scholarly journals A further analysis of Cuckoo Hashing with a Stash and Random Graphs of Excess r

2010 ◽  
Vol Vol. 12 no. 3 (Analysis of Algorithms) ◽  
Author(s):  
Reinhard Kutzelnigg
Keyword(s):  
Analysis Of Algorithms ◽  
Hash Table ◽  
Saddle Point Method ◽  
Access Time ◽  
Worst Case ◽  
Cuckoo Hashing ◽  
Original Algorithm ◽  
International Audience ◽  
Significant Probability ◽  
Table Data

Analysis of Algorithms International audience Cuckoo hashing is a hash table data structure offering constant access time, even in the worst case. As a drawback, the construction fails with small, but practically significant probability. However, Kirsch et al. (2008) showed that a constant-sized additional memory, the so called stash, is sufficient to reduce the failure rate drastically. But so far, using a modified insertion procedure that demands additional running time to look for an admissible key is required. As a major contribution of this paper, we show that the same bounds on the failure probability hold even without this search process and thus, the performance increases. Second, we extend the analysis to simplified cuckoo hashing, a variant of the original algorithm offering increased performance. Further, we derive some explicit asymptotic approximations concerning the number of usual resp. bipartite graphs related to the data structures. Using these results, we obtain much more precise asymptotic expansions of the success rate. These calculations are based on a generating function approach and applying the saddle point method. Finally, we provide numerical results to support the theoretical analysis.

scholarly journals Bipartite Random Graphs and Cuckoo Hashing

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Reinhard Kutzelnigg
Keyword(s):  
Saddle Point ◽  
Bipartite Graph ◽  
Random Graphs ◽  
Hash Table ◽  
Saddle Point Method ◽  
Asymptotic Results ◽  
Cuckoo Hashing ◽  
International Audience ◽  
Covering Tree ◽  
Number Of Cycles

International audience The aim of this paper is to extend the analysis of Cuckoo Hashing of Devroye and Morin in 2003. In particular we make several asymptotic results much more precise. We show, that the probability that the construction of a hash table succeeds, is asymptotically $1-c(\varepsilon)/m+O(1/m^2)$ for some explicit $c(\varepsilon)$, where $m$ denotes the size of each of the two tables, $n=m(1- \varepsilon)$ is the number of keys and $\varepsilon \in (0,1)$. The analysis rests on a generating function approach to the so called Cuckoo Graph, a random bipartite graph. We apply a double saddle point method to obtain asymptotic results covering tree sizes, the number of cycles and the probability that no complex component occurs.

scholarly journals Stochastic Flips on Dimer Tilings

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Thomas Fernique ◽  
Damien Regnault
Keyword(s):  
Fixed Point ◽  
Markov Process ◽  
Upper Bound ◽  
Numerical Experiments ◽  
Triangular Grid ◽  
Expected Number ◽  
Worst Case ◽  
Average Case ◽  
International Audience

International audience This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.

scholarly journals The Saddle Point Method for the Integral of the Absolute Value of the Brownian Motion

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Leonid Tolmatz
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method ◽  
Tail Asymptotics ◽  
Absolute Value ◽  
Exact Tail Asymptotics ◽  
International Audience ◽  
The Absolute

International audience The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.

scholarly journals Optimal Computer Crash Performance Precaution

2012 ◽  
Vol Vol. 14 no. 1 (Distributed Computing and...) ◽  
Author(s):  
Efraim Laksman ◽  
Hakan Lennerstad ◽  
Lars Lundberg
Keyword(s):  
Completion Time ◽  
Optimal Allocation ◽  
Parallel Program ◽  
Parallel Computer ◽  
Upper And Lower Bounds ◽  
Worst Case ◽  
Worst Case Ratio ◽  
International Audience ◽  
Computer Crash ◽  
The Cost

Distributed Computing and Networking International audience For a parallel computer system with m identical computers, we study optimal performance precaution for one possible computer crash. We want to calculate the cost of crash precaution in the case of no crash. We thus define a tolerance level r meaning that we only tolerate that the completion time of a parallel program after a crash is at most a factor r + 1 larger than if we use optimal allocation on m - 1 computers. This is an r-dependent restriction of the set of allocations of a program. Then, what is the worst-case ratio of the optimal r-dependent completion time in the case of no crash and the unrestricted optimal completion time of the same parallel program? We denote the maximal ratio of completion times f(r, m) - i.e., the ratio for worst-case programs. In the paper we establish upper and lower bounds of the worst-case cost function f (r, m) and characterize worst-case programs.

scholarly journals Digital search trees with m trees: Level polynomials and insertion costs

2011 ◽  
Vol Vol. 13 no. 3 (Analysis of Algorithms) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Analysis Of Algorithms ◽  
Approximate Counting ◽  
Search Trees ◽  
Digital Search Trees ◽  
International Audience ◽  
Digital Search

Analysis of Algorithms International audience We adapt a novel idea of Cichon's related to Approximate Counting to the present instance of Digital Search Trees, by using m (instead of one) such trees. We investigate the level polynomials, which have as coefficients the expected numbers of data on a given level, and the insertion costs. The level polynomials can be precisely described, thanks to formulae from q-analysis. The asymptotics of expectation and variance of the insertion cost are fairly standard these days and done with Rice's method.

scholarly journals Exponential bounds and tails for additive random recursive sequences

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Random Trees ◽  
Divide And Conquer ◽  
Recursive Sequences ◽  
Recursive Structures ◽  
International Audience ◽  
Exponential Bounds ◽  
The Moment

Analysis of Algorithms International audience Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method. The paper generalizes results of Rösler (% \citeyearNPRoesler:91, % \citeyearNPRoesler:92) and % \citeNNeininger:05 on subgaussian tails to more general classes of additive random recursive sequences. It also gives sufficient conditions for tail bounds of the form \exp(-a t^p) which are based on a characterization of \citeNKasahara:78.

scholarly journals On the tileability of polygons with colored dominoes

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Chris Worman ◽  
Boting Yang
Keyword(s):  
Analysis Of Algorithms ◽  
Time Algorithm ◽  
Orthogonal Polygon ◽  
Orthogonal Polygons ◽  
Domino Tiling ◽  
International Audience ◽  
Np Complete

Analysis of Algorithms International audience We consider questions concerning the tileability of orthogonal polygons with colored dominoes. A colored domino is a rotatable 2 × 1 rectangle that is partitioned into two unit squares, which are called faces, each of which is assigned a color. In a colored domino tiling of an orthogonal polygon P, a set of dominoes completely covers P such that no dominoes overlap and so that adjacent faces have the same color. We demonstrated that for simple layout polygons that can be tiled with colored dominoes, two colors are always sufficient. We also show that for tileable non-simple layout polygons, four colors are always sufficient and sometimes necessary. We describe an O(n) time algorithm for computing a colored domino tiling of a simple orthogonal polygon, if such a tiling exists, where n is the number of dominoes used in the tiling. We also show that deciding whether or not a non-simple orthogonal polygon can be tiled with colored dominoes is NP-complete.

scholarly journals On Compact Encoding of Pagenumber $k$ Graphs

2008 ◽  
Vol Vol. 10 no. 3 ◽  
Author(s):  
Cyril Gavoille ◽  
Nicolas Hanusse
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Constant Time ◽  
Worst Case ◽  
Encoding Scheme ◽  
Information Theoretic ◽  
Auxiliary Table ◽  
Minimum Number ◽  
International Audience

International audience In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;.

scholarly journals Finely homogeneous computations in free Lie algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Philippe Andary
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Simple Algorithm ◽  
Lexicographic Order ◽  
Worst Case ◽  
Free Lie Algebras ◽  
International Audience ◽  
Lyndon Word ◽  
The One ◽  
Lyndon Words

International audience We first give a fast algorithm to compute the maximal Lyndon word (with respect to lexicographic order) of \textitLy_α (A) for every given multidegree alpha in \textbfN^k. We then give an algorithm to compute all the words living in \textitLy_α (A) for any given α in \textbfN^k. The best known method for generating Lyndon words is that of Duval [1], which gives a way to go from every Lyndon word of length n to its successor (with respect to lexicographic order by length), in space and worst case time complexity O(n). Finally, we give a simple algorithm which uses Duval's method (the one above) to compute the next standard bracketing of a Lyndon word for lexicographic order by length. We can find an interesting application of this algorithm in control theory, where one wants to compute within the command Lie algebra of a dynamical system (letters are actually vector fields).

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