Covering point sets with two disjoint disks or squares

We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks C_R and C_B with disjoint interiors such that the number of red points covered by C_R plus the number of blue points covered by C_B is maximized. We give an algorithm to solve this problem in O(n^8/3log²n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axis-aligned unit squares S_R and S_B instead of unit disks can be solved in O(nlogn) time, which is optimal. If we do not restrict ourselves to axis-aligned squares, but require that both squares have a common orientation, we give a solution using O(n³logn) time.     

Proximity problems on line segments spanned by points

Finding the closest or farthest line segment (line) from a point are fundamental proximity problems. Given a set S of n points in the plane and another point q, we present optimal O(nlogn) time, O(n) space algorithms for finding the closest and farthest line segments (lines) from q among those spanned by the points in S. We further show how to apply our techniques to find the minimum (maximum) area triangle with a vertex at q and the other two vertices in S{q} in optimal O(nlogn) time and O(n) space. Finally, we give an O(nlogn) time, O(n) space algorithm to find the kth closest line from q and show how to find the k closest lines from q in O(nlogn+k) time and O(n+k) space.     

A quadratic distance bound on sliding between crossing-free spanning trees

Let S be a set of n points in the plane and let be the set of all crossing-free spanning trees of S. We show that it is possible to transform any two trees in into each other by O(n²) local and constant-size edge slide operations. Previously no polynomial upper bound for this task was known, but in [O. Aichholzer, F. Aurenhammer, F. Hurtado, Sequences of spanning trees and a fixed tree theorem, Comput. Geom.: Theory Appl. 21 (1–2) (2002) 3–20] a bound of O(n²logn) operations was conjectured.     

Sparse geometric graphs with small dilation

Given a set S of n points in , and an integer k such that 0k<n, we show that a geometric graph with vertex set S, at most n−1+k edges, maximum degree five, and dilation O(n/(k+1)) can be computed in time O(nlogn). For any k, we also construct planar n-point sets for which any geometric graph with n−1+k edges has dilation Ω(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position.     

Computing the Fréchet distance between piecewise smooth curves

We consider the Fréchet distance between two curves which are given as a sequence of m+n curved pieces. If these pieces are sufficiently well-behaved, we can compute the Fréchet distance in O(mnlog(mn)) time. The decision version of the problem can be solved in O(mn) time. The results are based on an analysis of the possible intersection patterns between circles and arcs of bounded curvature.     

Computing homotopic shortest paths efficiently

We give deterministic and randomized algorithms to find shortest paths homotopic to a given collection Π of disjoint paths that wind amongst n point obstacles in the plane. Our deterministic algorithm runs in time , and the randomized algorithm runs in expected time O(k_out+k_inlogn+n(logn)^1+ε). Here k_in is the number of edges in all the paths of Π, and k_out is the number of edges in the output paths.     

Visibility maps of segments and triangles in 3D

Let T be a set of n triangles in three-dimensional space, let s be a line segment, and let t be a triangle, both disjoint from T. We consider the subdivision of T based on (in)visibility from s; this is the visibility map of the segment s with respect to T. The visibility map of the triangle t is defined analogously. We look at two different notions of visibility: strong (complete) visibility, and weak (partial) visibility. The trivial Ω(n²) lower bound for the combinatorial complexity of the strong visibility map of both s and t is almost tight: we prove an O(n²(n)) upper bound for both structures, where (n) is the extremely slowly increasing inverse Ackermann function. Furthermore, we prove that the weak visibility map of s has complexity Θ(n⁵), and the weak visibility map of t has complexity Θ(n⁷). If T is a polyhedral terrain, the complexity of the weak visibility map is Ω(n⁴) and O(n⁵), both for a segment and a triangle. We also present efficient algorithms to compute all discussed structures.     

On the Fermat–Weber center of a convex object

We show that for any convex object Q in the plane, the average distance from the Fermat–Weber center of Q to the points in Q is at least Δ(Q)/7, where Δ(Q) is the diameter of Q, and that there exists a convex object P for which this distance is Δ(P)/6. We use this result to obtain a linear-time approximation scheme for finding an approximate Fermat–Weber center of a convex polygon Q.     

On finding widest empty curved corridors

An -siphon of width w is the locus of points in the plane that are at the same distance w from a 1-corner polygonal chain C such that is the interior angle of C. Given a set P of n points in the plane and a fixed angle , we want to compute the widest empty -siphon that splits P into two non-empty sets. We present an efficient O(nlog³n)-time algorithm for computing the widest oriented -siphon through P such that the orientation of a half-line of C is known. We also propose an O(n³log²n)-time algorithm for the widest arbitrarily-oriented version and an Θ(nlogn)-time algorithm for the widest arbitrarily-oriented -siphon anchored at a given point.     

Fast reconstruction of Delaunay triangulations

We present a new linear time algorithm to compute a good order for the point set of a Delaunay triangulation in the plane. Such a good order makes reconstruction in linear time with a simple algorithm possible. Similarly to the algorithm of Snoeyink and van Kreveld [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459–471], our algorithm constructs such orders in O(logn) phases by repeatedly removing a constant fraction of vertices from the current triangulation. Compared to [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459–471] we improve the guarantee on the number of removed vertices in each such phase. If we restrict the degree of the points (at the time they are removed) to 6, our algorithm removes at least 1/3 of the points while the algorithm from [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459–471] gives a guarantee of 1/10. We achieve this improvement by removing the points sequentially using a breadth first search (BFS) based procedure that—in contrast to [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459–471]—does not (necessarily) remove an independent set.

Besides speeding up the algorithm, removing more points in a single phase has the advantage that two consecutive points in the computed order are usually closer to each other. For this reason, we believe that our approach is better suited for vertex coordinate compression.

We implemented prototypes of both algorithms and compared their running time on point sets uniformly distributed in the unit cube. Our algorithm is slightly faster. To compare the vertex coordinate compression capabilities of both algorithms we round the resulting sequences of vertex coordinates to 16-bit integers and compress them with a simple variable length code. Our algorithm achieves about 14% better vertex data compression than the algorithm from [Proceedings of 5th European Symposium on Algorithms (ESA), 1997, pp. 459–471].     

Adaptive sampling for geometric problems over data streams

Geometric coordinates are an integral part of many data streams. Examples include sensor locations in environmental monitoring, vehicle locations in traffic monitoring or battlefield simulations, scientific measurements of earth or atmospheric phenomena, etc. This paper focuses on the problem of summarizing such geometric data streams using limited storage so that many natural geometric queries can be answered faithfully. Some examples of such queries are: report the smallest convex region in which a chemical leak has been sensed, or track the diameter of the dataset, or track the extent of the dataset in any given direction. One can also pose queries over multiple streams: for instance, track the minimum distance between the convex hulls of two data streams, report when datasets A and B are no longer linearly separable, or report when points of data stream A become completely surrounded by points of data stream B, etc. These queries are easily extended to more than two streams.

In this paper, we propose an adaptive sampling scheme that gives provably optimal error bounds for extremal problems of this nature. All our results follow from a single technique for computing the approximate convex hull of a point stream in a single pass. Our main result is this: given a stream of two-dimensional points and an integer r, we can maintain an adaptive sample of at most 2r+1 points such that the distance between the true convex hull and the convex hull of the sample points is O(D/r²), where D is the diameter of the sample set. The amortized time for processing each point in the stream is O(logr). Using the sample convex hull, all the queries mentioned above can be answered approximately in either O(logr) or O(r) time.     

The minimum-area spanning tree problem

Motivated by optimization problems in sensor coverage, we formulate and study the Minimum-Area Spanning Tree (mast) problem: Given a set of n points in the plane, find a spanning tree of of minimum “area”, where the area of a spanning tree is the area of the union of the n−1 disks whose diameters are the edges in . We prove that the Euclidean minimum spanning tree of is a constant-factor approximation for mast. We then apply this result to obtain constant-factor approximations for the Minimum-Area Range Assignment (mara) problem, for the Minimum-Area Connected Disk Graph (macdg) problem, and for the Minimum-Area Tour (mat) problem. The first problem is a variant of the power assignment problem in radio networks, the second problem is a related natural problem, and the third problem is a variant of the traveling salesman problem.     

The minimum weight triangulation problem with few inner points

We look at the computational complexity of 2-dimensional geometric optimization problems on a finite point set with respect to the number of inner points (that is, points in the interior of the convex hull). As a case study, we consider the minimum weight triangulation problem. Finding a minimum weight triangulation for a set of n points in the plane is not known to be NP-hard nor solvable in polynomial time, but when the points are in convex position, the problem can be solved in O(n³) time by dynamic programming. We extend the dynamic programming approach to the general problem and describe an exact algorithm which runs in O(6^kn⁵logn) time where n is the total number of input points and k is the number of inner points. If k is taken as a parameter, this is a fixed-parameter algorithm. It also shows that the problem can be solved in polynomial time if k=O(logn). In fact, the algorithm works not only for convex polygons, but also for simple polygons with k inner points.     

The minimum Manhattan network problem: Approximations and exact solutions

Given a set of points in the plane and a constant t1, a Euclidean t-spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two points is at most t. Such networks have applications in transportation or communication network design and have been studied extensively.

In this paper we study 1-spanners under the Manhattan (or L₁-) metric. Such networks are called Manhattan networks. A Manhattan network for a set of points is a set of axis-parallel line segments whose union contains an x- and y-monotone path for each pair of points. It is not known whether it is NP-hard to compute minimum Manhattan networks (MMN), i.e., Manhattan networks of minimum total length. In this paper we present an approximation algorithm for this problem. Given a set P of n points, our algorithm computes in O(nlogn) time and linear space a Manhattan network for P whose length is at most 3 times the length of an MMN of P.

We also establish a mixed-integer programming formulation for the MMN problem. With its help we extensively investigate the performance of our factor-3 approximation algorithm on random point sets.     

Algorithms for optimal area triangulations of a convex polygon

Given a convex polygon with n vertices in the plane, we are interested in triangulations of its interior, i.e., maximal sets of non-intersecting diagonals that subdivide the interior of the polygon into triangles. The MaxMin area triangulation is the triangulation of the polygon that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the triangulation that minimizes the area of the largest area triangle in the triangulation. We present algorithms that construct MaxMin and MinMax area triangulations of a convex polygon in O(n²logn) time and O(n²) space. The algorithms use dynamic programming and a number of geometric properties that are established within the paper.     

Farthest-point queries with geometric and combinatorial constraints

In this paper we discuss farthest-point problems in which a set or sequence S of n points in the plane is given in advance and can be preprocessed to answer various queries efficiently. First, we give a data structure that can be used to compute the point farthest from a query line segment in O(log²n) time. Our data structure needs O(nlogn) space and preprocessing time. To the best of our knowledge no solution to this problem has been suggested yet. Second, we show how to use this data structure to obtain an output-sensitive query-based algorithm for polygonal path simplification. Both results are based on a series of data structures for fundamental farthest-point queries that can be reduced to each other.     

Graph drawings with few slopes

The slope-number of a graph G is the minimum number of distinct edge slopes in a straight-line drawing of G in the plane. We prove that for Δ5 and all large n, there is a Δ-regular n-vertex graph with slope-number at least . This is the best known lower bound on the slope-number of a graph with bounded degree. We prove upper and lower bounds on the slope-number of complete bipartite graphs. We prove a general upper bound on the slope-number of an arbitrary graph in terms of its bandwidth. It follows that the slope-number of interval graphs, cocomparability graphs, and AT-free graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slope-number at most . Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.     

The Erdős–Nagy theorem and its ramifications

Given a simple polygon in the plane, a flip is defined as follows: consider the convex hull of the polygon. If there are no pockets do not perform a flip. If there are pockets then reflect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erdős introduced the problem of repeatedly flipping all the pockets of a simple polygon simultaneously and he conjectured that the polygon would become convex after a finite number of flips. In 1939 Béla Nagy proved that if at each step only one pocket is flipped the polygon will become convex after a finite number of flips. The history of this problem is reviewed, and a simple elementary proof is given of a stronger version of the theorem. Variants, generalizations, and applications of the theorem of interest in computational knot theory, polymer physics and molecular biology are discussed. Several results in the literature are improved with the application of the theorem. For example, Grünbaum and Zaks recently showed that even non-simple (self-crossing) polygons may be convexified in a finite number of suitable flips. Their flips each take Θ(n²) time to determine. A simpler proof of this result is given that yields an algorithm that takes O(n) time to determine each flip. In the context of knot theory Millet proposed an algorithm for convexifying equilateral polygons in 3-dimensions with a generalization of a flip called a pivot. Here Millet's algorithm is generalized so that it works also in dimensions higher than three and for polygons containing edges with arbitrary lengths. A list of open problems is included.     

Schematization of networks

We study the problem of computing schematized versions of network maps, like railroad or highway maps. Every path of the schematized map has two or three links with restricted orientations, and the schematized map must be topologically equivalent to the input map. Our approach can handle several types of schematizations, and certain additional constraints can be added, such as a minimum vertical distance between two paths. Our algorithm runs in O(nlogn) time, and experimental results showing the quality of the output are given.     

Geometric dilation of closed planar curves: New lower bounds

Given two points on a closed planar curve, C, we can divide the length of a shortest connecting path in C by their Euclidean distance. The supremum of these ratios, taken over all pairs of points on the curve, is called the geometric dilation of C. We provide lower bounds for the dilation of closed curves in terms of their geometric properties, and prove that the circle is the only closed curve achieving a dilation of π/2, which is the smallest dilation possible. Our main tool is a new geometric transformation technique based on the perimeter halving pairs of C.