Reconstruction of Weakly Simple Polygons from their Edges

Given n line segments in the plane, do they form the edge set of a weakly simple polygon; that is, can the segment endpoints be perturbed by at most epsilon, for any epsilon > 0, to obtain a simple polygon? While the analogous question for simple polygons can easily be answered in O(n log n) time, we show that it is NP-complete for weakly simple polygons. We give O(n)-time algorithms in two special cases: when all segments are collinear, or the segment endpoints are in general position. These results extend to the variant in which the segments are directed, and the counterclockwise traversal of a polygon should follow the orientation. We study related problems for the case that the union of the n input segments is connected. (i) If each segment can be subdivided into several segments, find the minimum number of subdivision points to form a weakly simple polygon. (ii) If new line segments can be added, find the minimum total length of new segments that creates a weakly simple polygon. We give worst-case upper and lower bounds for both problems

Recognizing Weakly Simple Polygons

We present an O(n log n)-time algorithm that determines whether a given planar n-gon is weakly simple. This improves upon an O(n^2 log n)-time algorithm by [Chang, Erickson, and Xu, SODA, 2015]. Weakly simple polygons are required as input for several geometric algorithms. As such, how to recognize simple or weakly simple polygons is a fundamental question

Maximum Area Axis-Aligned Square Packings

Given a point set S={s_1,...s_n} in the unit square U=[0,1]^2, an anchored square packing is a set of n interior-disjoint empty squares in U such that s_i is a corner of the ith square. The reach R(S) of S is the set of points that may be covered by such a packing, that is, the union of all empty squares anchored at points in S. It is shown that area(R(S))>= 1/2 for every finite set S subset U, and this bound is the best possible. The region R(S) can be computed in O(n log n) time. Finally, we prove that finding a maximum area anchored square packing is NP-complete. This is the first hardness proof for a geometric packing problem where the size of geometric objects in the packing is unrestricted

Reconfiguration of Polygonal Subdivisions via Recombination

Motivated by the problem of redistricting, we study area-preserving reconfigurations of connected subdivisions of a simple polygon. A connected subdivision of a polygon ?, called a district map, is a set of interior disjoint connected polygons called districts whose union equals ?. We consider the recombination as the reconfiguration move which takes a subdivision and produces another by merging two adjacent districts, and by splitting them into two connected polygons of the same area as the original districts. The complexity of a map is the number of vertices in the boundaries of its districts. Given two maps with k districts, with complexity O(n), and a perfect matching between districts of the same area in the two maps, we show constructively that (log n)^O(log k) recombination moves are sufficient to reconfigure one into the other. We also show that ?(log n) recombination moves are sometimes necessary even when k = 3, thus providing a tight bound when k = 3

Circumscribing Polygons and Polygonizations for Disjoint Line Segments

Given a planar straight-line graph G=(V,E) in R^2, a circumscribing polygon of G is a simple polygon P whose vertex set is V, and every edge in E is either an edge or an internal diagonal of P. A circumscribing polygon is a polygonization for G if every edge in E is an edge of P. We prove that every arrangement of n disjoint line segments in the plane has a subset of size Omega(sqrt{n}) that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to R^3. We show that it is NP-complete to decide whether a given graph G admits a circumscribing polygon, even if G is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization