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Metode Akra-Bazzi Sebagai Generalisasi Metode Master Dalam Menyelesaikan Relasi Rekurensi

2013
*
Jurnal Fourier
*

This research aims to know

doi:10.14421/fourier.2013.22.63-72
fatcat:emwxnfwdsjeszgihh2prg4qba4
*the**Akra*-*Bazzi*Method as an extension Method of*the**Master*. ... Note that*Akra*-*Bazzi*Method can solve a rekurensi devide-*and*-conquer with shorter calculation. ...*The**Master*Method is fairly powerfull*and*result in a closed form solution for devide-*and*-conquer recurrences with a special (but commonly-occuring) form [1] . ...##
###
Verifying Asymptotic Time Complexity of Imperative Programs in Isabelle
[article]

2018
*
arXiv
*
pre-print

In addition to

arXiv:1802.01336v1
fatcat:sencjmmrdzcdhalbbyjk2qmudm
*the*basic arguments, our framework is able to handle advanced techniques for time complexity analysis, such as*the*use of*the**Akra*-*Bazzi**theorem**and*amortized analysis. ... Various automation is built*and*incorporated into*the*auto2 prover to reason about separation logic with time credits,*and*to derive asymptotic behavior of functions. ... We thank Manuel Eberl for his impressive formalization of*the**Akra*-*Bazzi*method*and**the*functional correctness of*the*selection algorihtm,*and*Simon Wimmer for*the*formalization of*the*DP solution for ...##
###
Cache-Adaptive Analysis

2016
*
Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures - SPAA '16
*

Our techniques enable us to analyze a wide variety of algorithms -

doi:10.1145/2935764.2935798
dblp:conf/spaa/BenderDEFJLLM16
fatcat:z3vm6vlydbfktckpucxokknvde
*Master*-Method-style algorithms,*Akra*-*Bazzi*-style algorithms, collections of mutually recursive algorithms,*and*algorithms, such as FFT, ... , which is not covered by*the*above*theorems*. ...*The*same basic technique works for*Akra*-*Bazzi*-style algorithms*and*even for collections of mutually-recursive*Akra*-*Bazzi*-style algorithms (see*Theorem*6.10). ...##
###
A Real Elementary Approach to the Master Recurrence and Generalizations
[chapter]

2011
*
Lecture Notes in Computer Science
*

*The*

*master*

*theorem*provides a solution to a well-known divide-

*and*-conquer recurrence, called here

*the*

*master*recurrence. This paper proves two cook-book style generalizations of this

*master*

*theorem*. ...

*The*power

*and*simplicity of our approach comes from re-interpreting integer recurrences as real recurrences, with emphasis on elementary techniques

*and*real induction. ... For instance,

*the*summation rules for

*the*various growth-types are easily taught in introductory algorithms. Indeed, our perspectives have developed out of classroom teaching. ...

##
###
A simple master Theorem for discrete divide and conquer recurrences
[article]

2021
*
arXiv
*
pre-print

*The*aim of this note is to provide a

*Master*

*Theorem*for some discrete divide

*and*conquer recurrences: $$X_{n}=a_n+\sum_{j=1}^m b_j X_{\lfloor p_j n \rfloor},$$ where

*the*$p_i$'s belong to $(0,1)$. ...

*The*main novelty of this work is there is no assumption of regularity or monotonicity for $(a_n)$. ... of n≥1 |an| n s 0 +1

*and*our

*Master*

*Theorem*still applies. ...

##
###
Fast and Stable Pascal Matrix Algorithms
[article]

2017
*
arXiv
*
pre-print

We conduct numerical experiments which establish

arXiv:1711.08453v1
fatcat:mgibthhjzvethagdd2gmhrmgz4
*the*speed*and*stability of our algorithm, as well as*the*poor performance of*the*Toeplitz factorization algorithm. ... These algorithms use a recursive factorization of*the*triangular Pascal matrices*and*improve upon*the*cripplingly unstable $O(n log n)$ fast Fourier transform-based algorithms which involve a Toeplitz ... We apply*the**Akra*-*Bazzi**theorem*[3] to show that T n = Θ(kn log 2 n). ...##
###
A Master Theorem for Discrete Divide and Conquer Recurrences

2013
*
Journal of the ACM
*

*The*discrete nature of this recurrence (represented by

*the*floor function) introduces certain oscillations not captured by

*the*traditional

*Master*

*Theorem*, for example due to

*Akra*

*and*

*Bazzi*who primary ... We apply powerful techniques such as Dirichlet series, Mellin-Perron formula,

*and*(extended) Tauberian

*theorems*of Wiener-Ikehara to provide a complete

*and*precise solution to this basic computer science ... Then we recover

*the*asymptotics of T (x) directly by an application of

*the*Wiener-Ikehara

*theorem*: T (x) ∼ Cx s0 with C = a(−s 0 ) + g(−s 0 ) j b j p s0 j log(1/p j ) , which is in accordance with

*Akra*-

*Bazzi*...

##
###
A Master Theorem for Discrete Divide and Conquer Recurrences
[chapter]

2011
*
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms
*

*The*discrete nature of this recurrence (represented by

*the*floor function) introduces certain oscillations not captured by

*the*traditional

*Master*

*Theorem*, for example due to

*Akra*

*and*

*Bazzi*who primary ... We apply powerful techniques such as Dirichlet series, Mellin-Perron formula,

*and*(extended) Tauberian

*theorems*of Wiener-Ikehara to provide a complete

*and*precise solution to this basic computer science ... Then we recover

*the*asymptotics of T (x) directly by an application of

*the*Wiener-Ikehara

*theorem*: T (x) ∼ Cx s0 with C = a(−s 0 ) + g(−s 0 ) j b j p s0 j log(1/p j ) , which is in accordance with

*Akra*-

*Bazzi*...

##
###
On the Solution of Linear Recurrence Equations

1998
*
Computational optimization and applications
*

In this article, we present a general solution for linear divide-

doi:10.1023/a:1018373005182
fatcat:jxf5vbo4pzecdglmuwe76ol66m
*and*-conquer recurrences of*the*form Our approach handles more cases than*the**Master*method does[1]. ... This transform helps in mapping*the*sequence under consideration to*the*two dimensional plane where*the*solution becomes easier to obtain. ... of*the*Order transform to solve Equation 2 in*the*plane. • In*Theorem*4 we show that our results agree with*the**Master*method when*the*latter is applicable. • In Corollary 1 we summarize*the*results as ...##
###
Verified Tail Bounds for Randomized Programs
[chapter]

2018
*
Lecture Notes in Computer Science
*

We mechanize a

doi:10.1007/978-3-319-94821-8_33
fatcat:b3effzpfd5ck5bwd7haukjcyaa
*theorem*by Karp, along with several extensions, that provide an easy to use "cookbook" method for verifying tail bounds, much like*the*traditional "*Master**Theorem*" gives bounds for deterministic ...*The*development of type systems*and*static analyses that automatically bound*the*complexity of programs is an active area of research. ... Any opinions, findings*and*conclusions or recommendations expressed in this material are those of*the*authors*and*do not necessarily reflect*the*views of these organizations. ...##
###
Hierarchical Algorithm for Hidden Markov Model

2013
*
International Journal of Advanced Computer Science and Applications
*

*The*objective of this work is to reduce

*the*task of solving

*the*Forward algorithm, by offering faster improved algorithm which is based on divide

*and*conquer technique. ...

*The*Forward algorithm is an inference algorithm for hidden Markov models, which often leads to a very large hidden state space. ... Whereas,

*the*complexity of improved Forward algorithm can be calculated by using

*Akra*

*Bazzi*

*theorem*[16] (A generalization to

*the*well known

*Master*

*Theorem*[17] ) which allows calculating

*the*complexity ...

##
###
A new dichotomic algorithm for the uniform random generation of words in regular languages

2013
*
Theoretical Computer Science
*

When using floating point arithmetics, its bit-complexity is O(q log 2 n) in space

doi:10.1016/j.tcs.2012.07.025
fatcat:xuwakjtgp5b3vbxvx7sf5ushma
*and*O(qn log 2 n) in time, where n stands for*the*length of*the*word,*and*q stands for*the*number of states of a finite ... We implemented*the*algorithm*and*compared its behavior to*the*state-of-*the*-art algorithms, on a set of large automata from*the*VLTS benchmark suite. ...*Akra**and**Bazzi*, 1998) . ...##
###
Synthesis with Asymptotic Resource Bounds
[article]

2021
*
arXiv
*
pre-print

These typing rules are justified by

arXiv:2103.04188v2
fatcat:q32grfvsl5egxmitaj5okaldl4
*theorems*used in analysis of algorithms, such as*the**Master**Theorem**and**the**Akra*-*Bazzi*method. ... Prior methods for synthesis with a resource metric require*the*user to specify a concrete expression exactly describing resource usage, whereas our method uses big-O notation to specify*the*asymptotic ... Any opinions, findings,*and*conclusions or recommendations expressed in this publication are those of*the*authors,*and*do not necessarily reflect*the*views of*the*sponsoring entities. ...##
###
Continuous amortization and extensions: With applications to bisection-based root isolation

2016
*
Journal of symbolic computation
*

Figure 1 :

doi:10.1016/j.jsc.2016.01.007
fatcat:bee6vuww4zfx7bt3y2gonumqhm
*The*intervals J 1 , J 3 ,*and*J 5 all contribute to*the*local size bound at x because they contain*the*point x*and**the*stopping criterion is False on them. ... Here, w(J) denotes*the*length of*the*interval J. 2 In previous work (Burr et al. (2009), Sharma*and*Yap (2012),*and*Burr*and*Krahmer (2012) ), a local size bound was called a stopping function. ... algorithms,*the**master**theorem*(Cormen et al., 2001)*and**the**Akra*-*Bazzi**theorem*(*Akra**and**Bazzi*, 1998) provide formulae for computing*the*complexity of such algorithms. ...##
###
Optimal construction of a layer-ordered heap
[article]

2020
*
arXiv
*
pre-print

*The*layer-ordered heap (LOH) is a simple, recently proposed data structure used in optimal selection on $X+Y$, thealgorithm with

*the*best known runtime for selection on $X_1+X_2+\cdots+X_m$,

*and*

*the*fastest ... Here, we introduce a few algorithms for constructing LOHs, analyze their complexity,

*and*demonstrate that one algorithm is optimal for building a LOH of any rank $\alpha$. ... for this algorithm is solved by neither

*the*

*master*

*theorem*[3] nor

*the*more general

*Akra*-

*Bazzi*method [1] . ...

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