33,207 research outputs found
Mixed Map Labeling
Point feature map labeling is a geometric problem, in which a set of input
points must be labeled with a set of disjoint rectangles (the bounding boxes of
the label texts). Typically, labeling models either use internal labels, which
must touch their feature point, or external (boundary) labels, which are placed
on one of the four sides of the input points' bounding box and which are
connected to their feature points by crossing-free leader lines. In this paper
we study polynomial-time algorithms for maximizing the number of internal
labels in a mixed labeling model that combines internal and external labels.
The model requires that all leaders are parallel to a given orientation , whose value influences the geometric properties and hence the
running times of our algorithms.Comment: Full version for the paper accepted at CIAC 201
Visualizing Geophylogenies - Internal and External Labeling with Phylogenetic Tree Constraints
A geophylogeny is a phylogenetic tree where each leaf (biological taxon) has an associated geographic location (site). To clearly visualize a geophylogeny, the tree is typically represented as a crossing-free drawing next to a map. The correspondence between the taxa and the sites is either shown with matching labels on the map (internal labeling) or with leaders that connect each site to the corresponding leaf of the tree (external labeling). In both cases, a good order of the leaves is paramount for understanding the association between sites and taxa. We define several quality measures for internal labeling and give an efficient algorithm for optimizing them. In contrast, minimizing the number of leader crossings in an external labeling is NP-hard. We show nonetheless that optimal solutions can be found in a matter of seconds on realistic instances using integer linear programming. Finally, we provide several efficient heuristic algorithms and experimentally show them to be near optimal on real-world and synthetic instances
Many-to-One Boundary Labeling with Backbones
In this paper we study \emph{many-to-one boundary labeling with backbone
leaders}. In this new many-to-one model, a horizontal backbone reaches out of
each label into the feature-enclosing rectangle. Feature points that need to be
connected to this label are linked via vertical line segments to the backbone.
We present dynamic programming algorithms for label number and total leader
length minimization of crossing-free backbone labelings. When crossings are
allowed, we aim to obtain solutions with the minimum number of crossings. This
can be achieved efficiently in the case of fixed label order, however, in the
case of flexible label order we show that minimizing the number of leader
crossings is NP-hard.Comment: 23 pages, 10 figures, this is the full version of a paper that is
about to appear in GD'1
Multi-Sided Boundary Labeling
In the Boundary Labeling problem, we are given a set of points, referred
to as sites, inside an axis-parallel rectangle , and a set of pairwise
disjoint rectangular labels that are attached to 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 (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 crossing-free 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
Homotopical rigidity of polygonal billiards
Consider two -gons and . We say that the billiard flows in and
are homotopically equivalent if the set of conjugacy classes in the
fundamental group of which contain a periodic billiard orbit agrees with
the analogous set for . We study this equivalence relationship and compare
it to the equivalence relations, order equivalence and code equivalence,
introduced in \cite{BT1,BT2}. In particular we show if is a rational
polygon, and is homotopically equivalent to , then and are
similar, or affinely similar if all sides of are vertical and horizontal
A Roadmap to Reduce U.S. Food Waste By 20 Percent, Executive Summary 2016
The magnitude of the food waste problem is difficult to comprehend. The U.S. spends $218 billion a year -- 1.3% of GDP -- growing, processing, transporting, and disposing of food that is never eaten. The causes of food waste are diverse, ranging from crops that never get harvested, to food left on overfilled plates, to near-expired milk and stale bread.ReFED is a coalition of over 30 business, nonprofit, foundation, and government leaders committed to building a different future, where food waste prevention, recovery, and recycling are recognized as an untapped opportunity to create jobs, alleviate hunger, and protect the environment -- all while stimulating a new multi-billion dollar market opportunity. ReFED developed A Roadmap to Reduce U.S. Food Waste as a data-driven guide to collectively take action to reduce food waste at scale nationwide
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