## Quickselect and Dickman function (2000)

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Venue: | Combinatorics, Probability and Computing |

Citations: | 24 - 1 self |

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@ARTICLE{Hwang00quickselectand,

author = {Hsien-Kuei Hwang and Tsung-Hsi Tsai},

title = {Quickselect and Dickman function},

journal = {Combinatorics, Probability and Computing},

year = {2000},

volume = {11},

pages = {371}

}

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### Abstract

We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the m-th smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived. 1 Quickselect Quickselect is one of the simplest and e#cient algorithms in practice for finding specified order statistics in a given sequence. It was invented by Hoare [19] and uses the usual partitioning procedure of quicksort: choose first a partitioning key, say x; regroup the given sequence into two parts corresponding to elements whose values are less than and larger than x, respectively; then decide, according to the size of the smaller subgroup, which part to continue recursively or to stop if x is the desired order statistics; see Figure 1 for an illustration in terms of binary search trees. For more details, see Guibas [15] and Mahmoud [26]. This algorithm , although ine#cient in the worst case, has linear mean when given a sequence of n independent and identically distributed continuous random variables, or equivalently, when given a random permutation of n elements, where, here and throughout this paper, all n! permutations are equally likely. Let C n,m denote the number of comparisons used by quickselect for finding the m-th smallest element in a random permutation, where the first partitioning stage uses n 1 comparisons. Knuth [23] was the first to show, by some di#erencing argument, that E(C n,m ) = 2 (n + 3 + (n + 1)H n (m + 2)Hm (n + 3 -m)H n+1-m ) , n, where Hm = 1#k#m k -1 . A more transparent asymptotic approximation is E(C n,m ) (#), (#) := 2 #), # Part of the work of this author was done while he was visiting School of C...

### Citations

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Citation Context ...y likely. Let Cn,m denote the number of comparisons used by quickselect for finding the m-th smallest element in a random permutation, where the first partitioning stage uses n − 1 comparisons. Knut=-=h [23] was the first to sh-=-ow, by some differencing argument, that E(Cn,m) = 2 (n + 3 + (n + 1)Hn − (m + 2)Hm − (n + 3 − m)Hn+1−m) , for 1 ≤ m ≤ n, where Hm = � 1≤k≤m k−1 . A more transparent asymptotic appr... |

217 |
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Citation Context ... 1/u has the limiting density ρ(u) for u ≥ 1. The Dickman function plays an important role in analytic number theory, especially for problems in connection with the so-called psixylogy; see Tenenba=-=um [35]-=-, Hildebrand and Tenenbaum [17], and Moree [29] for further information and more instances. Besides its appearance and applications in number theory (see also Hirth [18]), the Dickman function also ar... |

55 |
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Citation Context ...ucible factors in random polynomials over finite fields (see Arratia et al. [2], Car [5], Knopfmacher and Manstavicius [22]), the size of the largest cycle in random permutations (see Shepp and Lloyd =-=[34],-=- Gourdon [11]), the sum of products of uniform random variables (see Goldie and Grübel [10], Devroye [7]), and allele frequencies in the infinitely-many neutral alleles diffusion model (see Watterson... |

45 | On the frequency of numbers containing prime factors of a certain relative magnitude, Ark - Dickman - 1930 |

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Citation Context ...(u) for u ≥ 1. The Dickman function plays an important role in analytic number theory, especially for problems in connection with the so-called psixylogy; see Tenenbaum [35], Hildebrand and Tenenbau=-=m [17]-=-, and Moree [29] for further information and more instances. Besides its appearance and applications in number theory (see also Hirth [18]), the Dickman function also arises in a large number of probl... |

35 |
Algorithm 65: Find
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(Show Context)
Citation Context ...f exchanges is also derived. 1 Quickselect Quickselect is one of the simplest and efficient algorithms in practice for finding specified order statistics in a given sequence. It was invented by Hoare =-=[19]-=- and uses the usual partitioning procedure of quicksort: choose first a partitioning key, say x; regroup the given sequence into two parts corresponding to elements whose values are less than and larg... |

35 |
Sorting: A Distribution Theory
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Citation Context ...up, which part to continue recursively or to stop if x is the desired order statistics; see Figure 1 for an illustration in terms of binary search trees. For more details, see Guibas [15] and Mahmoud =-=[26]-=-. This algorithm 1 , although inefficient in the worst case, has linear mean when given a sequence of n independent and identically distributed continuous random variables, or equivalently, when given... |

31 | Perpetuities with thin tails
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Citation Context ...opfmacher and Manstavicius [22]), the size of the largest cycle in random permutations (see Shepp and Lloyd [34], Gourdon [11]), the sum of products of uniform random variables (see Goldie and Grübel=-= [10]-=-, Devroye [7]), and allele frequencies in the infinitely-many neutral alleles diffusion model (see Watterson [36]). See also Arratia [1] and Arratia et al. [3] for a comprehensive survey on scale inva... |

29 |
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Citation Context ...anges used by quickselect or other versions of quickselect; see Section 5. Let Pn,m(y) = E(yCn,m ) denote the probability generating function of Cn,m. Then Pn,m satisfies the recurrence (see [21] and =-=[27]) Pn,m(y) = yn−1 ⎛ ⎝1 + n �-=- Pn−k,m−k(y) + � ⎞ Pk,m(y) ⎠ , (5) 1≤k<m m≤k<n for 1 ≤ m ≤ n, with the initial condition Pn,0(y) = δn,0, the Kronecker symbol. Proposition 3. For n ≥ 1, and, for 2 ≤ m ≤ n −... |

26 | Asymptotic distribution theory for Hoare’s selection algorithm
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Citation Context ...α) + 4α � 1 α log t dt + 4(1 − α) 1 − t � α 0 log(1 − t) t see also Paulsen [31]. Note that σ 2 (0) = σ 2 (1) = 1/2. The limiting distribution of Cn,m/n was studied independently by G=-=rübel and Rösler [14] a-=-nd Kodaj and Móri [24]; see also Grübel [12]. Although several (different) characterizations of the limiting distribution of Cn,m/n were derived, none of them is simple. Our aim of this paper is to ... |

25 |
Probability Theory, Fourth edition
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Citation Context ...ates that the moment sequence {gk} uniquely characterizes the distribution if � k g−1/(2k) 2k = ∞. The result (13) for m = 1 is thus proved by the Frechet-Shohat moment convergence theorem (see =-=Loève [25]). Note that the m-=-oment generating function G(z) = E(e W z ) satisfies Also uniformly in t. G(z) = � 1 0 G(xz)e x(1−x)z dx. � � �Tn,1(e it/n � � ) − G(it) � = o(1), (19) Proof of (5). Consider first t... |

23 | Analysis of Hoare’s FIND algorithm with median-of-three partition. Random Structures Algorithms 10
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(Show Context)
Citation Context ...z − 3)M ′ (z) + (z 2 − 6z + 6)M(z) = 6e z M(z), with the initial conditions M(0) = 1 and M ′ (0) = 2. Note that the mean is asymptotically the same as the quickselect proper; see Kirschenhofer=-= et al. [20], Gr�-=-�bel [13], Neininger [30] for related materials. Other Dickman “relatives”. Most examples we mentioned in the Introduction actually involve the Dickman function as the asymptotic distribution func... |

21 |
The stationary distribution of the infinitely-many neutral alleles diffusion model
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(Show Context)
Citation Context ... Gourdon [11]), the sum of products of uniform random variables (see Goldie and Grübel [10], Devroye [7]), and allele frequencies in the infinitely-many neutral alleles diffusion model (see Watterson=-= [36]-=-). See also Arratia [1] and Arratia et al. [3] for a comprehensive survey on scale invariant Poisson processes in which Dickman function appeared in several different forms. Our example of quickselect... |

14 |
algorithmique et géometrie des polynômes
- Gourdon, Combinatoire
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(Show Context)
Citation Context ... in random polynomials over finite fields (see Arratia et al. [2], Car [5], Knopfmacher and Manstavicius [22]), the size of the largest cycle in random permutations (see Shepp and Lloyd [34], Gourdon =-=[11])-=-, the sum of products of uniform random variables (see Goldie and Grübel [10], Devroye [7]), and allele frequencies in the infinitely-many neutral alleles diffusion model (see Watterson [36]). See al... |

13 |
On the median-of-k version of Hoare’s selection algorithm. Theor
- Grübel
- 1999
(Show Context)
Citation Context ... + (z 2 − 6z + 6)M(z) = 6e z M(z), with the initial conditions M(0) = 1 and M ′ (0) = 2. Note that the mean is asymptotically the same as the quickselect proper; see Kirschenhofer et al. [20], Gr�=-=�bel [13], Ne-=-ininger [30] for related materials. Other Dickman “relatives”. Most examples we mentioned in the Introduction actually involve the Dickman function as the asymptotic distribution function instead ... |

13 |
A generating functions approach for the analysis of grand averages for Multiple Quickselect. Random Structures Algorithms 13 189–209. MR1662782 Institut für Diskrete Mathematik und Geometrie Technische Universität Wien Wiedner Hauptstraße 8-10 A-1040 Wien
- Panholzer, Prodinger
- 1998
(Show Context)
Citation Context ...0). By the same approach, it can be shown that the multiple quickselect problems in Mahmoud and Smythe [28] introduce higher convolutions of the Dickman distribution; see also Panholzer and Prodinger =-=[32]-=-. Quickselect with median-of-three. If instead of choosing a random element to partition the random input, we take the median of three random elements as the partitioning key, then the total cost is i... |

12 |
Quicksort
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- 1980
(Show Context)
Citation Context ...≤ m ≤ n, where Vn,j(y) denotes the probability generating function of the number of exchanges used to partition the random permutation into two parts when the partitioning element is j (see Sedgew=-=ick [33]): � Vn,j(y) = � s � �� n−j -=-j−1 s s �n−1 j−1 � y s (1 ≤ j ≤ n), since � � � � j−1 n−1 s / j−1 is the probability that there are exactly s elements whose indices are less than the rank of the partition... |

9 | Logarithmic Combinatorial structures: A - Arratia, Barbour, et al. - 2003 |

9 | 2001a) Simulating perpetuities
- Devroye
(Show Context)
Citation Context ...Manstavicius [22]), the size of the largest cycle in random permutations (see Shepp and Lloyd [34], Gourdon [11]), the sum of products of uniform random variables (see Goldie and Grübel [10], Devroye=-= [7]-=-), and allele frequencies in the infinitely-many neutral alleles diffusion model (see Watterson [36]). See also Arratia [1] and Arratia et al. [3] for a comprehensive survey on scale invariant Poisson... |

9 |
The moments of FIND
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- 1997
(Show Context)
Citation Context ...2 α + 4(α 2 − α − 1) log α log(1 − α) − 2(1 − α) 2 log 2 (1 − α) − 4α log α − 4(1 − α) log(1 − α) + 2 3 π2α(1 − α) + 5α(1 − α) + 4α � 1 α log t dt + 4(1 −=-= α) 1 − t � α 0 log(1 − t) t see also Paulsen [31]. Note-=- that σ 2 (0) = σ 2 (1) = 1/2. The limiting distribution of Cn,m/n was studied independently by Grübel and Rösler [14] and Kodaj and Móri [24]; see also Grübel [12]. Although several (different)... |

8 | The Poisson-Dirichlet distribution and the scale-invariant Poisson process
- Arratia, Barbour, et al.
- 1999
(Show Context)
Citation Context ...random variables (see Goldie and Grübel [10], Devroye [7]), and allele frequencies in the infinitely-many neutral alleles diffusion model (see Watterson [36]). See also Arratia [1] and Arratia et al.=-= [3]-=- for a comprehensive survey on scale invariant Poisson processes in which Dickman function appeared in several different forms. Our example of quickselect is a new addition to this list of Dickman fun... |

8 |
A probabilistic approach to a differentialdifference equation arising in analytic number theory
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- 1973
(Show Context)
Citation Context ...e, yielding Cn,1 D ∼ n (1 + U1 + U1U2 + · · · ) , where the Uj’s are iid uniform [0, 1] random variables. The limiting distribution of the right-hand side is Dickman; see Arratia et al. [3], Ch=-=amayou [6], Devroye-=- [7]. For 2 ≤ m = o(n), we observe that the distribution of the sum � ξ(i, j) m≤i,j≤n is identical to the distribution of the number of comparisons used to find the smallest element in the se... |

8 |
Probabilistic analysis of multiple Quickselect
- Mahmoud, Smythe
- 1998
(Show Context)
Citation Context ...− n n � < x = e −2γ � x 0 � it 0 h + 2y h−1 h + 2 e v − 1 v � dv , (n ≥ 1). ρ(v)ρ(x − v) dv (x > 0). By the same approach, it can be shown that the multiple quickselect problems=-= in Mahmoud and Smythe [28]-=- introduce higher convolutions of the Dickman distribution; see also Panholzer and Prodinger [32]. Quickselect with median-of-three. If instead of choosing a random element to partition the random inp... |

8 |
Limit Laws for Random Recursive Structures and Algorithms
- Neininger
- 1999
(Show Context)
Citation Context ...)M(z) = 6e z M(z), with the initial conditions M(0) = 1 and M ′ (0) = 2. Note that the mean is asymptotically the same as the quickselect proper; see Kirschenhofer et al. [20], Grübel [13], Neining=-=er [30] for-=- related materials. Other Dickman “relatives”. Most examples we mentioned in the Introduction actually involve the Dickman function as the asymptotic distribution function instead of as the limiti... |

7 |
On random polynomials over finite fields
- Arratia, Barbour, et al.
- 1993
(Show Context)
Citation Context ...y (see also Hirth [18]), the Dickman function also arises in a large number of problems like the degree of the largest irreducible factors in random polynomials over finite fields (see Arratia et al. =-=[2]-=-, Car [5], Knopfmacher and Manstavicius [22]), the size of the largest cycle in random permutations (see Shepp and Lloyd [34], Gourdon [11]), the sum of products of uniform random variables (see Goldi... |

7 |
Hoare’s selection algorithm: a Markov chain approach
- Grübel
- 1998
(Show Context)
Citation Context ...log(1 − t) t see also Paulsen [31]. Note that σ 2 (0) = σ 2 (1) = 1/2. The limiting distribution of Cn,m/n was studied independently by Grübel and Rösler [14] and Kodaj and Móri [24]; see also =-=Grübel [12]. -=-Although several (different) characterizations of the limiting distribution of Cn,m/n were derived, none of them is simple. Our aim of this paper is to show that when 1 ≤ m = o(n), the limiting dist... |

6 | Probability Theory, 4th edn - Loeve - 1977 |

5 | Comparisons in Hoare’s Find algorithm - Kirschenhofer, Prodinger - 1998 |

4 |
On the central role of scale invariant Poisson processes on (0, ∞). Microsurveys in discrete probability
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(Show Context)
Citation Context ...of products of uniform random variables (see Goldie and Grübel [10], Devroye [7]), and allele frequencies in the infinitely-many neutral alleles diffusion model (see Watterson [36]). See also Arratia=-= [1]-=- and Arratia et al. [3] for a comprehensive survey on scale invariant Poisson processes in which Dickman function appeared in several different forms. Our example of quickselect is a new addition to t... |

4 | T.F.(1997) On the number of comparisons in Hoare’s algorithm “FIND - Kodaj, Móri |

3 | Theoremes de densite dans F q [X - Car - 1987 |

2 |
A principle of independence for binary tree searching
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(Show Context)
Citation Context ...he smaller subgroup, which part to continue recursively or to stop if x is the desired order statistics; see Figure 1 for an illustration in terms of binary search trees. For more details, see Guibas =-=[15]-=- and Mahmoud [26]. This algorithm 1 , although inefficient in the worst case, has linear mean when given a sequence of n independent and identically distributed continuous random variables, or equival... |

2 |
The convolution powers of the Dickman function
- Hensley
- 1986
(Show Context)
Citation Context ... the density. Let Z be a random variable with the Dickman distribution. Some known properties of this distribution are listed as follows. 1. The distribution of Z is infinitely divisible; see Hensley =-=[16]. 2. The moment generating func-=-tion of Z satisfies E(e sZ �� s e ) = exp v � − 1 dv = − v e−γ s exp the second equality holding for s ∈ C \ [0, ∞). 0 �� ∞ −s v −1 e −v � dv , (1) 3. The k-th cumulant ... |

2 |
Psixylogy and Diophantine Equations. Dissertation, Rijksuniversiteit te
- Moree
- 1993
(Show Context)
Citation Context ...he Dickman function plays an important role in analytic number theory, especially for problems in connection with the so-called psixylogy; see Tenenbaum [35], Hildebrand and Tenenbaum [17], and Moree =-=[29]-=- for further information and more instances. Besides its appearance and applications in number theory (see also Hirth [18]), the Dickman function also arises in a large number of problems like the deg... |

2 | A probabilistic approach to a dierential-dierence equation arising in analytic number theory - Chamayou - 1973 |

2 | The stationary distribution of the in neutral alleles diusion model - Watterson - 1976 |

1 |
Théorèmes de densité dans Fq[X
- Car
- 1987
(Show Context)
Citation Context ...so Hirth [18]), the Dickman function also arises in a large number of problems like the degree of the largest irreducible factors in random polynomials over finite fields (see Arratia et al. [2], Car =-=[5],-=- Knopfmacher and Manstavicius [22]), the size of the largest cycle in random permutations (see Shepp and Lloyd [34], Gourdon [11]), the sum of products of uniform random variables (see Goldie and Grü... |

1 | Exact simulation of random variables that are solutions of fixed-point equations
- Devroye, Neininger
- 2001
(Show Context)
Citation Context ...U and W being independent. W D = UW + U(1 − U), The distribution is uniquely characterized by its moments and not Dickman; see Figure 3. For simulations of the limit law W , see Devroye and Neininge=-=r [8]-=-. Our approach is to first prove the result (13) for m = 1, then to show that Wn,m and Wn,1 are close in distribution. 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.1 0.2 0.3 Figure 3: Histo... |

1 |
Probabilistic number theory, the GEM/Poisson-Dirichlet distribution and the arc-sine law
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- 1997
(Show Context)
Citation Context ...alled psixylogy; see Tenenbaum [35], Hildebrand and Tenenbaum [17], and Moree [29] for further information and more instances. Besides its appearance and applications in number theory (see also Hirth =-=[18]-=-), the Dickman function also arises in a large number of problems like the degree of the largest irreducible factors in random polynomials over finite fields (see Arratia et al. [2], Car [5], Knopfmac... |

1 | On the largest degree of an irreducible factor of a polynomial
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- 1997
(Show Context)
Citation Context ...on also arises in a large number of problems like the degree of the largest irreducible factors in random polynomials over finite fields (see Arratia et al. [2], Car [5], Knopfmacher and Manstavicius =-=[22])-=-, the size of the largest cycle in random permutations (see Shepp and Lloyd [34], Gourdon [11]), the sum of products of uniform random variables (see Goldie and Grübel [10], Devroye [7]), and allele ... |

1 |
On the number of comparisons
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- 1997
(Show Context)
Citation Context ... 4(1 − α) 1 − t � α 0 log(1 − t) t see also Paulsen [31]. Note that σ 2 (0) = σ 2 (1) = 1/2. The limiting distribution of Cn,m/n was studied independently by Grübel and Rösler [14] and K=-=odaj and Móri [24]; s-=-ee also Grübel [12]. Although several (different) characterizations of the limiting distribution of Cn,m/n were derived, none of them is simple. Our aim of this paper is to show that when 1 ≤ m = o... |

1 | On random polynomials over - Arratia, Barbour, et al. - 1993 |

1 | Analysis of Hoare's algorithm with median-of-three partition. Random Structures and Algorithms - Kirschenhofer, Martnez, et al. - 1997 |