||||
Then, it assigns $g[u]=(j-\sum_{v \in e \wedge \mathit{Visited}[v] = true} g[v]) \bmod 3$.
|
||||
Whenever it passes through a vertex $u$ from $e$,
|
||||
if $u$ has not yet been visited,
|
||||
it sets $\mathit{Visited}[u] = true$.
|
||||
% The value $g[i]=3$ is used to represent unassigned vertices.
|
||||
|
||||
If we stop the BDZ algorithm in the assigning step
|
||||
we obtain a PHF with range $[0,m-1]$.
|
||||
The PHF has the following form:
|
||||
$phf(x) = h_{i(x)}(x)$, where $x\in S$ and $i(x) = (g[h_0(x)] + g[h_1(x)] + g[h_2(x)]) \bmod 3$.
|
||||
In this case we do not need information for ranking and
|
||||
can set $g[i] = 0$ whenever $g[i]$ is equal to 3, where $0 \le i \le m-1$.
|
||||
Therefore, the range of the values stored in $g$ is narrowed
|
||||
from $[0,3]$ to $[0,2]$. By using arithmetic coding as block of
|
||||
values (see \cite{b08,bpz07} for details),
|
||||
or any compression technique that allows to perform
|
||||
random access in constant time to an array of compressed values \cite{fn07,gn06,sg06},
|
||||
we can store the resulting PHFs in $m\log 3 = c n\log 3$ bits,
|
||||
where $c \ge 1.23$. For $c = 1.23$, the space requirement is $1.95n$ bits.
|
||||
|
||||
|
||||
The {\em Ranking Step} in Figure~\ref{fig:overview}(c)
|
||||
outputs a data structure
|
||||
that permits to narrow the range of a PHF generated in the
|
||||
assigning step from $[0,m-1]$ to $[0,n-1]$ and thereby
|
||||
an MPHF is produced.
|
||||
The data structure allows to compute in constant time
|
||||
a function $\mathit{rank}\!\!:[0,m-1]\to [0,n-1]$
|
||||
that counts the number of assigned positions
|
||||
before a given position $v$ in $g$.
|
||||
For instance, $\mathit{rank}(4) = 2$ because
|
||||
the positions $0$ and $1$ are assigned
|
||||
since $g[0] \text{ and } g[1] \not = 3$.
|
||||
% and they come before position 4 in $g$.
|
||||
|
||||
For the implementation of the ranking step
|
||||
we have borrowed
|
||||
a simple and efficient implementation from
|
||||
\cite{dict-jour}.
|
||||
It requires $\epsilon \, m$ additional bits of space, where $0 < \epsilon < 1$,
|
||||
and is obtained by storing explicitly the
|
||||
$\mathit{rank}$ of every $k$th index in a rankTable, where $k
|
||||
=\lfloor\log(m)/\epsilon\rfloor$.
|
||||
The larger is $k$ the more compact is the resulting MPHF.
|
||||
Therefore, the users can tradeoff space for evaluation time
|
||||
by setting $k$ appropriately in the implementation.
|
||||
% In the implementation we let
|
||||
% $k$ to be set by the users so that they can trade off
|
||||
% space for evaluation time and vice-versa.
|
||||
We only allow values for $k$
|
||||
that are power of two (i.e., $k=2^{b_k}$ for some constant $b_k$) in order to replace the expensive
|
||||
division and modulo operations by
|
||||
bit-shift and bitwise ``and'' operations, respectively.
|
||||
We have used $k=256$
|
||||
in the experiments
|
||||
for generating more succinct MPHFs.
|
||||
We remark that it is still possible to obtain a more compact data structure by
|
||||
using the results presented in \cite{os07,rrr02}, but at the cost of a much more
|
||||
complex implementation.
|
||||
|
||||
We need to use an additional lookup table $T_r$
|
||||
to guarantee the constant evaluation time of $\mathit{rank}(u)$.
|
||||
Let us illustrate how $\mathit{rank}(u)$ is computed
|
||||
using both the rankTable and the lookup table $T_r$.
|
||||
We first look up
|
||||
the rank of the largest precomputed index
|
||||
$v\leq u$ in the rankTable,
|
||||
and use $T_r$ to count the number of assigned vertices from position
|
||||
$v$ to $u-1$.
|
||||
The lookup table $T_r$ allows us to count in constant time
|
||||
the number of assigned vertices in $\flat=\epsilon \log m$ bits,
|
||||
where $0 < \epsilon < 1$. Thus the actual evaluation time is $O(1/\epsilon)$.
|
||||
For simplicity and
|
||||
without loss of generality we let $\flat$ be a multiple of the number of
|
||||
bits $\beta$ used to encode each entry of $g$.
|
||||
As the values in $g$ come from the range $[0,3]$,
|
||||
then $\beta=2$ bits and we have tried $\flat = 8 \text{ and } 16$.
|
||||
We would expect that $\flat = 16$ should provide
|
||||
a faster evaluation time because we would need to carry out fewer lookups
|
||||
in $T_r$. But, for both values of $\flat$ the lookup table $T_r$ fits entirely in
|
||||
the CPU cache and we did not realize any significant difference in
|
||||
the evaluation times. Therefore we settle for $\flat=8$.
|
||||
We remark that each $r \ge 2$ requires
|
||||
a different lookup table $T_r$ that can be generated a priori.
|
||||
|
||||
|
||||
% To do this in $O(1/\epsilon)$ time
|
||||
% we use a lookup table $T_r$ that allows us to count
|
||||
% the number of assigned vertices in $\flat=\epsilon \log m$ bits
|
||||
% in constant time for any $0 < \epsilon < 1$.
|
||||
|
||||
|
||||
|
||||
% In general the PHFs or MPHFs are constructed based on random acyclic $r$-partite hypergraphs $G_r=(V,E)$,
|
||||
% where $V= V_0 \cup V_1 \cup \dots \cup V_{r-1}$ and $|V_i| = \eta = \lceil \frac{m}{r} \rceil$, where $0\leq i < r$.
|
||||
% The most efficient and compact functions are generated
|
||||
% when $r=3$ and $m=1.23n$. The value $1.23n$ is required to generate a
|
||||
% random acyclic $3$-partite hypergraph with high probability\footnote{Throughout this paper
|
||||
% we write ``with high probability'' to mean with probability
|
||||
% $1 - n^{-\delta}$ for $\delta > 0$.}~\cite{b08,bpz07}.
|
||||
|
||||
|
||||
% the family of linear transformations
|
||||
% presented in \cite{admp99}. A still faster option is the Jenkins function
|
||||
% proposed in \cite{j97}, which was used for all methods considered in this paper.
|
||||
|
||||
The resulting
|
||||
MPHFs have the following form:
|
||||
$h(x) = \mathit{rank}(\mathit{phf}(x))$.
|
||||
Then, we cannot get rid of
|
||||
the raking information by replacing the values 3 by 0 in the entries of $g$.
|
||||
% The array
|
||||
% $g$ is now representing a function $g:V\to \{0,1,2,3\}$
|
||||
% and $\mathit{rank}: V \to [0,n-1]$ is
|
||||
% now the cardinality of
|
||||
% $\{ u\in V \;\mid\; u<v \wedge g[u] \not = 3\}$.
|
||||
% Notice that a vertex $u$ is assigned if $g[u] \neq 3$.
|
||||
In this case each entry in the array $g$ is encoded
|
||||
with $2$ bits and we need $\epsilon m$ additional bits to compute function
|
||||
$\mathit{rank}$ in constant time. Then, the total space to store
|
||||
the resulting functions is $(2 + \epsilon)m = (2 + \epsilon)cn$ bits.
|
||||
By using $c = 1.23$ and $\epsilon = 0.125$
|
||||
we have obtained MPHFs that require approximately $2.62$ bits per key to be stored.
|
||||
|
||||
% Figure~\ref{prog:ram} presents a pseudo code for
|
||||
% the BDZ algorithm, showing how to implement the mapping,
|
||||
% assigning, and ranking steps. Next, it shows how to evaluate the PHF and the MPHF.
|
||||
% The MPHF algorithm uses a lookup table, which is also shown in the figure.
|
||||
%
|
||||
% \begin{figure}
|
||||
% \begin{center}
|
||||
% \vspace{-10mm}
|
||||
% \begin{lstlisting}[multicols=2]
|
||||
% @{\bf BDZ Algorithm}\\[1mm]@
|
||||
% @{\bf Input:} key set $S$, a constant $c \ge 1.23$, a constant $b_k$
|
||||
% and a family of ``good'' hash functions $\cal H$.\\[1mm]@
|
||||
% @{\bf Output:} an array $g$ with $m = \lceil cn \rceil$ 2-bit entries, and a rankTable with $(m >\!> b_k + 1)$ $\delta$-bit entries, where $\delta = 32 \text{ or } 64$ depending on the architecture. The operator $>\!>$ denotes the right shift of bits.\\[2mm]@
|
||||
% void @BDZ@ (@$S$@, @$\cal H$@, @$c$@, @$b_k$@, @$g$@, @rankTable@)@\\[2mm]@
|
||||
% // Mapping step
|
||||
% do
|
||||
% @$G.E = \emptyset$@;
|
||||
% select @$h'$@ at random from @$\cal H$@;
|
||||
% for @{\bf each}@ @$x \in S$@ do
|
||||
% @$H'$ = $h'(x)$@;
|
||||
% @$e$@ = @$\{h_0(x), h_1(x), h_2(x)\}$@;
|
||||
% addEdge (@$G$@, @$e$@);
|
||||
% @$\cal L$@ = isAcyclic(@$G$@);
|
||||
% while (@$G.E$@ is not empty);
|
||||
%
|
||||
% // Assigning step
|
||||
% for (@$u = 0$@; @$u < m$@; @$u$++@)
|
||||
% Visited[@$u$@] = @{\bf false}@;
|
||||
% @$g[u]$@ = @$3$@;
|
||||
% for (i = @$|{\cal L}|-1$@; i @$\ge 0$@; i@$--$@)
|
||||
% @$e$@ = @$\cal L$@[i];
|
||||
% sum = 0;
|
||||
% for (@$v$@ = 2; @$v \ge 0$@; @$v$@@$--$@)
|
||||
% if (not Visited[@$e[v]$@])
|
||||
% Visited[@$e[v]$@] = @{\bf true}@;
|
||||
% @$u$@ = @$e[v]$@;
|
||||
% @$j$@ = @$v$@;
|
||||
% else sum += @$g[e[v]]$@;
|
||||
% @g[u]@ = @$(j - \mathrm{sum}) \bmod 3$@;
|
||||
%
|
||||
% // Ranking step
|
||||
% sum = 0;
|
||||
% kmask = @$(2^{b_k}-1)$@;
|
||||
% for (i = 0; i < @$|g|$@; i++)
|
||||
% if((i & kmask) @==@ 0)
|
||||
% rankTable[i @$>\!> b_k$@] = sum;
|
||||
% if(@$g$@[i] @$\not = 3$@) sum++;
|
||||
%
|
||||
% @{\bf PHF Algorithm}\\[1mm]@
|
||||
% @{\bf Input:} a key $x \in S$, an array $g$ with $m = \lceil cn \rceil$ 2-bit entries, where $c \ge 1.23$, and the ``good'' hash functions $h'$ selected by the BDZ algorithm.\\[1mm]@
|
||||
% @{\bf Output:} the perfect hash function value for the key $x \in S$.\\[2mm]@
|
||||
% int phf (@$x$@, @$g$@, @$h'$@)
|
||||
% @$H'$@ = @$h'(x)$@;
|
||||
% @$e$@ = @$\{h_0(x), h_1(x), h_2(x)\}$@;
|
||||
% @$v$@ = @$(g[e[0]] + g[e[1]] + g[e[2]]) \bmod 3$@;
|
||||
% return @$e[v]$@;
|
||||
%
|
||||
% @{\bf Algorithm to Generate the Lookup Table}\\[1mm]@
|
||||
% @{\bf Input:} none\\[1mm]@
|
||||
% @{\bf Output:} the lookup table @$T_r$@ to be used by the mphf function. It counts the number of assigned
|
||||
% vertices in a single byte. As each entry in the array $g$ is encoded by 2 bits, then a single byte can store at most four 2-bit values. LS($i'$,2) stands for the value of the 2 least significant bits of $i'$.\\[2mm]@
|
||||
% void genLookupTable (@$T_r$@)
|
||||
% for (i = 0; i < 256; i++)
|
||||
% sum = 0;
|
||||
% @$i'$@ = i;
|
||||
% for (j = 0; j < 4; j++)
|
||||
% if(@$\text{LS}(i',2) \not = 3$@) sum++;
|
||||
% @$i'$@ = @$i' >\!> 2$@;
|
||||
% @$T_r[i]$@ = sum;
|
||||
%
|
||||
% @{\bf MPHF Algorithm}\\[1mm]@
|
||||
% @{\bf Input:} a key $x \in S$, an array $g$ with $m = \lceil cn \rceil$ 2-bit entries, where $c \ge 1.23$, the chosen ``good'' hash functions $h'$, a constant $b_k$ that makes $k=2^{b_k}$, the lookup table $T_r$ that counts the number of assigned vertices in a single byte, and a rankTable with $(m >\!> b_k + 1)$ $\delta$-bit entries, where $\delta = 32 \text{ or } 64$ depending on the architecture. The notation $g[i \to j]$ represents the values stored in the entries from $g[i]$ to $g[j]$ for $i\leq j$.\\[1mm]@
|
||||
% @{\bf Output:} the minimal perfect hash function value for the key $x \in S$.\\[2mm]@
|
||||
% int mphf (@$x$@, @$g$@, @$h'$@, @$b_k$@, @$T_r$@, @rankTable@)
|
||||
% @$u$@ = phf(@$x$@, @$g$@, @$h'$@);
|
||||
% j = @$u >\!> b_k$@; // @j@ = @$u$@/k
|
||||
% rank = rankTable[j];
|
||||
% i = j @$<\!< b_k$@; // @i@ = @j*k@
|
||||
% for(j = i + 4; j < u; i = j, j += 4)
|
||||
% rank += @$T_r[g[$@i @$\to$@ j@$]]$@;
|
||||
% for(j = j - 4; j < u; j ++)
|
||||
% if(@$g$@[j] @$\not =$@ 3) rank ++ ;
|
||||
% return rank;
|
||||
% \end{lstlisting}
|
||||
% \end{center}
|
||||
% \vspace{-6mm}
|
||||
% \caption{The BDZ algorithm and the resulting PHFs and MPHFs.}
|
||||
% \label{prog:ram}
|
||||
% \vspace{-7mm}
|
||||
% \end{figure}
|
||||
|
||||
$\eta$ ~~
|
||||
$\epsilon$ ~~
|
||||
$\varepsilon$
|
||||
12
tex/bdz/makefile
Executable file
12
tex/bdz/makefile
Executable file
@@ -0,0 +1,12 @@
|
||||
all:
|
||||
latex bdz.tex
|
||||
bibtex bdz
|
||||
latex bdz.tex
|
||||
latex bdz.tex
|
||||
dvips bdz.dvi -o bdz.ps
|
||||
run: clean all
|
||||
gv bdz.ps &
|
||||
html: clean all
|
||||
latex2html bdz.tex
|
||||
clean:
|
||||
rm bdz.dvi bdz.ps *.lot *.lof *.aux *.bbl *.blg *.log *.toc
|
||||
Reference in New Issue
Block a user