Euclidean Minimum Spanning Tree Research Articles

Testing Euclidean minimum spanning trees in the plane

Given a Euclidean graph G over a set P of n points in the plane, we are interested in verifying whether G is a Euclidean minimum spanning tree (EMST) of P or G differs from it in more than ϵ n edges. We assume that G is given in adjacency list representation and the point/vertex set P is given in an array. We present a property testing algorithm that accepts graph G if it is an EMST of P and that rejects with probability at least 2/3 if G differs from every EMST of P in more than ϵ, n edges. Our algorithm runs in O(√ n /ϵ ⋅ log 2 ( n /ϵ)) time and has a query complexity of O(√ n /ϵ ⋅ log ( n /ϵ)).

SAMPLING IN DYNAMIC DATA STREAMS AND APPLICATIONS

A dynamic geometric data stream is a sequence of m ADD/REMOVE operations of points from a discrete geometric space {1,…, Δ} d ?. ADD (p) inserts a point p from {1,…, Δ} d into the current point set P , REMOVE(p) deletes p from P . We develop low-storage data structures to (i) maintain ε-nets and ε-approximations of range spaces of P with small VC-dimension and (ii) maintain a (1 + ε)-approximation of the weight of the Euclidean minimum spanning tree of P . Our data structure for ε-nets uses [Formula: see text] bits of memory and returns with probability 1 – δ a set of [Formula: see text] points that is an e-net for an arbitrary fixed finite range space with VC-dimension [Formula: see text]. Our data structure for ε-approximations uses [Formula: see text] bits of memory and returns with probability 1 – δ a set of [Formula: see text] points that is an ε-approximation for an arbitrary fixed finite range space with VC-dimension [Formula: see text]. The data structure for the approximation of the weight of a Euclidean minimum spanning tree uses O ( log (1/δ)( log Δ/ε) O ( d )) space and is correct with probability at least 1 – δ. Our results are based on a new data structure that maintains a set of elements chosen (almost) uniformly at random from P .

Automatic Zonation of Urban Areas Based on the Similarity of H/V Spectral Ratios

Abstract In this article we propose an automatic procedure that performs thezonation of urban areas based on a set of horizontal-to-vertical ( H/V ) spectral ratiosfrom ambient noise recordings collected at many measurement points. The techniquesearches for the connected areas where the similarity among the spectral ratios ismaximized. The problem is posed as one of the optimal partitioning of the Delaunaytriangulation of the available measurement points. The technique explores and triesto partition some random variations of a Euclidean minimum spanning tree of thetriangulation. The optimization is performed using a genetic algorithm.Thetechniqueis applied to the zonation of the town of Vittorio Veneto in northeastern Italy and toa synthetic data set that tries to simulate the effects of some typical geological con-ditions.Introduction Seismic microzonation concerns the identification andmapping at local or site scales of areas having different po-tentials of hazardous earthquake effects, such as ground-shaking intensity, liquefaction, or landslide potential (Akiand Lee, 2003). It is commonly performed by experts takinginto account geophysical and geological information of dif-ferent natures. For ground-shaking intensity, a useful sup-port comes from the computation of horizontal-to-vertical(

Density-equalizing Euclidean minimum spanning trees for the detection of all disease cluster shapes

Existing disease cluster detection methods cannot detect clusters of all shapes and sizes or identify highly irregular sets that overestimate the true extent of the cluster. We introduce a graph-theoretical method for detecting arbitrarily shaped clusters based on the Euclidean minimum spanning tree of cartogram-transformed case locations, which overcomes these shortcomings. The method is illustrated by using several clusters, including historical data sets from West Nile virus and inhalational anthrax outbreaks. Sensitivity and accuracy comparisons with the prevailing cluster detection method show that the method performs similarly on approximately circular historical clusters and greatly improves detection for noncircular clusters.

Impact of Turbulent Flow and Mean Mixture Fraction Results on Mixing Model Behavior in Transported Scalar PDF Simulations of Turbulent Non-premixed Bluff Body Flames

Numerical simulation results are presented for three turbulent jet diffusion flames, stabilized behind a bluff body (Sydney Flames HM1-3). Interaction between turbulence and combustion is modeled with the transported joint-scalar PDF approach. The focus of the study is on the impact of the quality of simulation results in physical space on the behavior of two micro-mixing models in composition space: the Euclidean Minimum Spanning Tree (‘EMST’) model and the modified Curl coalescence dispersion (‘CD’) model. Profiles of conditional means and variances of thermo-chemical quantities, conditioned on the mixture fraction, are discussed in the recirculation region and in the neck zone behind. The impact of the flow and mixing fields in physical space on the mixing model behavior in composition space is strong for the CD model and increases as the turbulence – chemistry interaction becomes stronger. The EMST conditional profiles, on the contrary, are hardly affected.

Dual injection homogeneous charge compression ignition engine simulation using a stochastic reactor model

Multiple direct injection (MDI) is a promising strategy to enable fast-response ignition control as well as expansion of the homogeneous charge compression ignition (HCCI) engine operating window, thus realizing substantial reductions of soot and NOx emissions. The present paper extends a zero-dimensional-probability-density-function-based stochastic reactor model (SRM) for HCCI engines in order to incorporate MDI and an improved turbulent mixing model. For this, a simplistic spray model featuring injection, penetration, and evaporation sub-models is formulated, and mixing is described by the Euclidean minimal spanning tree (EMST) sub-model accounting for localness in composition space. The model is applied to simulate a gasoline HCCI engine, and the in-cylinder pressure predictions for single and dual injection cases show a satisfactory agreement with measurements. From the parametric studies carried out it is demonstrated that, as compared with single injection, the additional second injection contributes to prolonged heat release and consequently helps to prevent knock, thereby extending the operating range on the high load side. Tracking the phase space trajectories of individual stochastic particles provides significant insight into the influence of local charge stratification owing to direct injection on HCCI combustion.

Lower bounds for testing Euclidean Minimum Spanning Trees

The Euclidean Minimum Spanning Tree problem is to decide whether a given graph G = ( P , E ) on a set of points in the two-dimensional plane is a minimum spanning tree with respect to the Euclidean distance. Czumaj et al. [A. Czumaj, C. Sohler, M. Ziegler, Testing Euclidean Minimum Spanning Trees in the plane, Unpublished, Part II of ESA 2000 paper, downloaded from http://web.njit.edu/~czumaj/] gave a 1-sided-error non-adaptive property-tester for this task of query complexity O ˜ ( n ) . We show that every non-adaptive (not necessarily 1-sided-error) property-tester for this task has a query complexity of Ω ( n ) , implying that the test in [A. Czumaj, C. Sohler, M. Ziegler, Testing Euclidean Minimum Spanning Trees in the plane, Unpublished, Part II of ESA 2000 paper, downloaded from http://web.njit.edu/~czumaj/] is of asymptotically optimal complexity. We further prove that every adaptive property-tester has query complexity of Ω ( n 1 / 3 ) . Those lower bounds hold even when the input graph is promised to be a bounded degree tree.

INTERACTION BETWEEN CHEMISTRY AND MICRO-MIXING MODELING IN TRANSPORTED PDF SIMULATIONS OF TURBULENT NON-PREMIXED FLAMES

A comparative study is presented of the impact of chemistry modeling on the behavior of 2 micro-mixing models in transported scalar PDF simulations of turbulent non-premixed flames. The micro-mixing models are CD (modified Curl's Coalescence/Dispersion) and EMST (Euclidean Minimum Spanning Tree). A first order non-linear k-ε turbulence model is applied for the turbulent flow and mixing fields. Three C1 chemistry models are considered: 2 skeletal schemes containing 16 species and 31, resp. 41 elementary reactions, and one augmented reaction scheme (ARM), consisting of 9 independent species and 5 global reaction steps. The test cases considered are the piloted turbulent Delft Flame III jet diffusion flame and the bluff-body stabilized turbulent non-premixed Sydney HM1 flame. With EMST the chemistry model choice has a negligible effect on the micro-mixing model behavior or on the results in physical space. With CD on the other hand, larger differences appear.

Automatic Seismic Zonation Based on Stress-Field Uniformity Assessed from Focal Mechanisms

In this article we propose a new method for automatic seismic zonation based on focal mechanisms. The technique searches for the areas where the homogeneity of the stress field is maximized. The problem is posed as one of optimal partitioning of the graph corresponding to the Delaunay triangulation of the available epicenters. Two alternative techniques are used: the first one limits the search to the partitions of the Euclidean minimum spanning tree (emst) of the triangulation; the other one uses a number of spanning trees that are variations of the emst. The optimization is performed using a genetic algorithm. The technique is applied to seismic zonation in northeastern Italy by using a set of 219 focal mechanisms. The results are compared with those of a nonautomatic zonation used to investigate the distribution of the stress and strain fields and performed taking into account tectonic structures, faulting patterns, spatial distribution of seismicity, and other geophysical data. The proposed method is shown to be effective in finding areas of homogeneous stress and providing evidence for critical zones of complex behavior.

The minimum-area spanning tree problem

Motivated by optimization problems in sensor coverage, we formulate and study the Minimum-Area Spanning Tree ( mast) problem: Given a set P of n points in the plane, find a spanning tree of P of minimum “area”, where the area of a spanning tree T is the area of the union of the n − 1 disks whose diameters are the edges in T . We prove that the Euclidean minimum spanning tree of P is a constant-factor approximation for mast. We then apply this result to obtain constant-factor approximations for the Minimum-Area Range Assignment ( mara) problem, for the Minimum-Area Connected Disk Graph ( macdg) problem, and for the Minimum-Area Tour ( mat) problem. The first problem is a variant of the power assignment problem in radio networks, the second problem is a related natural problem, and the third problem is a variant of the traveling salesman problem.

Study of the performance of three micromixing models in transported scalar PDF simulations of a piloted jet diffusion flame (“Delft Flame III”)

Numerical simulation results are presented for a turbulent nonpremixed flame with local extinction and reignition. The transported scalar PDF approach is applied to the turbulence–chemistry interaction. The turbulent flow field is obtained with a nonlinear two-equation turbulence model. A C 1 skeletal scheme is used as the chemistry model. The performance of three micromixing models is compared: the interaction by exchange with the mean model (IEM), the modified Curl's coalescence/dispersion model (CD) and the Euclidean minimum spanning tree model (EMST). With the IEM model, global extinction occurs. With the standard value of model constant C ϕ = 2 , the CD model yields a lifted flame, unlike the experiments, while with the EMST model the correct flame shape is obtained. However, the conditional variances of the thermochemical quantities are underestimated with the EMST model, due to a lack of local extinction in the simulations. With the CD model, the flame becomes attached when either the value of C ϕ is increased to 3 or the pilot flame thermal power is increased by a factor of 1.5. With increased value of C ϕ better results for mixture fraction variance are obtained with both the CD and the EMST model. Lowering the value of C ϕ leads to better predictions for mean temperature with EMST, but at the cost of stronger overprediction of mixture fraction variance. These trends are explained as a consequence of variance production by macroscopic inhomogeneity and the specific properties of the micromixing models. Local time stepping is applied so that convergence is obtained more quickly. Iteration averaging reduces statistical error so that the limited number of 50 particles per cell is sufficient to obtain accurate results.

On Approximation Ratios of Minimum-Energy Multicast Routing in Wireless Networks

In the broadcasting of ad hoc wireless networks, energy conservation is a critical issue. Three heuristic algorithms were proposed in Wieselthier et al. (2000) for finding approximate minimum-energy broadcast routings: MST(minimum spanning tree), SPT(shortest-path tree), and BIP(broadcasting incremental power). Wan et al. (2001) characterized their performance in terms of approximation ratios. This paper points out some mistakes in the result of Wan et al. (2001), and proves that the upper bound of sum of squares of lengths of the edges in Euclidean MST in unit disk can be improved to 10.86, thus improves the approximation ratios of MST and BIP algorithm.

On the Euclidean Minimum Spanning Tree Problem

Given a weighted graph G(V;E), a minimum spanning tree for G can be obtained in linear time using a randomized algorithm or nearly linear time using a deterministic algorithm. Given n points in the plane, we can construct a graph with these points as nodes and an edge between every pair of nodes. The weight on any edge is the Euclidean distance between the two points. Finding a minimum spanning tree for this graph is known as the Euclidean minimum spanning tree problem (EMSTP). The minimum spanning tree algorithms alluded to before will run in time O(n2) (or nearly O(n2)) on this graph. In this note we point out that it is possible to devise simple algorithms for EMSTP in k- dimensions (for any constant k) whose expected run time is O(n), under the assumption that the points are uniformly distributed in the space of interest.CR Categories: F2.2 Nonnumerical Algorithms and Problems; G.3 Probabilistic Algorithms

Testing of mixing models for Monte Carlo probability density function simulations

Testing of mixing models widely used for Monte Carlo probability density function simulations of turbulent diffusion flames is performed using the data obtained from direct numerical simulations (DNS) that are specifically designed for the study of local flame extinction and reignition. In particular, the interaction by exchange with the mean (IEM) [J. Villermaux and J. C. Devillon, “Représentation de la coalescence et de la redispersion des domaines de ségrégation dans un fluide per modéle d’interaction phénoménologique,” in Proceedings of the Second International Symposia on Chemical Reaction Engineering (ISCRE, Netherlands, 1972), p. B1], the modified Curl [J. Janicka, W. Kolbe, and W. Kollmann, J. Non-Equilib. Thermodyn. 4, 47 (1979)], and the Euclidean minimum spanning tree (EMST) [S. Subramaniam and S. B. Pope, Combust. Flame 115, 487 (1998)] mixing models are tested. The tests are designed to examine the mixing model performance when implemented in both Reynolds-averaged simulations and large-eddy simulations. The exact value of the mixing frequency is taken from the DNS, so that the model performance can be more accurately determined. It is found that, in general, the EMST mixing model yields much better results than the IEM and the modified Curl mixing models.

Degree Frequencies in the Minimal Spanning Tree and Dimension Identification

We discuss the application of linear combinations of the degree frequencies in the minimal spanning tree to the problem of identifying the appropriate dimension for a data set from its interpoint distance matrix. This graph-theoretical methodology, of very low computational cost, can be of aid in the problem of Multidimensional Scaling and in dimensionality reduction. Results of Lee [Lee, S. (1999). The central limit theorem for euclidean minimal spanning trees II. Adv. Appl. Probability 31(4): 969–984] imply that the procedure proposed here is asymptotically consistent.

Approximating the Weight of the Euclidean Minimum Spanning Tree in Sublinear Time

We consider the problem of computing the weight of a Euclidean minimum spanning tree for a set of n points in $\mathbb R^d$. We focus on the setting where the input point set is supported by certain basic (and commonly used) geometric data structures that can provide efficient access to the input in a structured way. We present an algorithm that estimates with high probability the weight of a Euclidean minimum spanning tree of a set of points to within $1 + \eps$ using only $\widetilde{\O}(\sqrt{n} \, \text{poly} (1/\eps))$ queries for constant d. The algorithm assumes that the input is supported by a minimal bounding cube enclosing it, by orthogonal range queries, and by cone approximate nearest neighbor queries.

PDF modelling of turbulent non-premixed combustion with detailed chemistry

Although the significant advantage for the probability density function (PDF) methods of the exact treatment of chemical reactions in turbulent combustion problems, a detailed chemistry mechanism (e.g., the GRI mechanism) has not been implemented in the practical calculations by now due to the prohibitive computation of PDF methods. In this work, a detailed mechanism (GRI-Mech 3.0, consisting of 53 species and 325 elemental reactions) is firstly incorporated into the PDF calculation of a turbulent non-premixed jet flame (Sandia Flame D). The flow is formulated in the boundary layer form. The joint composition PDF closure level is applied and a multiple-time-scale (MTS) k– ε turbulence model is combined for the closure of turbulent transport terms. The molecular mixing process is modelled by the Euclidean minimum spanning tree (EMST) mixing model. The solutions are obtained by using the space marching algorithm for turbulence equations and node-based Monte Carlo particle method for PDF evolution equation. The chemical reaction source terms are integrated directly. Extensive comparisons between the predictions and the measurements are made, which involve radial profiles of mean and rms (root mean square), conditional mean, scatter plots of scalars and conditional PDF distribution etc. The flame structures are well represented by the present calculation, including intermediate species (e.g. CO and H 2) mass fractions, pollutant NO emission and local extinction.

Approximating geometric bottleneck shortest paths

In a geometric bottleneck shortest path problem, we are given a set S of n points in the plane, and want to answer queries of the following type: given two points p and q of S and a real number L, compute (or approximate) a shortest path between p and q in the subgraph of the complete graph on S consisting of all edges whose lengths are less than or equal to L. We present efficient algorithms for answering several query problems of this type. Our solutions are based on Euclidean minimum spanning trees, spanners, and the Delaunay triangulation. A result of independent interest is the following. For any two points p and q of S, there is a path between p and q in the Delaunay triangulation, whose length is less than or equal to 2 π/(3cos( π/6)) times the Euclidean distance | pq| between p and q, and all of whose edges have length at most | pq|.

Euclidean Bounded-Degree Spanning Tree Ratios

Let τK be the worst-case (supremum) ratio of the weight of the minimum degree-K spanning tree to the weight of the minimum spanning tree, over all finite point sets in the Euclidean plane. It is known that τ2 = 2 and τ5 = 1. In STOC ‘94, Khuller, Raghavachari, and Young established the following inequalities: 1.103 < τ3 \le 1.5 and 1.035 < τ4 \le 1.25. We present the first improved upper bounds: τ3 < 1.402 and τ4 < 1.143. As a result, we obtain better approximation algorithms for Euclidean minimum bounded-degree spanning trees. Let τK (d) be the analogous ratio in d-dimensional space. Khuller et al. showed thatτ3 (d) < 1.667 for any d. We observe that τ3 (d) < 1.633.

EFFICIENT CONSTRUCTION OF LOW WEIGHTED BOUNDED DEGREE PLANAR SPANNER

Given a set V of n points in a two-dimensional plane, we give an O(n log n)-time centralized algorithm that constructs a planar t-spanner for V, for [Formula: see text] such that the degree of each node is bounded from above by [Formula: see text], where 0&lt;α&lt;π/2 is an adjustable parameter. Here Cdel is the spanning ratio of the Delaunay triangulation, which is at most [Formula: see text]. We also show, by applying the greedy method in Ref. [14], how to construct a low weighted bounded degree planar spanner with spanning ratio ρ(α)2(1+∊) and the same degree bound, where ∊ is any positive real constant. Here, a structure is called low weighted if its total edge length is proportional to the total edge length of the Euclidean minimum spanning tree of V. Moreover, we show that our method can be extended to construct a planar bounded degree spanner for unit disk graphs with the adjustable parameter α satisfying 0&lt;α&lt;π/3. Previously, only centralized method6 of constructing bounded degree planar spanner is known, with degree bound 27 and spanning ratio t≃10.02. The distributed implementation of this centralized method takes O(n2) communications in the worst case. Our method can be converted to a localized algorithm where the total number of messages sent by all nodes is at most O(n).