Algorithms for Multi-Criteria Boundary Labeling

Abstract

We present new algorithms for labeling a set P of n points in the plane with labels that are aligned to one side of the bounding box of P . The points are connected to their labels by curves (leaders) that consist of two segments: a horizontal segment, and a second segment at a fixed angle with the first. Our algorithms find a collection of crossing-free leaders that minimizes the total number of bends, the total length, or any other 'badness' function of the leaders. A generalization to labels on two opposite sides of the bounding box of P is considered and an experimental evaluation of the performance is included.

External Labeling Techniques: A Taxonomy and Survey

“…Let us first assume that C A is an axis‐aligned rectangle. As already mentioned, Benkert et al [BHKN09] introduced boundary labelings with do‐leaders and preliminary observed that a crossing‐free solution might not always be feasible. For a general objective function, they proposed a dynamic programming approach, which in O ( n 5 ) time (in O ( n 14 ) time) determines whether there exist an optimal crossing‐free boundary labeling with do‐leaders, assuming that the ports of the labels are sliding while the positions of the labels are fixed along a single side (along two opposite sides, respectively) of the image region R ; in the positive case, the labeling can be reported without increasing the time complexity.…”

“…We assume | A | ≥ | F |, and that the cost function rates single labels. Similar to Benkert et al [BHKN09] we define the drawing area of a sub‐instance of that problem setting by a simple polygon specified by two point features p 1 , p 2 and two reference points a 1 , a 2 ; see Fig. 7a.…”

Dynamic one-sided boundary labeling

“…Instead of requiring that every frame of the animation must be a crossing-free labeling, we remove crossings only when the user stops moving the map view; this can be done in O(n log n) time [6]. Upon resumption of the movement, we keep the current order of the labels until the next break.…”

“…This problem is known to be NP-hard and many heuristics and approximation algorithms exist, see [19] and the extensive bibliography on map labeling maintained by Wolff and Strijk [20]. The (static) boundary labeling problem was first introduced as an algorithmic problem by Bekos et al [5] and subsequently studied in different settings for rectilinear and diagonal leader shapes with one or two bends and label positions on one, two, or four sides of the map [3,6,14]; placing the labels in multiple columns on one side of the map was also considered [4]. Still, all previous results assume that labels should be as large as possible.…”

Boundary-labeling algorithms for panorama images

“…Their main objective was to minimize the total leader length, but they also presented an algorithm for minimizing the number of bends in one-sided opo-labeling. Benkert et al [5] studied algorithms for oneand two-sided po-and do-labeling with arbitrary leaderdependent cost functions (including total length and number of bends); the algorithms were implemented and evaluated experimentally. Bekos et al [1] presented algorithms for combinations of more general octilinear leaders of types do, od, and pd and labels on one, two, and four sides of R. For uniform labels the algorithms are polynomial, whereas the authors showed NP-completeness for a variant involving non-uniform labels.…”