Optimal point removal in closed-2PM labeling

An optimal labeling where labels are disjoint axis-parallel equal-size squares is called 2PM labeling if the labels have maximum length each attached to its corresponding point on the middle of one of its horizontal edges. In a closed-2PM labeling, no two edges of labels containing points should intersect. Removing one point and its label, makes free room for its adjacent labels and may cause a global label expansion. In this paper, we construct several data structures in the preprocessing stage, so that any point removal event is handled efficiently. We present an algorithm which decides in O(lgn) amortized time whether a label removal leads to label expansion in which case a new optimal labeling for the remaining points is generated in O(n) amortized time.     

Optimal algorithm for a special point-labeling problem

A special class of map labeling problem is studied. Let P={p₁,p₂,…,p_n} be a set of point sites distributed on a 2D map. A label associated with each point p_i is an axis-parallel rectangle r_i of specified width. The height of all are same. The placement of r_i must contain p_i at its top-left or bottom-left corner, and it does not obscure any other point in P. The objective is to label the maximum number of points on the map so that the placed labels are mutually non-overlapping. We first consider a simple model for this problem. Here, for each point p_i, the corner specification (i.e., whether the point p_i would appear at the top-left or bottom-left corner of the label) is known a priori. We show that the time complexity of this problem is , and then propose an algorithm for this problem which runs in O(nlogn) time. If the corner specifications of the points in P are not known, our algorithm is a 2-approximation algorithm. Here we propose an efficient heuristic algorithm that is easy to implement. Experimental evidences show that it produces optimal solutions for most of the randomly generated instances and for all the standard benchmarks available in http://www.math-inf.uni-greifswald.de/map-labeling/.     

Greedy algorithms for generalized k-rankings of paths

A k-ranking of a graph is a labeling of the vertices with positive integers 1,2,…,k so that every path connecting two vertices with the same label contains a vertex of larger label. An optimal ranking is one in which k is minimized. Let Pn be a path with n vertices. A greedy algorithm can be used to successively label each vertex with the smallest possible label that preserves the ranking property. We seek to show that when a greedy algorithm is used to label the vertices successively from left to right, we obtain an optimal ranking. A greedy algorithm of this type was given by Bodlaender et al. in 1998 [1] which generates an optimal k-ranking of Pn. In this paper we investigate two generalizations of rankings. We first consider bounded (k,s)-rankings in which the number of times a label can be used is bounded by a predetermined integer s. We then consider kt-rankings where any path connecting two vertices with the same label contains t vertices with larger labels. We show for both generalizations that when G is a path, the analogous greedy algorithms generate optimal k-rankings. We then proceed to quantify the minimum number of labels that can be used in these rankings. We define the bounded rank number to be the smallest number of labels that can be used in a (k,s)-ranking and show for n?2, where i=⌊log₂(s)⌋+1. We define the kt-rank number, to be the smallest number of labels that can be used in a kt-ranking. We present a recursive formula that gives the kt-rank numbers for paths, showing for all an−1<j?an where {an} is defined as follows: a₁=1 and an=⌊((t+1)/t)an−1⌋+1.     

Labeling Points with Weights

Annotating maps, graphs, and diagrams with pieces of text is an important step in information visualization that is usually referred to as label placement. We define nine label-placement models for labeling points with axis-parallel rectangles given a weight for each point. There are two groups: fixed-position models and slider models. We aim to maximize the weight sum of those points that receive a label. We first compare our models by giving bounds for the ratios between the weights of maximum-weight labelings in different models. Then we present algorithms for labeling n points with unit-height rectangles. We show how an O(n\log n)-time factor-2 approximation algorithm and a PTAS for fixed-position models can be extended to handle the weighted case. Our main contribution is the first algorithm for weighted sliding labels. Its approximation factor is (2+\varepsilon), it runs in O(n ²/\varepsilon) time and uses O(n/\varepsilon) space. We show that other than for fixed-position models even the projection to one dimension remains NP-hard. For slider models we also investigate some special cases, namely (a) the number of different point weights is bounded, (b) all labels are unit squares, and (c) the ratio between maximum and minimum label height is bounded.     

Labeling Points with Weights

Annotating maps, graphs, and diagrams with pieces of text is an important step in information visualization that is usually referred to as label placement. We define nine label-placement models for labeling points with axis-parallel rectangles given a weight for each point. There are two groups: fixed-position models and slider models. We aim to maximize the weight sum of those points that receive a label. We first compare our models by giving bounds for the ratios between the weights of maximum-weight labelings in different models. Then we present algorithms for labeling n points with unit-height rectangles. We show how an O(n\log n)-time factor-2 approximation algorithm and a PTAS for fixed-position models can be extended to handle the weighted case. Our main contribution is the first algorithm for weighted sliding labels. Its approximation factor is (2+\varepsilon), it runs in O(n ²/\varepsilon) time and uses O(n/\varepsilon) space. We show that other than for fixed-position models even the projection to one dimension remains NP-hard. For slider models we also investigate some special cases, namely (a) the number of different point weights is bounded, (b) all labels are unit squares, and (c) the ratio between maximum and minimum label height is bounded.     

A new fast heuristic for labeling points

In the point site labeling problem, we are given a set P={p₁,p₂,…,pn} of point sites in the plane. The label of a point pi is an axis-parallel rectangle of specified size. The objective is to label the maximum number of points on the map so that the placed labels are mutually non-overlapping. Here, we investigate a special class of the point site labeling problem where (i) height of the labels of all the points are same but their lengths may differ, (ii) the label of a point pi touches the point at one of its four corners, and (iii) the label of one point does not obscure any other point in P. We describe an efficient heuristic algorithm for this problem which runs in time in the worst case. We run our algorithm as well as the algorithm Rules proposed by Wagner et al. on randomly generated point sets and on the available benchmarks. The results produced by our algorithm are almost the same as Rules in most of the cases. But our algorithm is faster than Rules in dense instance. We have also computed the optimum solutions for all the examples we have considered by designing an algorithm, which performs an exhaustive search in the worst case. We found that the exhaustive search algorithm runs reasonably fast for most of the examples we have considered.     

An optimal parallel algorithm for c-vertex-ranking of trees

For a positive integer c, a c-vertex-ranking of a graph G=(V,E) is a labeling of the vertices of G with integers such that, for any label i, deletion of all vertices with labels >i leaves connected components, each having at most c vertices with label i. The c-vertex-ranking problem is to find a c-vertex-ranking of a given graph using the minimum number of ranks. In this paper we give an optimal parallel algorithm for solving the c-vertex-ranking problem on trees in O(log₂n) time using linear number of operations on the EREW PRAM model.     

Practical Extensions of Point Labeling in the Slider Model*

This paper extends research by the authors together with Alexander Wolff on point label placement using a model where labels can be placed at any position that touches the point (the slider model). Such models have been shown to perform better than methods that allow only a fixed number of positions per label. The novelties in this paper include respecting other map features that must be avoided by the labels, and incorporating labels with different height. The result is an efficient and simple O((n+m)log(n+m)) time algorithm with a performance guarantee for label placement in the slider model. Here n is the number of points to be labeled and m is the combinatorial complexity of the map features that must be avoided. Due to its efficiency, the algorithm can be used in interactive and on-line mapping.     

Labeling Schemes for Dynamic Tree Networks

Distance labeling schemes are composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute the distance between any two vertices directly from their labels (without using any additional information). As applications for distance labeling schemes concern mainly large and dynamically changing networks, it is of interest to study distributed dynamic labeling schemes. The current paper considers the problem on dynamic trees, and proposes efficient distributed schemes for it. The paper first presents a labeling scheme for distances in the dynamic tree model, with amortized message complexity O(log² n) per operation, where n is the size of the tree at the time the operation takes place. The protocol maintains O(log² n) bit labels. This label size is known to be optimal even in the static scenario. A more general labeling scheme is then introduced for the dynamic tree model, based on extending an existing static tree labeling scheme to the dynamic setting. The approach fits a number of natural tree functions, such as distance, separation level, and flow. The main resulting scheme incurs an overhead of an O(log n) multiplicative factor in both the label size and amortized message complexity in the case of dynamically growing trees (with no vertex deletions). If an upper bound on n is known in advance, this method yields a different tradeoff, with an O(log² n/log log n) multiplicative overhead on the label size but only an O(log n/log log n) overhead on the amortized message complexity. In the fully dynamic model the scheme also incurs an increased additive overhead in amortized communication, of O(log² n) messages per operation.     

On constructing the relative neighborhood graphs in EuclideanK-dimensional spaces

In this paper, a new algorithm for constructing the relative neighborhood graph(RNG) of ann points set in Euclideank-dimensional space is presented, for fixedk≥3. The worst case running time of the algorithm isO(n ^2?a(k) (logn)^1?a(k) ), fora(k)=2^?(k+1), which is under the assumption that no three input points form an isosceles triangle. Previous algorithms needO(n ²) time. Our algorithm proceeds in two phases. First, a supergraph ofRNG withO(n) edges is constructed and then those edges which do not belong toRNG are eliminated.     

Smallest k-point enclosing rectangle and square of arbitrary orientation

Given a set of n points in 2D, the problem of identifying the smallest rectangle of arbitrary orientation, and containing exactly k(?n) points is studied in this paper. The worst case time and space complexities of the proposed algorithm are O(n²logn+nk(n−k)(n−k+logk)) and O(n), respectively. The algorithm is then used to identify the smallest square of arbitrary orientation, and containing exactly k points in O(n²logn+kn²(n−k)logn) time.     

A linear-time algorithm to construct a rectilinear Steiner minimal tree fork-extremal point sets

Ak-extremal point set is a point set on the boundary of ak-sided rectilinear convex hull. Given ak-extremal point set of sizen, we present an algorithm that computes a rectilinear Steiner minimal tree in timeO(k ⁴ n). For constantk, this algorithm runs inO(n) time and is asymptotically optimal and, for arbitraryk, the algorithm is the fastest known for this problem.     

基于马尔科夫随机场的散乱点云全局特征提取

为了精确提取点云数据中的特征信息,针对激光扫描获取的三维散乱点云数据,提出一种基于马尔科夫随机场（Markov random field, MRF）的散乱点云特征提取方法.首先,根据散乱点的曲率估计及阈值初始化点标号并判定稳定点,将稳定点标记存储在数组中;然后,将优化不稳定点的标号问题转化为随机场标号的能量函数问题,引用贝叶斯估计求后验概率分布函数及MAP-MRF（Maximum a posteriori-Markov random field）框架归约得到目标函数;最后,根据图割法α-expansion算法,利用标号调整过程中标号集相对能量变化得到不稳定点的最优标号集,将其与存储稳定点的数组综合,根据点标号提取特征点.实验结果表明,该方法简单、高效、无需人工调参,能够依据全局能量的变化自适应提取特征,特征提取结果令人满意.     

A note on maximum independent sets in rectangle intersection graphs

Finding the maximum independent set in the intersection graph of n axis-parallel rectangles is NP-hard. We re-examine two known approximation results for this problem. For the case of rectangles of unit height, Agarwal, van Kreveld and Suri [Comput. Geom. Theory Appl. 11 (1998) 209-218] gave a (1+1/k)-factor algorithm with an O(nlogn+n^2k−1) time bound for any integer constant k?1; we describe a similar algorithm running in only O(nlogn+nΔ^k−1) time, where Δ?n denotes the maximum number of rectangles a point can be in. For the general case, Berman, DasGupta, Muthukrishnan and Ramaswami [J. Algorithms 41 (2001) 443-470] gave a ⌈log_kn⌉-factor algorithm with an O(n^k+1) time bound for any integer constant k?2; we describe similar algorithms running in O(nlogn+nΔ^k−2) and n^O(k/logk) time.     

Reporting intersections among thick objects

Let E be a set of n objects in fixed dimension d. We assume that each element of E has diameter smaller than D and has volume larger than V. We give a new divide and conquer algorithm that reports all the intersecting pairs in O(nlogn+(D^d/V)(n+k)) time and using O(n) space, where k is the number of intersecting pairs. It makes use of simple data structures and primitive operations, which explains why it performs very well in practice. Its restriction to unit balls in low dimensions is optimal in terms of time complexity, space complexity and algebraic degree.     

Solving the Euclidean bottleneck matching problem byk-relative neighborhood graphs

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(n ²) time. We then use Gabow and Tarjan's bottleneck maximum cardinality matching algorithm for general graphs whose time-complexity isO((n logn)^0.5 m), wherem is the number of edges in the graph, to solve the bottleneck maximum cardinality matching problem in the 17RNG. This takesO(n ^1.5 log^0.5 n) time. The total time-complexity of our algorithm for the Euclidean bottleneck matching problem isO(n ² +n ^1.5 log^0.5 n).     

L1PMA: A Fortran 77 package for best L1 piecewise monotonic data smoothing

Fortran 77 software is presented for the calculation of a best L₁ approximation to n measurements that include random errors by requiring k−1 sign changes in the first divided differences of the approximation or equivalently k monotonic sections, alternately increasing and decreasing. A dynamic programming algorithm separates the measurements into optimal disjoint sections of adjacent data and applies to each section a single L₁ monotonic calculation. The most distinctive feature of the algorithm is that it terminates at a global minimum in at most n³+O(kn²) computer operations, although this calculation can exhibit O(n^k) local minima, because the optimal positions of the turning points are also unknowns of the optimization process. The arithmetic operations involved in this calculation are comparisons mainly spent in finding the medians of subranges of data during the monotonic calculations. The package employs techniques for median and for best L₁ monotonic approximation, while full details of these techniques are specified. The package has been applied and tested on a variety of data that have substantial differences and showed quadratic behaviour in n. Some numerical results demonstrate the performance of the method. Further, there is a commentary on the division of the code into subroutines. Driver programs and numerical examples with output are provided to help new users of the method. Besides that piecewise monotonicity is a property of a wide range of functions, an important application of the method is in estimating turning points of a function from some noisy measurements of its values.     

Finding the convex hull of a sorted point set in parallel

We present a parallel algorithm for finding the convex hull of a sorted planar point set. Our algorithm runs in O(log n) time using O(n/log n) processors in the CREW PRAM computational model, which is optimal. One of the techniques we use to achieve these optimal bounds is the use of a parallel data structure which we call the hull tree.     

Chromatic distribution of k-nearest neighbors of a line segment in a planar colored point set

Let P be a set of n colored points distributed arbitrarily in R². The chromatic distribution of the k-nearest neighbors of a query line segment ? is to report the number of points of each color among the k-nearest points of the query line segment. While solving this problem, we have encountered another interesting problem, namely the semicircular range counting query. Here a set of n points is given. The objective is to report the number of points inside a given semicircular range. We propose a simple algorithm for this problem with preprocessing time and space complexity O(n³), and the query time complexity O(logn). Finally, we propose the algorithm for reporting the chromatic distribution of k nearest neighbors of a query line segment. Using our proposed technique for semicircular range counting query, it runs in O(log²n) time.