PATRICIA tries

In typical random tries, internal nodes at successive levels may have only one descendant (corresponding to the extreme probabilities when binomial distribution assumes 0 and n), resulting in an increase in storage. Indeed, the expected number

μ

n of internal nodes under the initial condition

μ

1

=

0 is asymptotic to

(

h⁻¹

+ F [

^G

](

^{r log}1/pn

))

n (see Section 5.1). Thus the expected number of internal nodes with only one child is asymptotic to

(

h⁻¹

−

¹

+ F [

^G

](

^{r log}1/pn

))

n. In the symmetric case, the leading constant (neglecting the fluctuation term) is about 1

/

log 2

−

¹

≈

⁰

.

4427, about 44% extra space being needed, and this is the minimum when p varies between 0 and 1. The idea of PATRICIA¹ tries arose when there was a need to compress such a one-child-in-one-generation pattern; see[74,85]. When removing all such nodes, the resulting tree has n

−

1 internal nodes (for n external nodes). See [106] for an analysis connected to unary nodes of random tries, and [5,15,63,70,98]for other linear shape measures.

Under the same Bernoulli model, we can construct random PATRICIA tries by using the same rule for constructing an ordinary trie but compress all internal nodes with only one descendant. If Xn represents an additive shape parameter in a random PATRICIA trie of size n, then, for n



^2,

X_n

=

^{d X}I^_n

+

^X_n^∗_−I

This then translates into the recurrence for the moment-generating functions (assuming T_nindependent of X_n)

M_n

(

y

) = E 

e^Tny



1^k<n

π

_n^_,kM_k

(

y

)

M_n−k

(

y

) (

n



²

),

1 PATRICIA is the acronym of “practical algorithm to retrieve information coded in alphanumeric”.

with M0

(

y

) =

^M1

(

y

) =

1. It follows that the Poisson generating function

˜

_f1of

E(

^Xn

)

satisfies the functional equation

˜

f1

(

z

) = ˜

^f1

(

pz

) + ˜

^f1

(

qz

) + ˜

^g1

(

z

) −

^e−qzg

˜

1

(

pz

) −

^e−pzg

˜

1

(

qz

),

(49) with

˜

_f1

(

0

) = ˜

^f1^

(

0

) =

^{0, where} g

˜

1 represents the Poisson generating function of

E(

^Tn

)

. For convenience, we also assume

˜

g1

(

0

) = ˜

^g1

(

0

) =

^0.

The same tools we developed for tries readily apply to(49)and the same asymptotic pattern holds.

Theorem 7.1. Let 0

< θ < π /

2

, α <

1 and

β ∈ R

^.

(a) If more precisely

˜

^g1

∈ JS

_α,β^{, then}

E(

^Xn

)

n

=

^G1

(−

¹

)

h

+ F [

^G

](

^{r log}1/pn

) +

^o

(

1

),

where G1

(

s

) = M [˜

g1

(

z

) −

^e−qzg

˜

1

(

pz

) −

^e−pzg

˜

1

(

qz

) ;

^s

]

^.

(b) If

˜

^g1

∈ JS

^andg

˜

1

(

z

) =

^cz

+

^O

(|

^z

|

^α

(

log₊

|

^z

|)

^β

)

uniformly for

|

^arg

(

z

)|  θ

^{, then}

E(

^Xn

)

n

=

^c

hlog n

+

^d

h

+

^{p log2p}

+

^{q log2q}

2h²

+ F [

G1

](

r log_1/pn

) +

o

(

1

),

where G1

(

s

)

is the meromorphic continuation of

M [˜

^g1

(

z

) −

^e−qzg

˜

1

(

pz

) −

^e−pzg

˜

1

(

qz

);

^s

]

^{and d}

=

^lims→−¹

(

G1

(

s

) +

^c

/(

s

+

¹

))

.

Since the method of proof is the same as that ofTheorem 4.1, we omit the details.

For the variance of X_n, we have, using the same notations, V

˜

_X

(

z

) = ˜

^VX

(

pz

) + ˜

^VX

(

qz

) + ˜

^VT

(

z

) + ˜φ

0

(

z

) + ˜φ

1

(

z

) + ˜φ

2

(

z

),

where

˜φ

0

(

z

) = −

e^−qzg

˜

2

(

pz

) −

e^−pzg

˜

2

(

qz

) +

2

˜

g1

(

z

) 

e^−qzg

˜

1

(

pz

) +

e^−pzg

˜

1

(

qz

)

−

^2z

˜

g₁^

(

z

) 

qe^−qzg

˜

1

(

pz

) +

^pe−pzg

˜

1

(

qz

) −

^pe−qzg

˜

^₁

(

pz

) −

^qe−pzg

˜

₁^

(

qz

)

−

^z



qe^−qzg

˜

1

(

pz

) +

^pe−pzg

˜

1

(

qz

) −

^pe−qzg

˜

₁^

(

pz

) −

^qe−pzg

˜

^₁

(

qz

)

2

− 

e^−qzg

˜

1

(

pz

) +

^e−pz

˜

^g1

(

qz

)

2

,

and

˜φ

1

(

z

) = ˜

h2

(

z

) −

2

˜

g1

(

z

)  ˜

_f1

(

pz

) + ˜

f1

(

qz

)

−

2zg

˜

^₁

(

z

) 

p

˜

_f1^

(

pz

) +

q

˜

_f1^

(

qz

) +

2



e^−qz

˜

g1

(

pz

) +

e^−pzg

˜

1

(

qz

)  ˜

_f1

(

pz

) + ˜

f1

(

qz

)

−

^2z



qe^−qzg

˜

1

(

pz

) +

^pe−pzg

˜

1

(

qz

) −

^pe−qzg

˜

^₁

(

pz

) −

^qe−pz

˜

^g₁^

(

qz

) 

p

˜

f₁^

(

pz

) +

^q

˜

f₁^

(

qz

) ,

˜φ

₂

(

z

) =

^pqz

 ˜

_f1^

(

pz

) − ˜

^f1^

(

qz

)

2

.

Here_h

˜

₂ is given by h

˜

2

(

z

) =

2e^−z



n0

E(

Tn

) 

0jn

π

n,j

 E(

Xj

) + E(

Xn−^j

)

zⁿ

n

! −

2e^−z



n0



pⁿ

+

qⁿ

E(

Tn

)E(

Xn

)

^z

n

n

! .

Note that, byPropositions 3.2 and 3.3, ifg

˜

1

∈ JS

, then

˜

_f1

∈ JS

, which in turn implies, byProposition 3.5, that_h

˜

₂

∈ JS

. Consequently, if g

˜

1

∈ JS

andg

˜

2

∈ JS

, then both

˜

_f1

∈ JS

and

˜

_f2

∈ JS

. Thus our approach applies to

V(

^Xn

)

.

Theorem 7.2. Let 0

< θ < π /

2

, α <

1 and

β ∈ R

^{. Assume}g

˜

1

, ˜

g2

∈ JS

^andV

˜

T

(

z

) =

^O

( |

^z

|

^α

(

log₊

|

^z

|)

^β

)

for

|

^arg

(

z

) |  θ

^.

(a) If p

=

^q

=

¹

/

2, and^g

˜

1

∈ JS

_α,β^org

˜

1

∈ JS

_1,0^{, then}

V(

Xn

)

n

=

¹

log 2



k∈Z

G

(−

¹

+ χ

k

)

n^−χk

+

^o

(

1

),

where G

(

s

) = M [ ˜

^VT

(

z

) + ˜φ

0

(

z

) + ˜φ

1

(

z

) ;

^s

]

^.

(b) Assume p

=

^q.

(i) If^g

˜

1

∈ JS

_α,β^{, then}

V(

Xn

)

n

=

^G

(−

1

)

h

+ F [

^G

](

^{r log}1/pn

) +

^o

(

1

),

where G

(

s

) = Φ

1

(

s

) + Φ

2

(

s

)

with

Φ

1

(

s

) = M [ ˜

^VT

(

z

) + ˜φ

0

(

z

) + ˜φ

1

(

z

) ]

^and

Φ

2

(

s

)

is an analytic continuation of

M [ ˜φ

2

;

^s

]

^. (ii) Ifg

˜

1

(

z

) =

^z

+

^O

( |

^z

|

^α

(

log₊

|

^z

|)

^β

)

uniformly for

|

^arg

(

z

) |  θ

^{, then}

V(

^Xn

)

n

=

^{pq log2}

(

p

/

q

)

h³ log n

+

^d

h

+

^{p log2p}

+

^{q log2q}

2h²

+ F [

^G

](

^{r log}1/pn

) +

^o

(

1

).

Here G

(

s

) = Φ

1

(

s

)+Φ

2

(

s

)

with

Φ

1

(

s

)

as above,

Φ

2

(

s

)

is a meromorphic continuation of

M [ ˜φ

2

;

^s

]

^{and d}

=

^lims→−1

(

G

(

s

) +

pq log²

(

p

/

q

)/(

h²

(

s

+

¹

)))

.

The proof follows the same arguments as that ofTheorem 4.2and is omitted.

Consider the external path length, which satisfies(48)with Tn

=

n. In this case, we have

˜

g₁

(

z

) =

^z



1

−

^e−z

, ˜

g₂

(

z

) =

^z



1

−

^e−z

+

^z2

,

and

V

˜

T

(

z

) =

e^−z



z



1

−

e^−z

+

z²

(

1

−

z

)

e^−z

.

Also

˜φ

₁

(

z

) = −

^2zpq



(

zp

−

¹

)

e^−pz

˜

f₁^

(

pz

) + (

^zq

−

¹

)

e^−qz

˜

f₁^

(

qz

) + (

^zp

+

¹

)

e^−qz

˜

f₁^

(

pz

) + (

^zq

+

¹

)

e^−pz

˜

f₁^

(

qz

) +

2qze^−pz

˜

_f1

(

pz

) +

2pze^−qz

˜

_f1

(

qz

).

Observe that

G₁

(

s

) := M 

˜

g₁

(

z

) −

^e−qzg

˜

₁

(

pz

) −

^e−pzg

˜

₁

(

qz

);

^s



= −Γ (

^s

+

¹

) 

qp^−s−1

+

^pq−s−1

.

Thus, byTheorem 7.1,

E(

Xn

)

n

=

¹

hlog n

+ γ

h

+

^{p log2p}

+

q log²q

2h²

−

¹

+ F [

^G1

](

^{r log}1/pn

) +

^o

(

1

).

Now byTheorem 7.2, the variance satisfies

V(

Xn

)

n

=

^G

(−

1

)

h

+ F [

^G

](

^{r log}1/pn

) +

^o

(

1

),

where G

= Φ

1

+ Φ

2, as described inTheorem 7.2. Expressions can be derived for G. For brevity, consider only the symmetric case for which we have

G

(

s

) = Φ

1

(

s

) = Γ (

s

+

1

)



2^s+1

(

s

+

2

) −

^s2

+

^3s

+

⁶ 4



+

2^s+2



j1

( −

¹

)

^j

Γ (

s

+

^j

+

²

) (

j

−

¹

) !(

^2j

−

¹

) .

Note that the last series has the alternative form



j¹

(−

¹

)

^j

Γ (

s

+

^j

+

²

)

(

j

−

1

)!(

2^j

−

1

) = −Γ (

^s

+

³

) 

j¹ 1 2^j

(

1

+

2^−j

)

^3+s

.

Hence, the mean value of the periodic function is given by

1

+

³

4 log 2

+

² log 2



j¹ 1

2^j

(

1

+

2^−j

)

²

≈

⁰

.

361326059781678

. . .

which is the same as that obtained in[70]with a different expression (equating our expression with theirs gives the same identity(35)).

8. Conclusions

The prevalent appearance in diverse modeling contexts and high concentration of the binomial distribution make BSPs a distinctive subject full of featured properties and numerous extensions. Periodic oscillation is among the phenomena for which analytic tools proved to be a successful bridge between theory and practical observations. The analytic methodology developed in this paper, based specially on earlier works founded by Flajolet and his coauthors and aiming at clarifying the periodic oscillation of the variance, is itself easily amended for other circumstances, including particularly the case of quadratic shape measures such as the Wiener index (see [44]) or the analysis of partial-match queries (see[42]). The combination of Mellin analysis and Jacquet and Szpankowski’s analytic de-Poissonization (operated at the more abstract level of admissible functions) proves once again to be powerful tools for unriddling the intrinsic complexity of the asymptotic variance, and provides an efficient mechanical art of conjecturing and proving in more general contexts the structure of the variance. More developments will be discussed in a subsequent paper.

Acknowledgements

We thank both referees for their helpful and encouraging comments. The first author acknowledges partial support by the NSC under grant NSC-102-2115-M-009-002.

References

[1]D. Aldous, Probability distributions on cladograms, in: Random Discrete Structures, Minneapolis, MN, 1993, in: IMA Vol. Math. Appl., vol. 76, Springer, New York, 1996, pp. 1–18.

[2]C. Banderier, H.-K. Hwang, V. Ravelomanana, V. Zacharovas, Analysis of an exhaustive search algorithm in random graphs and the n^{c log n}-asymptotics, SIAM J. Discrete Math. (2014), submitted for publication.

[3]E. Biglieri, L. Györfi, Multiple Access Channels: Theory and Practice, NATO Security through Science Series – D: Information and Communication Security, IOS Press Inc., 2007.

[4]M.G.B. Blum, O. François, Minimal clade size and external branch length under the neutral coalescent, Adv. Appl. Probab. 37 (3) (2005) 647–662.

[5]J. Bourdon, M. Nebel, B. Vallée, On the stack-size of general tries, Theor. Inform. Appl. 35 (2) (2001) 163–185.

[6]R. Bradley, P. Strenski, Directed aggregation on the Bethe lattice: Scaling, mappings, and universality, Phys. Rev. B 31 (7) (1985) 4319.

[7]W.-M. Chen, H.-K. Hwang, Analysis in distribution of two randomized algorithms for finding the maximum in a broadcast communication model, J. Algorithms 46 (2) (2003) 140–177.

[8]C.A. Christophi, H.M. Mahmoud, Distribution of the size of random hash trees, pebbled hash trees and N-trees, Stat. Probab. Lett. 53 (3) (2001) 277–282.

[9]J. Clément, P. Flajolet, B. Vallée, Dynamical sources in information theory: a general analysis of trie structures, Algorithmica 29 (1–2) (2001) 307–369.

[10]L.-L. Cristea, H. Prodinger, Order statistics for the Cantor–Fibonacci distribution, Aequ. Math. 73 (1–2) (2007) 78–91.

[11]N.G. de Bruijn, D.E. Knuth, S.O. Rice, The average height of planted plane trees, in: Graph Theory and Computing, Academic Press, New York, 1972, pp. 15–22.

[12]R. De La Briandais, File searching using variable length keys, in: Papers Presented at the Western Joint Computer Conference, March 3–5, 1959, ACM, 1959, pp. 295–298.

[13]D.S. Dean, S.N. Majumdar, Phase transition in a generalized Eden growth model on a tree, J. Stat. Phys. 124 (6) (2006) 1351–1376.

[14]L. Devroye, Lecture Notes on Bucket Algorithms, Prog. Comput. Sci., vol. 6, Birkhäuser Boston Inc., Boston, MA, 1986.

[15]L. Devroye, Universal asymptotics for random tries and PATRICIA trees, Algorithmica 42 (1) (2005) 11–29.

[16]M. Drmota, B. Gittenberger, A. Panholzer, H. Prodinger, M.D. Ward, On the shape of the fringe of various types of random trees, Math. Methods Appl.

Sci. 32 (10) (2009) 1207–1245.

[17]M. Drmota, W. Szpankowski, A master theorem for discrete divide and conquer recurrences, in: Proceedings of the Twenty-Second Annual ACM–SIAM Symposium on Discrete Algorithms, Philadelphia, PA, 2011, SIAM, 2011, pp. 342–361.

[18]B. Eisenberg, On the expectation of the maximum of IID geometric random variables, Stat. Probab. Lett. 78 (2) (2008) 135–143.

[19]P. Erd ˝os, A. Hildebrand, A. Odlyzko, P. Pudaite, B. Reznick, The asymptotic behavior of a family of sequences, Pac. J. Math. 126 (2) (1987) 227–241.

[20]R. Fagin, J. Nievergelt, N. Pippenger, H.R. Strong, Extendible hashing – a fast access method for dynamic files, ACM Trans. Database Syst. 4 (3) (1979) 315–344.

[21]G. Fayolle, P. Flajolet, M. Hofri, On a functional equation arising in the analysis of a protocol for a multi-access broadcast channel, Adv. Appl. Probab.

18 (2) (1986) 441–472.

[22]G. Fayolle, P. Flajolet, M. Hofri, P. Jacquet, Analysis of a stack algorithm for random multiple-access communication, IEEE Trans. Inf. Theory 31 (2) (1985) 244–254.

[23]J.A. Fill, H.M. Mahmoud, W. Szpankowski, On the distribution for the duration of a randomized leader election algorithm, Ann. Appl. Probab. 6 (4) (1996) 1260–1283.

[24]P. Flajolet, On the performance evaluation of extendible hashing and trie searching, Acta Inform. 20 (4) (1983) 345–369.

[25]P. Flajolet, Approximate counting: a detailed analysis, BIT Numer. Math. 25 (1) (1985) 113–134.

[26]P. Flajolet, Évaluation de protocoles de communication: aspects mathématiques, in: Le Codage et la Transmission de l’Information: Journée annuelle de la Société Mathématique de France, Société Mathématique de France, 1988, pp. 1–22.

[27]P. Flajolet, Counting by coin tossings, in: M.J. Maher (Ed.), Advances in Computer Science – ASIAN 2004. Higher-Level Decision Making; Proceedings of the 9th Asian Computing Science Conference; Dedicated to Jean-Louis Lassez on the Occasion of His 5th Cycle Birthday, in: LNCS, vol. 3321, Springer, Berlin/Heidelberg, 2004, pp. 1–12.

[28]P. Flajolet, É. Fusy, O. Gandouet, F. Meunier, HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm, in: 2007 Conference on Analysis of Algorithms, AofA 07, Discrete Math. Theor. Comput. Sci. Proc., AH, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2007, pp. 127–145.

[29]P. Flajolet, X. Gourdon, P. Dumas, Mellin transforms and asymptotics: harmonic sums, Theor. Comput. Sci. 144 (1–2) (1995) 3–58.

[30]P. Flajolet, G.N. Martin, Probabilistic counting algorithms for data base applications, J. Comput. Syst. Sci. 31 (2) (1985) 182–209.

[31]P. Flajolet, M. Pelletier, M. Soria, On Buffon machines and numbers, in: Proceedings of the Twenty-Second Annual ACM–SIAM Symposium on Discrete Algorithms, Philadelphia, PA, 2011, SIAM, 2011, pp. 172–183.

[32]P. Flajolet, C. Puech, Partial match retrieval of multidimensional data, J. Assoc. Comput. Mach. 33 (2) (1986) 371–407.

[33]P. Flajolet, M. Régnier, R. Sedgewick, Some uses of the Mellin integral transform in the analysis of algorithms, in: Combinatorial Algorithms on Words, Maratea, 1984, in: NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., vol. 12, Springer, Berlin, 1985, pp. 241–254.

[34]P. Flajolet, M. Roux, B. Vallée, Digital trees and memoryless sources: from arithmetics to analysis, in: 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms, AofA’10, Discrete Math. Theor. Comput. Sci. Proc., AM, Assoc. Discrete Math.

Theor. Comput. Sci., Nancy, 2010, pp. 233–260.

[35]P. Flajolet, N. Saheb, The complexity of generating an exponentially distributed variate, J. Algorithms 7 (4) (1986) 463–488.

[36]P. Flajolet, R. Sedgewick, Mellin transforms and asymptotics: finite differences and Rice’s integrals, Theor. Comput. Sci. 144 (1–2) (1995) 101–124.

[37]P. Flajolet, R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009.

[38]P. Flajolet, D. Sotteau, A recursive partitioning process of computer science, in: A. Ballester, D. Cardús, E. Trillas (Eds.), Proceedings of the Second World Conference on Mathematics at the Service of Man, Las Palmas, Canary Islands, Spain, Universidad Politécnica de Las Palmas, 1982, pp. 25–30.

[39]P. Flajolet, J.-M. Steyaert, A branching process arising in dynamic hashing, trie searching and polynomial factorization, in: Automata, Languages and Programming, Aarhus, 1982, in: Lect. Notes Comput. Sci., vol. 140, Springer, Berlin, 1982, pp. 239–251.

[40]E. Fredkin, Trie memory, Commun. ACM 3 (September 1960) 490–499.

[41]M.L. Fredman, D.E. Knuth, Recurrence relations based on minimization, J. Math. Anal. Appl. 48 (1974) 534–559.

[42]M. Fuchs, The variance for partial match retrievals in k-dimensional bucket digital trees, in: 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms, AofA’10, Discrete Math. Theor. Comput. Sci. Proc., AM, Assoc. Discrete Math. Theor. Comput.

Sci., Nancy, 2010, pp. 261–275.

[43]M. Fuchs, The subtree size profile of plane-oriented recursive trees, in: ANALCO 11—Workshop on Analytic Algorithmics and Combinatorics, SIAM, Philadelphia, PA, 2011, pp. 85–92.

[44] M. Fuchs, C.-K. Lee, The Wiener index of random digital tries, 2012, submitted for publication.

[45]M. Fuchs, H. Prodinger, Words with a generalized restricted growth property, Indag. Math. 24 (4) (2013) 1024–1033 (special issue in memory of N.G.

de Bruijn).

[46]E. Gelenbe, R. Nelson, T. Philips, A. Tantawi, An approximation of the processing time for a random graph model of parallel computation, in: Proceed-ings of 1986 ACM Fall Joint Computer Conference, ACM ’86, Los Alamitos, CA, USA, 1986, IEEE Computer Society Press, 1986, pp. 691–697.

[47]M.T. Goodrich, D.S. Hirschberg, Improved adaptive group testing algorithms with applications to multiple access channels and dead sensor diagnosis, J. Comb. Optim. 15 (1) (2008) 95–121.

[48]P.J. Grabner, H. Prodinger, Asymptotic analysis of the moments of the Cantor distribution, Stat. Probab. Lett. 26 (3) (1996) 243–248.

[49]P.J. Grabner, H. Prodinger, Sorting algorithms for broadcast communications: mathematical analysis, Theor. Comput. Sci. 289 (1) (2002) 51–67.

[50]G.H. Hardy, Ramanujan. Twelve Lectures on Subjects Suggested by His Life and Work, Cambridge University Press, Cambridge, England, 1940.

[51]W.K. Hayman, A generalisation of Stirling’s formula, J. Reine Angew. Math. 196 (1956) 67–95.

[52]P. Hildebrandt, H. Isbitz, Radix exchange – an internal sorting method for digital computers, J. ACM 6 (2) (Apr. 1959) 156–163.

[53]F. Hubalek, H.-K. Hwang, W. Lew, H. Mahmoud, H. Prodinger, A multivariate view of random bucket digital search trees, J. Algorithms 44 (1) (2002) 121–158.

[54]D.R. Hush, C. Wood, Analysis of tree algorithms for rfid arbitration, in: 1998 IEEE Internat. Symp. Info. Th., IEEE, 1998, p. 107.

[55]H.-K. Hwang, M. Fuchs, V. Zacharovas, Asymptotic variance of random symmetric digital search trees, Discrete Math. Theor. Comput. Sci. 12 (2) (2010) 103–165.

[56]P. Jacquet, The part and try algorithm adapted to free channel access, Tech. Rep. RR-0436, INRIA, Rocquencourt, Aug. 1985.

[57]P. Jacquet, P. Muhlethaler, Marginal throughput of a stack algorithm for CSMA/CD random length packet communication when the load is over the channel efficiency, Tech. Rep. RR-1275, INRIA, Rocquencourt, Aug. 1990.

[58]P. Jacquet, M. Régnier, Limiting distributions for trie parameters, Rapports de Recherche 502, INRIA, Rocquencourt, 1986.

[59]P. Jacquet, M. Régnier, Trie partitioning process: limiting distributions, in: CAAP ’86, Nice, 1986, in: Lect. Notes Comput. Sci., vol. 214, Springer, Berlin, 1986, pp. 196–210.

[60]P. Jacquet, M. Régnier, Normal limiting distribution for the size and the external path length of tries, Tech. Rep. RR-0827, INRIA, Rocquencourt, Apr.

1988.

[61]P. Jacquet, W. Szpankowski, Asymptotic behavior of the Lempel–Ziv parsing scheme and [in] digital search trees, Theor. Comput. Sci. 144 (1–2) (1995) 161–197.

[62]P. Jacquet, W. Szpankowski, Analytical de-Poissonization and its applications, Theor. Comput. Sci. 201 (1–2) (1998) 1–62.

[63]S. Janson, Renewal theory in the analysis of tries and strings, Theor. Comput. Sci. 416 (2012) 33–54.

[64]S. Janson, W. Szpankowski, Analysis of an asymmetric leader election algorithm, Electron. J. Comb. 4 (1) (1997), Research Paper 17, 16 pp. (electronic).

[65]A.J.E.M. Janssen, M.J. de Jong, Analysis of contention tree algorithms, IEEE Trans. Inf. Theory 46 (6) (2000) 2163–2172.

[66]M.A. Kaplan, E. Gulko, Analytic properties of multiple-access trees, IEEE Trans. Inf. Theory 31 (2) (1985) 255–263.

[67]R. Kemp, The average number of registers needed to evaluate a binary tree optimally, Acta Inform. 11 (4) (1978/79) 363–372.

[68]P. Kirschenhofer, H. Prodinger, b-tries: a paradigm for the use of number-theoretic methods in the analysis of algorithms, in: Contributions to General Algebra, vol. 6, Hölder-Pichler-Tempsky, Vienna, 1988, pp. 141–154.

[69]P. Kirschenhofer, H. Prodinger, On some applications of formulae of Ramanujan in the analysis of algorithms, Mathematika 38 (1) (1991) 14–33.

[70]P. Kirschenhofer, H. Prodinger, W. Szpankowski, On the balance property of Patricia tries: external path length viewpoint, Theor. Comput. Sci. 68 (1) (1989) 1–17.

[71]P. Kirschenhofer, H. Prodinger, W. Szpankowski, On the variance of the external path length in a symmetric digital trie, Discrete Appl. Math. 25 (1–2) (1989) 129–143.

[72]P. Kirschenhofer, H. Prodinger, W. Szpankowski, Multidimensional digital searching and some new parameters in tries, Int. J. Found. Comput. Sci. 4 (1) (1993) 69–84.

[73]D.E. Knuth, The average time for carry propagation, Proc. K. Ned. Akad. Wet., Ser. A, Indag. Math. 40 (2) (1978) 238–242.

[74]D.E. Knuth, The Art of Computer Programming, vol. III, Sorting and Searching, 2nd ed., Addison-Wesley, 1998.

[75]G. Louchard, H. Prodinger, Asymptotics of the moments of extreme-value related distribution functions, Algorithmica 46 (3–4) (2006) 431–467.

[76]W.P. Maddison, M. Slatkin, Null models for the number of evolutionary steps in a character on a phylogenetic tree, Evolution (1991) 1184–1197.

[77]H. Mahmoud, P. Flajolet, P. Jacquet, M. Régnier, Analytic variations on bucket selection and sorting, Acta Inform. 36 (9–10) (2000) 735–760.

[78]H.M. Mahmoud, Evolution of Random Search Trees, Wiley–Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons Inc., New York, 1992.

[79]S.N. Majumdar, Traveling front solutions to directed diffusion-limited aggregation, digital search trees, and the Lempel–Ziv data compression algo-rithm, Phys. Rev. E 68 (2) (2003) 026103.

[80]J.L. Massey, Collision-resolution algorithms and random-access communications, in: G. Longo (Ed.), Multi-User Communication Systems, in: CISM Courses and Lectures, Springer, 1981, pp. 73–137.

[81]P. Mathys, P. Flajolet, Q -ary collision resolution algorithms in random-access systems with free or blocked channel access, IEEE Trans. Inf. Theory 31 (2) (1985) 217–243.

[82]R. Mellier, J.F. Myoupo, V. Ravelomanana, A non-token-based-distributed mutual exclusion algorithm for single-hop mobile ad hoc networks, in:

MWCN 2004, 2004, pp. 287–298.

[83]H. Mendelson, Analysis of extendible hashing, IEEE Trans. Softw. Eng. 6 (1982) 611–619.

[84]M.L. Molle, G.C. Polyzos, Conflict resolution algorithms and their performance analysis, Tech. Rep., Dept. CSE, UCSD, 1993.

[85]D.R. Morrison, Patricia—practical algorithm to retrieve information coded in alphanumeric, J. ACM 15 (4) (1968) 514–534.

[86]J.F. Myoupo, L. Thimonier, V. Ravelomanana, Average case analysis-based protocols to initialize packet radio networks, Wirel. Commun. Mob. Comput.

3 (4) (2003) 539–548.

[87]V. Namboodiri, L. Gao, Energy-aware tag anticollision protocols for rfid systems, IEEE Trans. Mob. Comput. 9 (1) (2010) 44–59.

[88]R. Neininger, L. Rüschendorf, A general limit theorem for recursive algorithms and combinatorial structures, Ann. Appl. Probab. 14 (1) (2004) 378–418.

[89]M. Nguyen-The, Distribution de valuations sur les arbres, PhD thesis, LIX, Ecole Polytechnique, 2004.

[90]P. Nicodème, B. Salvy, P. Flajolet, Motif statistics, Theor. Comput. Sci. 287 (2) (2002) 593–617.

[91]G. Park, H.-K. Hwang, P. Nicodème, W. Szpankowski, Profiles of tries, SIAM J. Comput. 38 (5) (2008/09) 1821–1880.

[92]W. Polasek, The Bernoullis and the origin of probability theory: Looking back after 300 years, Resonance 5 (8) (2000) 26–42.

[93]H. Prodinger, Combinatorial problems related to geometrically distributed random variables, in: Séminaire Lotharingien de Combinatoire, Gerolfingen, 1993, in: Prépubl. Inst. Rech. Math. Av. Univ., vol. 1993/34, Louis Pasteur, Strasbourg, 1993, pp. 87–95.

[94]H. Prodinger, How to select a loser, Discrete Math. 120 (1–3) (1993) 149–159.

[95]H. Prodinger, Periodic oscillations in the analysis of algorithms and their cancellations, J. Iran. Stat. Soc. 3 (2) (2004) 251–270.

[96]M. Régnier, P. Jacquet, New results on the size of tries, IEEE Trans. Inf. Theory 35 (1) (1989) 203–205.

[97]E.K. Ressler, Random list permutations in place, Inf. Process. Lett. 43 (5) (1992) 271–275.

[98]W. Schachinger, On the variance of a class of inductive valuations of data structures for digital search, Theor. Comput. Sci. 144 (1–2) (1995) 251–275.

[99]W. Schachinger, Limiting distributions for the costs of partial match retrievals in multidimensional tries, Random Struct. Algorithms 17 (3–4) (2000) 428–459.

[100]R. Sedgewick, Data movement in odd–even merging, SIAM J. Comput. 7 (3) (1978) 239–272.

[101]G. Shafer, The significance of Jacob Bernoulli’s Ars Conjectandi for the philosophy of probability today, J. Econom. 75 (1) (1996) 15–32.

[102]S.-H. Shiau, C.-B. Yang, A fast initialization algorithm for single-hop wireless networks, IEICE Trans. 88-B (11) (2005) 4285–4292.

[103]K. Simon, An improved algorithm for transitive closure on acyclic digraphs, Theor. Comput. Sci. 58 (1–3) (1988) 325–346.

[104]W. Szpankowski, Average Case Analysis of Algorithms on Sequences, Wiley-Interscience, New York, 2001.

[105]S. Wagner, On tries, contention trees and their analysis, Ann. Comb. 12 (4) (2009) 493–507.

[106]S. Wagner, On unary nodes in tries, in: 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algo-rithms, AofA’10, Discrete Math. Theor. Comput. Sci. Proc., AM, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010, pp. 577–589.

[107]C.-B. Yang, Reducing conflict resolution time for solving graph problems in broadcast communications, Inf. Process. Lett. 40 (6) (1991) 295–302.

在文檔中 An analytic approach to the asymptotic variance of trie statistics and related structures (頁 31-36)