Narrow Results

A rectilinear map consists of a set of mutually non-intersecting rectilinear (i.e., horizontal or vertical) line segments, and each segment is allowed to use a rectangular label of height B and length the same as the segment. Sliding labels are not restricted to any finite number of predefined positions but can slide and be placed at any position as long as it intersects the segment. This paper considers three versions of the problem of labeling a rectilinear map with sliding labels and presents efficient exact and approximation algorithms for them.

In this paper we propose and study two practical variations of the map labeling problem: Given a set S of n distinct (point) sites in the plane, label each site with: (1) a pair of non-intersecting squares of maximum possible size, (2) a pair of non-intersecting circles of maximum possible size (all the squares and circles are topologically open and are of uniform size). Almost nothing has been done before in this aspect, i.e., multi-label map labeling. We obtain constant-factor approximation algorithms for these problems. We also study bicriteria approximation schemes for these problems under a mild condition.

The automatic placement of text or symbol labels corresponding to graphical features is critical in several application areas such as cartography, geographical information systems, and graph drawing. In this paper we present a general framework for solving the problem of assigning text or symbol labels to a set of graphical features in two dimensional drawings or maps. Our approach does not favor the labeling of one type of graphical feature (such as a node, edge, or area) over another. Additionally, the labels are allowed to have arbitrary size and orientation. We also present a fast and simple technique, based on the general framework, for assigning labels to edges of graph drawings. We have implemented our techniques and have performed extensive experimentation on hierarchical and orthogonal drawings of graphs. The resulting label assignments are very practical and indicate the effectiveness of our approach.

In this paper, we consider a map labeling problem where the points to be labeled are restricted on a line. It is known that the 1d-4P and the 1d-4S unit-square label placement problem and the Slope-4P unit-square label placement problem can both be solved in linear time and the Slope-4S unit-square label placement problem can be solved in quadratic time in Ref. [8]. We extend the result to the following label placement problem: Slope-4P fixed-height (width) label or elastic label placement problem and present a linear time algorithm for it provided that the input points are given sorted. We further show that if the points are not sorted, the label placement problems have a lower bound of Ω(n log n), where n is the input size, under the algebraic computation tree model. Optimization versions of these point labeling problems are also considered.

Driven by the industrial challenge of labeling maps for GIS applications, we investigate the problem of computing a map region P such that a rectangular axis-parallel label L of a given size can be placed in it. The map region to be labeled is in general a non-convex n-gon which may contain holes. We first derive a new practical algorithm based on the sweep-line technique that determines the com set of Maximum Inscribed Rectangles (MIRs) in P in O(nk), where k is the size of the output, for the case when the polygon sides have an axis-parallel orientation. After the set of MIRs has been found, any subsequent query on label L placement runs in only O(logn) time. We then provide an algorithm to convert the general case to the axis-parallel case. Extensive experimentation in both laboratory and industrial settings confirms that the developed method is practical and highly efficient for processing large GIS data sets.