Randomized Algorithm

In this study, a new Hamiltonian loop construction algorithm is proposed for planar 3-regular diagrams of specific structures. Based on the original element theory and the perfect matching algorithm I proposed earlier, the algorithm establishes a deterministic Hamiltonian loop construction method by analyzing the local structural characteristics and global topological relationships of the graph. Theoretical analysis shows that for a specific structural plane 3-regular graph with a vertex number of 6k or 4i (k,i ≥ 1), the proposed algorithm can construct a Hamiltonian loop in linear time. The correctness of the algorithm is verified by rigorous mathematical proof, and its time complexity is significantly better than that of the Hamiltonian loop algorithm on the general graph. In addition, this study also reveals the intrinsic relationship between the Hamiltonian circuit, perfect matching, and graph coloring properties of this type of graph, which provides a new theoretical perspective for the study of the structural characteristics of specific graph classes. At the application level, the algorithm can be effectively applied to practical problems such as integrated circuit wiring optimization and network topology design. Experimental results show that compared with existing methods, the proposed algorithm has significant advantages in efficiency and scalability.

We propose a new nonparametric classification framework for numerical patterns, which can also be exploitable for exploratory data analysis. The key idea is approximating each class region by a family of convex geometric sets which can cover samples of the target class without containing any samples of other classes. According to this framework, we consider a combinatorial classifier based on a family of spheres, each of which is the minimum covering sphere for a subset of positive samples and does not contain any negative samples. We also present a polynomial-time exact algorithm and an incremental randomized algorithm to compute it. In addition, we discuss the soft-classification version and evaluate these algorithms by some numerical experiments.

We analyze a quasi-Monte Carlo method to solve the initial-value problem for a system of differential equations y (t) = f(t, y(t)). The function f is smooth in y and we suppose that f and D 1 y f are of bounded variation in t and that D 2 y f is bounded in a neighborhood of the graph of the solution. The method is akin to the second order Heun method of the Runge-Kutta family. It uses a quasi-Monte Carlo estimate of integrals. The error bound involves the square of the step size as well as the discrepancy of the point set used for quasi-Monte Carlo approximation. Numerical experiments show that the quasi-randomized method outperforms a recently proposed randomized numerical method.

This chapter reviews randomization algorithms developed in the last few years to solve a wide range of geometric optimization problems. We review a number of general techniques, including randomized binary search, randomized linear-programming algorithms, and random sampling. Next, we describe several applications of these techniques, including facility location, proximity problems, nearest neighbor searching, statistical estimators, and Euclidean TSP.

In this paper we present a truly practical and provably optimal O(n logh) time outputsensitive algorithm for the planar convex hull problem. The basic algorithm is similar to the algorithm presented in Chan, Snoeyink and Yap 2] where the median-nding step is replaced by an approximate median. We analyze two such schemes and show that for both methods, the algorithm runs in expected O(n log h) time. The expected number of comparisons can be made smaller than 5n logh for the upper-hull. We further show that the probability of deviation from expected running time approaches 0 rapidly with increasing values of n and h for any input. Our experiments suggest that this algorithm is a practical alternative to the worstcase O(n log n) algorithms like Graham's and especially faster for small output-sizes. Our approach bears some resemblance to a recent algorithm of Wenger 13] but our analysis is substantially di erent. The planar convex hull problem is perhaps the most studied problem in computational geometry and a large body of literature deals with computing convex hulls. Graham 5] was the rst to present an O(n log n) worst-case time algorithm. This algorithm is optimal as Yao 14] showed that (n log n) is the lower bound of the convex hull problem for the worst-case input. Some simple algorithms have O(n) expected time for known distributions of points such as uniform in a box, normal, etc. The rst output-sensitive algorithm was proposed by Chand and Kapur 3]. The two-dimensional version of their algorithm is known as the rope fence method and was independently reported by Jarvis 6]. The rope fence method takes O(nh) time to compute h extreme edges of the convex hull. Kirkpatrick and Seidel 8] proved an (n logh) lower bound when both input and output sizes are considered, so Yao's lower-bound is a special case when log h 2 (log n). They also proposed an O(n log h) optimal algorithm based on the prune-and-search technique developed by Dyer 4] and Megiddo 9]. However, it has high constants and is considered prohibitively complicated for implementation. Very recently, in 1], two O(n log h) algorithms have been proposed. One uses the linear-time median nding algorithm and the other uses a clever grouping technique. Although the latter algorithm does not have any expensive median-nding step it relies on a sophisticated logarithmic time tangent-nding routine.

The traditional zero-one principle for sorting networks states that "if a network with n input lines sorts all 2 n binary sequences into nondecreasing order, then it will sort any arbitrary sequence of n numbers into nondecreasing order". We generalize this to the situation when a network sorts almost all binary sequences and relate it to the behavior of the sorting network on arbitrary inputs. We also present an application to mesh sorting.

In this paper we focus on the problem of designing very fast parallel algorithms for the planar convex hull problem that achieve the optimal O(n log H) work-bound for input size n and output size H. Our algorithms are designed for the arbitrary CRCW PRAM model. We first describe a very simple ©(log n log H) time optimal deterministic algorithm for the planar hulls which is an improvement over the previously known ~(log 2 n) time algorithm for small outputs. For larger values of H, we can achieve a running time of ©(log n log log n) steps with optimal work. We also present a fast randomized algorithm that runs in expected time ©(log H-log log n) and does optimal O(nlogH) work. For logH = ~2(log log n), we can achieve the optimal running time of O(log H) while simultaneously keeping the work optimal. When log H is o(log n), our results improve upon the previously best known ©(log n) expected time randomized algorithm of Ghouse and Goodrich. The randomized algorithms do not assume any input distribution and the running times hold with high probability.

We address a version of the set-cover problem where we do not know the sets initially (and hence referred to as covert) but we can query an element to find out which sets contain this element as well as query a set to know the elements. We want to find a small set-cover using a minimal number of such queries. We present a Monte Carlo randomized algorithm that approximates an optimal set-cover of size OP T within O(log N ) factor with high probability using O(OP T • log 2 N ) queries where N is the number of element in the universal set. We apply this technique to the network discovery problem that involves certifying all the edges and non-edges of an unknown n-vertices graph based on layered-graph queries from a minimal number of vertices. By reducing it to the covert set-cover problem we present an O(log 2 n)competitive Monte Carlo randomized algorithm for the covert version of network discovery problem. The previously best known algorithm has a competitive ratio of Ω( √ n log n) and therefore our result achieves an exponential improvement.

The empty space around n disjoint line segments in the plane can be partitioned into n + 1 convex faces by extending the segments in some order. The dual graph of such a partition is the plane graph whose vertices correspond to the n+1 convex faces, and every segment endpoint corresponds to an edge between the two incident faces on opposite sides of the segment. We construct, for every set of n disjoint line segments in the plane, a convex partition whose dual graph is 2-edge connected. 1

We show that the abstract Voronoi diagram of n sites in the plane can be constructed in time O(n log n) by a randomized algorithm. This yields an alternative, but simpler, O(n log n) algorithm in many previously considered cases and the first O(n log n) algorithm in some cases, e.g., disjoint convex sites with the Euclidean distance function. Abstract Voronoi diagrams are given by a family of bisecting curves and were recently introduced by Klein [-13]. Our algorithm is based on Clarkson and Shor's randomized incremental construction technique [7].

This paper studies the point location problem in Delaunay triangulations without preprocessing and additional storage. The proposed procedure finds the query point simply by "walking through" the triangulation, after selecting a "good starting point" by random sampling. The analysis generalizes and extends a recent result for d = 2 dimensions by proving this procedure to take expected time close to O(nl/(d+l)) for point location in Delaunay triangulations of n random points in d = 3 dimensions. Empirical results in both two and three dimensions show that this procedure is efficient in practice.

We present a randomized algorithm for computing the Voronoi diagram of line segments using coarse-grained parallel machines. Operating on P processors, for any input of n line segments, this algorithm performs O((n log n)/P) local operations per processor, O(n/P) messages per processor, and O(1) communication phases, with high probability for n = (P 3+ε ).

A set of agents arrive simultaneously to a service facility. Each agent requires the facility for a certain length of time and incurs a cost for each unit of time spent in queue. Attached to each agent is an index, which is the ratio of her waiting cost rate to processing time. Eciency dictates that the agents be served in decreasing order of their indices. If monetary compensations are disallowed, fairness suggests that agents be served in a random order, each ordering of the agents being equally likely. The ecien t ordering is unfair to agents with low indices; the random service order is typically extremely inecien t. It is well-known that this gap can be bridged by using monetary compensations. This paper is motivated by the need to design compensation schemes that are fair to all the agents. To that end, we propose two solution concepts (RP and CRP cores), which serve to place upper bounds on the cost share of any coalition of agents. The solutions dier in the denition of the e...

We present a new Following Car Algorithm in Microscopic Urban Traffic Models which integrates some real-life factors that need to be considered, such as the effect of random distributions in the car speed, acceleration, entry of lane… Our architecture is based on Multi-Agent Randomized Systems (MARS) developed in earlier publications.

The AANDERAA-ROSENBERG conjecture states that every algorithm which decides whether some graph, which is represented by means' of its adjacencymatrix, has some non-trivial monotonic property, must in the worst case probe O(n 2 ) entries. We present a number of techniques which enable us to prove for several specific properties (e.g. connectedness) that in order to decide these properties in fact all edges must be probed by the algorithm. Moreover, we present some new examples of properties on undirected graphs, where not all the edges are needed, or even stronger, O(n) edges are sufficient.

The longest increasing circular subsequence (LICS) of a list is considered. A Monte-Carlo algorithm to compute it is given which has worst case execution time O(n 3/2 log n) and storage requirement O(n). It is proved that the expected length µ(n) of the LICS satisfies limn→∞ µ(n) Numerical experiments with the algorithm suggest that |µ(n) -2 √ n| = O(n 1/6 ).

In this paper, we introduced a statistical rule-based method to create rules for SpamAssassin to detect spams in different languages. The theoretical framework of generating and maintaining multilingual rules were also illustrated. The experiments were conducted against the dataset of three languages including Chinese, Vietnamese and English. The detecting achievement of multilingual rule was 89.5% for the true detection and only 3.8% for the failed alarm at the threshold of 2 while the true detection rate of single language rule was not over 61% and the failed alarm rate was up to 4.9%.

This paper describes new algorithms for approximately solving the concurrent multicommodity flow problem with uniform capacities. These algorithms are much faster than algorithms discovered previously. Besides being an important problem in its own right, the uniform-capacity concurrent flow problem has many interesting appli- cations. Leighton and Rao used uniform-capacity concurrent flow to find an approximately "sparsest cut" in a graph and thereby approximately solve a wide variety of graph problems, including minimum feedback arc set, minimum cut linear arrangement, and minimum area layout. However, their method appeared to be impractical as it required solving a large linear program. This paper shows that their method might be practical by giving an O(m log m) expected- time randomized algorithm for their concurrent flow problem on an m-edge graph. Raghavan and Thompson used uniform-capacity concurrent flow to solve approximately a channel width minimization problem in very large scale integration. An O (k 3/2 (m + n log n)) expected-time randomized algorithm and an O (k min {n, k} (m + n log n) log k) deterministic algorithm is given for this problem when the channel width is f2 (log n), where k denotes the number of wires to be routed in an n-node, m-edge network.

We consider the problem of incrementally designing a network to route demand to a single sink on an underlying metric space. We are given cables whose costs per unit length scale in a concave fashion with capacity. Under certain natural restrictions on the costs (called the Access Network Design constraints), we present a simple and efficient randomized algorithm that is competitive to the minimum cost solution when the demand points arrive online. In particular, if the order of arrival is a random permutation, we can prove a Ç´½µ competitive ratio. For the fully adverserial case, the algorithm is Ǵõ-competitive, where à is the number of different pipe types. Since the value of à is typically small, this improves the previous Ç´ÐÓ Ò ÐÓ ÐÓ Òµ-competitive algorithm which was based on probabilistically approximating the underlying metric by a tree metric. Our algorithm also improves the best known approximation ratio and running time for the offline version of this problem.

All previously known algorithms for solving the multicommodity ow problem with capacities are based on linear programming. The best of these algorithms 15] uses a fast matrix multiplication algorithm and takes O(k 3:5 n 3 m :5 log(nDU)) time for the multicommodity ow problem with integer demands and at least O(k 2:5 n 2 m :5 log(n ?1 DU)) time to nd an approximate solution, where k is the number of commodities, n and m denote the number of nodes and edges in the network, D is the largest demand, and U is the largest edge capacity. Substantially more time is needed to nd an exact solution. As a consequence, even multicommodity ow problems with just a few commodities are believed to be much harder than single-commodity maximum-ow or minimum-cost ow problems. In this paper, we describe the rst polynomial-time combinatorial algorithms for approximately solving the multicommodity ow problem. The running time of our randomized algorithm is (up to log factors) the same as the time needed to solve k single-commodity ow problems, thus giving the surprising result that approximately computing a k-commodity maximumow is not much harder than computing about k single-commodity maximum-ows in isolation. In fact, we prove that a (simple) k-commodity ow problem can be approximately solved by approximately solving O(k log 2 n) single-commodity minimum-cost ow problems. Our kcommodity algorithm runs in O(knm log 4 n) time with high probability. We also describe a deterministic algorithm that uses an O(k)-factor more time. Given any multicommodity ow problem as input, both algorithms are guaranteed to provide a feasible solution to a modi ed ow problem in which all capacities are increased by a (1 + )-factor, or to provide a proof that there is no feasible solution to the original problem. We also describe faster approximation algorithms for multicommodity ow problems with a special structure, such as those that arise in the \sparsest cut" problems studied in 8, 10, 9], and the uniform concurrent ow problems studied in 12, 9] if k p m.

Wireless sensor networks (WSNs) are emerging as an effective means for environment monitoring. This paper investigates a strategy for energy efficient monitoring in WSNs that partitions the sensors into covers, and then activates the covers iteratively in a round-robin fashion. This approach takes advantage of the overlap created when many sensors monitor a single area. Our work builds upon previous work in , where the model is first formulated. We have designed three approximation algorithms for a variation of the SET K-COVER problem, where the objective is to partition the sensors into covers such that the number of covers that include an area, summed over all areas, is maximized. The first algorithm is randomized and partitions the sensors, in expectation, within a fraction 1 -1 e (∼.63) of the optimum. We present two other deterministic approximation algorithms. One is a distributed greedy algorithm with a 1 2 approximation ratio and the other is a centralized greedy algorithm with a 1-1 e approximation ratio. We show that it is NP-Complete to guarantee better than 15 16 of the optimal coverage, indicating that all three algorithms perform well with respect to the best approximation algorithm possible in polynomial time, assuming P = N P . Simulations indicate that in practice, the deterministic algorithms perform far above their worst case bounds, consistently covering more than 72% of what is covered by an optimum solution. Simulations also indicate that the increase in longevity is proportional to the amount of overlap amongst the sensors. The algorithms are fast, easy to use, and according to simulations, significantly increase the longevity of sensor networks. The randomized algorithm in particular seems quite practical.

This paper presents two new randomized distributed algorithms for the generation of acyclic orientations upon anonymous distributed systems of arbitrary topology. Both algorithms, called Alg-Neighbors and Alg-Edges, make use of biased and unbiased dice having f ≥ 2 faces, and are analyzed in terms of correctness, expected time complexity and rate of convergence. First, the Alg-Neighbors algorithm is presented as a generalization of the Calabrese/França algorithm for dice (or coins) with 2 faces. It is proved that a convenient biasing function applied to all dice changes the expected time complexity from subexponential, i.e., O f f f −1 n−1 (for unbiased dice with f faces), to O(n), where n is the number of the system's nodes. Next, it is shown that the Alg-Edges algorithm is able to produce acyclic orientations in O(log f m) steps with high probability, where m denotes the total number of edges. Finally, a speed of convergence versus quality of acyclic orientation generation (associated number of colors) tradeoff is identified between Alg-Neighbors and Alg-Edges algorithms. Computational experiments were carried out in order to provide a more accurate description of the behavior of both algorithms.

We present an approach for applying symmetry reduction techniques to probabilistic model checking, a formal verification method for the quantitative analysis of systems with stochastic characteristics. We target systems with a set of non-trivial, but interchangeable, components such as those which commonly arise in randomised distributed algorithms or probabilistic communication protocols. We show, for three types of probabilistic models, that symmetry reduction, similarly to the non-probabilistic case, allows verification to instead be performed on a bisimilar quotient model which may be up to factorially smaller. We then propose an efficient algorithm for the construction of the quotient model using a symbolic implementation based on multi-terminal binary decision diagrams (MTBDDs) and, using four large case studies, demonstrate that this approach offers not only a dramatic increase in the size of probabilistic model which can be quantitatively analysed but also a significant decrease in the corresponding run-times.

We consider the problem of weighted rectilinear approximation on the plane and offer both exact algorithms and heuristics with provable performance bounds. Let S = {(p i , w i)} be a set of n points pi in the plane, with associated distance-modifying weights wi > 0. We present algorithms for finding the best fit to S among x-monotone rectilinear polylines R with a given number k < n of horizontal segments. We measure the quality of the fit by the greatest weighted vertical distance, i.e., the approximation error is max 1≤i≤n w i d v (p i , R), where d v (p i , R) is the vertical distance from pi to R. We can solve for arbitrary k optimally in O(n 2) or approximately in O(n log 2 n) time. We also describe a randomized algorithm with an O(n log 2 n) expected running time for the unweighted case and describe how to modify it to handle the weighted case in O(n log 3 n) expected time. All algorithms require O(n) space.

We consider the problem of weighted rectilinear approximation on the plane and offer both exact algorithms and heuristics with provable performance bounds. Let S = {(p i , w i)} be a set of n points pi in the plane, with associated distance-modifying weights wi > 0. We present algorithms for finding the best fit to S among x-monotone rectilinear polylines R with a given number k < n of horizontal segments. We measure the quality of the fit by the greatest weighted vertical distance, i.e., the approximation error is max 1≤i≤n w i d v (p i , R), where d v (p i , R) is the vertical distance from pi to R. We can solve for arbitrary k optimally in O(n 2) or approximately in O(n log 2 n) time. We also describe a randomized algorithm with an O(n log 2 n) expected running time for the unweighted case and describe how to modify it to handle the weighted case in O(n log 3 n) expected time. All algorithms require O(n) space.

Most algorithms of computational geometry are designed for the Real-RAM and non-degenerate inputs. We call such algorithms idealistic. Executing an idealistic algorithm with floating point arithmetic may fail. Controlled perturbation replaces an input x by a random nearby x in the δ-neighborhood of x and then runs the floating point version of the idealistic algorithm on x. The hope is that this will produce the correct result for x with constant probability provided that δ is small and the precision L of the floating point system is large enough. We turn this hope into a theorem for a large class of geometric algorithms and describe a general methodology for deriving a relation between δ and L. We exemplify the usefulness of the methodology by examples.

In analogy with ε-biased sets over Z n 2 , we construct explicit ε-biased sets over nonabelian finite groups G. That is, we find sets S ⊂ G such that E x∈S ρ(x) ≤ ε for any nontrivial irreducible representation ρ. Equivalently, such sets make G's Cayley graph an expander with eigenvalue |λ| ≤ ε. The Alon-Roichman theorem shows that random sets of size O(log |G|/ε 2) suffice. For groups of the form G = G 1 × • • • × G n , our construction has size poly(max i |G i |, n, ε −1), and we show that a set S ⊂ G n considered by Meka and Zuckerman that fools read-once branching programs over G is also ε-biased in this sense. For solvable groups whose abelian quotients have constant exponent, we obtain ε-biased sets of size (log |G|) 1+o(1) poly(ε −1). Our techniques include derandomized squaring (in both the matrix product and tensor product senses) and a Chernoff-like bound on the expected norm of the product of independently random operators that may be of independent interest.

Drisko proved that 2n − 1 matchings of size n in a bipartite graph have a rainbow matching of size n. For general graphs it is conjectured that 2n matchings suffice for this purpose (and that 2n − 1 matchings suffice when n is even). The known graphs showing sharpness of this conjecture for n even are called badges. We improve the previously best known bound from 3n − 2 to 3n − 3, using a new line of proof that involves analysis of the appearance of badges. We also prove a "cooperative" generalization: for t > 0 and n ≥ 3, any 3n − 4 + t sets of edges, the union of every t of which contains a matching of size n, have a rainbow matching of size n.

Given a directed acyclic graph (DAG) Gn = (Vn, E), a function on Gn is given by f : Vn → R. Such a function is monotone if f (x) ≤ f (y) for all (x, y) ∈ E. A local monotonicity reconstructor for Gn, introduced by Saks and Seshadhri (SICOMP 2010), is a randomized algorithm that, given access to an oracle for an almost monotone function f : Vn → R on Gn, can quickly evaluate a related function g : Vn → R which is guaranteed to be monotone. Furthermore, the reconstructor can be implemented in a distributed manner. Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitiveclosure-spanner (k-TC-spanner) of G is a directed graph H = (V, EH) that has (1) the same transitive-closure as G and (2) diameter at most k. Transitive-closure spanners are a common abstraction for applications in access control, property testing and data structures. In this paper, we show a connection between 2-TC-spanners of Gn and local monotonicity reconstructors for Gn. We show that an efficient local monotonicity reconstructor for Gn implies a sparse 2-TC-spanner of Gn, providing a new technique for proving lower bounds for local monotonicity reconstructors. We present tight upper and lower bounds on the size of the sparsest 2-TC-spanners of the directed hypercube and hypergrid, DAGs which are very-well studied in this area. These bounds imply lower bounds for local monotonicity reconstructors for the hypergrid (hypercube) that nearly match the known upper bounds.

We demonstrate that an algorithm proposed by Drineas et. al. in [7] to approximate the singular vectors/values of a matrix A, is not only of theoretical interest but also a fast, viable alternative to traditional algorithms. The algorithm samples a small number of rows (or columns) of the matrix, scales them appropriately to form a small matrix S and computes the singular value decomposition (SVD) of S, which is a good approximation to the SVD of the original matrix. We experimentally evaluate the accuracy and speed of this randomized algorithm using image matrices and three different sampling schemes. Our results show that our approximations of the singular vectors of A span almost the same space as the corresponding exact singular vectors of A.

We demonstrate that an algorithm proposed by Drineas et. al. in [7] to approximate the singular vectors/values of a matrix A, is not only of theoretical interest but also a fast, viable alternative to traditional algorithms. The algorithm samples a small number of rows (or columns) of the matrix, scales them appropriately to form a small matrix S and computes the singular value decomposition (SVD) of S, which is a good approximation to the SVD of the original matrix. We experimentally evaluate the accuracy and speed of this randomized algorithm using image matrices and three different sampling schemes. Our results show that our approximations of the singular vectors of A span almost the same space as the corresponding exact singular vectors of A.

A novel approach based on the application of an expressly tailored random search algorithm (Stochastic Gain Synthesis) to the design of a MiniUAV control system is presented in this paper. This methodology, joined to the well known Stochastic Robustness Analysis concept, may be applied to the design of control systems granting adequate level of robustness according to plant characterization and stability criteria. Due to the unconventional dynamics and its sensitivity to geometric, inertial and aerodynamic characteristics, control system design for MiniUAVs represents a challenging task whenever stability and performance requirements related to aircraft flying qualities have to be fulfilled. The robustness characteristics of the designed controller play a key role in granting adequate level of tolerance to plant uncertainties, environmental changes and flight condition variations within the planned mission profile.

The area of research on probabilistic and randomized methods for analysis and design of uncertain systems is fairly recent and is focused both on algorithmic as well as theoretical developments. In this paper a framework for randomization-based control design is presented and applied to a Mini-UAV platform. The proposed approach makes use of random search and uncertainty randomization for controller synthesis and probabilistic robustness analysis. Several structured uncertain parameters, related to the plant and to the operating conditions, are taken into account to design a robust flight control system. A selection criterion, based on estimated probability and its degradation function, is proposed in order to match stability and performance metrics fulfillment. Computational issues associated to the specific application, integration of a priori domain knowledge and human designer interaction with automated design are also addressed.

Mini-UAVs (Unmanned Aerial Vehicles) have been the subject of a large number of successful designs aimed to research, commercial and military purposes. A mini-UAV platform can be considered as a miniaturized aircraft so that classical design methodologies can be extended to this aerial vehicles category. However, due the presence of plant uncertainties, flight dynamic analysis and control system design are critical, and suitable robustness techniques need to be developed. In this paper, we present an innovative robust control strategy based on powerful tools provided by the theory of randomized algorithms. Within the context of the National Project Cofin 2004, the proposed methodology has been implemented for guidance/trajectory tracking and platform stabilization feedback loops of the autopilot system. The numerical results obtained are presented.

In this paper, we show an O(n + m) time Turing reduction from the tree pattern matching problem to another problem called the subset matching problem. Subsequent works have given efficient deterministic and randomized algorithms for the subset matching problem. Together, these works yield an O n log 2 m + m time deterministic algorithm and an O(n log n + m) time Monte Carlo algorithm for the tree pattern matching problem.

Given a graph G = (V, E) and two positive integers k and ∆, the ∆-closest phylogenetic k-th root problem (∆CP R k) is to find a (phylogenetic) tree T such that the degree of each internal node in T is at least three and at most ∆, the external nodes of T are exactly the elements of V , and the number of "kdisagreements" is minimized. In this paper we give an approximation algorithm for ∆CP R2 for any fixed ∆ > 3. The expected ratio of our algorithm is 3, if there is no vertex with degree more than ∆ − 2, and ∆ + 2, otherwise, which improves the best known previous ratio of 8 for ∆ ≤ 5.

This paper provides a formal framework for reasoning about randomized distributed algorithms. We generalize the notion of indistinguishability, the most useful tool in deterministic lower bounds, to apply to a probabilistic setting. We use the new notion to prove a lower bound on the probability at which it can be guaranteed that honest parties will not decide on a possibly bogus value. Moreover, we show that the bound is tight by providing a protocol that matches the bound. This completely characterizes the quality of decisions that protocols for a randomized multi-valued Consensus problem can guarantee in an asynchronous environment with Byzantine faults. Proving lower bounds for probabilistic protocols can be rather subtle. Probabilistic protocols have been around for at least four decades and are receiving a lot of attention with the emergence of blockchain systems. Nonetheless, we are aware of only a handful of randomized lower bounds. The formal tool that we develop in order to carry out our lower bound proof is of independent interest. It provides a rigorous yet accessible means to prove lower bounds for probabilistic protocols in general. CCS CONCEPTS • Theory of computation → Distributed algorithms; Semantics and reasoning; • Computing methodologies → Distributed computing methodologies.

This paper provides a formal framework for reasoning about randomized distributed algorithms. We generalize the notion of indistinguishability, the most useful tool in deterministic lower bounds, to apply to a probabilistic setting. We use the new notion to prove a lower bound on the probability at which it can be guaranteed that honest parties will not decide on a possibly bogus value. Moreover, we show that the bound is tight by providing a protocol that matches the bound. This completely characterizes the quality of decisions that protocols for a randomized multi-valued Consensus problem can guarantee in an asynchronous environment with Byzantine faults. Proving lower bounds for probabilistic protocols can be rather subtle. Probabilistic protocols have been around for at least four decades and are receiving a lot of attention with the emergence of blockchain systems. Nonetheless, we are aware of only a handful of randomized lower bounds. The formal tool that we develop in order to carry out our lower bound proof is of independent interest. It provides a rigorous yet accessible means to prove lower bounds for probabilistic protocols in general. CCS CONCEPTS • Theory of computation → Distributed algorithms; Semantics and reasoning; • Computing methodologies → Distributed computing methodologies.

Spammers continues to uses new methods and the types of email content becomes more difficult, text-based anti-spam methods are not good enough to prevent spam. Spam image making techniques are designed to bypass well-known image spam detection algorithms like optical character recognition (OCR) algorithm. As the use different methods to create image spam are increased, there must be an algorithm to prevent this type of spams. So using the different properties of images attached in the mail like width and height indicated in header of image, the aspect ratio of width and height, file size, image area, compression, owner, colors and file format of image. We can develop an algorithm which is used to detect the spam based images using the properties of the attached images. Also it is maintains a database for email addresses of spammers and blockthe email address based on result of detection process.

Low-rank matrix approximation is extremely useful in the analysis of data that arises in scientific computing, engineering applications, and data science. However, as data sizes grow, traditional low-rank matrix approximation methods, such as singular value decomposition (SVD) and column pivoting QR decomposition (CPQR), are either prohibitively expensive or cannot provide sufficiently accurate results. A solution is to use randomized low-rank matrix approximation methods such as randomized SVD , and randomized LU decomposition on extremely large data sets. In this paper, we focus on the randomized LU decomposition method. First, we employ a reorthogonalization procedure to perform the power iteration of the existing randomized LU algorithm to compensate for the rounding errors caused by the power method. Then we propose a novel randomized LU algorithm, called PowerLU, for the fixed low-rank approximation problem. PowerLU allows for an arbitrary number of passes of the input matrix, v ≥ 2. Recall that the existing randomized LU decomposition only allows an even number of passes. We prove the theoretical relationship between PowerLU and the existing randomized LU. Numerical experiments show that our proposed PowerLU is generally faster than the existing randomized LU decomposition, while remaining accurate. We also propose a version of PowerLU, called PowerLU FP, for the fixed precision low-rank matrix approximation problem. PowerLU FP is based on an efficient blocked adaptive rank determination Algorithm 4.1 proposed in this paper. We present numerical experiments that show that PowerLU FP can achieve almost the same accuracy and is faster than the randomized blocked QB algorithm by Martinsson and Voronin. We finally propose a single-pass algorithm based on LU factorization. Tests show that the accuracy of our single-pass algorithm is comparable with the existing single-pass algorithms.

Consider a geometric network G in the plane. The dilation between any two vertices x and y in G is the ratio of the shortest path distance between x and y in G to the Euclidean distance between them. The maximum dilation over all pairs of vertices in G is called the dilation of G. In this paper, a randomized algorithm is presented which, when given a polygonal cycle C on n vertices in the plane, computes in O(n log 3 n) expected time, the edge of C whose removal results in a polygonal path of smallest possible dilation. It is also shown that the edge whose removal gives a polygonal path of largest possible dilation can be computed in O(n log n) time. If C is a convex polygon, the running time for the latter problem becomes O(n). Finally, it is shown that a (1 - ϵ)-approximation to the dilation of every path C \{e}, for all edges e of C, can be computed in O(n log n) total time.

Data Fragments Allocation is an important issue in designing Distributed Database System. This problem is NP-complete, and thus requires fast heuristics and random algorithms to generate efficient solutions, so many algorithms for solving that have been reported in the literature. In this paper we used an object migration learning automaton-based algorithm. This approach is able to get suitable solutions in a reasonable amount of time even for moderate sized problems. Experimental results show that proposed algorithm has significant superiority over the several well-known methods.

Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g. , payment of royalties). Copies may be requested from IBM T.

x "I've never touched the hard stuff, only smoked grass a few times with the boys to be polite, and that's all, though ten is the age when the big guys come around teaching you all sorts to things. But happiness doesn't mean much to me, I still think life is better. Happiness is a mean son of a bitch and needs to be put in his place. Him and me aren't on the same team, and I'm cutting him dead. I've never gone in for politics, because somebody always stand to gain by it, but happiness is an even crummier racket, and their ought to be laws to put it out of business."-Momo, Emile Ajar 7.1 Vertical Decomposition σ Given a set S of n segments in the plane, and a subset R ⊆ S, let A | (R) denote the vertical decomposition of the plane, formed by the arrangement A (R) of the segments of R. This is the partition of the plane into interior disjoint vertical trapezoids formed by erecting vertical walls through each vertex of A | (R). Formally, a vertex of A | (R) is either an endpoint of a segment of R, or an intersection point of two of its segments. From each such vertex we shoot up (similarly, down) a vertical ray till it hits a segment of R, or it continues all the way to infinity. See figure on the right. Note, that a vertical trapezoid is defined by at most 4 segments: two segments defining its ceiling and floor, and two segments defining the two intersection points that induce the two vertical walls on its boundary. Of course, a vertical trapezoid might be degenerate, and thus defined by less segments (i.e., an unbounded vertical trapezoid, or a triangle). Vertical decomposition breaks the faces of the arrangement, that might be arbitrarily complicated, into entities (i.e., vertical trapezoids) of constant complexity. This make handling arrangements much easier computationally.

Spammers continues to uses new methods and the types of email content becomes more difficult, text-based anti-spam methods are not good enough to prevent spam. Spam image making techniques are designed to bypass well-known image spam detection algorithms like optical character recognition (OCR) algorithm. As the use different methods to create image spam are increased, there must be an algorithm to prevent this type of spams. So using the different properties of images attached in the mail like width and height indicated in header of image, the aspect ratio of width and height, file size, image area, compression, owner, colors and file format of image. We can develop an algorithm which is used to detect the spam based images using the properties of the attached images. Also it is maintains a database for email addresses of spammers and blockthe email address based on result of detection process.

Let $C$ be a depth-3 arithmetic circuit of size at most $s$, computing a polynomial $ f \in \mathbb{F}[x_1,\ldots, x_n] $ (where $\mathbb{F}$ = $\mathbb{Q}$ or $\mathbb{C}$) and the fan-in of the product gates of $C$ is bounded by $d$. We give a deterministic polynomial identity testing algorithm to check whether $f\equiv 0$ or not in time $ 2^d \text{ poly}(n,s) $.

Segmenting Arabic characters is one of the challenging and tedious tasks in the character recognition process. This work proposes an approach to construct Arabic characters segmentation based on Voronoi area. The proposed approach is constructed based on four existing algorithms, such as Voronoi Diagrams (VD), Divided-and-Conquer algorithm (DAC), Half Plane Intersection algorithm (HPI) and Incremental algorithm (I). The VD method performs the segmentation processes by obtaining extracting the line between the connected components, based on the neighbours graph. Whereas the DAC is a fundamental paradigm used for designing efficient algorithms, where the original problem is recursively divided into several simpler sub-problems of approximately equal size. Thenthen the solution of the original problem will be obtained, by merging the solutions of the subproblems. The HPI algorithm is based on Delauney Triangulation. The VD is constructed by HPI algorithm as follows: i): Connect connect Nearest nearest neighbors and; ii): draw the perpendicular bisector for each Delauney line. Meanwhile, the 'I' algorithm calculates the VD by incremental insertion of Voronoi regions, which makes it more efficient and numerically robust to produce the best structures and yield better results in the segmentation process. The proposed algorithm determines the neighbours graph by drawing a line from the centre of the connected components to trace the boundaries of the neighbours in a white background. If there is a gap between the connected components, then they are not considered as neighbours. The Euclidian distance is used as a base to draw line segment between the connected components, which is called as VD. In this research the IFN/ENIT1 dataset will be used. This dataset consists of 569 handwritten Arabic images of the Tunisian towns' names namesof Tunisian towns'. Several experiments has been carried out with the above mentioned base algorithms and compared in terms oftime,speed ofconstruction,the number ofvertices,and edges. The early results shows that in the static algorithm category, the performance of DAC is promising than the HPI in the static algorithm category, whereas the dynamic I algorithm consumes more time than static algorithms.