P. Flajolet and R. Sedgewick. The average case analysis of algorithms : Saddle point asymptotics. Technical Report RR-2376, Institut National de Recherche en Informatique et en Automatique.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... : 20) We must equilibrate the two terms of F 2 as b 1, so we set u = b w. This leads to (b w) Ai(b w) dw (21) F 3 ( 2 w w2 )b 2 2 ) b 1: We will now apply the powerful Saddle point method (see, for instance, Flajolet and Sedgewick [6]) We look for w such that F 4 (w ) 0, with F 4 (w) F 3 (w) b F 3 (w bz O( Now we set w = w y in F 3 . We derive F 1 F 3 O( So we nally set y = i = and integrating (21) on , we obtain the limiting expression e for ....

P. Flajolet and R. Sedgewick. The average case analysis of algorithms: Saddle point asymptotics. INRIA T.R., 2376, 1994.

.... b 2=3 Z i1 i1 e F3 (Ai 0 (b 2=3 w) Ai(b 2=3 w) 0 dw (14) where F 3 ( 2 2=3 z 1=3 p w w2 1=3 z 2=3 )b 2 2 2=3 z 1=3 b 4w 5z 1=3 2 2=3 32w 5=2 O( 1 b ) b 1: We will now apply the powerful Saddle point method (see, for instance, Flajolet and Sedgewick [6]) We look for w such that F 0 4 (w ) 0, with F 4 (w) F 3 (w) b 2 . This gives w 2 2=3 4z 2=3 1 4z b 5z 2 b 2 O( 1 b 3 ) and F 3 (w ) b 2 2 bz z 2 2 O( 1 b ) Now we set w = w y in F 3 . We derive F 1 F 3 z 2 2 2 2=3 ....

P. Flajolet and R. Sedgewick. The average case analysis of algorithms: Saddle point asymptotics. INRIA T.R., 2376, 1994.

....we could try a model based on Markov chains on urns. Indeed, an alternative proof of (6) can be obtained by using an urn model, as in Sevastyanov and Chistyakov, 39] and Chistyakov, 8] the Poissonization method and the standard saddle point method (see, for instance, Flajolet and Sedgewick, [16]) This will be the object of future work. 5 Conclusion Using various techniques from analysis and probability theory, we have analyzed the stochastic properties of the distinctness of classical compositions. The mean and variance have been derived for the Carlitz case. An open problem is to ....

Flajolet, P., Sedgewick, R. The Average Case Analysis of Algorithms: Saddle Point Asymptotics , INRIA T.R. 2376(1994).

....we could try a model based on Markov chains on urns. Indeed, an alternative proof of (3) can be obtained by using an urn model, as in Sevastyanov and Chistyakov, 34] and Chistyakov, 6] the Poissonization method and the standard saddle point method (see, for instance, Flajolet and Sedgewick, [12]) This will be the object of future work. 5 Conclusion Using various techniques from analysis and probability theory, we have analyzed the stochatic properties of the distinctness of classical compositions. The mean and variance have been derived for the Carlitz case. An open problem is to prove ....

Flajolet, P., Sedgewick, R. The Average Case Analysis of Algorithms: Saddle Point Asymptotics , INRIA T.R. 2376(1994).

....results with q replaced by q 1 2 and D n,k (q) counts direct sum decompositions into orthogonal subspaces. Example 4 (Tagged Permutations) A tagged permutation is a permutation written in one line form together with a distinguished increasing subsequence. Following Flajolet and Sedgewick [4], the generating function is given by h(x, y) 1 1 x exp # xy 1 x # , where the exponential variable x keeps track of permutation length and the ordinary variable y keeps track of distinguished subsequence length. Lifschitz and Pittel [11] and Flajolet and Sedgewick [4] obtained ....

....and Sedgewick [4] the generating function is given by h(x, y) 1 1 x exp # xy 1 x # , where the exponential variable x keeps track of permutation length and the ordinary variable y keeps track of distinguished subsequence length. Lifschitz and Pittel [11] and Flajolet and Sedgewick [4] obtained asymptotics for the coe#cients of h(x, 1) using real and complex analysis, respectively. Using Theorem 6(b) with f(x) x 1 x and C(r) 1, we see that f(x, y) exp # xy 1 x # is super admissible. One easily computes a f (r, s) s # r (1 r) 2 r 1 r # ,B f (r, s) s # r(1 r ....

P. Flajolet and R. Sedgewick, The average case analysis of algorithms: Saddle point asymptotics, INRIA Rpt. No. 2376 (1994).

....results with q replaced by q 1=2 and D n;k (q) counts direct sum decompositions into orthogonal subspaces. Example 4 (Tagged Permutations) A tagged permutation is a permutation written in one line form together with a distinguished increasing subsequence. Following Flajolet and Sedgewick [4], the generating function is given by h(x; y) 1 1 Gamma x exp ae xy 1 Gamma x oe ; where the exponential variable x keeps track of permutation length and the ordinary variable y keeps track of distinguished subsequence length. Lifschitz and Pittel [11] and Flajolet and Sedgewick [4] ....

....[4] the generating function is given by h(x; y) 1 1 Gamma x exp ae xy 1 Gamma x oe ; where the exponential variable x keeps track of permutation length and the ordinary variable y keeps track of distinguished subsequence length. Lifschitz and Pittel [11] and Flajolet and Sedgewick [4] obtained asymptotics for the coefficients of h(x; 1) using real and complex analysis, respectively. Using Theorem 6(b) with f(x) x 1 Gammax and C(r) 1, we see that f(x; y) exp n xy 1 Gammax o is super admissible. One easily computes a f (r; s) s r (1 Gammar) 2 r 1 Gammar ....

P. Flajolet and R. Sedgewick, The average case analysis of algorithms: Saddle point asymptotics, INRIA Rpt. No. 2376 (1994).

....y k 1 n;k ; where n;k is exponentially small. As is classically known, coefficients of large indices in large powers can be extracted by the saddle point method and, granted unicity of the saddle point on the contour (here jyj = 1) the consequence is a local limit law (LLL) We refer to [17, 20, 22, 23, 29] for this fact that is also an offspring of analytic approaches to local and central limit theorems originally stemming from the work of Daniels [8] Here, there is a saddle point at y = 1 and the argument establishes the local limit law, assuming the uniqueness condition (C) 2 1 0.5 0 0.5 1 ....

Flajolet, P., and Sedgewick, R. The average case analysis of algorithms: Saddle point asymptotics. Research Report 2376, Institut National de Recherche en Informatique et en Automatique, 1994. 55 pages.

No context found.

P. Flajolet and R. Sedgewick. The average case analysis of algorithms : Saddle point asymptotics. Technical Report RR-2376, Institut National de Recherche en Informatique et en Automatique.

No context found.

P. Flajolet and R. Sedgewick. The average case analysis of algorithms: saddle point asymptotics. Rapport de recherche n. 2376, INRIA Rocquencourt, October 1994.

No context found.

P. Flajolet and R. Sedgewick. The average case analysis of algorithms: Saddle point asymptotics. Research Report 2376, INRIA, 1994. 55 pages.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC