H.-K. Hwang and J.-M. Steyaert, On the number of heaps and the cost of heap construction, Colloquim on Mathematics andComputer Science: Algorithms, Trees, Combinatorics and Probabilities, Versailles, September 16 - 19,  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1 2 log 2 with k = 2ki : 8) In general, one might expect similar fractal fluctuations for the behaviour of sums of the type j0 f(fk=2 g) f being certain continuous function. Many such sums arise in the analysis of the number of heaps and the cost of constructing heaps, cf. [8]. Mellin transforms seem to be a pleasant tool for expliciting the inherent oscillating behaviours. To calculate the values of T n and S n : T n L n =2, we can use the following recurrences derived from (7) T n = T bn=2c T dn=2e L bn=2c bn=2c T 1 = 0; S n = S bn=2c S dn=2e ....

H.-K. Hwang and J.-M. Steyaert, On the number of heaps and the cost of heap constructions, submitted. 7

....in several problems. On the other hand, the balanced power of two rule appeared in considerably fewer problems, its usefulness being usually neglected. We briefly indicate some of the major problems in which this rule appeared. The associated recurrence is essentially the heap recurrence (see [9] or next section for details) 2 2n 3# f n 2 #log 2 2n 3# g n (n (with f 1 given) since it corresponds modulo shift to the sizes of the left and right sub heaps of a heap of size n. The recurrence was first (implicitly) studied by Knuth [10] cf. 9] It appeared as the solution to the ....

....essentially the heap recurrence (see [9] or next section for details) 2 2n 3# f n 2 #log 2 2n 3# g n (n (with f 1 given) since it corresponds modulo shift to the sizes of the left and right sub heaps of a heap of size n. The recurrence was first (implicitly) studied by Knuth [10] cf. [9]) It appeared as the solution to the recurrence with minimization (cf. Hammersley and Grimmett [7] f 1 : 0 and f n : min 1#j n f j f n j g n , 1) where g n is increasing and concave; cf. also [1, 14, 16] Karp and Glassey [4] independently discussed this rule referred to as ....

[Article contains additional citation context not shown here]

H.-K. Hwang and J.-M. Steyaert, On the number of heaps and the cost of heap construction, Theoretical Computer Science, to appear.

....2 log 2 with # k = 2k#i . 8) In general, one might expect similar fractal fluctuations for the behaviour of sums of the type j#0 f( k 2 ) f being certain continuous function. Many such sums arise in the analysis of the number of heaps and the cost of constructing heaps, cf. [8]. Mellin transforms seem to be a pleasant tool for expliciting the inherent oscillating behaviours. To calculate the values of T n and S n : T n L n 2, we can use the following recurrences derived from (7) # T n = T T L #n 2# T 1 = 0; and # S n = S S (n = odd) ....

H.-K. Hwang and J.-M. Steyaert, On the number of heaps and the cost of heap constructions, submitted. 7

....see Section 4. Both sequences (n) and (n) can be expressed completely in terms of g(n) as follows. For n 1 h 2 o 1 k i ; 7) n) hj k ) g 1 o i ; 8) see Hammersley and Grimmett [22] Hwang [31] Hwang and Steyaert [32]. From the recursive de nitions and these expressions, one easily derives the following bounds. Corollary 1. If g(n) is nondecreasing, then (n) 9) 5 ) n) g(n) 10) It is interesting to compare the corresponding terms in both bounds. Proposition ....

H.-K. Hwang and J.-M. Steyaert, On the number of heaps and the cost of heap construction, Colloquium on Mathematics and Computer Science, Versailles, September, 2002, accepted; available via the linl http://algo.stat.sinica.edu.tw.

....to F (n) and are much shorter; see Section 4. Both sequences #(n) and #(n) can be expressed completely in terms of g(n) as follows. For n 2 , 7) g ; 8) see Hammersley and Grimmett [22] Hwang [31] Hwang and Steyaert [32]. From the recursive definitions and these expressions, one easily derives the following bounds. Corollary 1. If g(n) is nondecreasing, then , 9) 5 ) 10) It is interesting to compare the corresponding terms in both bounds. Proposition 2. If g(n) is nondecreasing then ....

H.-K. Hwang and J.-M. Steyaert, On the number of heaps and the cost of heap construction, Colloquium on Mathematics and Computer Science, Versailles, September, 2002, accepted; available via the linl http://algo.stat.sinica.edu.tw.

....a direct consequence of our general assumptions on the moment generating function, we derive an asymptotic expression for moments of integral order. Then, we state an e#ective version of the central limit theorem of Haigh [16] by establishing the convergence rate) This theorem is borrowed from [21] where we applied it to derive the asymptotic normality of the cost of constructing a random heap. Examples from combinatorial enumerations, computer algorithms, probabilistic number theory, arithmetical semigroups and orthogonal polynomials will be discussed in the final section. Throughout this ....

....exponential convergence rate for M n (s) Remark. For Bender s central limit theorem [1, Thm 1] the convergence rate is O n , provided that the function A(s) in that theorem has bounded derivative for s near 0. 4 A classical central limit theorem The following theorem is borrowed from [21] where we used it to derive the asymptotic normality of the cost of constructing a random heap. Since the proof has already been given there, it is omitted here. Without the explicit error term it is due to Haight [16] The object of study here is sums of independent but not identically ....

[Article contains additional citation context not shown here]

H.-K. Hwang and J.-M. Steyaert. On the number of heaps and the cost of heap construction. Theoretical Computer Science, to appear.

....mergesort: a(n) 2 #log . Here Y (r, s) denotes the merge cost. Only the case (ii) needs some explanations: Indeed, since the underlying merge tree in queue mergesort is essentially a heap structure, the expression for a(n) can be easily verified. For asymptotics of such recurrences, see [9] for details. Note that if a(n) 1 we have an ine#cient implementation of the insertion sort. It seems of interest to ask the following question: fix Y , find su#cient conditions on a(n) so that X n is asymptotically normally distributed. Such considerations would provide further insight on the ....

H. K. Hwang and J.-M. Steyaert, On the number of heaps and the cost of heap construction, accepted for publication in Theoretical Computer Science.

....in several problems. On the other hand, the balanced power of two rule appeared in considerably fewer problems, its usefulness being usually neglected. We briefly indicate some of the major problems in which this rule appeared. The associated recurrence is essentially the heap recurrence (see [9] or next section for details) blog 2 2n=3c f blog 2 2n=3c g n (n 2) with f 1 given) since it corresponds modulo shift to the sizes of the left and right sub heaps of a heap of size n. The recurrence was first (implicitly) studied by Knuth [10] cf. 9] It appeared as the solution to ....

....the heap recurrence (see [9] or next section for details) blog 2 2n=3c f blog 2 2n=3c g n (n 2) with f 1 given) since it corresponds modulo shift to the sizes of the left and right sub heaps of a heap of size n. The recurrence was first (implicitly) studied by Knuth [10] cf. [9]) It appeared as the solution to the recurrence with minimization (cf. Hammersley and Grimmett [7] f 1 : 0 and for n 2 f n : min ff j f n Gammaj g g n ; 1) where g n is increasing and concave; cf. also [1, 14, 16] Karp and Glassey [4] independently discussed this rule referred to as ....

[Article contains additional citation context not shown here]

H.-K. Hwang and J.-M. Steyaert, On the number of heaps and the cost of heap construction, Theoretical Computer Science, to appear.

....by w) in an ordered set partition is (1 Gamma w(e Gamma 1) Obviously, e Gamma 1 is 1 regular. Theorem 5 applies. We can also consider other constraints like parity on the size of each block and on the number of blocks, cf. 7, p. 225 226] Many other examples can be found in [1, 13, 21] and [18, Ch. 7] 13 Example 5. Inversions in permutations. Given a permutation = 1 ; 2 ; n ) of f1; 2; 3; ng, a pair i ; j is called an inversion if i j and i j . Let Omega n denote the number of inversions in a random permutation of size n. Then it is easily seen ....

H.-K. Hwang and J.-M. Steyaert. On the number of heaps and the cost of heap construction. Theoretical Computer Science, to appear.

....mergesort: a(n) 2 dlog 2 ne1 . Here Y (r; s) denotes the merge cost. Only the case (ii) needs some explanations: Indeed, since the underlying merge tree in queue mergesort is essentially a heap structure, the expression for a(n) can be easily veri ed. For asymptotics of such recurrences, see [9] for details. Note that if a(n) 1 we have an inecient implementation of the insertion sort. It seems of interest to ask the following question: x Y , nd sucient conditions on a(n) so that X n is asymptotically normally distributed. Such considerations would provide further insight on the ....

H. K. Hwang and J.-M. Steyaert, On the number of heaps and the cost of heap construction, accepted for publication in Theoretical Computer Science.

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H.-K. Hwang and J.-M. Steyaert, On the number of heaps and the cost of heap construction, Colloquim on Mathematics andComputer Science: Algorithms, Trees, Combinatorics and Probabilities, Versailles, September 16 - 19,

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