Non-Crossing Geometric Steiner Arborescences

Motivated by the question of simultaneous embedding of several flow maps, we consider the problem of drawing multiple geometric Steiner arborescences with no crossings in the rectilinear and in the angle-restricted setting. When terminal-to-root paths are allowed to turn freely, we show that two rectilinear Steiner arborescences have a non-crossing drawing if neither tree necessarily completely disconnects the other tree and if the roots of both trees are "free". If the roots are not free, then we can reduce the decision problem to 2SAT. If terminal-to-root paths are allowed to turn only at Steiner points, then it is NP-hard to decide whether multiple rectilinear Steiner arborescences have a non-crossing drawing. The setting of angle-restricted Steiner arborescences is more subtle than the rectilinear case. Our NP-hardness result extends, but testing whether there exists a non-crossing drawing if the roots of both trees are free requires additional conditions to be fulfilled

MapSets: Visualizing embedded and clustered graphs

In addition to objects and relationships between them, groups or clusters of objects are an essential part of many real-world datasets: party affiliation in political networks, types of living organisms in the tree of life, movie genres in the internet movie database. In recent visualization methods, such group information is conveyed by explicit regions that enclose related elements. However, when in addition to fixed cluster membership, the input elements also have fixed positions in space (e.g., geo-referenced data), it becomes difficult to produce readable visualizations. In such fixed-clustering and fixed-embedding settings, some methods produce fragmented regions, while other produce contiguous (connected) regions that may contain overlaps even if the input clusters are disjoint. Both fragmented regions and unnecessary overlaps have a detrimental effect on the interpretation of the drawing. With this in mind, we propose MapSets: a visualization technique that combines the advantages of both methods, producing maps with non-fragmented and non-overlapping regions. The proposed method relies on a theoretically sound geometric algorithm which guarantees contiguity and disjointness of the regions, and also optimizes the convexity of the regions. A fully functional implementation is available in an online system and is used in a comparison with related earlier methods. © 2015, Brown University. All right reserved.National Science Foundation, NSF: 111597

MapSets: Visualizing embedded and clustered graphs

We describe MapSets, a method for visualizing embedded and clustered graphs. The proposed method relies on a theoretically sound geometric algorithm, which guarantees the contiguity and disjointness of the regions representing the clusters, and also optimizes the convexity of the regions. A fully functional implementation is available online and is used in a comparison with related earlier methods. © Springer-Verlag Berlin Heidelberg 2014

Shooting permanent rays among disjoint polygons in the plane

We present a data structure for ray shooting-and-insertion in the free space among disjoint polygonal obstacles with a total of $n$ vertices in the plane, where each ray starts at the boundary of some obstacle. The portion of each query ray between the starting point and the first obstacle hit is inserted permanently as a new obstacle. Our data structure uses O(n log n) space and preprocessing time, and it supports m successive ray shooting-and-insertion queries in O(n log2 n + m log m) total time. We present two applications for our data structure: (1) Our data structure supports efficient implementation of auto-partitions in the plane i.e. binary space partitions where each partition is done along the supporting line of an input segment. If n input line segments are fragmented into m pieces by an auto-partition, then it can now be implemented in O(n log2n+m log m) time. This improves the expected runtime of Patersen and Yao's classical randomized auto-partition algorithm for n disjoint line segments to O(n log2 n). (2) If we are given disjoint polygonal obstacles with a total of n vertices in the plane, a permutation of the reflex vertices, and a half-line at each reflex vertex that partitions the reflex angle into two convex angles, then the folklore convex partitioning algorithm draws a ray emanating from each reflex vertex in the prescribed order in the given direction until it hits another obstacle, a previous ray, or infinity. The previously best implementation (with a semi-dynamic ray shooting data structure) requires O(n3/2-e/2) time using O(n1+e) space. Our data structure improves the runtime to O(n log2 n)

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Watchman routes in the presence of convex obstacles

This thesis deals with the problem of computing shortest watchman routes in the presence of polygonal obstacles. Important recent results on watchman route problems are surveyed. An {dollar}O(n\sp3){dollar} algorithm for computing a shortest watchman route in the presence of a pair of convex obstacles is presented. Important open problems related to watchman route problems are discussed