Multi-sided Boundary Labeling

Abstract

In the Boundary Labeling problem, we are given a set of n points, referred to as sites, inside an axis-parallel rectangle R, and a set of n pairwise disjoint rectangular labels that are attached to R from the outside. The task is to connect the sites to the labels by non-intersecting rectilinear paths, so-called leaders, with at most one bend.In this paper, we study the Multi-Sided Boundary Labeling problem, with labels lying on at least two sides of the enclosing rectangle. We present a polynomial-time algorithm that computes a crossing-free leader layout if one exists. So far, such an algorithm has only been known for the cases in which labels lie on one side or on two opposite sides of R (here a crossing-free solution always exists). The case where labels may lie on adjacent sides is more difficult. We present efficient algorithms for testing the existence of a crossing-free leader layout that labels all sites and also for maximizing the number of labeled sites in a crossingfree leader layout. For two-sided boundary labeling with adjacent sides, we further show how to minimize the total leader length in a crossing-free layout.

External Labeling Techniques: A Taxonomy and Survey

“…Here it is no longer the case that a crossing‐free solution always exists. Kindermann et al [KNR∗16] presented an O ( n 2 )‐time dynamic programming algorithm to compute a valid crossing‐free solution if one exists, both for labels with fixed or sliding ports. Further, they presented an algorithm to maximize the number of labeled sites in a crossing‐free labeling in O ( n 3 log n ) time.…”

“…Multi‐sided boundary labeling Three‐sided labeling is explicitly considered only in two papers [KNR∗16, BCK∗18]. Kindermann et al [KNR∗16] showed that this problem can be reduced to splitting an instance into two 2‐sided subproblems.…”

“…Multi‐sided boundary labeling Three‐sided labeling is explicitly considered only in two papers [KNR∗16, BCK∗18]. Kindermann et al [KNR∗16] showed that this problem can be reduced to splitting an instance into two 2‐sided subproblems. This results in an O ( n 4 )‐time algorithm working in linear space for finding a crossing‐free solution (if one exists) or for maximizing the number of labeled features.…”

“…Similar dynamic programming approaches have been considered for other leader‐types; see Tables 1 and 2. Further, while most approaches specify instances merely based on leaders, few approaches also use other separators [KNR∗16, BCK∗18].…”

Pattern-Driven Navigation in 2D Multiscale Visualizations with Scalable Insets

“…Many papers focus on static and point-feature label placement, i.e., the static placement of a label directly adjacent to a data point. Instead, our tool requires dynamic, excentric [20] (i.e., labels are distant from their targets), and boundary [6,38] placement as insets are to be positioned interactively and out-of-place to avoid occluding other essential details in the visualization. Dynamic labeling [51,52,55], which has been formalized into the "consistent dynamic map labeling" and "active range optimization" problem by Been et al [4,5], describes label placement in the context of pan and zoom interfaces, where the position and size of labels are not fixed.…”