Computing discrete Fréchet distance
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Speeding up Pollard's rho method for computing discrete logarithms
Speeding up Pollard's Rho Method for Computing Discrete Logarithms
he Universitat Darmstadt Institut fur Theoretische Informatik Alexanderstra e 10, 64283 Darmstadt Germany teske@rmatik.tu-darmstadt.de ne a sequence (yi ) by yi+1 = f (yi ) for i = 0; 1; 2; : : :, with some starting value y0 . In this paper, we de ne and discuss new iterating functions for computing discrete logarithms with the rho method. We compare their performances in experiments with elliptic curve groups. Our experiments show that one of our newly de ned functions is expected to reduce the number of steps by a factor of approximately 0:8, in comparison with Pollard's originally used function, and we show that this holds independently of the size of the group order. For group orders large enough such that the run time for precomputation can be neglected, this means a real-time speed-up of more than 1:2.
he Universitat Darmstadt Institut fur Theoretische Informatik Alexanderstra e 10, 64283 Darmstadt Germany teske@rmatik.tu-darmstadt.de ne a sequence (yi ) by yi+1 = f (yi ) for i = 0; 1; 2; : : :, with some starting value y0 . In this paper, we de ne and discuss new iterating functions for computing discrete logarithms with the rho method. We compare their performances in experiments with elliptic curve groups. Our experiments show that one of our newly de ned functions is expected to reduce the number of steps by a factor of approximately 0:8, in comparison with Pollard's originally used function, and we show that this holds independently of the size of the group order. For group orders large enough such that the run time for precomputation can be neglected, this means a real-time speed-up of more than 1:2.
A Riemannian Framework for Tensor Computing
International Journal of Computer Vision 66(1), 41–66, 2006 c 2006 Springer Science + Business Media, Inc. Manufactured in The Netherlands. DOI: 10.1007/s11263-005-3222-z
42
Pennec, Fillard and Ayache
Pennec (1999, 2004) for general Riemannian manifolds, invariance properties leading in some case to a natural choice for the metric. In this paper, we show how this theory can be applied to tensors, leading to a new intrinsic computing framework for these geometric features with many important theoretical properties as well as practical computing properties. In the remaining of this section, we quickly investigate some connected works on tensors. Then, we summarize in Section 2 the main ideas of the statistical framework we developed on Riemannian manifolds. The aim is to exemplify the fact that choosing a Riemannian metric “automatically” determines a powerful framework to work on the manifold through the introduction of a few tools from differential geometry. In order to use this Riemannian framework on our tensor manifold, we propose in Section 3 an affine-invariant Riemannian metric on tensors. We demonstrate that it leads to very strong theoretical properties, as well as some important practical algorithms such as an intrinsic geodesic gradient descent. Section 4 focuses on the application of this framework to an important geometric data processing problem: interpolation of tensor values. We show that this problem can be tackled efficiently through a weighted mean optimization. However, if weights are easy to define for regularly sampled tensors (e.g. for linear or tri-linear interpolation), the problem proved to be more difficult for irregularly sampled values. With Section 5, we turn to tensors field computing, and more particularly filtering. If the Gaussian filtering may still be defined through weighted means, the partial differential equation (PDE) approach is slightly more complex. In particular, the metric of the tensor space has to be taken into account when computing the magnitude of the spatial gradient of the tensor field. Thanks to our Riemannian framework, we propose efficient numerical schemes to compute the gradient, its amplitude, and the Laplace-Beltrami operator used in linear diffusion. We also propose an adjustment of this manifold Laplacian that realizes an anisotropic filtering. Finally, Section 6 focuses on simple statistical approaches to regularize and restore missing values in tensor fields. Here, the use of the Riemannian distance inherited from the chosen metric is fundamental to define least-squares data attachment criteria for dense and sparsely distributed tensor fields that lead to simple implementation schemes in our intrinsic computing framework.
42
Pennec, Fillard and Ayache
Pennec (1999, 2004) for general Riemannian manifolds, invariance properties leading in some case to a natural choice for the metric. In this paper, we show how this theory can be applied to tensors, leading to a new intrinsic computing framework for these geometric features with many important theoretical properties as well as practical computing properties. In the remaining of this section, we quickly investigate some connected works on tensors. Then, we summarize in Section 2 the main ideas of the statistical framework we developed on Riemannian manifolds. The aim is to exemplify the fact that choosing a Riemannian metric “automatically” determines a powerful framework to work on the manifold through the introduction of a few tools from differential geometry. In order to use this Riemannian framework on our tensor manifold, we propose in Section 3 an affine-invariant Riemannian metric on tensors. We demonstrate that it leads to very strong theoretical properties, as well as some important practical algorithms such as an intrinsic geodesic gradient descent. Section 4 focuses on the application of this framework to an important geometric data processing problem: interpolation of tensor values. We show that this problem can be tackled efficiently through a weighted mean optimization. However, if weights are easy to define for regularly sampled tensors (e.g. for linear or tri-linear interpolation), the problem proved to be more difficult for irregularly sampled values. With Section 5, we turn to tensors field computing, and more particularly filtering. If the Gaussian filtering may still be defined through weighted means, the partial differential equation (PDE) approach is slightly more complex. In particular, the metric of the tensor space has to be taken into account when computing the magnitude of the spatial gradient of the tensor field. Thanks to our Riemannian framework, we propose efficient numerical schemes to compute the gradient, its amplitude, and the Laplace-Beltrami operator used in linear diffusion. We also propose an adjustment of this manifold Laplacian that realizes an anisotropic filtering. Finally, Section 6 focuses on simple statistical approaches to regularize and restore missing values in tensor fields. Here, the use of the Riemannian distance inherited from the chosen metric is fundamental to define least-squares data attachment criteria for dense and sparsely distributed tensor fields that lead to simple implementation schemes in our intrinsic computing framework.
Outlier Detection By Sampling With Accuracy Guarantees
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time, where the constant c is usually above three. In these domains, state-of-the-art algorithms may take days to detect DB-outliers, even in data sets of small sizes. The goal of this paper is to define an algorithm that can provide users with interactive-speed performance over the most expensive distance computations, giving the user the ability to “try out” various distance functions and queries during exploratory mining. The question we consider is: How can we reduce the required number of distance computations in DB-outlier mining? For certain distance measures and data sets, indexing and pruning techniques can be used to reduce the number of distance computations. Unfortunately, indexing [3] is not useful in the domains mentioned above due to high data dimensionality, and pruning [2] tends only to reduce the required number of distance computations to a few hundred or a few thousand per data point; as we will show experimentally, this is still too costly if every distance computation requires seconds of CPU time. In this paper, we consider a simple sampling algorithm for mining the kth nearest neighbor (kth-NN) DB-outliers [12]. Given integer parameters k and γ , kth-NN outliers are the top γ points whose distance to its kth-NN is greatest. Sampling has been considered before as an option for detecting outliers [8], but never along with a rigorous treatment of the effect on result quality. If the user is able to tolerate some rigorously-measured inaccuracy, our algorithm can give arbitrarily fast response times. Our algorithm works as follows. For each data point i, we randomly sample α points from the data set. Using the userspecified distance function, we find the kth-NN distance of point i in those α samples. After repeating the above process for each point, the sampling algorithm returns the γ points whose sampled kth-NN distance is greatest. Algorithm 1 presents the corresponding pseudo-code: Algorithm 1 Sampling Algorithm Input: A data set G of size n; k, specifying which NN distance as the criterion; α(α > k ), the sample size; γ , the number of outliers to return. Output: γ points as the kth-NN outliers 1: for each point i ∈ G do 2: Draw a random sample of size α from G (not including point i) 3: Calculate point i’s kth-NN distance in its sample 4: end for 5: Return the top γ points whose kth-NN distance in its sample is the greatest
Generalized-ICP
Two more modern alternatives are Iterative Dual Correspondence [15] and Metric-Based ICP [16]. IDC improves the point-matching process by maintaining two sets of correspondences. MbICP is designed to improve convergence with large initial orientation errors by explicitly putting a measure of rotational error as part of the distance metric to be minimized.
I. INTRODUCTION
Over the last decade, range images have grown in popularity and found increasing applications in fields including medical imaging, object modeling, and robotics. Because of occlusion and limited sensor range, most of these applications require accurate methods of combining multiple range images into a single model. Particularly in mobile robotics, the availability of range sensors capable of quickly capturing an entire 3D scene has drastically improved the state of the art. A striking illustration of this is the fact that virtually all competitors in the DARPA Grand Challenge relied on fast-scanning laser range finders as the primary input method for obstacle avoidance, motion planning, and mapping. Although GPS and IMUs are often used to calculate approximate displacements, they are not accurate enough to reliably produce precise positioning. In addition, there are many situation (tunnels, parking garages, tall buildings) which obstruct GPS reception and further decrease accuracy. To deal with this shortcoming, most applications rely on scan-matching of range data to refine the localization. Despite such wide usage, the typical approach to solving the scan-matching problem has remained largely unchanged since its introduction.
I. INTRODUCTION
Over the last decade, range images have grown in popularity and found increasing applications in fields including medical imaging, object modeling, and robotics. Because of occlusion and limited sensor range, most of these applications require accurate methods of combining multiple range images into a single model. Particularly in mobile robotics, the availability of range sensors capable of quickly capturing an entire 3D scene has drastically improved the state of the art. A striking illustration of this is the fact that virtually all competitors in the DARPA Grand Challenge relied on fast-scanning laser range finders as the primary input method for obstacle avoidance, motion planning, and mapping. Although GPS and IMUs are often used to calculate approximate displacements, they are not accurate enough to reliably produce precise positioning. In addition, there are many situation (tunnels, parking garages, tall buildings) which obstruct GPS reception and further decrease accuracy. To deal with this shortcoming, most applications rely on scan-matching of range data to refine the localization. Despite such wide usage, the typical approach to solving the scan-matching problem has remained largely unchanged since its introduction.
discrete意思
discrete意思离散数学（Discrete Mathematics）是数学的一个分支，主要研究离散对象及其关系、结构和操作等问题。
与连续数学相对应，离散数学主要关注于离散的数值和结构，而不是连续的数值和变化。
离散数学在计算机科学、信息科学、电子工程等领域具有重要的应用价值。
在离散数学中，研究的对象可以是离散的数值、集合、序列、图形等。
离散数学的主要内容包括集合论、逻辑、图论、代数结构、组合数学、概率论等。
下面将对离散数学的主要内容进行扩展描述。
1. 集合论（Set Theory）：集合论是离散数学的基础，研究的是集合及其运算、关系和性质等。
集合是离散数学中最基本的概念，集合论主要研究集合的性质和运算规则，如并集、交集、差集、补集等。
集合论的应用广泛，例如在数据库中用于数据的存储和查询，还在数学推理和证明中起着重要的作用。
2. 逻辑（Logic）：逻辑是研究推理和证明的学科，也是离散数学的重要组成部分。
逻辑主要包括命题逻辑、谓词逻辑和命题演算等。
命题逻辑研究的是命题及其逻辑关系，谓词逻辑研究的是谓词及其量词关系。
逻辑在计算机科学中有广泛的应用，例如在程序设计中用于控制流程和条件判断。
3. 图论（Graph Theory）：图论研究的是由节点和边组成的图形结构及其性质和应用。
图是离散数学中的重要概念，它可以用于描述各种实际问题，如网络拓扑、社交网络、路线规划等。
图论主要研究图的表示方法、图的遍历和连通性、图的最短路径和最小生成树等问题。
图论在计算机科学中有广泛的应用，例如在网络优化和算法设计中起着重要的作用。
4. 代数结构（Algebraic Structures）：代数结构是研究数学对象的代数性质和运算规则的学科。
代数结构主要包括群论、环论、域论等。
群论研究的是集合及其上的运算满足的代数规则，环论研究的是集合及其上的两种运算满足的代数规则，域论研究的是集合及其上的四种运算满足的代数规则。
代数结构在密码学、编码理论和计算机图形学等领域有广泛的应用。
计算机算法相关术语的英语词汇
计算机算法相关术语的英语词汇计算机英语字典Dictionaries堆Heap优先级队列Priority queue矩阵乘法Matrix multiplication贪心算法Greedy algorithm上界/下界Upper bound / Lower bound最好情况/最坏情况/平均情况Best case /Worst Case/ Average case 插入排序Insertion sort合并排序Merge sort堆排序Heap sort快速排序Quick sort动态规划DP (Dynamic Programming)背包问题Knapsack problem霍夫曼编码Huffman Coding迪杰斯特拉算法Dijkstras algorithm贝尔曼-福德算法Bellman-Ford algorithm弗洛伊德算法Floyd-Warshall algorithm回溯Back-TrackingN皇后问题N-Queen problem渐进增长Asymptotic growth(包含O-notation-notation -notation)线性规划Linear programming随机数生成Random number generation图的生成Generating graphs图论-多项式算法Graph Problems polynomial algorithm 连通分支Connected components最小生成树Minimum Spanning Tree最短路径Shortest pathNP问题Non-Deterministic Polynomial problem旅行商问题Traveling salesman problem同构Graph isomorphism压缩Text compression最长公共子串Longest Common Substring最短公共父串Shortest Common Superstring收敛速度Rate of convergence。
AI专用词汇
AI专⽤词汇LetterAAccumulatederrorbackpropagation累积误差逆传播ActivationFunction激活函数AdaptiveResonanceTheory/ART⾃适应谐振理论Addictivemodel加性学习Adversari alNetworks对抗⽹络AffineLayer仿射层Affinitymatrix亲和矩阵Agent代理/智能体Algorithm算法Alpha-betapruningα-β剪枝Anomalydetection异常检测Approximation近似AreaUnderROCCurve／AUCRoc曲线下⾯积ArtificialGeneralIntelligence/AGI通⽤⼈⼯智能ArtificialIntelligence/AI⼈⼯智能Associationanalysis关联分析Attentionmechanism注意⼒机制Attributeconditionalindependenceassumption属性条件独⽴性假设Attributespace属性空间Attributevalue属性值Autoencoder⾃编码器Automaticspeechrecognition⾃动语⾳识别Automaticsummarization⾃动摘要Aver agegradient平均梯度Average-Pooling平均池化LetterBBackpropagationThroughTime通过时间的反向传播Backpropagation/BP反向传播Baselearner基学习器Baselearnin galgorithm基学习算法BatchNormalization/BN批量归⼀化Bayesdecisionrule贝叶斯判定准则BayesModelAveraging／BMA贝叶斯模型平均Bayesoptimalclassifier贝叶斯最优分类器Bayesiandecisiontheory贝叶斯决策论Bayesiannetwork贝叶斯⽹络Between-cla ssscattermatrix类间散度矩阵Bias偏置/偏差Bias-variancedecomposition偏差-⽅差分解Bias-VarianceDilemma偏差–⽅差困境Bi-directionalLong-ShortTermMemory/Bi-LSTM双向长短期记忆Binaryclassification⼆分类Binomialtest⼆项检验Bi-partition⼆分法Boltzmannmachine玻尔兹曼机Bootstrapsampling⾃助采样法／可重复采样／有放回采样Bootstrapping⾃助法Break-EventPoint／BEP平衡点LetterCCalibration校准Cascade-Correlation级联相关Categoricalattribute离散属性Class-conditionalprobability类条件概率Classificationandregressiontree/CART分类与回归树Classifier分类器Class-imbalance类别不平衡Closed-form闭式Cluster簇/类/集群Clusteranalysis聚类分析Clustering聚类Clusteringensemble聚类集成Co-adapting共适应Codin gmatrix编码矩阵COLT国际学习理论会议Committee-basedlearning基于委员会的学习Competiti velearning竞争型学习Componentlearner组件学习器Comprehensibility可解释性Comput ationCost计算成本ComputationalLinguistics计算语⾔学Computervision计算机视觉C onceptdrift概念漂移ConceptLearningSystem/CLS概念学习系统Conditionalentropy条件熵Conditionalmutualinformation条件互信息ConditionalProbabilityTable／CPT条件概率表Conditionalrandomfield/CRF条件随机场Conditionalrisk条件风险Confidence置信度Confusionmatrix混淆矩阵Connectionweight连接权Connectionism连结主义Consistency⼀致性／相合性Contingencytable列联表Continuousattribute连续属性Convergence收敛Conversationalagent会话智能体Convexquadraticprogramming凸⼆次规划Convexity凸性Convolutionalneuralnetwork/CNN卷积神经⽹络Co-oc currence同现Correlationcoefficient相关系数Cosinesimilarity余弦相似度Costcurve成本曲线CostFunction成本函数Costmatrix成本矩阵Cost-sensitive成本敏感Crosse ntropy交叉熵Crossvalidation交叉验证Crowdsourcing众包Curseofdimensionality维数灾难Cutpoint截断点Cuttingplanealgorithm割平⾯法LetterDDatamining数据挖掘Dataset数据集DecisionBoundary决策边界Decisionstump决策树桩Decisiontree决策树／判定树Deduction演绎DeepBeliefNetwork深度信念⽹络DeepConvolutionalGe nerativeAdversarialNetwork/DCGAN深度卷积⽣成对抗⽹络Deeplearning深度学习Deep neuralnetwork/DNN深度神经⽹络DeepQ-Learning深度Q学习DeepQ-Network深度Q⽹络Densityestimation密度估计Density-basedclustering密度聚类Differentiab leneuralcomputer可微分神经计算机Dimensionalityreductionalgorithm降维算法D irectededge有向边Disagreementmeasure不合度量Discriminativemodel判别模型Di scriminator判别器Distancemeasure距离度量Distancemetriclearning距离度量学习D istribution分布Divergence散度Diversitymeasure多样性度量／差异性度量Domainadaption领域⾃适应Downsampling下采样D-separation（Directedseparation）有向分离Dual problem对偶问题Dummynode哑结点DynamicFusion动态融合Dynamicprogramming动态规划LetterEEigenvaluedecomposition特征值分解Embedding嵌⼊Emotionalanalysis情绪分析Empiricalconditionalentropy经验条件熵Empiricalentropy经验熵Empiricalerror经验误差Empiricalrisk经验风险End-to-End端到端Energy-basedmodel基于能量的模型Ensemblelearning集成学习Ensemblepruning集成修剪ErrorCorrectingOu tputCodes／ECOC纠错输出码Errorrate错误率Error-ambiguitydecomposition误差-分歧分解Euclideandistance欧⽒距离Evolutionarycomputation演化计算Expectation-Maximization期望最⼤化Expectedloss期望损失ExplodingGradientProblem梯度爆炸问题Exponentiallossfunction指数损失函数ExtremeLearningMachine/ELM超限学习机LetterFFactorization因⼦分解Falsenegative假负类Falsepositive假正类False PositiveRate/FPR假正例率Featureengineering特征⼯程Featureselection特征选择Featurevector特征向量FeaturedLearning特征学习FeedforwardNeuralNetworks/FNN前馈神经⽹络Fine-tuning微调Flippingoutput翻转法Fluctuation震荡Forwards tagewisealgorithm前向分步算法Frequentist频率主义学派Full-rankmatrix满秩矩阵Func tionalneuron功能神经元LetterGGainratio增益率Gametheory博弈论Gaussianker nelfunction⾼斯核函数GaussianMixtureModel⾼斯混合模型GeneralProblemSolving通⽤问题求解Generalization泛化Generalizationerror泛化误差Generalizatione rrorbound泛化误差上界GeneralizedLagrangefunction⼴义拉格朗⽇函数Generalized linearmodel⼴义线性模型GeneralizedRayleighquotient⼴义瑞利商GenerativeAd versarialNetworks/GAN⽣成对抗⽹络GenerativeModel⽣成模型Generator⽣成器Genet icAlgorithm/GA遗传算法Gibbssampling吉布斯采样Giniindex基尼指数Globalminimum全局最⼩GlobalOptimization全局优化Gradientboosting梯度提升GradientDescent梯度下降Graphtheory图论Ground-truth真相／真实LetterHHardmargin硬间隔Hardvoting硬投票Harmonicmean调和平均Hessematrix海塞矩阵Hiddendynamicmodel隐动态模型H iddenlayer隐藏层HiddenMarkovModel/HMM隐马尔可夫模型Hierarchicalclustering层次聚类Hilbertspace希尔伯特空间Hingelossfunction合页损失函数Hold-out留出法Homo geneous同质Hybridcomputing混合计算Hyperparameter超参数Hypothesis假设Hypothe sistest假设验证LetterIICML国际机器学习会议Improvediterativescaling/IIS改进的迭代尺度法Incrementallearning增量学习Independentandidenticallydistributed/i.i.d.独⽴同分布IndependentComponentAnalysis/ICA独⽴成分分析Indicatorfunction指⽰函数Individuallearner个体学习器Induction归纳Inductivebias归纳偏好I nductivelearning归纳学习InductiveLogicProgramming／ILP归纳逻辑程序设计Infor mationentropy信息熵Informationgain信息增益Inputlayer输⼊层Insensitiveloss不敏感损失Inter-clustersimilarity簇间相似度InternationalConferencefor MachineLearning/ICML国际机器学习⼤会Intra-clustersimilarity簇内相似度Intrinsicvalue固有值IsometricMapping/Isomap等度量映射Isotonicregression等分回归It erativeDichotomiser迭代⼆分器LetterKKernelmethod核⽅法Kerneltrick核技巧K ernelizedLinearDiscriminantAnalysis／KLDA核线性判别分析K-foldcrossvalidationk折交叉验证／k倍交叉验证K-MeansClusteringK–均值聚类K-NearestNeighb oursAlgorithm/KNNK近邻算法Knowledgebase知识库KnowledgeRepresentation知识表征LetterLLabelspace标记空间Lagrangeduality拉格朗⽇对偶性Lagrangemultiplier拉格朗⽇乘⼦Laplacesmoothing拉普拉斯平滑Laplaciancorrection拉普拉斯修正Latent DirichletAllocation隐狄利克雷分布Latentsemanticanalysis潜在语义分析Latentvariable隐变量Lazylearning懒惰学习Learner学习器Learningbyanalogy类⽐学习Learn ingrate学习率LearningVectorQuantization/LVQ学习向量量化Leastsquaresre gressiontree最⼩⼆乘回归树Leave-One-Out/LOO留⼀法linearchainconditional randomfield线性链条件随机场LinearDiscriminantAnalysis／LDA线性判别分析Linearmodel线性模型LinearRegression线性回归Linkfunction联系函数LocalMarkovproperty局部马尔可夫性Localminimum局部最⼩Loglikelihood对数似然Logodds／logit对数⼏率Lo gisticRegressionLogistic回归Log-likelihood对数似然Log-linearregression对数线性回归Long-ShortTermMemory/LSTM长短期记忆Lossfunction损失函数LetterM Machinetranslation/MT机器翻译Macron-P宏查准率Macron-R宏查全率Majorityvoting绝对多数投票法Manifoldassumption流形假设Manifoldlearning流形学习Margintheory间隔理论Marginaldistribution边际分布Marginalindependence边际独⽴性Marginalization边际化MarkovChainMonteCarlo/MCMC马尔可夫链蒙特卡罗⽅法MarkovRandomField马尔可夫随机场Maximalclique最⼤团MaximumLikelihoodEstimation/MLE极⼤似然估计／极⼤似然法Maximummargin最⼤间隔Maximumweightedspanningtree最⼤带权⽣成树Max-P ooling最⼤池化Meansquarederror均⽅误差Meta-learner元学习器Metriclearning度量学习Micro-P微查准率Micro-R微查全率MinimalDescriptionLength/MDL最⼩描述长度Minim axgame极⼩极⼤博弈Misclassificationcost误分类成本Mixtureofexperts混合专家Momentum动量Moralgraph道德图／端正图Multi-classclassification多分类Multi-docum entsummarization多⽂档摘要Multi-layerfeedforwardneuralnetworks多层前馈神经⽹络MultilayerPerceptron/MLP多层感知器Multimodallearning多模态学习Multipl eDimensionalScaling多维缩放Multiplelinearregression多元线性回归Multi-re sponseLinearRegression／MLR多响应线性回归Mutualinformation互信息LetterN Naivebayes朴素贝叶斯NaiveBayesClassifier朴素贝叶斯分类器Namedentityrecognition命名实体识别Nashequilibrium纳什均衡Naturallanguagegeneration/NLG⾃然语⾔⽣成Naturallanguageprocessing⾃然语⾔处理Negativeclass负类Negativecorrelation负相关法NegativeLogLikelihood负对数似然NeighbourhoodComponentAnalysis/NCA近邻成分分析NeuralMachineTranslation神经机器翻译NeuralTuringMachine神经图灵机Newtonmethod⽜顿法NIPS国际神经信息处理系统会议NoFreeLunchTheorem ／NFL没有免费的午餐定理Noise-contrastiveestimation噪⾳对⽐估计Nominalattribute列名属性Non-convexoptimization⾮凸优化Nonlinearmodel⾮线性模型Non-metricdistance⾮度量距离Non-negativematrixfactorization⾮负矩阵分解Non-ordinalattribute⽆序属性Non-SaturatingGame⾮饱和博弈Norm范数Normalization归⼀化Nuclearnorm核范数Numericalattribute数值属性LetterOObjectivefunction⽬标函数Obliquedecisiontree斜决策树Occam’srazor奥卡姆剃⼑Odds⼏率Off-Policy离策略Oneshotlearning⼀次性学习One-DependentEstimator／ODE独依赖估计On-Policy在策略Ordinalattribute有序属性Out-of-bagestimate包外估计Outputlayer输出层Outputsmearing输出调制法Overfitting过拟合／过配Oversampling过采样LetterPPairedt-test成对t检验Pairwise成对型PairwiseMarkovproperty成对马尔可夫性Parameter参数Parameterestimation参数估计Parametertuning调参Parsetree解析树ParticleSwarmOptimization/PSO粒⼦群优化算法Part-of-speechtagging词性标注Perceptron感知机Performanceme asure性能度量PlugandPlayGenerativeNetwork即插即⽤⽣成⽹络Pluralityvoting相对多数投票法Polaritydetection极性检测Polynomialkernelfunction多项式核函数Pooling池化Positiveclass正类Positivedefinitematrix正定矩阵Post-hoctest后续检验Post-pruning后剪枝potentialfunction势函数Precision查准率／准确率Prepruning预剪枝Principalcomponentanalysis/PCA主成分分析Principleofmultipleexplanations多释原则Prior先验ProbabilityGraphicalModel概率图模型ProximalGradientDescent/PGD近端梯度下降Pruning剪枝Pseudo-label伪标记LetterQQuantizedNeu ralNetwork量⼦化神经⽹络Quantumcomputer量⼦计算机QuantumComputing量⼦计算Quasi Newtonmethod拟⽜顿法LetterRRadialBasisFunction／RBF径向基函数RandomFo restAlgorithm随机森林算法Randomwalk随机漫步Recall查全率／召回率ReceiverOperatin gCharacteristic/ROC受试者⼯作特征RectifiedLinearUnit/ReLU线性修正单元Recurr entNeuralNetwork循环神经⽹络Recursiveneuralnetwork递归神经⽹络Referencemodel参考模型Regression回归Regularization正则化Reinforcementlearning/RL强化学习Representationlearning表征学习Representertheorem表⽰定理reproducingke rnelHilbertspace/RKHS再⽣核希尔伯特空间Re-sampling重采样法Rescaling再缩放Residu alMapping残差映射ResidualNetwork残差⽹络RestrictedBoltzmannMachine/RBM受限玻尔兹曼机RestrictedIsometryProperty/RIP限定等距性Re-weighting重赋权法Robu stness稳健性/鲁棒性Rootnode根结点RuleEngine规则引擎Rulelearning规则学习LetterS Saddlepoint鞍点Samplespace样本空间Sampling采样Scorefunction评分函数Self-Driving⾃动驾驶Self-OrganizingMap／SOM⾃组织映射Semi-naiveBayesclassifiers半朴素贝叶斯分类器Semi-SupervisedLearning半监督学习semi-SupervisedSupportVec torMachine半监督⽀持向量机Sentimentanalysis情感分析Separatinghyperplane分离超平⾯SigmoidfunctionSigmoid函数Similaritymeasure相似度度量Simulatedannealing模拟退⽕Simultaneouslocalizationandmapping同步定位与地图构建SingularV alueDecomposition奇异值分解Slackvariables松弛变量Smoothing平滑Softmargin软间隔Softmarginmaximization软间隔最⼤化Softvoting软投票Sparserepresentation稀疏表征Sparsity稀疏性Specialization特化SpectralClustering谱聚类SpeechRecognition语⾳识别Splittingvariable切分变量Squashingfunction挤压函数Stability-plasticitydilemma可塑性-稳定性困境Statisticallearning统计学习Statusfeaturefunction状态特征函Stochasticgradientdescent随机梯度下降Stratifiedsampling分层采样Structuralrisk结构风险Structuralriskminimization/SRM结构风险最⼩化S ubspace⼦空间Supervisedlearning监督学习／有导师学习supportvectorexpansion⽀持向量展式SupportVectorMachine/SVM⽀持向量机Surrogatloss替代损失Surrogatefunction替代函数Symboliclearning符号学习Symbolism符号主义Synset同义词集LetterTT-Di stributionStochasticNeighbourEmbedding/t-SNET–分布随机近邻嵌⼊Tensor张量TensorProcessingUnits/TPU张量处理单元Theleastsquaremethod最⼩⼆乘法Th reshold阈值Thresholdlogicunit阈值逻辑单元Threshold-moving阈值移动TimeStep时间步骤Tokenization标记化Trainingerror训练误差Traininginstance训练⽰例／训练例Tran sductivelearning直推学习Transferlearning迁移学习Treebank树库Tria-by-error试错法Truenegative真负类Truepositive真正类TruePositiveRate/TPR真正例率TuringMachine图灵机Twice-learning⼆次学习LetterUUnderfitting⽋拟合／⽋配Undersampling⽋采样Understandability可理解性Unequalcost⾮均等代价Unit-stepfunction单位阶跃函数Univariatedecisiontree单变量决策树Unsupervisedlearning⽆监督学习／⽆导师学习Unsupervisedlayer-wisetraining⽆监督逐层训练Upsampling上采样LetterVVanishingGradientProblem梯度消失问题Variationalinference变分推断VCTheoryVC维理论Versionspace版本空间Viterbialgorithm维特⽐算法VonNeumannarchitecture冯·诺伊曼架构LetterWWassersteinGAN/WGANWasserstein⽣成对抗⽹络Weaklearner弱学习器Weight权重Weightsharing权共享Weightedvoting加权投票法Within-classscattermatrix类内散度矩阵Wordembedding词嵌⼊Wordsensedisambiguation词义消歧LetterZZero-datalearning零数据学习Zero-shotlearning零次学习。
计算机常用术语
第一部分、计算机算法常用术语中英对照Data Structures 基本数据结构Dictionaries 字典Priority Queues 堆Graph Data Structures 图Set Data Structures 集合Kd-Trees 线段树Numerical Problems 数值问题Solving Linear Equations 线性方程组Bandwidth Reduction 带宽压缩Matrix Multiplication 矩阵乘法Determinants and Permanents 行列式Constrained and Unconstrained Optimization 最值问题Linear Programming 线性规划Random Number Generation 随机数生成Factoring and Primality Testing 因子分解/质数判定Arbitrary Precision Arithmetic 高精度计算Knapsack Problem 背包问题Discrete Fourier Transform 离散Fourier变换Combinatorial Problems 组合问题Sorting 排序Searching 查找Median and Selection 中位数Generating Permutations 排列生成Generating Subsets 子集生成Generating Partitions 划分生成Generating Graphs 图的生成Calendrical Calculations 日期Job Scheduling 工程安排Satisfiability 可满足性Graph Problems -- polynomial 图论-多项式算法Connected Components 连通分支Topological Sorting 拓扑排序Minimum Spanning Tree 最小生成树Shortest Path 最短路径Transitive Closure and Reduction 传递闭包Matching 匹配Eulerian Cycle / Chinese Postman Euler回路/中国邮路Edge and Vertex Connectivity 割边/割点Network Flow 网络流Drawing Graphs Nicely 图的描绘Drawing Trees 树的描绘Planarity Detection and Embedding 平面性检测和嵌入Graph Problems -- hard 图论-NP问题Clique 最大团Independent Set 独立集Vertex Cover 点覆盖Traveling Salesman Problem 旅行商问题Hamiltonian Cycle Hamilton回路Graph Partition 图的划分Vertex Coloring 点染色Edge Coloring 边染色Graph Isomorphism 同构Steiner Tree Steiner树Feedback Edge/Vertex Set 最大无环子图Computational Geometry 计算几何Convex Hull 凸包Triangulation 三角剖分Voronoi Diagrams Voronoi图Nearest Neighbor Search 最近点对查询Range Search 范围查询Point Location 位置查询Intersection Detection 碰撞测试Bin Packing 装箱问题Medial-Axis Transformation 中轴变换Polygon Partitioning 多边形分割Simplifying Polygons 多边形化简Shape Similarity 相似多边形Motion Planning 运动规划Maintaining Line Arrangements 平面分割Minkowski Sum Minkowski和Set and String Problems 集合与串的问题Set Cover 集合覆盖Set Packing 集合配置String Matching 模式匹配Approximate String Matching 模糊匹配Text Compression 压缩Cryptography 密码Finite State Machine Minimization 有穷自动机简化Longest Common Substring 最长公共子串Shortest Common Superstring 最短公共父串DP——Dynamic Programming——动态规划recursion ——递归第二部分、编程词汇A2A integration A2A整合abstract 抽象的abstract base class (ABC)抽象基类abstract class 抽象类abstraction 抽象、抽象物、抽象性access 存取、访问access level访问级别access function 访问函数account 账户action 动作activate 激活active 活动的actual parameter 实参adapter 适配器add-in 插件address 地址address space 地址空间address-of operator 取地址操作符ADL (argument-dependent lookup)ADO(ActiveX Data Object)ActiveX数据对象advancedaggregation 聚合、聚集algorithm 算法alias 别名align 排列、对齐allocate 分配、配置allocator分配器、配置器angle bracket 尖括号annotation 注解、评注API (Application Programming Interface) 应用(程序)编程接口app domain (application domain)应用域application 应用、应用程序application framework 应用程序框架appearance 外观append 附加architecture 架构、体系结构archive file 归档文件、存档文件argument引数(传给函式的值)。
两类冠图的 Laplacian 谱
两类冠图的 Laplacian 谱卢鹏丽;苗玉芳【摘要】The spectra contain a lot of information concerning a graph.The corona graphs are complex and the com-putation of their spectra is more complex.In this paper, two classes of corona graphs were defined:the corona-ver-tex of the subdivisional graph of G1and G2 , denoted byG1◇G2 , and the corona-edge of the subdivisional graph of G1 and G2 , denoted by G1☆G2 .Using the block matrix, the coronal and Kronecker product, the Laplacian spectra of G1◇G2 and G1☆G2 were determined in terms of the corresponding spectra of G1 and G2 .By using the Laplacian spectra, the number of spanning trees and Kirchhoff index of G1◇G2 and G1☆G2 are also obtained.%图的谱蕴含着图的许多信息。
冠图是一种比较复杂的图，冠图的谱更加难以计算。
文中定义了两类冠图，分别是：图G1和G2的剖分图的冠点图G1◇G2和剖分图的冠边图G1☆G2。
应用分块矩阵、矩阵的coronal、克罗内克积证明了两类冠图的Laplacian谱可以表示为原图G1和G2的Laplacian谱；并给出了两类冠图的生成树数目以及Kirchhoff指数。
常用计算机英语词汇3000例(含音标)之欧阳育创编
常用计算机英语词汇3000例（含音标）（仅供参考）abbreviate[ə'bri:vieit]vt.缩写，省略abbreviation[əbri:vi'eiʃən]n.缩短，省略，简称abend异常结束Abnormal[æb'nɔ:məl]异常abort[ə'bɔ:t]异常中止aboveboard[ə'bʌv'bɔ:d]ad. 照直，公开的absence['æbsəns]n.缺少，没有absolute['æbsəlu:t]绝对absorption[əb'sɔ:pʃən]吸收abstract['æbstrækt]抽象acceleration[ækselə'reiʃən]加速度accelerator[æk'seləreitə]n.加速装置，加速剂accent['æksənt,æk'sent]强调accept[ək'sept]接受accepting接收access['ækses]访问accessible[æk'sesəbl]可存取accessor存取元accessory[æk'sesəri]附件accidentally[æksi'dentəli]ad. 偶然地accommodate[ə'kɔmədeit]容纳accordingtoa.按照，根据account[ə'kaunt]帐户accounting[ə'kauntiŋ]会计accumulate[ə'kju:mjuleit]累加accumulator[ə'kju:mjuleitə]累加器accuracy['ækjurəsi]准确度accurately['ækjuritli]准确地achieve[ə'tʃi:v]vt.完成ACK确认符acknowledge[ək'nɔlidʒ]确认acknowledgement[ək'nɔlidʒmənt]确认acknowledgment[ək'nɔlidʒmənt]n.接收(收妥)，承认acoustic[ə'ku:stik]声音的acquire[ə'kwaiə]获取acquisition['ækwi'ziʃən]获取acronym['ækrənim]字首组合词across[ə'krɔs]跨越action['ækʃən]动作activate['æktiveit]激活activation[ækti'veiʃən]激活active['æktiv]活动的activity[æk'tiviti]活动actual['æktjuəl]实际的actuator致动器adaptability[ə'dæptə'biliti]适应性adapter[ə'dæptə]适配器add[æd]添加addend[ə'dend]加数adder['ædə]加法器addition[ə'diʃən]n.加法，增加additional[ə'diʃənl]附加的additionallyad. 另外，又additive['æditiv]添加的addRess[ə'dRes]地址addRessability可寻址性addRessee[æDRe'si:]被访地址addRessing寻址adequate['ædikwit]a.足够的，充分的adjacency[ə'dʒeisnsi]邻近adjacent[ə'dʒeisənt]邻近的adjust[ə'dʒʌst]调整adjustment[ə'dʒʌstmənt]调整administration[ədminis'treiʃən]管理administrative[əd'ministrətiv]管理的administrator[əd'ministreitə]管理员advance[əd'va:ns]v.进展advanced[əd'va:nst]高级的affect[ə'fekt]vt.影响，改变，感动affected[ə'fektid]a.受了影响的again[ə'gein,ə'gen]再次aggregate['ægrigeit]聚集aid[eid]n.帮助，辅助程序airflow['/əflou]气流aisle[ail]通道alarm[ə'la:m]警报alert[ə'lə:t]报警algebraic代数algorithm['ælgəriðm]算法alias['eiliæs]别名aliasing别名判别align[ə'lain]v.定位，对齐aligneda.对齐的，均衡的aligner调整器alignment[ə'lainmənt]数据对齐all[ɔ:l]全部allocate['æləkeit]分配allocation[ælə'keiʃən]分配allocator分配器allow[ə'lau]允许allowable[ə'lauəbl]a.容许的，承认的allowance[ə'lauəns]允许alloweda.容许的ally['ælai,ə'lai]v.联合，与...关联alpha['ælfə]n.希腊字母α，未知数alphabet['ælfəbit]字母alphabetic[ælfə'betik]字母的alphabetical[ælfə'betikəl]a.字母(表)的，ABC的alphabetically[ælfə'betikli]ad. 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交替地，轮流地alternative[ɔ:l'tə:nətiv]替代项amber['æmbə]琥珀色amount[ə'maunt]金额amp安培ampere['æmp/ə]安培ampersand['æmpəsænd]n.＆号(and)amplifier['æmplifait]放大器amplitude['æmplitju:d]振幅analog['ænəlɔg]模拟analogue['ænəlɔg]模拟Download[daʊn'ləʊd][,daʊn'lod]下载analysis[ə'næləsis]分析analyst['ænəlist]分析员analyze['ænəlaiz]分析analyzer分析机ancestor['ænsistə]祖先AnD[ænd,ənd]与angle['æŋgl]角度Angstrom['æŋstrəm]埃animate['ænimeit]动画绘制Animation[ænə'meiʃən]动画annotate['ænəuteit]注释annotation[ænou'teiʃən]注释announce[ə'nauns]vt.发表，宣布announcement[ə'naunsmənt]宣布announciator报警器anode['ænəud]阳极another[ə'nʌðə]另一个AnSIn.美国国家标准协会answering['a:nsəriŋ]应答antialiasing排斥假名anticipate[æn'tisipeit]vt.预先考虑，抢...先antistatic防静电aperture['æpətjuə]小孔apostrophe[ə'pɔstrəfi]单引号appear[ə'piə]出现append[ə'pend]附加appendix[ə'pendiks]n.附录Apple['æpl]n.苹果公司applicable['æplikəbl]a.可适用的，合适的application[æpli'keiʃən]应用（程序）applied[ə'plaid]a.适用的，外加的apply[ə'plai]应用appropriate[ə'prəupriit]适当的appropriatelyad. 适当地apron['eiprən]附表architecture['a:kitektʃə]体系结构archive归档area['əriə]区argument['a:gju:mənt]参数arithmetic[ə'riθmətik]算术arrange[ə'reindʒ]排列arrangement[ə'reindʒmənt]排列array[ə'rei]数组arrow['ærəu]箭头article['a:tikl]物件ascend[ə'send]上升ascending[ə'sendiŋ]升序ASCIIn.美国信息交换标准码assemble[ə'sembl]汇编assembled组装assembler汇编程序assembly[ə'sembli]汇编assertion[ə'sə:ʃən]断言asset['æset]资产Assign[ə'sain]分配assigneda.指定的，赋值的assignment[ə'sainmənt]赋值assist[ə'sist]辅助assistance[ə'sistəns]辅助associate[ə'səuʃieit]v.相联，联想，关联associated关联的association[əsəusi'eiʃən]关联associativity关联性assortment[ə'sɔ:tmənt]n.种类，花色品种assume[ə'sju:m]假设assumed[ə'sju:md]a.假定的asterisk['æstərisk]星号async异步asynchronous异步atom['ætəm]原子atomicity[ætə'misiti]原子性attach[ə'tætʃ]连接attacheda.附加的attachment[ə'tætʃmənt]连接附件attempt[ə'tempt]试图attention[ə'tenʃən]n.注意(信号) attenuation[ətenju'eiʃən]衰减attenuator衰减器attractive[ə'træktiv]吸引人的attribute['ætribju:t]属性audio['ɔ:diou]声频audit['ɔ:dit]审查augend['ɔ:dʒend]被加数augment[ɔ:g'ment,'ɔ:gmənt]v.增加，添加，扩充authentication[ɔ:θenti'keiʃən]认证authenticator[ɔ:'θentikeitə]认证器author['ɔ:θə]n.程序设计者，作者authority[ɔ:'θɔriti]权限authorization[ɔ:θərai'zeiʃən]权限authorize['ɔ:θəraiz]授权authorized['ɔ:θəraizd]特许的auto['ɔ:təu]a.自动的autodialer自动拔号器autoindexn.自动变址(数)automata[ɔ:'tɔmətə]自动机automatic[ɔ:tə'mætik]自动的automatically[ɔ:tə'mætikəli]ad. 自动地，机械地automation[ɔ:tə'meiʃən]自动化automaton[ɔ:'tɔmətən]自动机autonumber自动号autopush自动推autostart自动启动auxiliary[ɔ:g'ziljəri]辅助的availability[əveilə'biliti]可用性available[ə'veiləbl]可用的average['ævəridʒ]平均avoid[ə'vɔid]vt.避免，取消，无效babble['bæbl]串音backbone['bækbəun]主干backend后端backflush逆算法background['bækgraund]后台backlog待办事项backscrolling反卷backslash反斜线backspace['bækspeis]退格backtab退格制表backup['bækʌp]备份backward['bækwəd]反向badge[bædʒ]标记balance['bæləns]平衡band[bænd]区bandwidth带宽bank[bæŋk]存储单元banking['bæŋkiŋ]出界bar[ba:]条base[beis]基数baseband基带baseline基线basename基名basic['beisik]基本basis['beisis]基础bass[beis]低音部batch[bætʃ]批处理battery['bætəri]电池baud波特beacon['bi:kən]信标beam[bi:m]束beep[bi:p]n.蜂鸣声，嘀嘀声beeper蜂鸣器begin[bi'gin]开始behavior[bi'heivjə]行为bel贝尔bell[bel]响铃bellfast快贝尔below[bi'ləu]下面benchmark基准beyond[bi'jɔnd]prep.超过，那边bias['baiəs]偏离bibliography[bibli'ɔgrəfi]书目bid[bid]请求bidder['bidə]请求者bill[bil]票据billing['biliŋ]开票binary['bainəri]二进制bind[baind]联接binder['baində]联接器binding['baindiŋ]联接bionics[bai'ɔniks]仿生学BIOSn.基本输入/输出系统bipolar[bai'poulə]双极性的biquinary[bi'kwinəri]二五混合进制（的）bistable双稳态的bit[bit]位bitmap位图blank[blæŋk]空格blanket['blæŋkit]涂层b link[bliŋk]闪烁blinking闪烁blip[blip]（缩微胶卷画面上的光点）标志blit位块传送block[blɔk]块blocking['blɔkiŋ]分块blower['blouə]鼓风机board[bɔ:d]n.板，插件板body['bɔdi]主体boldface['bouldfeis]黑体book[buk]书籍bookmark['bukma:k]书签Boolean布尔boot[bu:t]引导Bootable可引导bootstrap['bu:tstræp]自举border['bɔ:də]边框borrow['bɔrəu]借位bottleneck['bɔtlnek]瓶颈bottom['bɔtəm]底部bounce[bauns]弹回bound[baund]界限boundary['baundəri]边界box[bɔks]框bpi位／英寸bps位／秒brace[breis]花括号braces花括号bracket['brækit]方括号bracketed['brækətid]a.加括号的brackets方括号branch[bra:ntʃ]分支break[breik]中断breakpoint断点breve[bri:v]短音符号bridge[bridʒ]网桥bridging[bridʒiŋ]桥接bright[brait]明亮brightness[braitnis]亮度broadband宽频带broadcast['brɔ:dka:st]广播browse[brauz]浏览Browser['brauzə]浏览器bucket['bʌkit]存储桶buffer['bʌfə]缓冲区buffering缓冲bug[bʌg]错误build[bild]构建bulb[bʌlb]灯泡bumper['bʌmpə]保险杆burst[bə:st]脉冲串bus[bʌs]总线business['biznis]商业bussback反馈busy['bizi]忙button['bʌtn]按钮buyer['baiə]采购人员buzzer['bʌzə]蜂鸣器bypass['bai-pa:s]旁路byte[bait]字节cable['keibl]电缆cabling布线cache[kæʃ]高速缓存caching高速缓存CAD[kæd]计算机辅助设计cage[keidʒ]盒子calculate['kælkjuleit]计算calculation[kælkju'leiʃən]n.计算，统计，估计calculator['kælkjuleitə]计算器calendar['kælində]日历calibrate['kælibreit]校准call[kɔ:l]调用callback['kɔ:lbæk]回叫caller['kɔ:lə]调用程序calling['kɔ:liŋ]调用callout调出camcorder便携式摄像机cancel['kænsəl]取消candidate['kændidit]候选cannot['kænɔt]不能canvas['kænvəs]画布capability[keipə'biliti]能力capable['keipəbl]有能力的capacitor[kə'pæsitə]电容capacity[kə'pæsiti]能力capital['kæpitl]大写capitalizeda.大写的caption[kæpʃən]图表说明capture['kæptʃə]捕捉card[ka:d]卡caret['kærət]插入记号carousel[kæru'zel]n.圆盘传送带carriage['kæridʒ]托架carrier['kæriə]载波carry['kæri]进位cartridge['ka:tridʒ]盒式磁带cascade[kæs'keid]级联case[keis]n.情况，场合cash[kæʃ]n.现金cassette[ka:'set]盒式磁带cast[ka:st]强制转型catalog['kætəlɔg]编目catalogue['kætəlɔg]编目catch[kætʃ]捕捉category['kætigəri]种类cation['kætaiən]n.正离子，阳离子cause[kɔ:z]原因caution['kɔ:ʃən]注意CD光盘，激光唱片cedilla[si'dilə]变音符cell[sel]单元Celsius摄氏center['sentə]中心centering['sentəriŋ]向中对齐centerline中线centimeter厘米central['sentrəl]中央的centrex中央交换机century['sentʃuri]n.世纪certain['sə:tən]a.确实的，确定的certainty['sə:tnti]n.必然，确实certification[sə:tifi'keiʃən]确证chain[tʃein]链chained[tʃeind]链接chaining链接chamber['tʃeimbə]箱change[tʃeindʒ]更改channel['tʃænl]通道channelizing沟道效应chapter['tʃæptə]章char[tʃa:]字符character['kæriktə]字符characteristic[kæriktə'ristik]特性charge[tʃa:dʒ]费用charging充电chart[tʃa:t]图表chassis['ʃæsi]机壳check[tʃek]检查checker['tʃekə]检查器checkout检出checkpoint['tʃekpɔint]检查点child[tʃaild]儿子节点children['tʃildRən]子女chip[tʃip]芯片choice[tʃɔis]选项choose[tʃu:z]v.挑选，选择，选定chord[kɔ:d]弦chunk[tʃʌŋk]n.厚块，大部分ciphertext密码文本circle['sə:kl]n.圆，圈，循环，周期circuit['sə:kit]电路circular['sə:kjulə]循环circumflex['sə:kəmfleks]弯曲circumstance['sə:kəmstəns]n.情况，环境，细节citation[sai'teiʃən]引用city['siti]城市clamp[klæmp]夹class[kla:s]类classify['klæsifai]分类clause[klɔ:z]子句clear[kliə]清除clearinghouse['kliəriŋhaus]清除库clerk[kla:k,klə:k]职员click[klik]单击client['klaiənt]客户clip[klip]裁剪clipboard['klipbɔ:d]裁剪板cliping裁剪clipper['klipə]n.剪刀clock[klɔk]时钟clocking定时clockwise['klɔkwaiz]顺时针方向Close[kləuz,kləus]关闭closed[kləuzd]a.关闭的，闭迹closely['kləusli]a.精密地，仔细地cluster['klʌstə]群集coalesce[kouə'les]结合coating涂层coax[kəuks]同轴coaxial同轴的code[kəud]代码coder['koudə]编码器codeset代码集coding编码coefficient[kəui'fiʃənt]系数coexist['kouig'zist]共存coexistence['kouig'zistəns]共存coffret传输接口collapse[kə'læps]崩溃collate[kɔ'leit]整理collation[kɔ'leiʃn,kə-]整理collator[kɔ'leitə]整理器collection[kə'lekʃən]堆集collision[kə'liʒ(ə)n]冲突colon['koulən]冒号colour['kʌlə]颜色column['kɔləm]列combination[kɔmbi'neiʃən]组合combine[kəm'bain]组合comboboxn.组合框comma['kɔmə]逗号command[kə'ma:nd]命令comment['kɔment]注解commercial[kə'mə:ʃəl]a.商业的，经济的commit[kə'mit]落实commitment[kə'mitmənt]落实commodity[kə'mɔditi]商品common['kɔmən]公共的communicate[kə'mju:nikeit]通信communication[kə'mju:nikeiʃən]n.通信compact['kɔmpækt,kəm'pækt]压缩compaction压缩company['kʌmpəni]公司comparand比较字comparator['kɔmpəreitə]比较器compare[kəm'p/ə]比较comparison[kəm'pærisn]比较compatibility兼容性compatible[kəm'pætəbl]兼容的compilation[kɔmpi'leiʃən]编译compile[kəm'pail]编译compiler[kəm'pailə]编译器complement['kəmplimənt]补码complementer反相器complete[kəm'pli:t]完成completely[kəm'pli:tli]ad. 十分，完全，彻底completion[kəm'pli:ʃən]完成complex['kɔmpleks]复杂的complexities复杂性complexity[kəm'pleksiti]复杂程度complicated['kɔmplikeitid]v.使复杂化，使混乱component[kəm'pəunənt]部件components部件compose[kəm'pəuz]组成composing[kəm'pouziŋ]组成composite['kɔmpəzit]合成composition[kɔmpə'ziʃən]组合compress[kəm'pres]压缩compression[kɔm'preʃən]压缩compressor[kəmpresə]压缩器comprise[kəm'praiz]vt.包括，由...组成compute[kəm'pju:t]计算computer[kəm'pju:tə]计算机computing计算concatenate[kɔn'kætineit]并置concatenation[kɔn'kætə'neiʃən]并置concentration[kɔnsən'treiʃən]集中concentrator['kɔnsentreitə]集线器concept['kɔnsept]n.概念concordance[kən'kɔ:dəns]重要语汇索引concurrent[kən'kʌrənt]同时的condense[kən'dens]压缩condition[kən'diʃən]条件conditional[kən'diʃənl]条件的conditioning调节conductor[kən'dʌktə]导线conduit['kɔndit]护线管confidential[kɔnfi'denʃəl]机密configuration[kənfigju'reiʃən]配置configurator配置程序configure[kən'figə]配置confirm[kən'fə:m]确认confirmation[kɔnfə'meiʃən]确认conflict['kɔnflikt,kən'flikt]v.冲突，碰头conflicting[kən'fliktiŋ]冲突conform[kən'fɔ:m]vi.遵从，符合confuse[kən'fju:z]vt.使混乱，干扰congestion[kən'dʒestʃən]拥塞conjunction[kən'dʒʌŋkʃən]与connect[kə'nekt]连接connected[kə'nektid]连接connection[kə'nekʃən]连接connective[kə'nektiv]连接词connectivity连通性connector[kə'nektə]连接器consecutive[kən'sekjutiv]连续的consent[kən'sent]插座consequently['kɔnsikwəntli]ad. 因此，从而consider[kən'sidə]考虑consideration[kənsidə'reiʃən]n.考虑，研究，讨论considered[kən'sidəd]a.考虑过的，被尊重的consist[kən'sist]vi.符合，包括consistency[kən'sistənsi]一致性consistent[kən'sistənt]一致的console[kən'səul]控制台consolidate[kən'sɔlideit]合并constn.常数constant['kɔnstənt]常量constantly['kɔnstəntli]ad. 不变地，经常地constraint[kən'streint]约束constructing构造constructor构造成员consult[kən'sʌlt]v.咨询，顾问consumable[kəm'sju:məbl]消费品consume[kən'sju:m]v.消耗，使用contact['kɔntækt,kən'tækt]联系contain[kən'tein]包含container[kən'teinə]容器containment包含content['kɔntent,kən'tent]内容contention[kən'tenʃən]争用context['kɔntekst]上下文contextual[kən'tekstʃuəl]上下文的contiguous[kən'tigjuəs]相连的continue[kən'tinju(:)]继续continued[kən'tinju:d]接上页continuously[kən'tinjuəsli]ad. 连续不断地contractor[kən'træktə]承包方contrast['kɔntræst,kən'træst]反差control[kən'trəul]控制controllability可控制性controlled[kən'trould]a.受控制的，受操纵的controller[kən'troulə]控制器convegence收敛convenience[kən'vi:njəns]n.方便，便利convenient[kən'vi:njənt]a.方便的，便利的convention[kən'venʃən]约定conventional[kən'venʃənl]a.常规的，习惯的converg收敛converged收敛的conversation[kɔnvə'seiʃən]对话conversational[kɔnvə'seiʃənl]会话式conversion[kən'və:ʃən]转换convert[kən'və:t,'kɔnvə:t]转换converted[kən'və:tid]转换的converter转换器coordinate[kəu'ɔdinit]坐标coordinator[kou'ɔ:dineitə]协调程序coprocessor协处理器copy['kɔpi]复制copying复制copyright['kɔpirait]n.版权cord[kɔ:d]n.绳子，电线core[kɔ:]核心coresidency共存corner['kɔ:nə]角corona[kə'rounə]电晕correct[kə'rekt]正确correction[kə'rekʃən]n.校正，修正correctly[kə'rektli]正确correlator相关因子correspond[kɔris'pɔnd]vi.通信(联系) corresponding[kɔris'pɔndiŋ]相应的corrupt[kə'rʌpt]v.有毛病的corrupted毁坏的cosine['kousain]余弦cost[kɔst]成本costing['kɔ:stiŋ]成本法Coulomb[ku:'lɔm]库仑count[kaunt]计数counter['kauntə]计数器counterclockwise[kauntəklɔkwaiz]反时针方向country['kʌntri]国家coupler['kʌplə]耦合器coupling['kʌpliŋ]耦合courier['kuriə]信使course[kɔ:s]n.过程，航向，课程cover['kʌvə]盖CpU控制处理部件crank[kræŋk]曲柄crash[kræʃ]崩溃craze[kreiz]n.开裂create[kri(:)'eit]建立creation[kri'eiʃən]n.创造，创作creator[kri(:)'eitə]建立者credentials凭证credit['kredit]信用crew[kru:]组员criteria[krai'tiriə]标准criterion[krai'tiəriən]n.标准，判据，准则critical['kritikəl]临界的crop[krɔp]v.切，剪切cross[krɔs]跨crossfoot交叉结算crowding拥挤cryogenics低温学cryotron低温管cryptographic[kriptə'græfik]密码cryptography[krip'tɔgrəfi]密码术CUA公共用户存取访问culling['kʌliŋ]挑选cumulative['kjumjuleitiv]累积的current['kʌrənt]当前currently['kʌrəntli]ad. 目前，现在cursor['kə:sə]光标curtate卡片部分cushion['kuʃən]缓冲器custom['kʌstəm]a.用户customation定制customer['kʌstəmə]用户customize定制cut[kʌt]剪下cutoff['kʌtɔf]截止cutout['kʌtaut]开口cybernetics['saibə:'netiks]控制论cycle['saikl]循环cyclinder柱面cyrill西里尔daemon守护程序daily[deili]a.每日的，日常的damage['dæmidʒ]损坏damping阻尼dark[da:k]黑暗DASD直接存取存储器data[deitə]数据database数据库datagram数据报date[deit]日期datum[deitəm]数据day[dei]天db分贝deactivate[di:'æktiveit]释放deactivated释放deactivation释放deadlock[dedlɔk]死锁deal[di:l]v.处理，分配，交易dealer[di:lə]经销商deallocate释放dearly[diəli]ad. 极，非常，昂贵地death[deθ]n.毁灭，消灭debit[debit]借额deblock解块deblocking解块debug[di:'bʌg]调试debugger调试器deca十（词头）deci十分之一（词头）decibel[desibel]分贝decide[di'said]v.(使)判定，判断decimal[desiməl]十进制decipher[di'saifə]译码decision[di'siʒən]判定deck[dek]叠declaration[deklə'reiʃən]说明declarative[di'klærətiv]说明的declarator说明符declare[di'kl/ə]说明declared[di'kl/əd]a.承认的，申报的decode[di:'koud]译码decoder译码器decollate[di'kɔleit]分开decompression还原deconcentration[di:kɔnsən'treiʃən]分散deconcentrator分线器decrease[di'kri:s,'di:kri:s]v.减少，降低，缩短decreasing递减decrement[dekrimənt]减decrypt译码decurl去卷曲dedicate[dedikeit]专用dedicated[dedikeitid]专用的dedication[dedi'keiʃən]专用deduction[di'dʌkʃən]扣除default[di'fɔ:lt]缺省defect[di'fekt]缺陷defective[di'fektiv]a.故障的，有毛病的definable[di'fainəbl]a.可定义的，可确定的define[di'fain]定义definition[defi'niʃən]定义限定deflect[di'flekt]转向degrade[di'greid]v.降低，减少，递降degree[di'gri:]度delay[di'lei]延迟deleave拆散delegation[deli'geiʃən]授权delete[di'li:t]删除deletion[di'li:ʃən]n.删去(部分)，删除delimit[di:'limit]定界delimiter定界符delinquent[di'liŋkwənt]拖欠帐款deliver[di'livə]传递（回叫）delivery[di'livəri]传递demand[di'ma:nd]需求demangling识别解码demodulate[di:'mɔdjuleit]解调demodulation[di:mɔdju'leiʃən]解调demodulator解调器demographic[di:məgræfik]人口统计的demon[di:mən]精灵程序demonstrate[demənstreit]v.论证，证明，证实demonstration[demən'streiʃən]示范demount[di:'maunt]卸下demultiplex多路分用demultiplexer多路分用器denary[di:nəri]十进制的denormal非正常表示的denote[di'nəut]表示density[densiti]密度deny[di'nai]否认department[di'pa:tmənt]n.部门，门类，系depend[di'pend]取决于dependent[di'pendənt]从属dependents从属depress[di'pres]按下depth[depθ]深度deque双队列dequeue离队derivation[deri'veiʃən]派生descend[di'send]下降descendant[di'sendənt]子代describe[dis'kraib]描述descrieda.被看到的，被发现的description[dis'kripʃən]n.描述descriptor描述信息块deselect取消选择deserialize串行变并行deserializer串并转换器design[di'zain]设计designate[dezigneit]指定designateda.指定的，特指的designation[dezig'neiʃən]指定designlist设计表desirable[di'zaiərəbl]a.所希望的，称心的desire[di'zaiə]v.期望desired期望的desk[desk]n.书桌，控制台，面板desktop桌面desposition支配权destage离台destination[desti'neiʃən]目的地destroy[dis'trɔi]毁坏destructor[di'strʌktə]析构成员detached[di'tætʃt]分离的detail[di:teil]细节detailed[di:teild]详细的detect[di'tekt]检测detent[di:'tent]稳定装置deter[di'tə:]vt.阻止，拦住，妨碍determinant行列式值determine[di'tə:min]确定develop[di'veləp]开发developer[di'veləpə]开发者developing开发development[di'veləpmənt]n.开发，研制deviation[di:vi'eiʃən]偏差device[di'vais]设备diacritic[daiə'kritik]发音符号diacritical[daiə'kritikl]a.区分的，辩别的diaeresis分音符diagnose[daiəgnəuz]诊断diagnostic[daiəg'nɔstik]诊断的diagnostics['daiəg'nɔstiks]诊断diagonal[dai'ægənl]对角的diagonally[dai'ægənli]ad. 斜(对) diagram['daiəgræm]图表dial[daiəl]拨号dialing拨号dialog[daiəlɔg]对话dibit双位dictionary[dikʃənəri]字典differ[difə]vi.不同，不一致difference[difrəns]差different[difrənt]不同的differentiate[difə'renʃieit]v.区别，分辨differentiator微分器digest[di'dʒest,'daidʒest]整理digit[didʒit]数字digital[didʒitl]数字digitize[didʒitaiz]数字化dimension[di'menʃən]维dimensionaln....维的dimensionality[di'menʃə'næliti,dai-]维数diode[daioud]二极管diphthong[difθɔŋ]元音连字dipping浸渍法direct[di'rekt]直接的direction[di'rekʃən]指导directly[di'rektli]ad. 直接地，立即director[di'rektə]导向器directory[di'rektəri]目录dirid目录标识符dirname目录名Dirve驱动器disable[dis'eibl]禁用disabled[dis'eibld]禁用的disappear[disə'piə]vi.消失disassembler反汇编器disc[disk]磁盘discard[dis'ka:d]v.删除，废除，放弃discarded废弃的discipline[disiplin]规程disconnect[diskə'nekt]断开disconnection[diskə'nekʃn]断开discrete[di'skrit]离散的discriminant判别式discriminator[dis'krimineitə]鉴别器disjunction[dis'dʒʌŋkʃən]析取disk[disk]磁盘diskette软盘dismount[dis'maunt]卸下dispatch[dis'pætʃ]分派dispatcher[di'spætʃə]调度器dispersant[dis'pə:sənt]分散剂displacement[dis'pleismənt]位移display[dis'plei]显示dispose[dis'pəuz]配置disposition[dispə'ziʃən]配置disregard[disri'ga:d]vt.轻视,把..忽略不计dissector[di'sektə]析象器distinction[dis'tiŋkʃən]n.区别，相异，特性distinguish[dis'tiŋgwiʃ]v.区别，辨识distort[dis'tɔ:t]失真distortion[dis'tɔ:ʃən]失真distribute[dis'tribju:t]vt.分布，配线，配给distributed分布的distribution[distri'bju:ʃən]分发dithering抖动ditto[ditəu]同上diversion[dai'və:ʃən]转换divide[di'vaid]v.除dividend[dividend]被除数division[di'viʒən]部分divisor[di'vaizə]除数DLL动态连接库DOcument['dɔkjumənt]文档DOcumentation[dɔkjumen'teiʃən]文档编制DOcumenting记录DOmain[də'mein]域DOS磁盘操作系统DOt[dɔt]点DOuble['dʌbl]a.两倍的，成双的DOublet['dʌblit]二位字节DOubleword双字DOwn[daun]向下DOwnload卸载DOwnstream[daunstri:m]顺流DRaft[DRa:ft]草稿DRag[DRæg]拖曳DRawable['DRɔ:əbl]可绘制的DRift[DRift]漂移DRive[DRaiv]驱动器DRiver['DRaivə]驱动程序DRop[DRɔp]放下DRum[DRʌm]磁鼓DRyer['DRaiə]干燥器DUal['dju:əl]a.对偶的，双的DUe[dju:]到期DUmb[dʌm]哑的DUmmy['dʌmi]哑的DUmp[dʌmp]转储DUodecimal[dju:ə'desiml]十二进制的DUplex双工DUplicate['dju:plikit]重复DUplication[dju:pli'keiʃən]复制DUplicator['dju:plikeitə]复印机DUrability['djuərə'biliti]耐用性DUration[djuə'reiʃən]持续时间DUring['djuəriŋ]在期间DYnamic[dai'næmik]动态DYnamicizer动态转换器each[i:tʃ]每个echo['ekəu]回显edge[edʒ]边edit['edit]编辑editing编辑edition[i'diʃən]版本editor['editə]编辑器effect[i'fekt]n.效率，作用，效能effective[i'fektiv]a.有效的effectivity有效性efficiency[i'fiʃənsi]效率efficiently[i'fiʃəntli]ad. 有效地eigenvalue['aigənvælju:]特征值eject[i:'dʒekt]弹出edaboration[ilæbə'reiʃən]加工edapsedvi.经过edectronic[ilek'trɔnik]电子的edectronics[ilek'trɔniks]电子学edement['edimənt]元素ediminate[i'limineit]消去edlipse[i'lips]椭圆edlipsis[i'lipsis]n.省略符号，省略(法) embedded嵌入的embedding嵌入emboldening增亮embossment[im'bɔsmənt]凸起emphasis['emfəsis]强调emphasize['emfəsaiz]v.强调，着重，增强empty['empti]a.空，零，未占用emulate['emjuleit]仿真emulation[emju'leiʃən]仿真emulator['emjuleitə]仿真器emulsion[i'mʌʃən]感光乳剂enable[i'neibl]允许encapsulation封装encipher[in'saifə]加密器enclave[en'cleiv]程序集enclose[in'kləuz]vt.封闭，密封，包装enclosure[in'kləuʒə]安装箱encode[in'koud]编码encoder[in'koudə]编码器encounter[in'kauntə]遇到encrypt加密encryption加密end[end]末端endian字节存储次序endpoint端点engineering[endʒi'niəriŋ]工程enhance[in'ha:ns]增强enlargement放大enqueue排队enquiry询问enroll[in'rəul]报名ensure[in'ʃuə]保证enter['entə]输入enterprise['entəpraiz]企业entire[in'taiə]a.总体entirely[in'taiəli]ad. 完全地，彻底地entity['entiti]实体entrance['entrəns]入口entropy['entrəpi]熵entry['entri]项目enumerator[i'nju:məreitə]枚举符environ[in'vairən]vt.围绕，包围environment[in'vaiərənmənt]环境environmental[invaiərən'mntl]a.周围的，环境的epilogue['epilɔg]结尾epoch['i:pɔk]戳记equal['i:kwəl]等于equalization[i:kwəlai'zeiʃən]均衡equalizer['i:kwəlaizə]均衡器equally['i:kwəli]ad. 相等地，相同地equation[i'kweiʃən]方程式equipment[i'kwipmənt]装置equivalence[i'kwivələns]等价equivalent[i'kwivələnt]等价的equivocation[ikwivə'keiʃən]条件信息量总平均值erase[i'reiz]擦除eraser[i'reizə]擦除器ergoromics人类工程学erlang厄兰error['erə]错误escalation[eskə'leiʃən]逐步上升escape[is'keip]跳出esoteric[esəu'terik]机密的especially[is'peʃəli]ad. 特别(是)，尤其essentially[i'senʃəli]ad. 实质上，本来establish[is'tæbliʃ]建立establishment[is'tæbliʃmənt]建立ethernet以太网evaluate[i'væljueit]估计event[i'vent]事件every['evri]每个examine[ig'zæmin]v.检验，考试，审查example[ig'za:mpl]例子exceed[ik'si:d]超过exceededa.过度的，非常的except[ik'sept]prep.除...之外，除非exception[ik'sepʃən]异常excerpt['eksə:pt]摘录exchange[iks'tʃeindʒ]交换exclamation[eksklə'meiʃən]n.惊叹(号) exclude[iks'klu:d]排除exclusive[iks'klu:siv]互斥的executablea.可执行的execute['eksikju:t]执行execution[eksi'kju:ʃən]执行exerciser['eksəsaizə]试验程序exhaust[ig'zɔ:st]v.取尽，用完exist[ig'zist]存在existing现存的exit['eksit]退出expand[iks'pænd]扩充expander[iks'pændə]扩充器expanding[iks'pændiŋ]a.扩展的，扩充的expansion[iks'pænʃən]n.展开，展开式expect[iks'pekt]vt.期望，期待，盼望expenditure[iks'penditʃ]支出expenses费用experience[iks'piəriəns]vt.试验experiment[iks'perimənt]n.实验，试验(研究) experimentation[eksperimen'teiʃən]n.实验(工作，法)expertise[ekspə'ti:z]专门知识expire[iks'paiə]v.终止，期满explain[iks'plein]说明explanation[eksplə'neiʃən]说明explanatory[iks'plænətəri]a.解释(性)的expletive[ek'spli:tiv]虚词explicitlyad. 明显地，显式地exponent[iks'pounənt]指数exponential[ekspou'nenʃəl]指数的exponentiation取幂export[eks'pɔ:t,'ekspɔ:t]调出exporter[iks'pɔ:tə]出口服务器exposing曝光exposure[iks'pəuʒə]曝光度express[iks'pres]a.快速的expression[iks'preʃən]表达式expunge[eks'pʌndʒ]vt.擦除，删掉extend[iks'tend]扩展extension[iks'tenʃən]扩充extent[iks'tent]范围external[eks'tə:nl]外部的extra['ekstrə]a.特别的，额外的extract[iks'trækt,'ekstrækt]抽取extremely[iks'tri:mli]ad. 极端地，非常extremity[iks'tremiti]极限eyecatcher识别序列face[feis]n.面，表面facilities设施facility[fə'siliti]n.设施，装备，便利facsimile[fæk'simili]传真factor['fæktə]因子factorial[fæk'tɔ:riəl]阶乘factoring因式分解factorization['fæktərai'zeiʃn]因式分解fail[feil]失败failure['feiljə]失败fallback撤退false[fɔ:ls]a.假(布尔值)，错误familiar[fə'miljə]a.熟悉的，惯用的familiarize[fəmiljəraiz]vt.使熟悉，使通俗化family['fæmili]系列fancy['fænsi]n.想象(的),精制的fast[fa:st]快速fastbackn.快速返回fasten['fa:sn]固定fastselect快速选择father['fa:ðə]n.父，上层(树节点的) fault[fɔ:lt]故障feature['fi:tʃə]特征成份feed[fi:d]馈送feedback['fi:dbæk]反馈feedboard供纸板feeder['fi:də]输电线feminine['feminin]阴性ferrite['ferait]铁氧体ferromagnetics磁学fetch[fetʃ]取fiber['faibə]纤维field[fi:ld]字段figure['figə]图file[fail]文件filename文件名fileset文件集filespace文件空间filetab文件标记filingn.（文件的）整理汇集fill[fil]填充filler['filə]填充符film[film]薄膜filter['filtə]筛选程序final['fainl]最后的finally['fainəli]ad. 终于，最后find[faind]寻找finish['finiʃ]完成。
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Computing Discrete Fr´echet Distance
Thomas Eiter and Heikki Mannila Christian Doppler Labor fu¨r Expertensyteme
Technische Universit¨at Wien Paniglgasse 16, 1040 Wien E-mail: eiter@vexpert.dbai.tuwien.ac.at
∗Authors’ addresses: Thomas Eiter, Information Systems Department, Technical University of Vienna, Paniglgasse 16, A-1040 Vienna, Austria; Heikki Mannila, Department of Computer Science, University of Helsinki, P.O. Box 26 (Teollisuuskatu 23), SF-00014 University of Helsinki, Finland. email: eiter@vexpert.dbai.tuwien.ac.at, mannila@cs.helsinki.fi
Let P : [0, n] → V be a polygonal curve. We denote the sequence (P (0), P (1), . . . , P (n)) of endpoints of the line segments of P by σ(P ). Let P and Q be polygonal curves and σ(P ) = (u1, . . . , up) and σ(Q) = (v1, . . . , vq) the corresponding sequences. A coupling L between P and Q is a sequence
In this paper we describe a discrete variation of the Fr´echet distance for polygonal curves. The variation is called the coupling distance δdF . It is based on the idea of
(ua1 , vb1 ), (ua2 , vb2 ), . . . , (uam , vbm )
of distinct pairs from σ(P ) × σ(Q) such that a1 = 1, b1 = 1, am = p, bm = q, and for all i = 1, . . . , q we have ai+1 = ai or ai+1 = ai + 1, and bi+1 = bi or bi+1 = bi. Thus, a coupling has to respect the order of the points in P and Q. The length ||L|| of the coupling L is the length of the longest link in L, that is,
On the basis of these results, the following way of approximately computing the Fr´echet distance between two arbitrary curves suggests itself: First compute proper polygonal approximations to the curves and then compute their coupling distance, rather than computing their exact Fr´echet distance.
2
It is immediate that δdF (P, P ) = 0, δdF (P, Q) = δdF (Q, P ); furthermore, one can check that δdF (P, Q) = 0 implies P = Q and that δdF (P, Q) ≤ δdF (P, R) + δdF (R, Q). We thus have the following.
We also briefly address variants of the discrete Fr´echet distance, and find an interesting relation to a measure of distance between logical theories.
2 Discrete Fr´echet distance
April 25, 1994
CD-TR 94/64
Computing Discrete Fr´echet Distance∗
Thomechnical University of Vienna
Heikki Mannila
Dept. of Computer Science University of Helsinki
δF (f, g) = inf max d(f (α(t)), g(β(t))),
α,β t∈[0,1]
where α (resp. β) is an arbitrary continuous nondecreasing function from [0, 1] onto [a, b] (resp. [a , b ]).
1
looking at all possible couplings between the end points of the line segments of the polygonal curves.
We show that δdF provides good approximations to δF . Specifically, we show that δdF is an upper bound for δF , and that the difference between these measures is bounded by the length of the longest edge of the polygonal curves. We also show that δdF can be computed in O(pq) time using a very simple algorithm.
In computing the Fr´echet distance between arbitrary curves one typically approximates the curves by polygonal curves. A polygonal curve is a curve P : [0, n] → V , where n is a positive integer, such that for each i ∈ {0, 1, . . . , n − 1}, the restriction of P to the interval [i, i + 1] is affine, that is P (i + λ) = (1 − λ)P (i) + λP (i + 1).
A fundamental study on the computational properties of the Fr´echet distance was done by Alt and Godau [1]. They give an algorithm that computes the exact Fr´echet distance between two polygonal curves in time O(pq log2 pq), where p and q are the number of segments on the polygonal curves. The algorithm is fairly involved, as it uses the parametric search technique.
keywords: Fr´echet distance, metrics, curves, polygonal curves
1 Introduction
Given two curves in a metric space, the Fr´echet distance δF between them can be defined intuitively as follows. A man is walking a dog on a leash: the man can move on one curve, the dog on the other; both may vary their speed, but backtracking is not allowed. What is the length of the shortest leash that is sufficient for traversing both curves? The Fr´echet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves. Therefore it is often better than the well-known Hausdorff distance. This distance function was introduced by Fr´echet in 1906 [6].
Thomas Eiter and Heikki Mannila Christian Doppler Labor fu¨r Expertensyteme
Technische Universit¨at Wien Paniglgasse 16, 1040 Wien E-mail: eiter@vexpert.dbai.tuwien.ac.at
∗Authors’ addresses: Thomas Eiter, Information Systems Department, Technical University of Vienna, Paniglgasse 16, A-1040 Vienna, Austria; Heikki Mannila, Department of Computer Science, University of Helsinki, P.O. Box 26 (Teollisuuskatu 23), SF-00014 University of Helsinki, Finland. email: eiter@vexpert.dbai.tuwien.ac.at, mannila@cs.helsinki.fi
Let P : [0, n] → V be a polygonal curve. We denote the sequence (P (0), P (1), . . . , P (n)) of endpoints of the line segments of P by σ(P ). Let P and Q be polygonal curves and σ(P ) = (u1, . . . , up) and σ(Q) = (v1, . . . , vq) the corresponding sequences. A coupling L between P and Q is a sequence
In this paper we describe a discrete variation of the Fr´echet distance for polygonal curves. The variation is called the coupling distance δdF . It is based on the idea of
(ua1 , vb1 ), (ua2 , vb2 ), . . . , (uam , vbm )
of distinct pairs from σ(P ) × σ(Q) such that a1 = 1, b1 = 1, am = p, bm = q, and for all i = 1, . . . , q we have ai+1 = ai or ai+1 = ai + 1, and bi+1 = bi or bi+1 = bi. Thus, a coupling has to respect the order of the points in P and Q. The length ||L|| of the coupling L is the length of the longest link in L, that is,
On the basis of these results, the following way of approximately computing the Fr´echet distance between two arbitrary curves suggests itself: First compute proper polygonal approximations to the curves and then compute their coupling distance, rather than computing their exact Fr´echet distance.
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It is immediate that δdF (P, P ) = 0, δdF (P, Q) = δdF (Q, P ); furthermore, one can check that δdF (P, Q) = 0 implies P = Q and that δdF (P, Q) ≤ δdF (P, R) + δdF (R, Q). We thus have the following.
We also briefly address variants of the discrete Fr´echet distance, and find an interesting relation to a measure of distance between logical theories.
2 Discrete Fr´echet distance
April 25, 1994
CD-TR 94/64
Computing Discrete Fr´echet Distance∗
Thomechnical University of Vienna
Heikki Mannila
Dept. of Computer Science University of Helsinki
δF (f, g) = inf max d(f (α(t)), g(β(t))),
α,β t∈[0,1]
where α (resp. β) is an arbitrary continuous nondecreasing function from [0, 1] onto [a, b] (resp. [a , b ]).
1
looking at all possible couplings between the end points of the line segments of the polygonal curves.
We show that δdF provides good approximations to δF . Specifically, we show that δdF is an upper bound for δF , and that the difference between these measures is bounded by the length of the longest edge of the polygonal curves. We also show that δdF can be computed in O(pq) time using a very simple algorithm.
In computing the Fr´echet distance between arbitrary curves one typically approximates the curves by polygonal curves. A polygonal curve is a curve P : [0, n] → V , where n is a positive integer, such that for each i ∈ {0, 1, . . . , n − 1}, the restriction of P to the interval [i, i + 1] is affine, that is P (i + λ) = (1 − λ)P (i) + λP (i + 1).
A fundamental study on the computational properties of the Fr´echet distance was done by Alt and Godau [1]. They give an algorithm that computes the exact Fr´echet distance between two polygonal curves in time O(pq log2 pq), where p and q are the number of segments on the polygonal curves. The algorithm is fairly involved, as it uses the parametric search technique.
keywords: Fr´echet distance, metrics, curves, polygonal curves
1 Introduction
Given two curves in a metric space, the Fr´echet distance δF between them can be defined intuitively as follows. A man is walking a dog on a leash: the man can move on one curve, the dog on the other; both may vary their speed, but backtracking is not allowed. What is the length of the shortest leash that is sufficient for traversing both curves? The Fr´echet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves. Therefore it is often better than the well-known Hausdorff distance. This distance function was introduced by Fr´echet in 1906 [6].