Two-dimensional closest pair problem: A closer look

Developing algorithms that produce approximate solutions is always interesting when we are generating the final solution for a problem. Progressive algorithms report a partial solution to the user which approximates the final solution in some specific steps. Thus, the user can stop the algorithm if the error of the partial solution is tolerable in terms of the application. In this paper, we study the closest pair problem under the Euclidean metric. A progressive algorithm is designed for the closest pair problem, which consists of steps and spends time in each step. In step r, the error of the partial solution is bounded by , where α is the ratio of the maximum pairwise distance and the minimum pairwise distance of points.

We present a simple greedy algorithm, QuickCP, for finding the closest-pair of points on a plane. It is based on the observation that if two points are close to each other, then it is very likely that their sorted projections to x-axis and/or to y-axis will reflect that closeness. Therefore we order our search starting from the pairs with closest x-projections (and closest y-projections) to find the closest pair quickly and make a fast exit. Empirical data up to \(2^{26}\) (over 60 million points) indicates that this approach quickly detects the closest pair and it is, on average 70 % faster than the classical divide-and-conquer algorithm registering, to the best of our knowledge, the fastest recorded solution times in the literature for the planar closest-pair problem. A second contribution of our work is that QuickCP can be used as part of the divide-and-conquer algorithms to handle small sub-problems. Measurements on data up to \(2^{26}\) points show that this co-operation speeds up the basic divide-and-conquer algorithm on average 50 %.

An important query for spatial database research is to find the closest pair of objects in a given space. Existing work assumes two objects of the closest pair come from two different data sets indexed by R-trees. The closest pair in the whole space will be found via an optimzed R-tree join technique. However, this technique doesn’t perform well when the two data sets are identical. And it doesn’t work when the search range is some area other than the whole space. In this paper, we address the closest pair problem within the same data set. Further more, we propose a practical extension to the closest pair problem to involve a query range. The problem now becomes finding the closest pair of objects among those inside a given range. After extending the existing techniques to solve the new problem, we proposed two index structures based on augmenting the R-tree and we also give algorithms for maintaining these structrures. Experimental results show that our structures are more robust than earlier approaches.

We improve the famous divide-and-conquer algorithm by Bentley and Shamos for the planar closest-pair problem. For n points on the plane, our algorithm keeps the optimal O(n log n) time complexity a...

Point location problems for n points in d-dimensional Euclidean space (and ep spaces more generally) have typically had two kinds of running-time solutions: (Nearly-Linear) less than dpoly(d) · n logO(d) n time, or (Barely-Subquadratic) f(d) · n2−1/Θ(d) time, for various f. For small d and large n, running times are generally feasible, while the times are generally infeasible, requiring essentially quadratic time. For example, in the Euclidean metric, finding a Closest Pair among n points in Rd is nearly-linear, solvable in 2o(d) · n logO(1) n time, while the known algorithms for finding a Furthest Pair (the diameter of the point set) are only barely-subquadratic, requiring Ω(n2--1/Θ(d)) time. Why do these proximity problems have such different time complexities? Is there a barrier to obtaining nearly-linear algorithms for problems which are currently only barely-subquadratic? We give a novel exact and deterministic self-reduction for the Orthogonal Vectors problem on n vectors in {0, 1}d to n vectors in Zω(log d) that runs in 2o(d) time. As a consequence, barely-subquadratic problems such as Euclidean diameter, Euclidean bichromatic closest pair, and incidence detection do not have O(n2−ϵ) time algorithms (in Turing models of computation) for dimensionality d = ω(log log n)2, unless the popular Orthogonal Vectors Conjecture and the Strong Exponential Time Hypothesis are false. That is, while the poly-log-log-dimensional case of Closest Pair is solvable in n1+o(1) time, the poly-log-log-dimensional case of Furthest Pair can encode difficult large-dimensional problems conjectured to require n2−o(1) time. We also show that the All-Nearest Neighbors problem in ω(log n) dimensions requires n2−o(1) time to solve, assuming either of the above conjectures.

Given a set of n points in some d-dimensional Euclidean space, each point colored with one of k( ≥ 2) colors, a bichromatic closest (resp., farthest) pair is a closest (resp., farthest) pair of points of different colors. We present efficient algorithms to compute a bichromatic closest pair and a bichromatic farthest pair. We consider both static, and dynamic versions with respect to color flips. We also give some combinatorial bounds on the multiplicities of extreme distances in this setting.KeywordsSpan TreeComplete Bipartite GraphGeometric GraphPattern Recognition LetterConvex QuadrilateralThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

K-nearest neighbors (KNN) has been successfully used for recommendation, but querying neighbors of high quality is nearly impossible when the feature space is small or has limited training data. However, due to privacy requirements and government policies, directly transferring data from one data owner to another is not workable. Therefore, we propose a novel KNN approach, secured federated KNN (SF-KNN), that takes privacy requirements into consideration and builds a federated model to gain global neighbors with joint parties, in order to improve the model performance. Specifically, it empowers the parties to train high-quality models with little data. More importantly, it makes cross-domain training possible. We implement SF-KNN on Euclidean and cosine metrics using user-based and item-based methods. In our experiment, we evaluate the proposed SF-KNN on three data sources, MovieLens, Netflix, and Amazon, and several diverse domains, movies, books, clothes, jewellery and food, by comparing it against various baselines. The experiment results indicate that SF-KNN is able to learn more precise neighbors than a local KNN trained by parties individually. In general, it outperforms the local KNN on all of the datasets, reaching 15% average accuracy gain on the Euclidean metric and 8% of it on the cosine metric when simulating 10 parties across all data sources.

We consider 1-local online routing on a special class of geometric graphs called Delaunay triangulations (DTs). A geometric graph G = (V, E) of a point set consists of a set of points in the plane and edges between them, where each edge weighs as the Euclidean distance between it’s end-points. DTs are one of the useful classes of these graphs because of some good properties which can help during the navigation process, therefore over the years DTs have been widely proposed as network topologies for several times.In this paper, we present a new memoryless online routing (MOR) algorithm for DTs which is simple, elegant, and easy to implement, while having an acceptable performance.The set of MOR algorithms are suitable for cases where we want to find a path using only local information, our proposed algorithm is memoryless or 1-local, in k-local routing, we find a path between a source vertex s to a destination vertex t while our knowledge at each step is limited to the locations of s and t, the location of current vertex and it’s k-neighborhood vertices.We also evaluate and compare the perforamnce of our prpopsed algorithm with existing MOR algorithms. Our experimental results implied that our proposed algorithm has an acceptable performance in both Euclidean and link metrics and it outperforms all of the existing MOR algorithms in Euclidean metric, and some of them in the link metric as well. Finally, we pose two open problems to solve in the future.

A finite metric space, (S,d) , contains a finite set of points and a distance function on pairs of points. A contraction is an embedding, h, of a finite metric space (S, d) into Rd where for any u, v E S, the Euclidean (&) distance between h(u) and h(v) is no more than d(u, v). The distortion of the embedding is the maximum over pairs of the ratio of d(u, w) and the Euclidean distance between h(u) and h(v). Bourgain showed that any graphical metric could be embedded with distortion O(logn). Linial, London and Rabinovich and Aumman and Rabani used such embeddings to prove an O(log k) approximate max-flow min-cut theorem for k commodity flow problems. A generalization of embeddings that preserve distances between pairs-of points are embeddings that preserve volumes of larger sets. In particular, A (k, c)-volume respecting embedding of n-points in any metric space is a contraction where every subset of k points has within an ck-’ factor of its maximal possible k l-dimensional volume. Feige invented these embeddings in devising a polylogarithmic approximation algorithm for the bandwidth problem using these embeddings. Feige’s methods have subsequently been used by Vempala for approximating versions of the VLSI layout problem. Feise showed that a (k, O(10,g~‘~ n,/m)) volume r&ecting embedding‘ eksted.” Be -recently found improved (k, 0( mdk log k + log n)) volume respecting embeddings. For metrics arising from planar graphs (planar metrics), we give (k,O(m)) volume respecting contractions. As a corollary, we give embeddings for Permission to makkr digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. XC’99 Miami Beach Florida Copyright ACM 1999 I-581 13-068-6/99/06...$5.00 planar metrics with distortion O(e). This gives rise to an O(e)-approximate max-flow min-cut theorem for multicommodity flow problems in planar graphs. We also give an improved bound for volume respecting embeddings for Euclidean metrics. In particular, we give an (k,O(dog klog D)) volume respecting embedding where D is the ratio of the largest distance to the smallest distance in the metric. Our results give improvements for Feige’s and Vempala’s approximation algorithms for planar and Euclidean metrics. For volume respecting embeddings, our embeddings do not degrade very fast when preserving the volumes of large subsets. This may be useful in the future for approximation algorithms or if volume .respecting embeddings prove to be of independent interest.

ABSTRACTPrevious studies demonstrate the effectiveness of the Euclidean distance shape descriptor for characterizing individual tree crowns derived from Light Detection and Ranging (LiDAR) data. This paper evaluates the effectiveness of the longitude–latitude shape descriptor for tree crown shape characterization in comparison with the Euclidean distance descriptor using LiDAR data for 43 individual tree crowns. The three-dimensional (3D) shape signatures obtained from longitude–latitude and Euclidean distance measures of the tree crowns are compared with 3D shape signatures of simulated ellipsoid and hemisphere tree crowns. Kolmogorov–Smirnov test (K–S test) and Mean Square Error (MSE) are used to measure the similarity of longitude–latitude and Euclidean distance signatures. The experimental results from all 43 examined trees suggest that the longitude–latitude shape descriptor can determine tree crown shapes with different leaf traits, while Euclidean distance is only able to characterize the shape of evergreen trees. Furthermore, the longitude–latitude shape descriptor, unlike the Euclidean distance shape descriptor, is not sensitive to the number of LiDAR points from the tree crown.

The performance of the cellular computer proposed by Mego has been investigated by programming a number of associative search algorithms and analyzing the time and space required for their execution. The present work describes the results of three different analysis techniques applied to algorithms for nearest neighbor and closest pair problems for files of n points in k-dimensional space.Brute force methods are described for solving the nearest neighbor problem. If the initial program expression is suitably placed in memory, analysis of the most parallel brute force algorithm yields complexity results of O(k) time and O(kn) space. These bounds are shown to be asymptotically optimal with respect to the problem and the machine.Brute force, semi-parallel, and divide-and-conquer solutions are described for solving the closest pair problem. Analyses of the best semi-parallel algorithms yield complexity results of O(kn) time and space, which are shown to be asymptotically optimal.

Given n vectors with dimension m in Boolean domain, how to find two vectors whose pairwise Hamming distance is minimum? This problem is known as the Closest Pair Problem. If these vectors are generated uniformly at random except two of them are correlated with Pearson-correlation coefficient, then the problem is called the Light Bulb Problem. In this work, we propose a novel coding-based scheme for the Closest Pair Problem. We design both randomized and deterministic algorithms, which achieve the best-known running time when the length of input vectors m is small and the minimum distance is very small compared to m. When applied to the Light Bulb Problem, our result yields state-of-the-art deterministic running time when the Pearson-correlation coefficient is very large. Specifically, when it is greater than 0.9933, our deterministic algorithm runs faster than the previously best deterministic algorithm (Alman, SOSA 2019).

In any subspace of the real line Rwith the usual Euclidean metric d(x, y) = Ix-Yl, every triangle is degenerate. In R2 or R3 with the usual Euclidean metrics, a triangle is degenerate if and only if its are collinear. With our intuition of a degenerate triangle having collinear vertices extended to arbitrary metric spaces, we might expect that a metric space in which every triangle is degenerate must be linear. It might be reasonable to expect that any linear metric space is isometric to a subset of R with the usual metric. When classifying all metric spaces that have only degenerate triangles, we find that there are such metric spaces other than (isometric images of) subspaces of R. These other spaces, however, all have precisely four points and all are of the same form. In the final section, we illustrate that the usual topology on R2 can not be generated by any metric in which all triangles are degenerate. metric space (M, p) is a set of points M with a metric, or distance function, p: M x M o [O, oo) that satisfies some natural properties we expect of distances: p(x,y) = p(y,x) foranyx,y cM, p(x,y) = O if andonlyif x =y, and p(x, y) + Ply, z) 2 p(x, z) for any x, y, z c M (triangle inequality). If (M, p) and (N, b) are two metric spaces, a function f: M o N such that p(x,y) = b(f(x), f(y)) for any x,y c M is called an isometry. Isometries are always one-to-one. The metric spaces (M, p) and (N, b) are isometric if there exists an isometry from M onto N. Generally, one does not distinguish between isometric metric spaces. If (M, p) is a given metric space, we say that {x1, x2, X3} c M forms a degenerate triangle if p(xi, x;) + p(xj, Xk) = p(xi, Xk) for some permutation i, j, k of the indices {1, 2, 3}. Thus, {x1, x2, X3} forms a degenerate triangle if the triangle inequality is an equality for some permutation of the points x1, x2, and X3. We want to study metric spaces in which every triangle is degenerate, that is, metric spaces such that every 3-element subset forms a degenerate triangle. For brevity, such a metric space is called a degenerate space, and its metric is called a degenerate metric. Triangles with fewer than three distinct are clearly degenerate, and in what follows, we assume that all triangles have three distinct vertices, unless otherwise noted. With this understanding, we can make statements such as A four-point space has exactly four distinct triangles. The real line with the usual Euclidean metric d(x, y) = Ix-yl is denoted by R. It is a familiar fact that R is a degenerate space. Any subspace of a degenerate space is a degenerate space. Given any degenerate space M, one might suspect that it must be isomorphic to a subspace of R, and might attempt to construct an isometry from M into R. As we will see, the construction of such an isometry requires that if {y, o, p} and {o, p, x} are (degenerate) triangles in M with longest sides {y, p} and {o, x} respectively, then triangle {y, o, x} must have longest side {y, x}. Surprisingly, this

We consider the application of optical computational operations to problems that arise in various geometric applications, such as computer graphics, image processing, pattern matching. etc. We show how to implement, using only a constant number of basic optical operations, a variety of algorithms that solve fairly complex tasks of this sort. We discuss the methodology that we propose, list the geometric algorithms that we have developed to date, and illustrate the approach in two more detailed applications, motivated by computer graphics and pattern matching: drawing all possible lines connecting pairs in a given set of points, and computing the closest pair (orkclosest pairs) of points in a given set.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)Faster Deterministic and Las Vegas Algorithms for Offline Approximate Nearest Neighbors in High DimensionsJosh Alman, Timothy M. Chan, and Ryan WilliamsJosh Alman, Timothy M. Chan, and Ryan Williamspp.637 - 649Chapter DOI:https://doi.org/10.1137/1.9781611975994.39PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We present a deterministic, truly subquadratic algorithm for offline (1 + ε)-approximate nearest or farthest neighbor search (in particular, the closest pair or diameter problem) in Hamming space in any dimension d ≤ nδ, for a sufficiently small constant δ > 0. The running time of the algorithm is roughly for nearest neighbors, or for farthest. The algorithm follows from a simple combination of expander walks, Chebyshev polynomials, and rectangular matrix multiplication. We also show how to eliminate errors in the previous Monte Carlo randomized algorithm of Alman, Chan, and Williams [FOCS’16] for offline approximate nearest or farthest neighbors, and obtain a Las Vegas randomized algorithm with expected running time . Finally, we note a simplification of Alman, Chan, and Williams' method and obtain a slightly improved Monte Carlo randomized algorithm with running time . As one application, we obtain improved deterministic and randomized (1 + ε)-approximation algorithms for MAX-SAT. Previous chapter Next chapter RelatedDetails Published:2020eISBN:978-1-61197-599-4 https://doi.org/10.1137/1.9781611975994Book Series Name:ProceedingsBook Code:PRDA20Book Pages:xxii + 3011