Mn Log Research Articles

FAST ITERATION ALGORITHM FOR INTEGRAL EQUATIONS OF THE FIRST KIND ARISING IN 2D DIFFRACTION BY SOFT OBSTACLES

We propose a new iteration numerical algorithm to solve boundary integral equations of the first kind arising in the 2D scattering by soft obstacles. The main idea is to operate on each iteration step with an integral equation, which has a convolution kernel, by changing the full kernel with a special averaging procedure. The practical convergence of the algorithm is demonstrated by some examples for three different geometries. If M is the number of iterations then the computational cost of the algorithm is MN log (N).

Set Covering with Our Eyes Closed

Given a universe $U$ of $n$ elements and a weighted collection $\mathscr{S}$ of $m$ subsets of $U$, the universal set cover problem is to a priori map each element $u \in U$ to a set $S(u) \in \mathscr{S}$ containing $u$ such that any set $X{\subseteq U}$ is covered by $S(X)=\cup_{u\in XS(u)$. The aim is to find a mapping such that the cost of $S(X)$ is as close as possible to the optimal set cover cost for $X$. (Such problems are also called oblivious or a priori optimization problems.) Unfortunately, for every universal mapping, the cost of $S(X)$ can be $\Omega(\sqrt{n})$ times larger than optimal if the set $X$ is adversarially chosen. In this paper we study the performance on average, when $X$ is a set of randomly chosen elements from the universe: we show how to efficiently find a universal map whose expected cost is $O(\log mn)$ times the expected optimal cost. In fact, we give a slightly improved analysis and show that this is the best possible. We generalize these ideas to weighted set cover and show similar guarantees to (nonmetric) facility location, where we have to balance the facility opening cost with the cost of connecting clients to the facilities. We show applications of our results to universal multicut and disc-covering problems and show how all these universal mappings give us algorithms for the stochastic online variants of the problems with the same competitive factors.

Elevated cord serum manganese level is associated with a neonatal high ponderal index

BackgroundThe effects of low-level prenatal manganese (Mn) exposure on neonatal growth remain unclear. The level of fetal Mn that may be considered “safe” has never been examined. MethodsA multicenter study including 1377 mother-infant pairs was conducted from 2008 through 2009 in Shanghai. Mn concentrations were determined for both the cord and maternal serum, as well as neonatal birth weight and birth length. The ponderal index (PI) was calculated as (birth weight g/birth length cm3)×100, and a ponderal index ≥3.17 was defined as a high ponderal index (HPI). ResultsThe median serum Mn concentration was 4.0μg/L in the cord blood, and was 2.8μg/L in maternal blood. Of 1377 infants, 135 (9.8%) had a HPI. After adjusting for potential confounders, cord serum Mn was not associated with birth weight. However, there was a linear relationship between the cord serum Mn and the birth length (adjusted ß=−0.5, 95% CI=−0.7 to −0.2, p<0.0001). Additionally, a nonlinear relationship was observed between the cord serum Mn and the ponderal index, and between the cord serum Mn and HPI. The ponderal index and the prevalence of HPI increased with Mn levels above 5.0μg/L (Log Mn ≥0.7). A high level of Mn in the cord (≥5.0μg/L) was associated with a higher ponderal index (adjusted ß=0.2, 95% CI=0.1 to 0.2, p<0.001) and a high risk of HPI (adjusted OR=3.3, 95% CI=1.8–6.0, p<0.001). ConclusionsHigher prenatal Mn exposure, even at a low level, is associated with a higher prevalence of HPI in a nonlinear pattern. Cord serum Mn levels less than 5.0μg/L may be considered safe with respect to neonatal ponderal index assessment.

Online Bounds on Balancing Two Independent Criteria with Replication and Reallocation

We study the online bicriteria load balancing problem in this paper. We choose a system of distributed homogeneous file servers located in a cluster as the scenario and propose three online approximate solutions for balancing their loads and required storage spaces upon placements. We first revisit the best existing solution for simple placement (i.e., without replication and reallocation), and rewrite it in our first algorithm by imposing some flexibilities. Our second algorithm is to apply document replication. The upper bound of load is significantly reduced, without sacrificing that of the storage space. This upper bound contains at least one special case which can never be outperformed by any online simple placement algorithms. Lastly, we show that there exists an online algorithm which allows very little document reallocation, but gives an upper bound result on the load and storage space, which is never reachable by any online algorithms for simple placement. The time complexities of the first two algorithms are in O(log M), and the last algorithm runs in O(log MN) time, where M is the number of servers, and N is the number of existing documents.

A New Algorithm for the Open-Pit Mine Production Scheduling Problem

For the purpose of production scheduling, open-pit mines are discretized into three-dimensional arrays known as block models. Production scheduling consists of deciding which blocks should be extracted, when they should be extracted, and what to do with the blocks once they are extracted. Blocks that are close to the surface should be extracted first, and capacity constraints limit the production in each time period. Since the 1960s, it has been known that this problem can be cast as an integer programming model. However, the large size of some real instances (3–10 million blocks, 15–20 time periods) has made these models impractical for use in real planning applications, thus leading to the use of numerous heuristic methods. In this article we study a well-known integer programming formulation of the problem that we refer to as C-PIT. We propose a new decomposition method for solving the linear programming relaxation (LP) of C-PIT when there is a single capacity constraint per time period. This algorithm is based on exploiting the structure of the precedence-constrained knapsack problem and runs in O (mn log n) in which n is the number of blocks and m a function of the precedence relationships in the mine. Our computations show that we can solve, in minutes, the LP relaxation of real-sized mine-planning applications with up to five million blocks and 20 time periods. Combining this with a quick rounding algorithm based on topological sorting, we obtain integer feasible solutions to the more general problem where multiple capacity constraints per time period are considered. Our implementation obtains solutions within 6% of optimality in seconds. A second heuristic step, based on local search, allows us to find solutions within 3% in one hour on all instances considered. For most instances, we obtain solutions within 1–2% of optimality if we let this heuristic run longer. Previous methods have been able to tackle only instances with up to 150,000 blocks and 15 time periods.

COMPUTING THE DISCRETE FRÉCHET DISTANCE WITH IMPRECISE INPUT

We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2O(d2) m2n2 log 2(mn) the minimum Fréchet distance between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log 3(mn) + (m2+n2) log (mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L∞ distance, we give an O(dmn log (dmn))-time algorithm. We also give efficient O(dmn)-time algorithms to approximate the maximum Fréchet distance, as well as the minimum and maximum Fréchet distance under translation. These algorithms achieve constant factor approximation ratios in "realistic" settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).

Note on the prime divisors of Farey fractions

Let P1(n) ≥ P2(n) ≥· · · be the prime divisors of a natural number n arranged in the non-increasing order. The limit distribution of the sequences (log Pi(mn)/ log(mn), i ≥ 1) for m/n ꞓ 2 (lambda1; lambda2), n ≤ x, are considered. It is proved that under some conditions on lambdai the limit distribution of the sequences exists and is closely related to the Poisson–Dirichlet distribution.

An O( mn log ( nU)) time algorithm to solve the feasibility problem

The phase I maximum flow and most positive cut methods are used to solve the feasibility problem. Both of these methods take one maximum flow computation. Thus the feasibility problem can be solved using maximum flow algorithms. Let n and m be the number of nodes and arcs, respectively. In this paper, we present an algorithm to solve the feasibility problem with integer lower and upper bounds. The running time of our algorithm is O( mn log ( nU)), where U is the value of maximum upper bound. Our algorithm improves the O( m 2 log ( nU))-time algorithm in [12]. Hence the current algorithm improves the running time in [12] by a factor of n. Sleator and Goldberg’s algorithm is one of the well-known maximum flow algorithms, which runs in O( mn log n) time, see [5]. Under similarity assumption [11], our algorithm runs in O( mn log n) time, which is the running time of Sleator and Goldberg’s algorithm. The merit of our algorithm is that, in the case of infeasibility of the given network, it not only diagnoses infeasibility but also presents some information that is useful to modeler in estimating the maximum cost of adjusting the infeasible network.

Constrained decompositions of integer matrices and their applications to intensity modulated radiation therapy

AbstractWe consider combinatorial optimization problems arising in radiation therapy. Given a matrix I with non‐negative integer entries, we seek a decomposition of I as a weighted sum of binary matrices having the consecutive ones property, such that the total sum of the coefficients is minimized. The coefficients are restricted to be non‐negative integers. Here, we investigate variants of the problem with additional constraints on the matrices used in the decomposition. Constraints appearing in the application include the interleaf motion and interleaf distance constraints. The former constraint was previously studied by Baatar et al. [Discr Appl Math 152 (2005), 6–34] and Kalinowski [Discr Appl Math 152 (2005), 52–88]. The latter constraint was independently considered by Kumar [Working paper (2007)] in the case where coefficients of the decomposition are not restricted to be integers. For both constraints, we prove that finding an optimal decomposition reduces to finding a maximum value potential in an auxiliary network with integer arc lengths and no negative length cycle. This allows us to simplify and unify the previous approaches. Moreover, we give an O (MN + KM) algorithm to solve the problem under the interleaf distance constraint, where M and N, respectively, denote the number of rows and columns of the matrix I and K is the number of matrices used in the decomposition. We also give an O (MN log M + KM) algorithm for solving the problem under the interleaf motion constraint and hence improve on previous results. Finally, we show the problem can still be solved in O (MN log M + KM) time when both constraints are considered simultaneously. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

Dynamic evolution of economically preferred facilities

Abstract In a sustained development scenario, it is often the case that an investment is to be made over time in facilities that generate benefits. The benefits result from joint synergies between the facilities expressed as positive utilities specific to some subsets of facilities. As incremental budgets to finance fixed facility costs become available over time, additional facilities can be opened. The question is which facilities should be opened in order to guarantee that the overall benefit return over time is on the highest possible trajectory. This problem is common in situations such as ramping up a communication or transportation network where the facilities are hubs or service stations, or when introducing new technologies such as alternative fuels for cars and the facilities are fueling stations, or when expanding the production capacity with new machines, or when facilities are functions in a developing organization that is forced to make choices of where to invest limited funding. An intuitive strategy frequently used to evolve the set of facilities is a greedy approach that picks additional facilities which provide a highest rate of return on each budget increment. Such strategy is shown to have adverse impact by locking the system into future suboptimal solutions with total benefit that can be arbitrarily small compared to the optimal evolution sequence. It is shown here that the degree of suboptimality of such greedy strategy is extreme in that it can lead to the loss of the majority of future benefits. The main result here is the generation of the entire efficient frontier of optimal solution sets for different budget levels. This frontier contains information that guides the schedule of the optimal evolution of the set of facilities in future expansions. It is demonstrated that the efficient frontier forms a concave envelope, and the configurations on the frontier’s breakpoints are nested. The efficient frontier is generated by efficient and strongly polynomial algorithms. For n potential facility locations and for subsets of positive benefits of total size m we find the breakpoints of the efficient frontier using a parametric minimum cut procedure in time O ( nm log n 2 m ) , and all N configurations on the efficient frontier in time O( mn log n + N ).

The Pseudoflow Algorithm: A New Algorithm for the Maximum-Flow Problem

We introduce the pseudoflow algorithm for the maximum-flow problem that employs only pseudoflows and does not generate flows explicitly. The algorithm solves directly a problem equivalent to the minimum-cut problem—the maximum blocking-cut problem. Once the maximum blocking-cut solution is available, the additional complexity required to find the respective maximum-flow is O(m log n). A variant of the algorithm is a new parametric maximum-flow algorithm generating all breakpoints in the same complexity required to solve the constant capacities maximum-flow problem. The pseudoflow algorithm has also a simplex variant, pseudoflow-simplex, that can be implemented to solve the maximum-flow problem. One feature of the pseudoflow algorithm is that it can initialize with any pseudoflow. This feature allows it to reach an optimal solution quickly when the initial pseudoflow is “close” to an optimal solution. The complexities of the pseudoflow algorithm, the pseudoflow-simplex, and the parametric variants of pseudoflow and pseudoflow-simplex algorithms are all O(mn log n) on a graph with n nodes and m arcs. Therefore, the pseudoflow-simplex algorithm is the fastest simplex algorithm known for the parametric maximum-flow problem. The pseudoflow algorithm is also shown to solve the maximum-flow problem on s, t-tree networks in linear time, where s, t-tree networks are formed by joining a forest of capacitated arcs, with nodes s and t adjacent to any subset of the nodes.

The Chebyshev fast Gauss and nonuniform fast Fourier transforms and their application to the evaluation of distributed heat potentials

We present a method for the fast and accurate computation of distributed heat potentials in two dimensions. The distributed source is assumed to be given in terms of piecewise space–time Chebyshev polynomials. We discretize uniformly in time, whereas in space the polynomials are defined on the leaf nodes of a quadtree. The quadtree can vary at each time step. We combine a product integration rule with fast algorithms (fast heat potentials, nonuniform FFT, fast Gauss transform) to obtain a high-order accurate method with optimal complexity. If N is the number of time steps, M is the maximum number of leaf nodes over all the time steps and the input contains a qth-order polynomial representation of f, then, our method requires O ( q 3 MN log M ) work to evaluate the heat potential at arbitrary MN space–time target locations. The overall convergence rate of the method is of order q. We present numerical experiments for q = 4, 8, and 16, and we verify the theoretical convergence rate of the method. When the solution is sufficiently smooth, the 16th-order variant results in significant computational savings, even in the case in which we require only a few digits of accuracy.

An Efficient Algorithm for the Nearly Equitable Edge Coloring Problem

An edge coloring of a multigraph is nearly equitable if, among the edges incident to each vertex, the numbers of edges colored with any two colors differ by at most two. It has been proved that the problem of finding a nearly equitable edge coloring can be solved in O(m/k) time, where m and k are the numbers of edges and given colors, respectively. In this paper, we present a recursive algorithm that runs in O (mn log (m/(kn) + 1)) time, where n is the number of vertices. This algorithm improves the bestknown worst-case time complexity. When k = O(1), the time complexity of all known algorithms is O(m), which implies that this time complexity remains to be the best for more than twenty years since 1982 when Hilton and de Werra gave a constructive proof for the existence of a nearly equitable edge coloring for any multigraph. Our result is the first that improves this time complexity when m/n grows to infinity; e.g., m = n for an arbitrary constant θ > 1. Submitted: July 2008 Reviewed: September 2008 Revised: October 2008 Accepted: October 2008 Final: November 2008 Published: December 2008 Article type: Regular paper Communicated by: D. Wagner This research was supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan. A preliminary version of this paper appeared in Proceedings of ISAAC [10]. E-mail addresses: sharryx@al.cm.is.nagoya-u.ac.jp (Xuzhen Xie) yagiura@nagoya-u.jp (Mutsunori Yagiura) takao@al.cm.is.nagoya-u.ac.jp (Takao Ono) hirata@is.nagoya-u.ac.jp (Tomio Hirata) zwick@tau.ac.il (Uri Zwick) 384 X. Xie et al. An Efficient Algorithm for Nearly Equitable Edge Coloring

Fast neighborhood search for two- and three-dimensional nesting problems

In this paper we present a new heuristic solution method for two-dimensional nesting problems. It is based on a simple local search scheme in which the neighborhood is any horizontal or vertical translation of a given polygon from its current position. To escape local minima we apply the meta-heuristic method Guided Local Search. The strength of our solution method comes from a new algorithm which is capable of searching the neighborhood in polynomial time. More precisely, given a single polygon with m edges and a set of polygons with n edges the algorithm can find a translation with minimum overlap in time O( mn log( mn)). Solutions for standard test instances are generated by an implementation and a comparison is done with recent results from the literature. The solution method is very robust and most of the best solutions found are also the currently best results published. Our approach to the problem is very flexible regarding problem variations and special constraints, and as an example we describe how it can handle materials with quality regions. Finally, we generalize the algorithm for the fast neighborhood search and present a solution method for three-dimensional nesting problems.

An $\tilde{O}(m^{2}n)$ Algorithm for Minimum Cycle Basis of Graphs

We consider the problem of computing a minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over $\mathbb{F}_{2}$ generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Minimum cycle basis are useful in a number of contexts, e.g. the analysis of electrical networks and structural engineering. The previous best algorithm for computing a minimum cycle basis has running time O(m ω n), where ω is the best exponent of matrix multiplication. It is presently known that ω<2.376. We exhibit an O(m 2 n+mn 2log n) algorithm. When the edge weights are integers, we have an O(m 2 n) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(m ω ) time. For any ε>0, we also design an 1+ε approximation algorithm. The running time of this algorithm is O((m ω /ε)log (W/ε)) for reasonably dense graphs, where W is the largest edge weight.

A polynomial algorithm to compute the minimum degree spanning trees of directed acyclic graphs with applications to the broadcast problem

In this paper, we focus on the directed minimum degree spanning tree problem and the minimum time broadcast problem. Firstly, we propose a polynomial time algorithm for the minimum degree spanning tree problem in directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove that the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in O ( mn log n ) time, where m and n are the numbers of edges and vertices of the graph, respectively. Moreover, we apply the new algorithm to the minimum time broadcast problem. Two consequences for directed acyclic graphs are: (1) the problem under the vertex-disjoint paths mode can be approximated within a factor of O ( log n / log OPT ) of the optimum in O ( mn log n ) -time; (2) the problem under the edge-disjoint paths mode can be approximated within a factor of O ( Δ * / log Δ * ) of the optimum in O ( mn log n ) -time, where Δ * is the minimum k such that there is a spanning tree of the graph with maximum degree k.

A pixel-based regularization approach to inverse lithography

Inverse lithography attempts to synthesize the input mask which leads to the desired output wafer pattern by inverting the forward model from mask to wafer. In this article, we extend our earlier framework for image prewarping to solve the mask design problem for coherent, incoherent, and partially coherent imaging systems. We also discuss the synthesis of three variants of phase shift masks (PSM); namely, attenuated (or weak) PSM, 100% transmission PSM, and strong PSM with chrome. A new two-step optimization strategy is introduced to promote the generation and placement of assist bar features. The regularization framework is extended to guarantee that the estimated PSM have only two or three (allowable) transmission values, and the aerial-image penalty term is introduced to boost the aerial image contrast and keep the side-lobes under control. Our approach uses the pixel-based mask representation, a continuous function formulation, and gradient-based iterative optimization techniques to solve the inverse problem. The continuous function formulation allows analytic calculation of the gradient in O( MNlog ( MN)) operations for an M × N pattern making it practically feasible. We also present some results for coherent and incoherent imaging systems with very low k 1 values to demonstrate the effectiveness of our approach.

The two‐median problem on Manhattan meshes

AbstractWe investigate the two‐median problem on a mesh with M columns and N rows (M ≥ N), under the Manhattan (L1) metric. We derive exact algorithms with respect to m, n, and r, the number of columns, rows, and vertices, respectively, that contain requests. Specifically, we give an O(mn2 log m) time, O(r) space algorithm for general (nonuniform) meshes (assuming m ≥ n). For uniform meshes, we give two algorithms both using O(MN) space. One is an O(MN2) time algorithm, while the other is an algorithm running in O(MN log N) time with high probability and in O(MN2) time in the worst case assuming the weights are independent and identically distributed random variables satisfying certain natural conditions. These improve upon the previously best‐known algorithm that runs in O(mn2r) time. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 49(3), 226–233 2007

A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options Under Levy Processes

A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within exponentially Levy models, including the exponentially affine jump-diffusion models. For an M -times exercisable Bermudan option, the overall complexity is O(MN log(N)) with N grid points used to discretise the price of the underlying asset. It is shown how to price American options efficiently by applying Richardson extrapolation to the prices of Bermudan options.

Gapped Permutation Pattern Discovery for Gene Order Comparisons

The list of species whose complete DNA sequence have been read is growing steadily, and it is believed that comparative genomics is in its early days. Permutations patterns (groups of genes in some "close" proximity) on gene sequences of genomes across species is being studied under different models, to cope with this explosion of data. The challenge is to (intelligently and efficiently) analyze the genomes in the context of other genomes. In this paper, we present a generalized model that uses three notions, gapped permutation patterns (with gap g), genome clusters, via quorum, K>1, parameter, and, possible multiplicity in the patterns. The task is to automatically discover all permutation patterns (with possible multiplicity), that occur with gap g in at least K of the given m genomes. We present (log mN (I) + /Sigma/log/Sigma/N (O)) time algorithm where m is the number of sequences, each defined on Sigma, N (I) is the size of the input and N (O) is the size of the maximal gene clusters that appear in at least K of the m genomes.