Labeling weighted points with bounded sliding labels

Information Processing Letters, 2004

A special class of map labeling problem is studied. Let P = {p 1 , p 2 , . . . , 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 r i , i = 1, 2, . . . , n 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 (n log n), and then propose an algorithm for this problem which runs in O(n log n) 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 .

Mathematical Programming, 2003

We investigate the NP -hard label number maximization problem (LNM): Given a set of rectangular labels Λ, each of which belongs to a point feature in the plane, the task is to find a labeling for a largest subset Λ P of Λ. A labeling is a placement such that none of the labels overlap and each λ ∈ Λ P is placed according to a labeling model: In the discrete models, the label must be placed so that the labeled point coincides with an element of a predefined subset of corners of the rectangular label, in the continuous or slider models, the point must lie on a subset of boundaries of the label. Obviously, the slider models allow a continuous movement of a label around its point feature, leading to a significantly higher number of labels that can be placed.

Computational Geometry, 1999

Information Processing Letters, 2009

In the point site labeling problem, we are given a set P={p 1 ,p 2 , ,p n } of point sites in the plane. The label of a point p i 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 ...

2005

In the map labeling problem, we are given a set P = {p 1 , p 2 , . . . , p n } of point sites distributed on a 2D map. The label of a point p i is an axis-parallel rectangle of specified size. The objective is to label maximum number of points on the map so that the placed labels are mutually non-overlapping. Here, we investigate a special class of map labeling problem where (i) the height of the label of each point is the same but its length may be different from the others, (ii) the label of a point p i touches the point at one of its four corners and (iii) it does not obscure any other point in P . We describe an efficient heuristic algorithm for this problem which runs in O(n √ n) time in the worst case. We run our algorithm as well as the algorithm proposed in [14] on the available benchmarks . The results produced by our algorithm is same as that of in most of the cases, and is one less in few cases. But the time taken by our algorithm is much less than .

2004

Given a set of labeled points forming a valid map labeling, we are interested in a fast update of the labels if a point shaped object moves on an unknown path in the map. In this paper, there are labels that assumed to be axis-parallel, unit-length, and square-shaped, each attached to one point in the middle of one of its edges. We assume that a moving object can freely move on the map and sends notifications about its new positions.

Discrete Optimization, 2006

We are given a edge-weighted undirected graph G = (V, E) and a set of labels/colors C = {1, 2, . . . , p}. A non-empty subset C v ⊆ C is associated with each vertex v ∈ V . A coloring of the vertices is feasible if each vertex v is colored with a color of C v . A coloring uniquely defines a subset E ⊆ E of edges having different colored endpoints. The problem of finding a feasible coloring which defines a minimum weight E is, in general, NP-hard. In this work we first propose polynomial time algorithms for some special cases, namely when the input graph is a tree, a cactus or with bounded tree-width. Then, an implicit enumeration scheme for finding an optimal coloring in the general case is described and computational results are presented.

International Journal of Computational Geometry & Applications, 2002

We deal with a map-abeling problem, named LOFL (Left-part Ordered Flexible Labeling), to label a set of points in a plane in the presence of polygonal obstacles. The label for each point is selected from a set of rectangles with various shapes satisfying the left-part ordered property, and is placed near to the point after scaled by a scaling factor σ which is common to all points. In this paper, we give an optimal O((n + m) log (n + m)) algorithm to decide the feasibility of LOFL for a fixed scaling factor σ, and an O((n + m) log 2 (n + m)) time algorithm to find the largest feasible scaling factor σ, where n is the number of points and m is the total number of edges of the polygonal obstacles.

Information Processing Letters, 1998

Given a rectilinear map consisting of n disjoint line segments, the corresponding map labeling problem is to place a maximum width rectangle at each segment using one of the three natural ways. In a recent paper, it is shown that if all segments are horizontal then the problem can be solved in optimal O(n log n) time. For the general problem a factor-2 approximate solution and a Polynomial Time Approximation Scheme are also proposed. In this paper, we show that the general problem is polynomially solvable with a nontrivial use of 2SAT and the solution can be even generalized to the case of allowing k natural placements for each segment, where k is any fixed constant. We believe this technique can be also used to solve other geometric packing problems. @ 1998 Elsevier Science B.V.

Scope and Purpose-The purpose of this paper is to present a labeling algorithm to solve the assignment problem. The Hungarian method for solving the assignment problem, as stated in most text books, has an ambiguity and may result in cycling when programmed on a digital computer. We present an easy to use labeling algorithm and its implementation in QuickBasic 2.0 for solving the assignment problem. Abstract-This paper describes a simple labeling algorithm to solve the assignment problem. The labeling approach is based on the maximum-flow formulation of the problem of finding the fewest number of lines to cover all zeros in the reduced assignment matrix. The approach is useful for educational purposes as well as programming environments. The labeling procedure can be used without requiring familiarity with the maximum-flow problem.