SOUZA, R. M.; OLIVEIRA, F. S.; PINTO, P. E. D.

doi:10.5540/tema.2019.020.03.0457

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Articles • TEMA (São Carlos) 20 (3) • Sep-Dec 2019 • https://doi.org/10.5540/tema.2019.020.03.0457 copy

Uma Nova Proposta para a Obtenção da Complexidade de Pior Caso do ShellSort

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

The worst-case time-complexity of the ShellSort, a comparison sorting algorithm, depends on a sequence of steps given as input. Each step consists of an integer representing an index offset of the pairs of elements that should be compared during the sorting of an input array. Such a complexity is known only to some specific sequences. In this work, we use a relation between the ShellSort and Frobenius number to present a new algorithm that provides an upper bound on the number of comparisons that ShellSort performs, for any given array and sequence of steps. We apply this algorithm, together with an empirical complexity analysis, to study sequences whose worst-case complexities are known through analytical method. We show that the empirical approach succeeded in determining such complexities. Based on such positive results, we extend the study to sequences for which the worst-case complexities are unknown.

Key words
complexity of algorithms; empirical method; ShellSort

Autor, Ano	Passos ${p_{1}, \dots, p_{k}}$	Pior caso
Shell, 1959^[16][16] D.L. Shell. A high-speed sorting procedure. Communications of the ACM, 2(7) (1959), 30–32.	${⌊ n / 2^{i} ⌋ : 0 < i \leq ⌊ {log}_{2} n ⌋}$	$Θ (n^{2})$
	Ex.: ${1, 2, 5, 10, 21}$ , para $n = 43$	$Θ (n^{2})$
Frank e Lazarus, 1960^[5][5] R.M. Frank & R.B. Lazarus. A high-speed sorting procedure. Communications of the ACM, 3(1) (1960), 20–22.	${2 \cdot ⌊ n / 2^{i} ⌋ + 1 : 1 < i \leq ⌊ {log}_{2} n ⌋}$	$Θ (n^{3 / 2})$
	Ex.: ${1, 3, 5, 11, 21}$ , para $n = 43$	$Θ (n^{3 / 2})$
Hibbard, 1963^[8][8] T.N. Hibbard. An empirical study of minimal storage sorting. Communications of the ACM, 6(5) (1963), 206–213.	${2^{i} - 1 : 0 < i \leq ⌊ {log}_{2} n ⌋}$	$Θ (n^{3 / 2})$
	${1, 3, 7, 15, 31, \dots}$	$Θ (n^{3 / 2})$
Papernov e Stasevich, 1965^[12][12] A.A. Papernov & G.V. Stasevich. A method of information sorting in computer memories. Problemy Peredachi Informatsii, 1(3) (1965), 81–98.	${1} \cup {2^{i - 1} + 1 : 1 < i \leq ⌊ {log}_{2} n ⌋}$	$Θ (n^{3 / 2})$
	${1, 3, 5, 9, 17, \dots}$	$Θ (n^{3 / 2})$
Pratt, 1972^[14][14] V.R. Pratt. “Shellsort and sorting networks”. Ph.D. thesis, Universidade de Stanford (1972).	${q = 2^{r} 3^{s} : r, s \geq 0 ∣ q < n}$	$Θ (n {log}^{2} n)$
	${1, 2, 3, 4, 6, 8, 9, 12, \dots}$	$Θ (n {log}^{2} n)$
Knuth, 1973^[10][10] D.E. Knuth. “The Art of Computer Programming 3: Sorting and Searching”. Addison-Wesley, EUA (1998).	${q = (3^{i} - 1) / 2 : i > 0 ∣ q \leq ⌈ n / 3 ⌉}$	$Θ (n^{3 / 2})$
	${1, 4, 13, 40, 121, \dots}$	$Θ (n^{3 / 2})$
Incerpi e Sedgewick, 1983^[9][9] J. Incerpi & R. Sedgewick. Improved upper bounds on Shellsort. In “Anais do 24º Annual Symposium on Foundations of Computer Science, 1983”. IEEE (1983), pp. 48–55.	${p_{1} = 1} \cup {p_{i + 1} = p_{i - r + 1} q_{r} :$	$O (n^{1 + \sqrt{\frac{8 \ln (5 / 2)}{\ln n}}})$
	$i > 0$ , onde
	$r$ tal que $(\binom{r}{2}) < i \leq (\binom{r + 1}{2})$ ,
	onde $(\binom{r}{s}) = 0$ , se $r < s$ , e
	$q_{r}$ é o menor inteiro $\geq (5 / 2)^{r}$
	tal que $M D C (q_{r}, q_{r - 1}, \dots, q_{1}) = 1}$
	${q_{1} = 3, q_{2} = 7, q_{3} = 16, q_{4} = 41, \dots}$
	${1, 3, 7, 21, 48, \dots}$
Sedgewick, 1986^[15][15] R. Sedgewick. A new upper bound for ShellSort. Journal of Algorithms, 7(2) (1986), 159–173.	${1} \cup {q = 4^{i} + 3 \cdot 2^{i - 1} + 1 :$	$O (n^{4 / 3})$
	$i > 0 ∣ q < n}$
	${1, 8, 23, 77, 281, \dots}$
Gonnet e Baeza-Yates, 1991^[6][6] G.H. Gonnet & R.A. Baeza-Yates. “Handbook of Algorithms and Data Structures: in Pascal and C”. Addison-Wesley, EUA (1991).	${⌊ (0, 45454)^{i} \cdot n ⌋ :$	em aberto
	$0 < i \leq ⌊ {log}_{2, 2} n ⌋}$
	Ex.: ${1, 3, 8, 19}$ , para $n = 43$
Tokuda, 1992^[19][19] N. Tokuda. An improved Shellsort. Anais do 12º IFIP World Computer Congress on Algorithms, Software, Architecture-Information Processing, 1 (1992), 449–457.	${q = ⌈ (9^{i} - 4^{i}) / (5 \cdot 4^{i - 1}) ⌉ :$	em aberto
	$i > 0 ∣ q < n}$
	${1, 4, 9, 11, 26, \dots}$
Ciura, 2001^[2][2] M. Ciura. Best increments for the average case of shellsort. In “Fundamentals of Computation Theory”. Springer (2001), pp. 106–117.	${1, 4, 10, 23, 57, 132, 301, 701, 1750}$	em aberto
	(sequência finita)	em aberto

ShellSort							InsertionSort
$p_{3} = 4$	$\underset{̲}{𝟓}$	$6$	$1$	$4$	$\underset{̲}{𝟑}$	$2$	$\underset{̲}{5}$	$\underset{̲}{6}$	$1$	$4$	$3$	$2$
$p_{3} = 4$	$3$	$\underset{̲}{𝟔}$	$1$	$4$	$5$	$\underset{̲}{𝟐}$	$5$	$\underset{̲}{𝟔}$	$\underset{̲}{𝟏}$	$4$	$3$	$2$
$p_{2} = 2$	$\underset{̲}{𝟑}$	$2$	$\underset{̲}{𝟏}$	$4$	$5$	$6$	$\underset{̲}{𝟓}$	$\underset{̲}{𝟏}$	$6$	$4$	$3$	$2$
	$1$	$\underset{̲}{2}$	$3$	$\underset{̲}{4}$	$5$	$6$	$1$	$5$	$\underset{̲}{𝟔}$	$\underset{̲}{𝟒}$	$3$	$2$
	$1$	$2$	$\underset{̲}{3}$	$4$	$\underset{̲}{5}$	$6$	$1$	$\underset{̲}{𝟓}$	$\underset{̲}{𝟒}$	$6$	$3$	$2$
	$1$	$2$	$3$	$\underset{̲}{4}$	$5$	$\underset{̲}{6}$	$\underset{̲}{1}$	$\underset{̲}{4}$	$5$	$6$	$3$	$2$
$p_{1} = 1$	$\underset{̲}{1}$	$\underset{̲}{2}$	$3$	$4$	$5$	$6$	$1$	$4$	$5$	$\underset{̲}{𝟔}$	$\underset{̲}{𝟑}$	$2$
	$1$	$\underset{̲}{2}$	$\underset{̲}{3}$	$4$	$5$	$6$	$1$	$4$	$\underset{̲}{𝟓}$	$\underset{̲}{𝟑}$	$6$	$2$
	$1$	$2$	$\underset{̲}{3}$	$\underset{̲}{4}$	$5$	$6$	$1$	$\underset{̲}{𝟒}$	$\underset{̲}{𝟑}$	$5$	$6$	$2$
	$1$	$2$	$3$	$\underset{̲}{4}$	$\underset{̲}{5}$	$6$	$\underset{̲}{1}$	$\underset{̲}{3}$	$4$	$5$	$6$	$2$
	$1$	$2$	$3$	$4$	$\underset{̲}{5}$	$\underset{̲}{6}$	$1$	$3$	$4$	$5$	$\underset{̲}{𝟔}$	$\underset{̲}{𝟐}$
							$1$	$3$	$4$	$\underset{̲}{𝟓}$	$\underset{̲}{𝟐}$	$6$
							$1$	$3$	$\underset{̲}{𝟒}$	$\underset{̲}{𝟐}$	$5$	$6$
							$1$	$\underset{̲}{𝟑}$	$\underset{̲}{𝟐}$	$4$	$5$	$6$
							$\underset{̲}{1}$	$\underset{̲}{2}$	$3$	$4$	$5$	$6$

	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
Início	V	F	F	F	F	F	F	F	F	F	F	F	F
$5$	V	F	F	F	F	V	F	F	F	F	V	F	F
$7$	V	F	F	F	F	V	F	V	F	F	V	F	V
$9$	V	F	F	F	F	V	F	V	F	V	V	F	V
	$13$	$14$	$15$	$16$	$17$	$18$	$19$	$20$	$21$	$22$	$23$
Início	F	F	F	F	F	F	F	F	F	F	F
$5$	F	F	V	F	F	F	F	V	F	F	F
$7$	F	V	V	F	V	F	V	V	F	V	F
$9$	F	V	V	V	V	V	V	V	V	V	V
Retorno	13

Sequência	Analítico	Empírico
Shell, 1959 ^[16][16] D.L. Shell. A high-speed sorting procedure. Communications of the ACM, 2(7) (1959), 30–32.	$Θ (n^{2})$
${1, 2, 4, 8, 16, 32, 64, 128, 256, 512, \dots}$		$O (n^{2})$
${1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, \dots}$		$O (n^{1, 5})$
Frank e Lazarus, 1960 ^[5][5] R.M. Frank & R.B. Lazarus. A high-speed sorting procedure. Communications of the ACM, 3(1) (1960), 20–22.	$Θ (n^{1, 5})$
${1, 3, 5, 9, 17, 33, 65, 129, 257, 513, \dots}$		$O (n^{1, 5})$
${1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, \dots}$		$O (n^{1, 5})$
Hibbard, 1963 ^[8][8] T.N. Hibbard. An empirical study of minimal storage sorting. Communications of the ACM, 6(5) (1963), 206–213.	$Θ (n^{1, 5})$
${1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, \dots}$		$O (n^{1, 5})$
Papernov e Stasevich, 1965 ^[12][12] A.A. Papernov & G.V. Stasevich. A method of information sorting in computer memories. Problemy Peredachi Informatsii, 1(3) (1965), 81–98.	$Θ (n^{1, 5})$
${1, 3, 5, 9, 17, 33, 65, 129, 257, 513, \dots}$		$O (n^{1, 5})$
Pratt, 1972 ^[14][14] V.R. Pratt. “Shellsort and sorting networks”. Ph.D. thesis, Universidade de Stanford (1972).	$Θ (n {log}^{2} n)$
${1, 2, 3, 4, 6, 8, 9, 12, 16, 18, \dots}$		$O (n {log}^{2} n)$
Knuth, 1973 ^[10][10] D.E. Knuth. “The Art of Computer Programming 3: Sorting and Searching”. Addison-Wesley, EUA (1998).	$Θ (n^{1, 5})$
${1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524, \dots}$		$O (n^{1, 5})$
Incerpi e Sedgewick, 1983 ^[9][9] J. Incerpi & R. Sedgewick. Improved upper bounds on Shellsort. In “Anais do 24º Annual Symposium on Foundations of Computer Science, 1983”. IEEE (1983), pp. 48–55.	$O (n^{1, 597})$
${1, 3, 7, 21, 48, 112, 336, 861, 1968, 4592, \dots}$		$O (n^{1, 411})$
Sedgewick, 1986 ^[15][15] R. Sedgewick. A new upper bound for ShellSort. Journal of Algorithms, 7(2) (1986), 159–173.	$O (n^{1, 333 \dots})$
${1, 8, 23, 77, 281, 1073, 4193, 16577, 65921, \dots}$		$O (n^{1, 329})$
Gonnet e Baeza-Yates, 1991 ^[6][6] G.H. Gonnet & R.A. Baeza-Yates. “Handbook of Algorithms and Data Structures: in Pascal and C”. Addison-Wesley, EUA (1991).	em aberto
${1, 2, 5, 12, 27, 60, 133, 293, 645, 1420, \dots}$		$O (n^{1, 145})$ ,
		$O (n {log}^{3} n)$
${1, 4, 11, 26, 59, 132, 292, 644, 1419, 3124, \dots}$		$O (n^{1, 129})$
		$O (n {log}^{3} n)$
Tokuda, 1992 ^[19][19] N. Tokuda. An improved Shellsort. Anais do 12º IFIP World Computer Congress on Algorithms, Software, Architecture-Information Processing, 1 (1992), 449–457.	em aberto
${1, 4, 9, 20, 46, 103, 233, 525, 1182, 2660, \dots}$		$O (n^{1, 255})$ ,
		$O (n {log}^{5, 5} n)$
Ciura, 2001 ^[2][2] M. Ciura. Best increments for the average case of shellsort. In “Fundamentals of Computation Theory”. Springer (2001), pp. 106–117.	em aberto
${1, 4, 10, 23, 57, 132, 301, 701, 1750, 3937, \dots}$		$O (n^{1, 372})$

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[1] *Autor correspondente: Raquel Marcolino de Souza – E-mail: raquelmarcolino25@gmail.com – https://orcid.org/0000-0002-8834-9172