Broadcast in the rendezvous model

In many large, distributed or mobile networks, broadcast algorithms are used to update information stored at the nodes. In this paper, we propose a new model of communication based on rendezvous and analyze a multi-hop distributed algorithm to broadcast a message in a synchronous setting. In the rendezvous model, two neighbors u and v can communicate if and only if u calls v and v calls u simultaneously. Thus nodes u and v obtain a rendezvous at a meeting point. If m is the number of meeting points, the network can be modeled by a graph of n vertices and m edges. At each round, every vertex chooses a random neighbor and there is a rendezvous if an edge has been chosen by its two extremities. Rendezvous enable an exchange of information between the two entities. We get sharp lower and upper bounds on the time complexity in terms of number of rounds to broadcast: we show that, for any graph, the expected number of rounds is between ln n and O (n²). For these two bounds, we prove that there exist some graphs for which the expected number of rounds is either O (ln (n)) or Ω (n²). For specific topologies, additional bounds are given.     

Scalable 2D Convex Hull and Triangulation Algorithms for Coarse Grained Multicomputers

In this paper we describe scalable parallel algorithms for building the convex hull and a triangulation ofncoplanar points. These algorithms are designed for thecoarse grained multicomputermodel:pprocessors withO(n/p)⪢O(1) local memory each, connected to some arbitrary interconnection network. They scale over a large range of values ofnandp, assuming only thatn⩾p^1+ε(ε>0) and require timeO((T_sequential/p)+T_s(n, p)), whereT_s(n, p) refers to the time of a global sort ofndata on approcessor machine. Furthermore, they involve only a constant number of global communication rounds. Since computing either 2D convex hull or triangulation requires timeT_sequential=Θ(n log n) these algorithms either run in optimal time,Θ((n log n)/p), or in sort time,T_s(n, p), for the interconnection network in question. These results become optimal whenT_sequential/pdominatesT_s(n, p) or for interconnection networks like the mesh for which optimal sorting algorithms exist.     

Coarse-Grained Parallel Geometric Search

We present a parallel algorithm for solving thenext element search problemon a set of line segments, using a BSP-like model referred to as thecoarse grained multicomputer(CGM). The algorithm requiresO(1) communication rounds (h-relations withh=O(n/p)),O((n/p) log n) local computation, andO((n/p) log p) memory per processor, assumingn/p⩾p. Our result implies solutions to the point location, trapezoidal decomposition, and polygon triangulation problems. A simplified version for axis-parallel segments requires onlyO(n/p) memory per processor, and we discuss an implementation of this version. As in a previous paper by Develliers and Fabri (Int. J. Comput. Geom. Appl.6(1996), 487–506), our algorithm is based on a distributed implementation of segment trees which are of sizeO(n log n). This paper improves onop. cit.in several ways: (1) It studies the more general next element search problem which also solves, e.g., planar point location. (2) The algorithms require onlyO((n/p) log n) local computation instead ofO(log p*(n/p) log n). (3) The algorithms require onlyO((n/p) log p) local memory instead ofO((n/p) log n).     

Energy-efficient topology control algorithm for maximizing network lifetime in wireless sensor networks with mobile sink

Uneven energy consumption is an inherent problem in wireless sensor networks characterized by multi-hop routing and many-to-one traffic pattern. Such unbalanced energy dissipation can significantly reduce network lifetime. In this paper, we study the problem of prolonging network lifetime in large-scale wireless sensor networks where a mobile sink gathers data periodically along the predefined path and each sensor node uploads its data to the mobile sink over a multi-hop communication path. By using greedy policy and dynamic programming, we propose a heuristic topology control algorithm with time complexity O(n(m + n log n)), where n and m are the number of nodes and edges in the network, respectively, and further discuss how to refine our algorithm to satisfy practical requirements such as distributed computing and transmission timeliness. Theoretical analysis and experimental results show that our algorithm is superior to several earlier algorithms for extending network lifetime.     

A Parallel Implementation of Buchberger's Algorithm overZpforp ≤ 31991

Gröbner bases of ideals of polynomials are known to have many applications. They have been applied to problems in commutative algebra, statistics, graph theory, robotics and differential equations. Their use as a research tool, however, is limited by their computational complexity. These two facts have inspired numerous attempts to parallelize Buchberger's algorithm to compute them.In this paper, we describe a parallel implementation developed on the Cray T3D using the extensions to C provided by ac. The program is based on the publicly available package Macaulay which computes Gröbner bases of homogeneous ideals overZ_pfor primesp ≤  31991. The efficiency is nearly 100% on up to 16 processors for moderately sized problems. Above 16 processors, the efficiency drops.     

Generalized Strong Pseudoprime Tests and Applications

We describe probabilistic primality tests applicable to integers whose prime factors are all congruent to 1 mod r where r is a positive integer;r =  2 is the Miller–Rabin test. We show that if ν rounds of our test do not find n ≠  =  (r +  1)²composite, then n is prime with probability of error less than (2 r) ^− ν. Applications are given, first to provide a probabilistic primality test applicable to all integers, and second, to give a test for values of cyclotomic polynomials.     

DFT study of the chlorine promotion effect on the ethylene adsorption over iron clusters

This work explores how electronic perturbations induced by chlorine atoms can enhance the activity of iron toward ethylene. The metal clusters include Fe_n (n = 2–4), in which each adatom (Cl) has an inclination to be adsorbed at the bridge site with electrostatic interaction. Ethylene adsorption over pure and chlorine-doped Fe_nCl_m (n,m ≤ 4) clusters is analyzed using density functional theory (DFT) calculations, in π and di-σ adsorption modes. One of the interesting features is that the adsorption mode of ethylene changes by going from trimers to tetramers. Ethylene never orients toward di-σ mode for FeFe bond in Fe₂ and Fe₃ series, while this orientation is preferred in tetramers. Our results demonstrate that the progressive change in the ethylene adsorption could not be sustained with increasing portion of chlorine in metal cluster. In this study, we attempt to provide a sensible justification for this phenomenon by the natural bond orbital (NBO) and quantum theory of atoms-in-molecules (QTAIM) analyses.     

Composing Power Series Over a Finite Ring in Essentially Linear Time

Fix a finite commutative ringR. Letuandvbe power series overR, withv(0) = 0. This paper presents an algorithm that computes the firstnterms of the compositionu(v), given the firstnterms ofuandv, inn^1 + o(1)ring operations. The algorithm is very fast in practice whenRhas small characteristic.     

Highly fault-tolerant cycle embeddings of hypercubes

The hypercube Q_n is one of the most popular networks. In this paper, we first prove that the n-dimensional hypercube is 2n − 5 conditional fault-bipancyclic. That is, an injured hypercube with up to 2n − 5 faulty links has a cycle of length l for every even 4 ⩽ l ⩽ 2ⁿ when each node of the hypercube is incident with at least two healthy links. In addition, if a certain node is incident with less than two healthy links, we show that an injured hypercube contains cycles of all even lengths except hamiltonian cycles with up to 2n − 3 faulty links. Furthermore, the above two results are optimal. In conclusion, we find cycles of all possible lengths in injured hypercubes with up to 2n − 5 faulty links under all possible fault distributions.     

Efficient and exact quantum compression

We present a divide and conquer based algorithm for optimal quantum compression/decompression, using O(n(log⁴n)log log n) elementary quantum operations. Our result provides the first quasi-linear time algorithm for asymptotically optimal (in size and fidelity) quantum compression and decompression. We also outline the quantum gate array model to bring about this compression in a quantum computer. Our method uses various classical algorithmic tools to significantly improve the bound from the previous best known bound of O(n³) for this operation.     

On reliability of the folded hypercubes

In this paper, we explore the 2-extra connectivity and 2-extra-edge-connectivity of the folded hypercube FQ_n. We show that κ₂(FQ_n) = 3n − 2 for n ⩾ 8; and λ₂(FQ_n) = 3n − 1 for n ⩾ 5. That is, for n ⩾ 8 (resp. n ⩾ 5), at least 3n − 2 vertices (resp. 3n − 1 edges) of FQ_n are removed to get a disconnected graph that contains no isolated vertices (resp. edges). When the folded hypercube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.     

On identifying codes that are robust against edge changes

Assume that G = (V, E) is an undirected graph, and C ⊆ V. For every v ∈ V, denote I_r(G; v) = {u ∈ C: d(u,v) ≤ r}, where d(u,v) denotes the number of edges on any shortest path from u to v in G. If all the sets I_r(G; v) for v ∈ V are pairwise different, and none of them is the empty set, the code C is called r-identifying. The motivation for identifying codes comes, for instance, from finding faulty processors in multiprocessor systems or from location detection in emergency sensor networks. The underlying architecture is modelled by a graph. We study various types of identifying codes that are robust against six natural changes in the graph; known or unknown edge deletions, additions or both. Our focus is on the radius r = 1. We show that in the infinite square grid the optimal density of a 1-identifying code that is robust against one unknown edge deletion is 1/2 and the optimal density of a 1-identifying code that is robust against one unknown edge addition equals 3/4 in the infinite hexagonal mesh. Moreover, although it is shown that all six problems are in general different, we prove that in the binary hypercube there are cases where five of the six problems coincide.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

Fast congruence closure and extensions

Congruence closure algorithms for deduction in ground equational theories are ubiquitous in many (semi-)decision procedures used for verification and automated deduction. In many of these applications one needs an incremental algorithm that is moreover capable of recovering, among the thousands of input equations, the small subset that explains the equivalence of a given pair of terms. In this paper we present an algorithm satisfying all these requirements. First, building on ideas from abstract congruence closure algorithms, we present a very simple and clean incremental congruence closure algorithm and show that it runs in the best known time O(n  log  n). After that, we introduce a proof-producing union-find data structure that is then used for extending our congruence closure algorithm, without increasing the overall O(n  log  n) time, in order to produce a k-step explanation for a given equation in almost optimal time (quasi-linear in k). Finally, we show that the previous algorithms can be smoothly extended, while still obtaining the same asymptotic time bounds, in order to support the interpreted functions symbols successor and predecessor, which have been shown to be very useful in applications such as microprocessor verification.     

Finding extreme supported solutions of biobjective network flow problems: An enhanced parametric programming approach

We address the problem of determining a complete set of extreme supported efficient solutions of biobjective minimum cost flow (BMCF) problems. A novel method improving the classical parametric method for this biobjective problem is proposed. The algorithm runs in O(Nn(m + nlogn)) time determining all extreme supported non-dominated points in the outcome space and one extreme supported efficient solution associated with each one of them. Here n is the number of nodes, m is the number of arcs and N is the number of extreme supported non-dominated points in outcome space for the BMCF problem. The memory space required by the algorithm is O(n + m) when the extreme supported efficient solutions are not required to be stored in RAM. Otherwise, the algorithm requires O(N + m) space. Extensive computational experiments comparing the performance of the proposed method and a standard parametric network simplex method are presented.     

Neuro-fuzzy model based on digital images for the monitoring of coffee bean color during roasting in a spouted bed

An adaptive-network-based fuzzy inference system based on color image analysis was used to estimate coffee bean moisture content during roasting in a spouted bed. The neuro-fuzzy model described the grain moisture changes as a function of brightness (L^*), browning index (BI) and the distance to a defined standard (ΔE). An image-capture device was designed to monitor color variations in the L^*a^*b^* space for high temperatures samples taken from the reactor. The proposed model was composed of three Gaussian-type fuzzy sets based on the scatter partition method. The neuro-fuzzy model was trained with the Back-propagation algorithm using experimental measurements at three air temperature levels (400, 450 and 500 °C). The performance of the neuro-fuzzy model resulted better compared to conventional methods obtaining a coefficient of determination > 0.98, a root mean square error < 0.002 and a modified Schwarz–Rissanen information criterion < 0. The simplicity of the model and its robustness against changes in the input variables make it suitable for monitoring on-line the roasting process.     

Primary Decomposition of Lattice Basis Ideals

We study the primary decomposition of lattice basis ideals. These ideals are binomial ideals with generators given by the elements of a basis of a saturated integer lattice. We show that the minimal primes of such an ideal are completely determined by the sign pattern of the basis elements, while the embedded primes are not. As a special case we examine the ideal generated by the 2  ×  2 adjacent minors of a generic m × n matrix. In particular, we determine all minimal primes in the 3  × n case. We also present faster ways of computing a generating set for the associated toric ideal from a lattice basis ideal.     

A study of nonlinear dispersive equations with solitary-wave solutions having compact support

With the use of Adomian decomposition method, the prototypical, genuinely nonlinear K(m,n) equation, u_t+(u^m)_x+(uⁿ)_xxx=0, which exhibits compactons — solitons with finite wavelength — is solved exactly. Two numerical illustrations, K(2,2) and K(3,3), are investigated to illustrate the pertinent features of the proposed scheme. The technique is presented in a general way so that it can be used in nonlinear dispersive equations.     

Fast LLL-type lattice reduction

We modify the concept of LLL-reduction of lattice bases in the sense of Lenstra, Lenstra, Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982) 515–534 towards a faster reduction algorithm. We organize LLL-reduction in segments of the basis. Our SLLL-bases approximate the successive minima of the lattice in nearly the same way as LLL-bases. For integer lattices of dimension n given by a basis of length 2^O(n), SLLL-reduction runs in O (n^{5 +ε}) bit operations for every ε > 0, compared to O (n^{7 +ε}) for the original LLL and to O (n^{6 +ε}) for the LLL-algorithms of Schnorr, A more efficient algorithm for lattice reduction, Journal of Algorithm, 9 (1988) 47–62 and Storjohann, Faster Algorithms for Integer Lattice Basis Reduction. TR 249, Swiss Federal Institute of Technology, ETH-Zurich, Department of Computer Science, Zurich, Switzerland, July 1996. We present an even faster algorithm for SLLL-reduction via iterated subsegments running in O (n³log n) arithmetic steps. Householder reflections are shown to provide better accuracy than Gram–Schmidt for orthogonalizing LLL-bases in floating point arithmetic.     

PetRBF — A parallel O(N) algorithm for radial basis function interpolation with Gaussians

We have developed a parallel algorithm for radial basis function (rbf) interpolation that exhibits O(N) complexity, requires O(N) storage, and scales excellently up to a thousand processes. The algorithm uses a gmres iterative solver with a restricted additive Schwarz method (rasm) as a preconditioner and a fast matrix-vector algorithm. Previous fast rbf methods — achieving at most O(NlogN) complexity — were developed using multiquadric and polyharmonic basis functions. In contrast, the present method uses Gaussians with a small variance with respect to the domain, but with sufficient overlap. This is a common choice in particle methods for fluid simulation, our main target application. The fast decay of the Gaussian basis function allows rapid convergence of the iterative solver even when the subdomains in the rasm are very small. At the same time we show that the accuracy of the interpolation can achieve machine precision. The present method was implemented in parallel using the petsc library (developer version). Numerical experiments demonstrate its capability in problems of rbf interpolation with more than 50 million data points, timing at 106 s (19 iterations for an error tolerance of 10^? 15) on 1024 processors of a Blue Gene/L (700 MHz PowerPC processors). The parallel code is freely available in the open-source model.