Fast Deterministic Algorithm Research Articles

Fast Approximation Algorithm for Restricted Euclidean Bottleneck Steiner Tree Problem

Bottleneck Steiner tree problem asks to find a Steiner tree for n terminals with at most k Steiner points such that the length of the longest edge in the tree is minimized. The problem has applications in the design of wireless communication networks. In this paper we study a restricted version of the bottleneck Steiner tree problem in the Euclidean plane which requires that only degree-2 Steiner points are possibly adjacent in the optimal solution. We first show that the problem is NP-hard and cannot be approximated within unless P=NP, and provide a fast polynomial time deterministic approximation algorithm with performance ratio .

A fast algorithm for kinematic chain isomorphism identification based on dividing and matching vertices

Kinematic chain isomorphism identification is a crucial issue in mechanism topology and an important application of Graph Isomorphism to mechanisms. In this paper, a kinematic chain is uniquely represented by a graph, and a fast deterministic algorithm called the Dividing and Matching Algorithm (DMA) is proposed. First, the vertices of each graph are divided by the degree. Then, vertex connection properties in a sub-graph and between sub-graphs are explored. Accordingly the expanded square degree and the correlation degree are proposed, based on which, the Dividing Vertex Algorithm (DVA) is developed to divide vertices into sets. Moreover, it is proved that only the vertices from the corresponding sets between two graphs are possible to be bijective or matched, which avoids exhaustive search. Eventually, a backtracking procedure is employed to match the vertices between corresponding sets by calling up DVA repeatedly. DMA detects whether the adjacency matrices of two graphs can be adjusted to be equivalent by changing the orders of vertices. Justifications for the reliability of each part of DMA are provided. The experiments and comparisons with existing algorithms show the effectiveness and efficiency of DMA.

Fast deterministic single-exposure coherent diffractive imaging at sub-Ångström resolution

In coherent diffractive imaging (CDI), conventional image-forming optics are replaced by the solution of an inverse scattering problem which obtains the exit surface wave from the diffraction pattern. To date CDI implementations typically use nonlinear iterative solutions, often using more than one diffraction pattern, and have been mainly implemented in the optical and x-ray regimes. Here we present single-exposure CDI reconstructions of a region of a cerium dioxide nanocrystal, illuminated by a coherent electron probe, in which the positions of both cerium and the lighter oxygen atoms are recovered simultaneously. We employ a fast deterministic algorithm based on the iterative solution of a set of linear equations with input data obtained from the inverse Fourier transform of the diffraction pattern. The solution of the linear equations proceeds using a conjugate gradient least-squares method based on fast Fourier transforms. This allows regularization of the inversion problem by iteration number, making it robust to experimental noise. This approach is successful where standard nonlinear phase retrieval techniques for the same data set are not.

A Finite Population Model of Molecular Evolution: Theory and Computation

This article is concerned with the evolution of haploid organisms that reproduce asexually. In a seminal piece of work, Eigen and coauthors proposed the quasispecies model in an attempt to understand such an evolutionary process. Their work has impacted antiviral treatment and vaccine design strategies. Yet, predictions of the quasispecies model are at best viewed as a guideline, primarily because it assumes an infinite population size, whereas realistic population sizes can be quite small. In this paper we consider a population genetics-based model aimed at understanding the evolution of such organisms with finite population sizes and present a rigorous study of the convergence and computational issues that arise therein. Our first result is structural and shows that, at any time during the evolution, as the population size tends to infinity, the distribution of genomes predicted by our model converges to that predicted by the quasispecies model. This justifies the continued use of the quasispecies model to derive guidelines for intervention. While the stationary state in the quasispecies model is readily obtained, due to the explosion of the state space in our model, exact computations are prohibitive. Our second set of results are computational in nature and address this issue. We derive conditions on the parameters of evolution under which our stochastic model mixes rapidly. Further, for a class of widely used fitness landscapes we give a fast deterministic algorithm which computes the stationary distribution of our model. These computational tools are expected to serve as a framework for the modeling of strategies for the deployment of mutagenic drugs.

Distributed deterministic edge coloring using bounded neighborhood independence

We study the edge-coloring problem in the message-passing model of distributed computing. This is one of the most fundamental problems in this area. Currently, the best-known deterministic algorithms for (2Δ −1)-edge-coloring requires O(Δ) + log* n time (Panconesi and Rizzi in Distrib Comput 14(2):97–100, 2001), where Δ is the maximum degree of the input graph. Also, recent results of Barenboim and Elkin (2010) for vertex-coloring imply that one can get an O(Δ)-edge-coloring in $${O(\Delta^{\epsilon}\cdot \log n)}$$ time, and an $${O(\Delta^{1 + \epsilon})}$$ -edge-coloring in O(log Δ log n) time, for an arbitrarily small constant $${\epsilon > 0}$$ . In this paper we devise a significantly faster deterministic edge-coloring algorithm. Specifically, our algorithm computes an O(Δ)-edge-coloring in $${O(\Delta^{\epsilon}) + \log* n}$$ time, and an $${O(\Delta^{1 + \epsilon})}$$ -edge-coloring in O(log Δ) + log* n time. This result improves the state-of-the-art running time for deterministic edge-coloring with this number of colors in almost the entire range of maximum degree Δ. Moreover, it improves it exponentially in a wide range of Δ, specifically, for 2 Ω(log*n) ≤ Δ ≤ polylog(n). In addition, for small values of Δ (up to log1 - δ n, for some fixed δ > 0) our deterministic algorithm outperforms all the existing randomized algorithms for this problem. Also, our algorithm is the first O(Δ)-edge-coloring algorithm that has running time o(Δ) + log* n, for the entire range of Δ. All previous (deterministic and randomized) O(Δ)-edge-coloring algorithms require $${\Omega(\min \{\Delta, \sqrt{\log n}\ \})}$$ time. On our way to these results we study the vertex-coloring problem on graphs with bounded neighborhood independence. This is a large family of graphs, which strictly includes line graphs of r-hypergraphs (i.e., hypergraphs in which each hyperedge contains r or less vertices) for r = O(1), and graphs of bounded growth. We devise a very fast deterministic algorithm for vertex-coloring graphs with bounded neighborhood independence. This algorithm directly gives rise to our edge-coloring algorithms, which apply to general graphs. Our main technical contribution is a subroutine that computes an O(Δ/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p 2) + log* n time, for a parameter p, 1 ≤ p ≤ Δ. In all previous efficient distributed routines for m-defective p-coloring the product m· p is super-linear in Δ. In our routine this product is linear in Δ, and this enables us to speed up the algorithm drastically.

Deterministic Algorithm Computing All Generators: Application in Cryptographic Systems Design

Primitive elements play important roles in the Diffie-Hellman protocol for establishment of secret communication keys, in the design of the ElGamal cryptographic system and as generators of pseudo-random numbers. In general, a deterministic algorithm that searches for primitive elements is currently unknown. In information-hiding schemes, where a primitive element is the key factor, there is the freedom in selection of a modulus. This paper provides a fast deterministic algorithm, which computes every primitive element in modular arithmetic with special moduli. The algorithm requires at most O(log2p) digital operations for computation of a generator. In addition, the accelerated-descend algorithm that computes small generators is described in this paper. Several numeric examples and tables illustrate the algorithms and their properties.

Fast arithmetic in unramified p-adic fields

Let p be prime and Z p n a degree n unramified extension of the ring of p-adic integers Z p . In this paper we give an overview of some very fast deterministic algorithms for common operations in Z p n modulo p N . Combining existing methods with recent work of Kedlaya and Umans about modular composition of polynomials, we achieve quasi-linear time algorithms in the parameters n and N, and quasi-linear or quasi-quadratic time in log p , for most basic operations on these fields, including Galois conjugation, Teichmüller lifting and computing minimal polynomials.

On neighbor discovery in cognitive radio networks

A cognitive radio node is a radio device capable of operating over multiple channels. As a result, a network consisting of one or more cognitive radio nodes can adapt to varying channel availability in its geographical region by dynamically changing the channel (or channels) nodes are using for communication. We investigate the problem of designing a fast deterministic algorithm for neighbor discovery in an undirected network consisting of one or more cognitive radio nodes when different nodes may have different subsets of channels available for communication at their location. We say that a neighbor discovery algorithm is fast if its time-complexity is polynomial in actual network-size. We prove that it is impossible to devise a fast algorithm to solve the neighbor discovery problem if nodes are neither size-aware nor collision-aware. When nodes are collision-aware, we present a fast algorithm for neighbor discovery with time-complexity of O ( p m log n ) , where p denotes the actual number of nodes present in the network, n denotes the size of space used for assigning labels to the nodes, and m denotes the number of channels over which nodes can operate. Our algorithm has the desirable property that every node knows when the neighbor discovery has completed and, moreover, all nodes terminate at the same time. When nodes are size-aware, we use a gossiping algorithm for a single-channel network with large labels that was proposed recently to derive a fast adaptive algorithm for neighbor discovery. We also investigate the neighbor discovery problem when the network may be directed. Finally, we describe a way to speed up neighbor discovery when nodes are collision-aware and channel availability sets of neighboring nodes do not vary significantly.

Optimum Receiver Array Design for Magnetic Induction Tomography

Magnetic induction tomography (MIT) is an imaging modality that aims at mapping the distribution of the electrical conductivity inside the body. Eddy currents are induced in the body by magnetic induction and the resulting fields are measured by an array of receiver coils. In MIT, the location of the receivers affects the quality of the image reconstruction. In this paper, a fast deterministic algorithm was applied to obtain optimum receiver array designs for a given specific excitation. The design strategy is based on the iterative exclusion of receiver locations, which yield poor conductivity information, from the space spanning all possible locations until a feasible design is reached. The applicability of "regionally focused" MIT designs that increase the image resolution at a particular region was demonstrated. Currently used design geometries and the corresponding reconstructed images were compared to the images obtained by optimized designs. The eigenvalue analysis of the Hessian matrix showed that the algorithm tends to maintain identical conductivity information content sensed by the receivers. Although the method does not guarantee finding the optimum design globally, the results demonstrate the practical usability of this algorithm in MIT experimental designs.

A panoply of quantum algorithms

This paper's aim is to explore improvements to, and applications of, a fundamental quantum algorithm invented by Grover[1]. Grover's algorithm is a basic tool that can be applied to a large number of problems in computer science, creating quantum algorithms that are polynomially faster than fastest known and fastest possible classical algorithms that solve the same problems. Our goal in this paper is to make these techniques readily accessible to those without a strong background in quantum physics: we achieve this by providing a set of tools, each of which makes use of Grover's algorithm or similar techniques, which can be used as subroutines in many quantum algorithms. The tools we provide are carefully constructed: they are easy to use, and in many cases they are asymptotically faster than the best tools previously available. The tools we build on include algorithms by Boyer, Brassard, Hoyer and Tapp[2], Buhrman, Cleve, de Witt and Zalka[3] and Durr and Hoyer[4]. After creating our tools, we create several new quantum algorithms, each of which is faster than the fastest known deterministic classical algorithm that accomplishes the same aim, and some of which are faster than the fastest possible deterministic classical algorithm. These algorithms solve problems from the fields of graph theory and computational geometry, and some employ dynamic programming techniques. We discuss a breadth-first search that is faster than Θ(edges) (the classical limit) in a dense graph, maximum-points-on-a-line in O(N3/2 lgN) (faster than the fastest classical algorithm known), as well as several other algorithms that are similarly illustrative of solutions in some class of problem. Through these new algorithms we illustrate the use of our tools, working to encourage their use and the study of quantum algorithms in general.

Smoothing of Piecewise Linear Paths

We present an anytime-capable fast deterministic greedy algorithm for smoothing piecewise linear paths consisting of connected linear segments. With this method, path points with only a small influence on path geometry (i.e. aligned or nearly aligned points) are successively removed. Due to the removal of less important path points, the computational and memory requirements of the paths are reduced and traversing the path is accelerated. Our algorithm can be used in many different applications, e.g. sweeping, path finding, programming-by-demonstration in a virtual environment, or 6D CNC milling. The algorithm handles points with positional and orientational coordinates of arbitrary dimension.

Mathematical Complexity of Running Filters on Semi-Groups and Related Problems

This paper proves that the mathematical complexity of running filters on semi-groups is C(p)=3-(6/(p+1)) operations per sample, where p is the filter window. On other algebraic structures, the mathematical complexity of the filtering can be lower. Two such cases are investigated: the running filtering on max/min lattice and on additions group. They correspond to max/min and moving average filters. It is shown that the algorithms developed for semi-groups are of interest in such cases, too. Thus, the fastest deterministic algorithm for max/min filtering known so far is based on an algorithm for running filtering on semi-groups. First, an optimized version is proposed in this paper. Then, further improvement for data-dependent max/min algorithms is proposed. Finally, the case of moving average and exponential smoothing is investigated. Their implementation by using the algorithms for semi-groups is of lower complexity than the classical recursive implementation for the particular case of small size windows (p=3 and p=4).

Deterministic Rendezvous in Graphs

Two mobile agents having distinct identifiers and located in nodes of an unknown anonymous connected graph, have to meet at some node of the graph. We seek fast deterministic algorithms for this rendezvous problem, under two scenarios: simultaneous startup, when both agents start executing the algorithm at the same time, and arbitrary startup, when starting times of the agents are arbitrarily decided by an adversary. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of steps since the startup of the later agent until rendezvous is achieved. We first show that rendezvous can be completed at cost O(n + log l) on any n-node tree, where l is the smaller of the two identifiers, even with arbitrary startup. This complexity of the cost cannot be improved for some trees, even with simultaneous startup. Efficient rendezvous in trees relies on fast network exploration and cannot be used when the graph contains cycles. We further study the simplest such network, i.e., the ring. We prove that, with simultaneous startup, optimal cost of rendezvous on any ring is Θ(D log l), where D is the initial distance between agents. We also establish bounds on rendezvous cost in rings with arbitrary startup. For arbitrary connected graphs, our main contribution is a deterministic rendezvous algorithm with cost polynomial in n, τ and log l, where τ is the difference between startup times of the agents. We also show a lower bound Ω (n2) on the cost of rendezvous in some family of graphs. If simultaneous startup is assumed, we construct a generic rendezvous algorithm, working for all connected graphs, which is optimal for the class of graphs of bounded degree, if the initial distance between agents is bounded.

Binary State Pattern Clustering: A Digital Paradigm for Class and Biomarker Discovery in Gene Microarray Studies of Cancer

Class and biomarker discovery continue to be among the preeminent goals in gene microarray studies of cancer. We have developed a new data mining technique, which we call Binary State Pattern Clustering (BSPC) that is specifically adapted for these purposes, with cancer and other categorical datasets. BSPC is capable of uncovering statistically significant sample subclasses and associated marker genes in a completely unsupervised manner. This is accomplished through the application of a digital paradigm, where the expression level of each potential marker gene is treated as being representative of its discrete functional state. Multiple genes that divide samples into states along the same boundaries form a kind of gene-cluster that has an associated sample-cluster. BSPC is an extremely fast deterministic algorithm that scales well to large datasets. Here we describe results of its application to three publicly available oligonucleotide microarray datasets. Using an alpha-level of 0.05, clusters reproducing many of the known sample classifications were identified along with associated biomarkers. In addition, a number of simulations were conducted using shuffled versions of each of the original datasets, noise-added datasets, as well as completely artificial datasets. The robustness of BSPC was compared to that of three other publicly available clustering methods: ISIS, CTWC and SAMBA. The simulations demonstrate BSPC's substantially greater noise tolerance and confirm the accuracy of our calculations of statistical significance.

Asynchronous deterministic rendezvous in graphs

Two mobile agents (robots) having distinct labels and located in nodes of an unknown anonymous connected graph have to meet. We consider the asynchronous version of this well-studied rendezvous problem and we seek fast deterministic algorithms for it. Since in the asynchronous setting, meeting at a node, which is normally required in rendezvous, is in general impossible, we relax the demand by allowing meeting of the agents inside an edge as well. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of edge traversals of both agents until rendezvous is achieved. If agents are initially situated at a distance D in an infinite line, we show a rendezvous algorithm with cost O ( D | L min | 2 ) when D is known and O ( ( D + | L max | ) 3 ) if D is unknown, where | L min | and | L max | are the lengths of the shorter and longer label of the agents, respectively. These results still hold for the case of the ring of unknown size, but then we also give an optimal algorithm of cost O ( n | L min | ) , if the size n of the ring is known, and of cost O ( n | L max | ) , if it is unknown. For arbitrary graphs, we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost O ( D | L min | ) if the topology of the graph and the initial positions are known to agents.

An O ( VE ) algorithm for ear decompositions of matching-covered graphs

Our main result is an O(nm) -time (deterministic) algorithm for constructing an ear decomposition of a matching-covered graph, where n and m denote the number of nodes and edges. The improvement in the running time comes from new structural results that give a sharpened version of Lovász and Plummer's Two-Ear Theorem. Our algorithm is based on O(nm) -time algorithms for two other fundamental problems in matching theory, namely, finding all the allowed edges of a graph, and finding the canonical partition of an elementary graph. To the best of our knowledge, no faster deterministic algorithms are known for these two fundamental problems.

Design of discrete coefficient FIR filters by a fast entropy-directed deterministic annealing algorithm

We present an entropy-directed deterministic annealing optimization algorithm and show its applicability to the problem of designing digital filters with discrete coefficients, each implemented as a sum of signed power-of-two terms and additional general hardware constraints. The algorithm is based on analogies from statistical mechanics and is related to the well-known mean field annealing algorithm. It utilizes estimates of conditional entropy to prune the problem during the optimization, thereby reducing the computational time by 30 to 50%. In conjunction with a scheme to compute the value of the objective function as a sequence of updates, this approach leads to a very fast algorithm. As an application example demonstrating the potential of the new method, we consider the design of digital filters with discrete coefficients consisting of a minimum number of signed power-of-two terms.

Deterministic sorting in O( nloglog n) time and linear space

We present a fast deterministic algorithm for integer sorting in linear space. Our algorithm sorts n integers in the range {0,1,2,…, m−1} in linear space in O( nloglog n) time. This improves our previous result [Y. Han, Inform. and Comput. 170 (1) (2001) 81–94] which sorts in O( nloglog nlogloglog n) time and linear space. This also improves previous best deterministic sorting algorithm [A. Andersson, T. Hagerup, S. Nilsson, R. Raman, in: Proc. 1995 Symposium on Theory of Computing (1995) 427–436; Y. Han, X. Shen, in: Proc. 1995 International Computing and Combinatorics Conference, in: Lecture Notes in Comput. Sci. 959 (1995) 324–333] which sorts in O( nloglog n) time but uses O( m ε ) space. Our results also improves the result of Andersson et al. [A. Andersson, T. Hagerup, S. Nilsson, R. Raman, in: Proc. 1995 Symposium on Theory of Computing (1995) 427–436] which sorts in O( nloglog n) time and linear space but uses randomization.

Improved Fast Integer Sorting in Linear Space

We present a fast deterministic algorithm for integer sorting in linear space. Our algorithm sorts n integers in the range {0, 1, 2, …, m−1} in linear space in O(n log log n log log log n) time. When log m≥log2+ϵn, ϵ>0, we can further achieve O(n log log n) time. This improves the O(n(log log n)2) time bound given in M. Thorup (1998) in “Proc. 1998 ACM-SIAM Symp. on Discrete Algorithms (SODA'98),” pp. 550–555). This result is obtained by combining our new technique with that of Thorup's. Signature sorting (A. Andersson, T. Hagerup, S. Nilsson, and R. Raman, 1995, in “Proc. 1995 Symposium on Theory of Computing,” pp. 427–436), A. Andersson's result (1996, in “Proc. 1996 IEEE Symp. on Foundations of Computer Science,” pp. 135–141), R. Raman's result (1996, Lecture Notes in Computer Science, Vol. 1136, pp. 121–137, Springer-Verlag Berlin/New York), and our previous result (Y. Han and X. Shen, 1999, in “Proc. 1999 Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'99),” Baltimore, MD, January, pp. 419–428) are also used for the design of our algorithms. We provide an approach and techniques which are totally different from previous approaches and techniques for the problem. As a consequence our technique can be extended to apply to nonconservative sorting and parallel sorting. Our nonconservative sorting algorithm sorts n integers in {0, 1, …, m−1} in time O(n(log log n)2/(log k+log log log n)) using word length k log(m+n), where k≤log n. Our EREW parallel algorithm sorts n integers in {0, 1, …, m−1} in O((log n)2) time and O(n(log log n)2/log log log n) operations provided log m=Ω((log n)2).

Solving an Inverse Diffusion Problem for Magnetic Resonance Dosimetry by a Fast Regularization Method

An inverse diffusion problem that appears in Magnetic Resonance dosimetry is studied. The problem is shown to be equivalent to a deconvolution problem with a known kernel. To cope with the singularity of the kernel, nonlinear regularization functionals are considered which can provide regular solutions, reproduce steep gradients and impose positivity constraints. A fast deterministic algorithm for solving the involved non-convex minimization problem is used. Accurate restorations on real 256×256 images are obtained by the algorithm in a few minutes on a 266-MHz PC that allow to precisely quantitate the relative absorbed dose.