Let @math and @math be sets of @math red and @math blue points in the plane, respectively, with @math . Let @math be a perfect matching between points from @math and @math , using @math straight line segments to match the points, that is, each point is an endpoint of exactly one line segment, and each line segment has one red and one blue endpoint. We forbid line segments to cross. Denote the length of a longest line segment in @math with @math , which we also call the [ ] value, bn of @math . We aim to find a matching under given constraints that minimizes @math . Any such matching is called a [ ] bottleneck matching of @math .
A variant of the bichromatic case is the so-called bicolored (or multicolored, when there are arbitrary many colors) case, where only the points of the same color are allowed to be matched. Abu-Affash, Bhore and Carmi in @cite_12 examined bicolored matchings that minimize the number of crossings between edges matching different color sets. They presented an algorithm to compute a bottleneck matching of points in convex position among all matchings that have no crossings of this kind.
@cite_12
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Let (X ) be a set of multicolored points in the plane such that no three points are collinear and each color appears on at most ( |X| 2 ) points. We show the existence of a non-crossing properly colored geometric perfect matching on (X ) (if (|X| ) is even), and the existence of a non-crossing properly colored geometric spanning tree with maximum degree at most (3 ) on (X ). Moreover, we show the existence of a non-crossing properly colored geometric perfect matching in the plane lattice. In order to prove these our results, we propose an useful lemma that gives a good partition of a sequence of multicolored points. This paper presents a framework of layered graph matching for integrating graph partition and matching. The objective is to find an unknown number of corresponding graph structures in two images. We extract discriminative local primitives from both images and construct a candidacy graph whose vertices are matching candidates (i.e., a pair of primitives) and whose edges are either negative for mutual exclusion or positive for mutual consistence. Then we pose layered graph matching as a multicoloring problem on the candidacy graph and solve it using a composite cluster sampling algorithm. This algorithm assigns some vertices into a number of colors, each being a matched layer, and turns off all the remaining candidates. The algorithm iterates two steps: 1) Sampling the positive and negative edges probabilistically to form a composite cluster, which consists of a few mutually conflicting connected components (CCPs) in different colors and 2) assigning new colors to these CCPs with consistence and exclusion relations maintained, and the assignments are accepted by the Markov Chain Monte Carlo (MCMC) mechanism to preserve detailed balance. This framework demonstrates state-of-the-art performance on several applications, such as multi-object matching with large motion, shape matching and retrieval, and object localization in cluttered background. Motivated by a crane assignment problem, we consider a Euclidean bipartite matching problem with edge-crossing constraints. Specifically, given n red points and n blue points in the plane, we want to construct a perfect matching between red and blue points that minimizes the length of the longest edge, while imposing a constraint that no two edges may cross each other. We show that the problem cannot be approximately solved within a factor less than 1:277 in polynomial time unless P = NP. We give simple dynamic programming algorithms that solve our problem in two special cases, namely (1) the case where the red and blue points form the vertices of a convex polygon and (2) the case where the red points are collinear and the blue points lie to one side of the line through the red points. A majority of the existing stereo matching algorithms assume that the corresponding color values are similar to each other. However, it is not so in practice as image color values are often affected by various radiometric factors such as illumination direction, illuminant color, and imaging device changes. For this reason, the raw color recorded by a camera should not be relied on completely, and the assumption of color consistency does not hold good between stereo images in real scenes. Therefore, the performance of most conventional stereo matching algorithms can be severely degraded under the radiometric variations. In this paper, we present a new stereo matching measure that is insensitive to radiometric variations between left and right images. Unlike most stereo matching measures, we use the color formation model explicitly in our framework and propose a new measure, called the Adaptive Normalized Cross-Correlation (ANCC), for a robust and accurate correspondence measure. The advantage of our method is that it is robust to lighting geometry, illuminant color, and camera parameter changes between left and right images, and does not suffer from the fattening effect unlike conventional Normalized Cross-Correlation (NCC). Experimental results show that our method outperforms other state-of-the-art stereo methods under severely different radiometric conditions between stereo images. Given a set of pointsV in the plane, the Euclidean bottleneck matching problem is to match each point with some other point such that the longest Euclidean distance between matched points, resulting from this matching, is minimized. To solve this problem, we definek-relative neighborhood graphs, (kRNG) which are derived from Toussaint's relative neighborhood graphs (RNG). Two points are calledk-relative neighbors if and only if there are less thank points ofV which are closer to both of the two points than the two points are to each other. AkRNG is an undirected graph (V,E r k) whereE r k is the set of pairs of points ofV which arek-relative neighbors. We prove that there exists an optimal solution of the Euclidean bottleneck matching problem which is a subset ofE r 17. We also prove that ¦E r k¦ < 18kn wheren is the number of points in setV. Our algorithm would construct a 17RNG first. This takesO(n2) time. We then use Gabow and Tarjan's bottleneck maximum cardinality matching algorithm for general graphs whose time-complexity isO((n logn)0.5m), wherem is the number of edges in the graph, to solve the bottleneck maximum cardinality matching problem in the 17RNG. This takesO(n1.5 log0.5n) time. The total time-complexity of our algorithm for the Euclidean bottleneck matching problem isO(n2 +n1.5 log0.5n). We consider computing a maximum non-crossing matching in convex bipartite graphs. For a convex bipartite graph of n vertices and m edges, we present an O ( n log n ) time algorithm for finding a maximum non-crossing matching in the graph. The previous best algorithm takes O ( m + n log n ) time (Malucelli et?al., 1993). Since m = ? ( n 2 ) in the worst case, our result improves the previous work. This paper presents a novel approach for matching 2D points between a video projector and a digital camera. Our method is motivated by camera-projector applications for which the projected image needs to be warped to prevent geometric distortion. Since the warping process often needs geometric information on the 3D scene that can only be obtained from triangulation, we propose a technique for matching points in the projector to points in the camera based on arbitrary video sequences. The novelty of our method lies in the fact that it does not require the use of pre-designed structured light patterns as is usually the case. The back bone of our application lies in a function that matches activity patterns instead of colors. This makes our method robust to pose, to severe photometric and geometric distortions. It also does not require calibration of the color response curve of the camera-projector system. We present quantitative and qualitative results with synthetic and real life examples, and compare the proposed method with the scale invariant feature transform (SIFT) method and with a state-of-the-art structured light technique. We show that our method performs almost as well as structured light methods and significantly outperforms SIFT when the contrast of the video captured by the camera has been degraded. Let A and B be two sets of n objects in d , and let Match be a (one-to-one) matching between A and B . Let min(Match ), max(Match ), and Σ(Match) denote the length of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of Match , respectively. Bottleneck matching— a matching that minimizes max(Match )— is suggested as a convenient way for measuring the resemblance between A and B . Several algorithms for computing, as well as approximating, this resemblance are proposed. The running time of all the algorithms involving planar objects is roughly O(n 1.5 ) . For instance, if the objects are points in the plane, the running time of the exact algorithm is O(n 1.5 log n ) . A semidynamic data structure for answering containment problems for a set of congruent disks in the plane is developed. This data structure may be of independent interest. Given a set of n men represented by n points lying on a line, and n women represented by n points lying on another parallel line, with each person having a list that ranks some people of opposite gender as his her acceptable partners in strict order of preference. In this problem, we want to match people of opposite genders to satisfy people’s preferences as well as making the edges not crossing one another geometrically. A noncrossing blocking pair w.r.t. a matching M is a pair (m, w) of a man and a woman such that they are not matched with each other but prefer each other to their own partners in M, and the segment (m, w) does not cross any edge in M. A weakly stable noncrossing matching (WSNM) is a noncrossing matching that does not admit any noncrossing blocking pair. In this paper, we prove the existence of a WSNM in any instance by developing an (O(n^2) ) algorithm to find one in a given instance. A perfect straight-line matching @math on a finite set @math of points in the plane is a set of segments such that each point in @math is an endpoint of exactly one segment. @math is non-crossing if no two segments in @math cross each other. Given a perfect straight-line matching @math with at least one crossing, we can remove this crossing by a flip operation. The flip operation removes two crossing segments on a point set @math and adds two non-crossing segments to attain a new perfect matching @math . It is well known that after a finite number of flips, a non-crossing matching is attained and no further flip is possible. However, prior to this work, no non-trivial upper bound on the number of flips was known. If @math (resp. @math ) is the maximum length of the longest (resp. shortest) sequence of flips starting from any matching of size @math , we show that @math and @math (resp. @math and @math ). We present a deterministic distributed algorithm that computes a (2&#x03B4;-1)-edge-coloring, or even list-edge-coloring, in any n-node graph with maximum degree &#x03B4;, in O(log^8 &#x03B4; &#x22C5; log n) rounds. This answers one of the long-standing open questions of distributed graph algorithms from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were 2^ O(&#x221A; log n ) by Panconesi and Srinivasan [STOC92] and &#x00D5;(&#x221A; &#x03B4; ) + O(log^* n) by Fraigniaud, Heinrich, and Kosowski [FOCS16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of (2&#x03B4;-1)-edge-coloring to poly(loglog n) rounds.The key technical ingredient is a deterministic distributed algorithm for hypergraph maximal matching, which we believe will be of interest beyond this result. In any hypergraph of rank r &#x2014; where each hyperedge has at most r vertices &#x2014; with n nodes and maximum degree &#x03B4;, this algorithm computes a maximal matching in O(r^5 log^ 6+log r &#x03B4; &#x22C5; log n) rounds.This hypergraph matching algorithm and its extensions also lead to a number of other results. In particular, we obtain a polylogarithmic-time deterministic distributed maximal independent set (MIS) algorithm for graphs with bounded neighborhood independence, hence answering Open Problem 5 of Barenboim and Elkins book, a ((log &#x03B4; &#x03B5;)^ O(log 1 &#x03B5;) )-round deterministic algorithm for (1+&#x03B5;)-approximation of maximum matching, and a quasi-polylogarithmic-time deterministic distributed algorithm for orienting &#x03BB;-arboricity graphs with out-degree at most (1+&#x03B5;)&#x03BB; , for any constant &#x03B5; 0, hence partially answering Open Problem 10 of Barenboim and Elkins book. We introduce a new distance between two distributions that we call the Earth Mover's Distance (EMD), which reflects the minimal amount of work that must be performed to transform one distribution into the other by moving "distribution mass" around. This is a special case of the transportation problem from linear optimization, for which efficient algorithms are available. The EMD also allows for partial matching. When used to compare distributions that have the same overall mass, the EMD is a true metric, and has easy-to-compute lower bounds. In this paper we focus on applications to image databases, especially color and texture. We use the EMD to exhibit the structure of color-distribution and texture spaces by means of Multi-Dimensional Scaling displays. We also propose a novel approach to the problem of navigating through a collection of color images, which leads to a new paradigm for image database search. We introduce a general technique for "colorizing" greyscale images by transferring color between a source, color image and a destination, greyscale image. Although the general problem of adding chromatic values to a greyscale image has no exact, objective solution, the current approach attempts to provide a method to help minimize the amount of human labor required for this task. Rather than choosing RGB colors from a palette to color individual components, we transfer the entire color "mood" of the source to the target image by matching luminance and texture information between the images. We choose to transfer only chromatic information and retain the original luminance values of the target image. Further, the procedure is enhanced by allowing the user to match areas of the two images with rectangular swatches. We show that this simple technique can be successfully applied to a variety of images and video, provided that texture and luminance are sufficiently distinct. The images generated demonstrate the potential and utility of our technique in a diverse set of application domains. Graph coloring is one of the most famous computational problems with applications in a wide range of areas such as planning and scheduling, resource allocation, and pattern matching. So far coloring problems are mostly studied on static graphs, which often stand in stark contrast to practice where data is inherently dynamic and subject to discrete changes over time. A temporal graph is a graph whose edges are assigned a set of integer time labels, indicating at which discrete time steps the edge is active. In this paper we present a natural temporal extension of the classical graph coloring problem. Given a temporal graph and a natural number ∆, we ask for a coloring sequence for each vertex such that (i) in every sliding time window of ∆ consecutive time steps, in which an edge is active, this edge is properly colored (i.e. its endpoints are assigned two different colors) at least once during that time window, and (ii) the total number of different colors is minimized. This sliding window temporal coloring problem abstractly captures many realistic graph coloring scenarios in which the underlying network changes over time, such as dynamically assigning communication channels to moving agents. We present a thorough investigation of the computational complexity of this temporal coloring problem. More specifically, we prove strong computational hardness results, complemented by efficient exact and approximation algorithms. Some of our algorithms are linear-time fixed-parameter tractable with respect to appropriate parameters, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH). We propose a new algorithm for color transfer between images that have perceptually similar semantic structures. We aim to achieve a more accurate color transfer that leverages semantically meaningful dense correspondence between images. To accomplish this, our algorithm uses neural representations for matching. Additionally, the color transfer should be spatially variant and globally coherent. Therefore, our algorithm optimizes a local linear model for color transfer satisfying both local and global constraints. Our proposed approach jointly optimizes matching and color transfer, adopting a coarse-to-fine strategy. The proposed method can be successfully extended from one-to-one to one-to-many color transfer. The latter further addresses the problem of mismatching elements of the input image. We validate our proposed method by testing it on a large variety of image content. We propose a new algorithm for color transfer between images that have perceptually similar semantic structures. We aim to achieve a more accurate color transfer that leverages semantically-meaningful dense correspondence between images. To accomplish this, our algorithm uses neural representations for matching. Additionally, the color transfer should be spatially-variant and globally coherent. Therefore, our algorithm optimizes a local linear model for color transfer satisfying both local and global constraints. Our proposed approach jointly optimize matching and color transfer, adopting a coarse-to-fine strategy. The proposed method can be successfully extended from "one-to-one" to "one-to-many" color transfers. The latter further addresses the problem of mismatching elements of the input image. We validate our proposed method by testing it on a large variety of image content. Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call minimum conflict-free coloring (min-CF-coloring). In its general form, the input of the min-CF-coloring problem is a set system @math , where each @math is a subset of X. The output is a coloring @math of the sets in @math that satisfies the following constraint: for every @math there exists a color @math and a unique set @math such that @math and @math . The goal is to minimize the number of colors used by the coloring @math . Min-CF-coloring of general set systems is not easier than the classic graph coloring problem. However, in view of our motivation, we consider set systems induced by simple geometric regions in the plane. In particular, we study disks (both congruent and noncongruent), axis-parallel rectangles (with a constant ratio between the smallest and largest rectangle), regular hexagons (with a constant ratio between the smallest and largest hexagon), and general congruent centrally symmetric convex regions in the plane. In all cases we have coloring algorithms that use O(log n) colors (where n is the number of regions). Tightness is demonstrated by showing that even in the case of unit disks, @math colors may be necessary. For rectangles and hexagons we also obtain a constant-ratio approximation algorithm when the ratio between the largest and smallest rectangle (hexagon) is a constant. We also consider a dual problem of CF-coloring points with respect to sets. Given a set system @math , the goal in the dual problem is to color the elements in X with a minimum number of colors so that every set @math contains a point whose color appears only once in S. We show that O(log |X|) colors suffice for set systems in which X is a set of points in the plane and the sets are intersections of X with scaled translations of a convex region. This result is used in proving that O(log n) colors suffice in the primal version. Kernelized sorting is a method for aligning objects across two domains by considering within-domain similarity, without a need to specify a cross-domain similarity measure. In this paper we present the Convex Kernelized Sorting method where, unlike in the previous approaches, the cross-domain object matching is formulated as a convex optimization problem, leading to simpler optimization and global optimum solution. Our method outputs soft alignments between objects, which can be used to rank the best matches for each object, or to visualize the object matching and verify the correct choice of the kernel. It also allows for computing hard one-to-one alignments by solving the resulting Linear Assignment Problem. Experiments on a number of cross-domain matching tasks show the strength of the proposed method, which consistently achieves higher accuracy than the existing methods. Convex empirical risk minimization is a basic tool in machine learning and statistics. We provide new algorithms and matching lower bounds for differentially private convex empirical risk minimization assuming only that each data point's contribution to the loss function is Lipschitz and that the domain of optimization is bounded. We provide a separate set of algorithms and matching lower bounds for the setting in which the loss functions are known to also be strongly convex.Our algorithms run in polynomial time, and in some cases even match the optimal non-private running time (as measured by oracle complexity). We give separate algorithms (and lower bounds) for (aepsi;, 0)- and (aepsi;,a#x03B4;)-differential privacy, perhaps surprisingly, the techniques used for designing optimal algorithms in the two cases are completely different. Our lower bounds apply even to very simple, smooth function families, such as linear and quadratic functions. This implies that algorithms from previous work can be used to obtain optimal error rates, under the additional assumption that the contributions of each data point to the loss function is smooth. We show that simple approaches to smoothing arbitrary loss functions (in order to apply previous techniques) do not yield optimal error rates. In particular, optimal algorithms were not previously known for problems such as training support vector machines and the high-dimensional median. We consider an online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflict-free, in the sense that in every interval I there is a color that appears exactly once in I. We present several deterministic and randomized algorithms for achieving this goal, and analyze their performance, that is, the maximum number of colors that they need to use, as a function of the number n of inserted points. We first show that a natural and simple (deterministic) approach may perform rather poorly, requiring Ω(√n) colors in the worst case. We then modify this approach, to obtain an efficient deterministic algorithm that uses a maximum of Θ(log2 n) colors. Next, we present two randomized solutions. The first algorithm requires an expected number of at most O(log2 n) colors, and produces a coloring which is valid with high probability, and the second one, which is a variant of our efficient deterministic algorithm, requires an expected number of at most O(log n log log n) colors but always produces a valid coloring. We also analyze the performance of the simplest proposed algorithm when the points are inserted in a random order, and present an incomplete analysis that indicates that, with high probability, it uses only O(log n) colors. Finally, we show that in the extension of this problem to two dimensions, where the relevant ranges are disks, n colors may be required in the worst case. The average-case behavior for disks, and cases involving other planar ranges, are still open.