Listing A.6: d-cuckoo.cpp
1 // //////////////////////////////////////////////////////////
// d − Cuckoo Hashing L i b r a r y // by Reinhard K u t z e l n i g g
// bug r e p o r t s and comments t o k u t z e l n i g g @ d m g . t u w i e n . ac . a t
5 // www . dmg . t u w i e n . ac . a t / k u t z e l n i g g
// l a s t u p d a t e d : $Date : 2008/06/29 0 7 : 2 4 : 1 8 $ // d i s t r i b u t e d under t h e c o n d i t i o n s o f t h e LGPL
// //////////////////////////////////////////////////////////
10 template<c l a s s S> H a s h d c u c k<S>: : H a s h d c u c k (i n t m, i n t d , bool s e p a r a t e , i n t maxloop ) {
H a s h d c u c k : : m = m;
H a s h d c u c k : : maxloop = maxloop ;
15 H a s h d c u c k : : s e p a r a t e = s e p a r a t e ; i f( d > 1 2 6 )
d = 1 2 6 ;
H a s h d c u c k : : d = d ; n = 0 ;
20 t a b l e = new S[m ] ; s t a t u s = new i n t[m ] ; hv = new uint[ d ] ;
f o r (i n t i =0 ; i<m ; i ++)
25 {
s t a t u s [ i ] = −1;
} }
30
template<c l a s s S> H a s h d c u c k<S>: : ˜ H a s h d c u c k ( ) {
d e l e t e[ ] t a b l e ; d e l e t e[ ] s t a t u s ;
35 d e l e t e[ ] hv ; }
template<c l a s s S> i n t H a s h d c u c k<S>: : s e a r c h (S s )
40 {
f o r (i n t i =0 ; i<d ; i ++) {
i f ( s t a t u s [ h ( s , i )]>=0 && t a b l e [ h ( s , i )]== s ) return i +1;
45 }
return −d ; }
50 template<c l a s s S> i n t H a s h d c u c k<S>: : i n s e r t (S s ) {
S p , q=s ; i n t i , j ; i n t n r q ;
55 i n t n r p ; hv [ 0 ] = h ( q , 0 ) ;
i f ( s t a t u s [ hv [ 0 ] ] == −1)
{ // f i r s t i n s p e c t e d c e l l i s empty
60 t a b l e [ hv [ 0 ] ] = q ; s t a t u s [ hv [ 0 ] ] = 0 ; n++;
return 1 ; }
65
A Selected C++ code listings
f o r ( j =1 ; j !=d ; j ++) {
hv [ j ] = h ( q , j ) ;
i f ( s t a t u s [ hv [ j ] ] == −1 | | t a b l e [ hv [ j ]]== q )
70 { // f o u n d an empty c e l l o r q a l r e a d y i n t a b l e t a b l e [ hv [ j ] ] = q ;
s t a t u s [ hv [ j ] ] = j ; n++;
return 1+ j ;
75 }
}
// a l l p o s s i b l e c e l l s f o r q a r e o c c u p i e d p = t a b l e [ hv [ 0 ] ] ; // p w i l l b e k i c k e d o u t n r p = s t a t u s [ hv [ 0 ] ] ;
80 t a b l e [ hv [ 0 ] ] = q ; s t a t u s [ hv [ 0 ] ] = 0 ; q = p ;
n r q = ( n r p +1) % d ;
85 f o r ( i =1; i<maxloop ; i ++) {
f o r ( j=n r q ; j != n r p ; j =( j +1)%d ) {
hv [ j ] = h ( q , j ) ;
90 i f ( s t a t u s [ hv [ j ] ] ==−1 | | t a b l e [ hv [ j ]]== q ) { // f o u n d an empty c e l l o r q a l r e a d y i n t a b l e
t a b l e [ hv [ j ] ] = q ; s t a t u s [ hv [ j ] ] = j ; n++;
95 i n t s t e p s = j − n r q + 1 ;
i f ( s t e p s < 0 ) s t e p s += d ;
return i∗( d−1)+1+ s t e p s ; }
100 }
// a l l p o s s i b l e c e l l s f o r q a r e o c c u p i e d p = t a b l e [ hv [ n r q ] ] ; // p w i l l b e k i c k e d o u t
n r p = s t a t u s [ hv [ n r q ] ] ; t a b l e [ hv [ n r q ] ] = q ;
105 s t a t u s [ hv [ n r q ] ] = n r q ; q = p ;
n r q = ( n r p +1) % d ; }
return −maxloop∗( d−1)+1;
110 // r e h a s h r e q u i r e d , i n s e r t i o n n o t p o s s i l e }
template<c l a s s S> void H a s h d c u c k<S>: : d e l (S s )
115 {
i n t r = s e a r c h ( s ) ; i f ( r >= 0 )
{
s t a t u s [ h ( s , r ) ] = −1;
120 n−−;
} }
125 template<c l a s s S> void H a s h d c u c k<S>: : o u t ( s t d : : o s t r e a m∗ s t r e a m ) {
i n t i , j ;
∗s t r e a m << ”∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗\n” ;
130 ∗s t r e a m << ”n=” << n << ” m=” <<m<< ” d=” << d << ”\n” ;
A.6 d-cuckoo.cpp
f o r ( i =0; i<m && i<100; i +=7) {
f o r ( j=i ; j<i +7 && j<m; j ++)
135 {
stream−>width ( 1 2 ) ;
∗s t r e a m ; }∗s t r e a m << ”\n” ;
140
f o r ( j=i ; j<i +7 && j<m; j ++) {
stream−>width ( 1 2 ) ; i f ( s t a t u s [ j ] < 0 )
145 ∗s t r e a m << ” ” ;
e l s e
∗s t r e a m << t a b l e [ j ] ; }∗s t r e a m << ”\n” ;
150 f o r ( j=i ; j<i +7 && j<m; j ++) {
stream−>width ( 1 2 ) ;
∗s t r e a m << s t a t u s [ j ] ; }
155 ∗s t r e a m << ”\n\n” ; } }
160 template<c l a s s S> uint∗ H a s h d c u c k<S>: : o c c u p a n c y ( ) {
uint i ; i n t p ; uint∗ o ;
165 o = new uint[ d ] ; f o r ( i =0 ; i<d ; i ++)
o [ i ] = 0 ;
f o r ( i =0 ; i<m ; i ++) {
170 p = s t a t u s [ i ] ; i f ( p >= 0 )
o [ p]++;
}
return o ;
175 }
// /////////////////////////////////////////////////////
// /////////////////////////////////////////////////////
180
H a s h d c u c k p o l : : H a s h d c u c k p o l
(i n t m, i n t d , bool s e p a r a t e , i n t maxloop , RandBase∗ random ) : H a s h d c u c k<uint>(m, d , s e p a r a t e , maxloop )
185 {
a = new uint[ d ] ; b =new uint[ d ] ;
i f ( s e p a r a t e )
190 M = m/d ;
e l s e M = m;
f o r (i n t i =0 ; i<d ; i ++)
195 {
a [ i ] = random−>g e t (1< <31);
b [ i ] = random−>g e t (1< <31);
A Selected C++ code listings
} }
200
H a s h d c u c k p o l : : ˜ H a s h d c u c k p o l ( ) {
d e l e t e[ ] a ;
205 d e l e t e[ ] b ; }
// /////////////////////////////////////////////////////
210 // /////////////////////////////////////////////////////
H a s h d c u c k c w : : H a s h d c u c k c w
(i n t m, i n t d , bool s e p a r a t e , i n t maxloop , RandBase∗ random )
215 : H a s h d c u c k<uint>(m, d , s e p a r a t e , maxloop ) {
i f ( s e p a r a t e ) M = m/d ; e l s e
220 M = m;
r = new i n t∗[ d ] ;
f o r (i n t j =0 ; j<d ; j ++) {
r [ j ] = new i n t[ 1 2 8 ] ;
225 f o r (i n t i =0 ; i<128 ; i ++) {
r [ j ] [ i ] = random−>g e t (1< <31);
} }
230 }
H a s h d c u c k c w : : ˜ H a s h d c u c k c w ( ) {
235 f o r (i n t j =0 ; j<d ; j ++) {
d e l e t e[ ] r [ j ] ; }
d e l e t e[ ] r ;
240 }
i n l i n e i n t H a s h d c u c k c w : : h (uint s , i n t nr ) {
245 i n t addr = −1;
uint h = 0 ;
f o r (uint i =0 ; i<8 ; i ++) {
h ˆ= r [ nr ] [ addr += 1 + ( s & 0xF ) ] ;
250 s >>= 4 ; }
return h % M + s e p a r a t e∗nr∗M; ; }
255
// /////////////////////////////////////////////////////
// /////////////////////////////////////////////////////
260 H a s h d c u c k s i m : : H a s h d c u c k s i m
(i n t m, i n t d , i n t n end , bool s e p a r a t e , RandBase∗ random )
: H a s h d c u c k<uint>( s e p a r a t e ? d∗(m/d ) : m, d , s e p a r a t e , d∗n e n d +1) {
A.6 d-cuckoo.cpp
H a s h d c u c k s i m : : n e n d = n e n d ;
265 hv = new uint∗[ d ] ; i f( ! s e p a r a t e ) {
f o r (i n t j =0 ; j<d ; j ++)
270 {
hv [ j ] = new uint[ n e n d ] ; f o r (i n t i =0 ; i<n e n d ; i ++) {
hv [ j ] [ i ] = random−>g e t (m) ;
275 }
} } e l s e {
280 f o r (i n t j =0 ; j<d ; j ++) {
hv [ j ] = new uint[ n e n d ] ; f o r (i n t i =0 ; i<n e n d ; i ++) {
285 hv [ j ] [ i ] = random−>g e t (m/d ) + j∗(m/d ) ; } }
} }
290
H a s h d c u c k s i m : : ˜ H a s h d c u c k s i m ( ) {
f o r (i n t j =0 ; j<d ; j ++)
295 {
d e l e t e[ ] hv [ j ] ; }
d e l e t e[ ] hv ; }
A Selected C++ code listings
Listing A.7: tables.hpp
1 // //////////////////////////////////////////////////////////
// ( B i p a r t i t e ) D i s j o i n t S e t−F o r e s t L i b r a r y // by Reinhard K u t z e l n i g g
// bug r e p o r t s and comments t o k u t z e l n i g g @ d m g . t u w i e n . ac . a t
5 // www . dmg . t u w i e n . ac . a t / k u t z e l n i g g
// l a s t u p d a t e d : $Date : 2007/09/24 0 7 : 5 7 : 5 1 $ // d i s t r i b u t e d under t h e c o n d i t i o n s o f t h e LGPL
// //////////////////////////////////////////////////////////
10 // R e f e r e n c e s :
// T. C. Cormen , C. E . L e i s e r s o n , and R. L . R i v e s t , // I n t r o d u c t i o n t o A l g o r i t h m s , MIT P r e s s , London , 2 0 0 0 .
#i f n d e f TABLES HPP
15#define TABLES HPP
#include <m a l l o c . h>
c l a s s T a b l e s
20 {
bool b i p a r t i t e ; i n t ∗s e t 1 ; i n t ∗s i z e 1 ; bool ∗c y c l i c 1 ;
25 i n t ∗s e t 2 ; i n t ∗s i z e 2 ; bool ∗c y c l i c 2 ; public:
30 T a b l e s (i n t m, i n t m2= 0 ) ;
˜ T a b l e s ( ) ;
// r e t u r n s t h e p a r e n t o f u i n t p a r e n t (i n t u ) ;
35
// u n i o n s t h e components w i t h r o o t s u and v ( new r o o t v ) void merge (i n t u , i n t v ) ;
// r e t u r n s t h e s i z e o f t h e component c o n t a i n i n g u , o n l y f o r r o o t n o d e s !
40 i n t g e t s i z e (i n t u ) ;
// s e t s t h e s i z e o f t h e component w i t h r o o t u t o s void s e t s i z e (i n t u , i n t s ) ;
45 // r e t u r n s i f u i s i n a c y c l e , u s e o n l y f o r r o o t n o d e s ! i n t g e t c y c l i c (i n t u ) ;
// marks t h e component w i t h r o o t u a s c y c l i c void s e t c y c l i c (i n t u ) ;
50
// r e t u r n s t h e r o o t node o f u i n t f i n d s e t (i n t u ) ;
void∗ operator new( s i z e t s ) { return m a l l o c ( s ) ; }
55 void operator d e l e t e(void∗ p ) { f r e e ( p ) ; } };
#e n d i f /∗TABLES HPP ∗/
A.8 tables.cpp
Listing A.8: tables.cpp
1 // //////////////////////////////////////////////////////////
// B i p a r t i t e D i s j o i n t S e t−F o r e s t L i b r a r y // by Reinhard K u t z e l n i g g
// bug r e p o r t s and comments t o k u t z e l n i g g @ d m g . t u w i e n . ac . a t
5 // www . dmg . t u w i e n . ac . a t / k u t z e l n i g g
// l a s t u p d a t e d : $Date : 2008/07/07 1 4 : 5 7 : 4 9 $ // d i s t r i b u t e d under t h e c o n d i t i o n s o f t h e LGPL
// //////////////////////////////////////////////////////////
10#include ” t a b l e s . hpp”
T a b l e s : : T a b l e s (i n t m, i n t m2) {
i f (m2<= 0 )
15 b i p a r t i t e = f a l s e; e l s e
b i p a r t i t e = true; s e t 1 = new i n t[m+ 1 ] ;
20 s i z e 1 = new i n t[m+ 1 ] ; c y c l i c 1 =new bool[m+ 1 ] ; f o r (i n t j =1 ; j<m+1 ; j ++) {
25 s e t 1 [ j ] = j ; s i z e 1 [ j ] = 1 ; c y c l i c 1 [ j ] = f a l s e; }
30 i f( b i p a r t i t e ) {
s e t 2 = new i n t[ m2+ 1 ] ; s i z e 2 = new i n t[ m2+ 1 ] ; c y c l i c 2 = new bool[ m2+ 1 ] ;
35
f o r (i n t j =1 ; j<m2+1 ; j ++) {
s e t 2 [ j ] =−j ; s i z e 2 [ j ] = 1 ;
40 c y c l i c 2 [ j ] = f a l s e; } }
}
45 T a b l e s : : ˜ T a b l e s ( ) {
d e l e t e[ ] s e t 1 ; d e l e t e[ ] s i z e 1 ; d e l e t e[ ] c y c l i c 1 ;
50
i f ( b i p a r t i t e ) {
d e l e t e[ ] s e t 2 ; d e l e t e[ ] s i z e 2 ;
55 d e l e t e[ ] c y c l i c 2 ; } }
60 i n t T a b l e s : : p a r e n t (i n t u ) {
i f ( u > 0 )
{ // f i r s t t a b l e return s e t 1 [ u ] ;
65 }
A Selected C++ code listings
// s e c o n d t a b l e return s e t 2 [−u ] ; }
70
void T a b l e s : : merge (i n t u , i n t v ) {
i f ( u > 0 )
{ // f i r s t t a b l e
75 s e t 1 [ u ] = v ; return; }
// s e c o n d t a b l e s e t 2 [−u ] = v ;
80 }
i n t T a b l e s : : g e t s i z e (i n t u ) {
85 i f ( u>0)
{ // f i r s t t a b l e return s i z e 1 [ u ] ; }
// s e c o n d t a b l e
90 return s i z e 2 [−u ] ; }
void T a b l e s : : s e t s i z e (i n t u , i n t s )
95 {
i n t h ; i f ( u>0)
{ // f i r s t t a b l e
100 h = s i z e 1 [ u ] ; s i z e 1 [ u ] = s ; return; }
// s e c o n d t a b l e
105 h = s i z e 2 [−u ] ; s i z e 2 [−u ] = s ; }
110 i n t T a b l e s : : g e t c y c l i c (i n t u ) {
i f ( u>0)
{ // f i r s t t a b l e return c y c l i c 1 [ u ] ;
115 }
return c y c l i c 2 [−u ] ; }
120 void T a b l e s : : s e t c y c l i c (i n t u ) {
i n t h ; i f ( u>0)
125 { // f i r s t t a b l e h = c y c l i c 1 [ u ] ;
c y c l i c 1 [ u ] = true; return;
}
130 // s e c o n d t a b l e h = c y c l i c 2 [−u ] ;
A.8 tables.cpp
c y c l i c 2 [−u ] = true; }
135
i n t T a b l e s : : f i n d s e t (i n t u ) { // s t a n d a r d v e r s i o n
while ( p a r e n t ( u ) != u )
140 u = p a r e n t ( u ) ; return u ;
} /∗
145 i n t T a b l e s : : f i n d s e t ( i n t u ) { // s h o r t e n p a t h l e n g t h
i n t v = p a r e n t ( u ) ; i n t w = u ;
i f ( v != u )
150 {
w = f i n d s e t ( v ) ; merge ( u , w ) ; }
r e t u r n w ;
155 }
∗/
Appendix B
Selected Maple worksheets
In this chapter, we present some of the most important Maple worksheets used to perform the calculations decribed in the previous chapters. All files (and some others) are included on the attached CD-ROM.
In general, each worksheet starts with the derivation of the saddle point theorem that is required to obtain the desired results. After this initial calculation, the actual derivation of special results start. At this point, is should be easy to exchange the involved functions and hence adopt the worksheet to perform different calculations that are however based on the same saddle point method.
(1.1)
substitution(1.2) (2.1)
integration
B.1 asym main one.mw
B.1 asym main one.mw
(3.2) (41)
(3.1)
output expansion definition of funtions f and g, calculation of cummulants
(4.1) (4.2)
B Selected Maple worksheets
(5.2)
(4.3) (5.1) (5.3)
asymptotic (main result) additional calculations
(6.1) probabilities
B.1 asym main one.mw
(1.3)
(1.2)
(1.1)
calculations for variable − substitution (1.4)
(1.3) substitution
B Selected Maple worksheets
B.2 asym main.mw
(3.1)
integration (5.1)
definition of funtions f and g, calculation of cummulants
B.2 asym main.mw
(5.2) (6.1)
asymptotic (main result)
(6.3)
(6.2) (7.1)
additional calculations
B Selected Maple worksheets
probabilities
B.2 asym main.mw
(1.3)
(1.1) (1.2)
calculations for variable substitution
(1.3) (1.4) variable substitution
B Selected Maple worksheets
B.3 asym main asymmetric.mw
(3.1)
integration
(4.1) (5.1)(4.4)(4.2) (4.3)
calculate x0 and y0 (saddle point) definition of funtions f and g, calculation of cummulants
B.3 asym main asymmetric.mw
(5.3)
(5.2)
(6.1) (6.3)(6.2) (6.4)
asymptotic (main result)
B Selected Maple worksheets
(6.6)
(6.5) (7.1)
calculate failure probability
(7.2) (7.3) calculate max. n depending on c
B.3 asym main asymmetric.mw
(8.2)
(8.1) probabilities
B Selected Maple worksheets
Index
Brent’s variation, 4, 94, 104
Cauchy’s Integral Formula, 20, 22, 26 chaining
Index
with chaining, 5, 6, 14 direct, 5
separate, 5
with open addressing, 2, 6 hypergraph, 10
insertion operation, 85 key, 1
Lagrange inversion, 39
Lagrange Inversion Theorem, 17 Laplace transform, 19
Laplace’s method, 21
moment generating function, 19 multigraph, 33
polynomial hash function, 14 probe sequence, 2
probing linear, 3 quadratic, 3 uniform, 3 random number, 109 saddle point, 20
asymmetric cuckoo hashing, 44 method, 20
simplified cuckoo hashing, 36 standard cuckoo hashing, 41 search operation, 99
skip list, 1
Stirling’s formula, 21 tail
completition, 21 integrals, 21 tree
bipartite, 38
rooted labelled, 38 unrooted labelled, 38 function, 34
rooted labelled, 34 unrooted labelled, 34
Bibliography
Milton Abramowitz and Irena A. Stegun, editors. Handbook of Mathematical Functions.
Dover Publications, Washington, DC, 1970.
Tsiry Andriamampianina and Vlady Ravelomanana. Enumeration of connected uniform hypergraphs. In FPSAC 05, 2005.
Nikolas Askitis. Efficient Data Structures for Cache Architectures. PhD thesis, School of Computer Science and Information Technology, RMIT University, Australia, August 2007.
Yossi Azar, Andrei Z. Broder, Anna R. Karlin, and Eli Upfal. Balanced allocations.
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John R. Black, Charles U. Martel, and Hongbin Qi. Graph and hashing algorithms for modern architectures: Design and performance. In WAE ’92, Saarbr¨ucken, Germany, August 20-22, 1998, Proceedings, pages 37–48. Max-Planck-Institut f¨ur Informatik, 1998.
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ACM, 16(2):105–109, 1973.
Andrei Z. Broder and Michael Mitzenmacher. Using multiple hash functions to improve ip lookups. InINFOCOM, pages 1454–1463, 2001.
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Bibliography
John B. Conway. Functions of One Complex Variable. Springer, New York, second edition, 1978.
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduc-tion to Algorithms. MIT Press, Cambridge, Mass., London, England, second ediIntroduc-tion, 2001.
Zbigniew J. Czech, George Havas, and Bohdan S. Majewski. Perfect hashing. Theoretical Computer Science, 182(1-2):1–143, 1997.
Jurek Czyzowicz, Wojciech Fraczak, and Feliks Welfed. Notes about d-cuckoo hashing.
Technical Report RR 06/05-1, Universit´e du Qu´ebec en Outaouais, 2006.
Ketan Dalal, Luc Devroye, Ebrahim Malalla, and Erin McLeis. Two-way chaining with reassignment. SIAM J. Comput., 35(2):327–340, 2005.
Luc Devroye and Pat Morin. Cuckoo hashing: Further analysis. Information Processing Letters, 86(4):215–219, 2003.
Reinhard Diestel. Graph Theory. Springer, Berlin, 2005.
Martin Dietzfelbinger and Christoph Weidling. Balanced allocation and dictionaries with tightly packed constant size bins. Theoretical Computer Science, 380(1-2):47–68, 2007.
Martin Dietzfelbinger and Philipp Woelfel. Almost random graphs with simple hash functions. In STOC ’03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 629–638. ACM, 2003.
Martin Dietzfelbinger, Joseph Gil, Yossi Matias, and Nicholas Pippenger. Polynomial hash functions are reliable (extended abstract). InICALP ’92, volume 623 of LNCS, pages 235–246. Springer, 1992. ISBN 3-540-55719-9.
Martin Dietzfelbinger, Anna R. Karlin, Kurt Mehlhorn, Friedhelm Meyer auf der Heide, Hans Rohnert, and Robert Endre Tarjan. Dynamic perfect hashing: Upper and lower bounds. SIAM J. Comput., 23(4):738–761, 1994.
Martin Dietzfelbinger, Torben Hagerup, Jyrki Katajainen, and Martti Penttonen. A reliable randomized algorithm for the closest-pair problem. J. Algorithms, 25(1):19–51, 1997.
Michael Drmota. A bivariate asymptotic expansion of coefficients of powers of generating functions. European Journal of Combinatorics, 15(2):139–152, 1994.
Michael Drmota and Reinhard Kutzelnigg. A precise analysis of cuckoo hashing.preprint, 2008.
Michael Drmota and Mich`ele Soria. Marking in combinatorial constructions: Generating functions and limiting distributions. Theoretical Computer Science, 144(1&2):67–99, 1995.
Michael Drmota and Mich`ele Soria. Images and preimages in random mappings. SIAM Journal on Discrete Mathematics, 10(2):246–269, 1997.
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Table of Symbols and Notations
Symbol Interpretation
z real part of the complex number z z imaginary part of the complex number z P(A) probability of an eventA
P(A|B) probability of an eventA, conditioned on the occurrence of eventB Eη expectation of a random variable η
Vη variance of a random variable η
δi,j Kronecker’s delta,δi,j =
1 ifiequals j 0 otherwise x floor ofx, greatest integerksatisfying k≤x {x} fractional part ofx (i.e. x− x)
⊕ bitwise exclusive or operation xk x(x+ 1)(x+ 2). . .(x+k−1) xk x(x−1)(x−2). . .(x−k+ 1) f(n) =O(g(n)) limn→∞ f(n)g(n) = 0
f(n) =O(g(n)) |f(n)| ≤c|g(n)|holds for all suff. largenand a constant c >0 f(n) = Θ(g(n)) c1|g(n)| ≤ |f(n)| ≤c2|g(n)|for suff. largenand const. c1, c2 >0 f(n) = Ω(g(n)) |f(n)| ≥c|g(n)|holds for all suff. largenand a constant c >0
x bold font indicates vectors, x= (x1, . . . , xd) xk multipower,xk=xk11. . . xkdd
[xn]a(x) n-th coefficient of the seriesa(x) =
n≥0anxn t(x) generating function of rooted labelled trees t(x)˜ generating function of unrooted labelled trees t(x, y) generating function of rooted labelled bipartite trees t(x, y)˜ generating function of unrooted labelled bipartite trees
List of Figures
1.1 An associative array implemented by a hash table. . . 2
1.2 Collision resolution by open addressing. . . 5
1.3 Collision resolution by chaining. . . 6
1.4 An evolving cuckoo hash table. . . 9
1.5 Insertion procedure in case of a tree component. . . 10
1.6 Insertion procedure in case of a unicyclic component. . . 11
3.1 The modulus of the function ez/z3 close to the origin. . . 22
4.1 Asymptotic expansion of the failure probability . . . 47
5.1 The modulus of f eζσ, eiτ−1 e−ζσe1−iτ in the relevant area. . . 54
7.1 Bounds for the expected number of steps per insertion. . . 89
7.2 Bounds for the insertion using asymmetric cuckoo hashing. . . 95
8.1 Additional search costs in simplified cuckoo hashing. . . 101
8.2 Comparison of successful search . . . 105
8.3 Comparison of unsuccessful search . . . 105
A.1 Hierarchy of the C++ classes. . . 112
List of Tables
0.1 Overview of results concerning cuckoo hashing. . . viii
1.1 Asymptotic approximations of the cost of search operations. . . 6
2.1 “Dictionary” of constructions of unlabelled combinatorial configurations. . 17
2.2 “Dictionary” of constructions of labelled combinatorial configurations. . . 17
4.1 Number of failures . . . 48
5.1 Number of failures, critical case . . . 59
5.2 Average number of edges at the occurrence of the first bicycle. . . 59
6.1 Number of trees of sizes one, two, and five for ε= 0.4. . . 79
6.2 Number of trees of sizes one, two, and five for ε= 0.2. . . 80
6.3 Number of trees of sizes one, two, and five for ε= 0.1. . . 81
6.4 Number of trees of sizes one, two, and five for ε= 0.06. . . 82
6.5 Number of trees of sizes one, two, and five for ε= 0.04. . . 83
6.6 Number of cycles and number of nodes contained in cyclic components. . 84
7.1 Construction cost . . . 96
7.2 Average maximum number of steps of an insertion . . . 97
7.3 Maximum number of steps of an insertion . . . 98
8.1 Successful search . . . 106
A.1 Classes of the hash library. . . 112
List of Listings
7.1 Optimal insertion for cuckoo hashing. . . 93
8.1 Optimal search for cuckoo hashing. . . 100
9.1 Dynamic disjoint-set data structure. . . 108
A.1 base.hpp . . . 113
A.2 cuckoo.hpp . . . 114
A.3 cuckoo.cpp . . . 116
A.4 cuckoo main.cpp . . . 122
A.5 d-cuckoo.hpp . . . 127
A.6 d-cuckoo.cpp . . . 129
A.7 tables.hpp . . . 134
A.8 tables.cpp . . . 135
B.1 asym main one.mw . . . 139
B.2 asym main.mw . . . 142
B.3 asym main asymmetric.mw . . . 146
Lebenslauf
Ich wurde am 18. Mai 1980, als Sohn von Rosemarie Kutzelnigg, geborene Otter, und Alfred Kutzelnigg, in Vorau geboren. Von 1986 bis 1990 besuchte ich die Volksschule in Vorau, und von 1990 bis 1994 die Hauptschule, ebenfalls in meinem Geburtsort. An-schließend besuchte ich von 1994 bis 1999 die h¨ohere technische Bundeslehranstalt f¨ur Elektrotechnik in Pinkafeld, mit Schwerpunkt Steuerungs- und Regelungstechnik. Dort legte ich am 16. Juni 1999 die Reifepr¨ufung mit ausgezeichneten Erfolg ab. In den dar-auf folgenden Monaten leiste ich den Pr¨asenzdienst beim ¨osterreichischen Bundesheer in Pinkafeld ab.
Im September 2000 immatrikulierte ich an der Technischen Universit¨at Wien und be-gann mit dem Studium der Technischen Mathematik, Studienrichtung Mathematische Computerwissenschaften. Dieses schloss ich mit der Ablegung der zweiten Diplompr¨ufung am 23. November 2005 mit ausgezeichnetem Erfolg ab. W¨ahrend des Studiums war ich im Sommersemester 2003 f¨ur das Institut f¨ur Computergraphik und Algorithmen, und ab dem darauf folgenden Wintersemester f¨ur das Institut f¨ur Analysis und Scientific Com-puting, als Tutor t¨atig.
Im M¨arz 2006 inskribierte ich das Doktoratsstudium der technischen Wissenschaften.
Gleichzeitig nahm ich ein Anstellung als Projektassistent im von Prof. Michael Drmota geleiteten FWF-Projekt S9604
”Analytic and Probabilistic Methods in Combinatorics“
an, welches dem nationalen Forschungsnetzwerk
”Analytic Combinatorics and Probabi-listic Number Theory“ zugeh¨ort. Nach einem schweren Sportunfall im M¨arz 2008 und mehrmonatigen Krankenstand, wechselte ich im Juli 2008 zum Forschungsprojekt
” Net-work Models, Governance and R&D collaboration netNet-works“, Projekt Nummer 028875 des EU FP6-NEST-Adventure Programms. F¨ur dieses, an der TU Wien ebenfalls von Prof. Michael Drmota geleitete Projekt, bin ich derzeit als Projektassistent t¨atig.