Many-to-One Boundary Labeling with Backbones

Abstract

Abstract. In this paper we study many-to-one boundary labeling with backbone leaders. In this model, a horizontal backbone reaches out of each label into the feature-enclosing rectangle. Feature points associated with this label are linked via vertical line segments to the backbone. We present algorithms for label number and leader-length minimization. If crossings are allowed, we aim to minimize their number. This can be achieved efficiently in the case of fixed label order. We show that the corresponding problem in the case of flexible label order is NP-hard.

External Labeling Techniques: A Taxonomy and Survey

“…Further, some methods use a multicriteria objective function (P2.3) taking into account leader lengths, but also separation between callouts and displacement of features or labels [YDGM17, FP99, AHS05, BRL09]. Other objective functions (P2.4) are either arbitrary quality measures based on a single leader [BHKN09], finding a feasible labeling at all [LWY09], maximizing the number of labeled features [KNR∗16], maximizing the label size [BKPS06a], maximizing the number of internal labels in a mixed model [LN10, BKPS11, LNS16], or minimizing the number of leaders in many‐to‐one labeling [BCF∗15, Lin10, LKY08].…”

“…They proved that minimizing crossings in 1‐sided many‐to‐one boundary labeling with po‐leaders is NP‐hard, but they also presented a greedy heuristic iteratively assigning label positions with locally fewest leader crossings. Bekos et al [BCF∗15] investigated a different labeling style using hyperleaders (recall Section 2.2). Here a po‐hyperleader for k sites consists of k vertical p‐segments connecting the sites to a single horizontal o‐segment, which connects to the label.…”

“…Then each side is solved with their 1‐sided algorithm. The hyperleader labeling by Bekos et al [BCF∗15] is also considered in a 2‐sided setting, but using duplicated labels that are vertically aligned on both sides. This actually makes the problem algorithmically simpler because now each hyperleader spans the entire width of the image region and splits the instance into two parts.…”

“…Publications: [BKPS06a], [BKSW07], [BHKN09], [LWY09], [LN10], [LPT∗11], [FHS∗12], [BCK∗18], [HPL14], [KLW14], [GHN15], [BCF∗15], [KNR∗16], [LNS16], [FS16], [NNR17].…”

“…In the many‐to‐one boundary labeling setting, Lin et al [LKY08] prove that the problem of minimizing the number of crossings between leaders can be expressed as a specific instance of the NP‐hard T wo ‐L ayer C rossing M inimization problem [EW94], according to which the goal is to compute a two‐layer drawing of a given bipartite graph with minimum number of crossings. Bekos et al [BCF∗15] proposed a reduction from the F ixed ‐L inear C rossing N umber problem [MNKF90] to prove that the problem of minimizing the number of crossings between leaders remains NP‐hard, even in the case where several leaders to the same label can share a common (horizontal) segment. Recall that the fixed‐linear crossing number problem asks for a linear embedding of a given graph, in which the order of the vertices is fixed along a line ℒ and the edges must be drawn either completely above or completely below ℒ such that the number of edge‐crossings is minimum.…”