Planar Bichromatic Bottleneck Spanning Trees

Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8?2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck ? can be converted to a planar bichromatic spanning tree of bottleneck at most 8?2 ?

Spanners for Geometric Intersection Graphs

Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.Comment: 16 pages, 5 figures, Late

Geometric Planar Networks on Bichromatic Points

We study four classical graph problems – Hamiltonian path, Traveling salesman, Minimum spanning tree, and Minimum perfect matching on geometric graphs induced by bichromatic ( Open image in new window and Open image in new window ) points. These problems have been widely studied for points in the Euclidean plane, and many of them are NP -hard. In this work, we consider these problems in two restricted settings: (i) collinear points and (ii) equidistant points on a circle. We show that almost all of these problems can be solved in linear time in these constrained, yet non-trivial settings.acceptedVersio

Flip Distance to some Plane Configurations

We study an old geometric optimization problem in the plane. Given a perfect matching M on a set of n points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip operation removes two crossing edges from M and adds two non-crossing edges. Let f(M) and F(M) denote the minimum and maximum lengths of a flip sequence on M, respectively. It has been proved by Bonnet and Miltzow (2016) that f(M)=O(n^2) and by van Leeuwen and Schoone (1980) that F(M)=O(n^3). We prove that f(M)=O(n Delta) where Delta is the spread of the point set, which is defined as the ratio between the longest and the shortest pairwise distances. This improves the previous bound for point sets with sublinear spread. For a matching M on n points in convex position we prove that f(M)=n/2-1 and F(M)={{n/2} choose 2}; these bounds are tight. Any bound on F(*) carries over to the bichromatic setting, while this is not necessarily true for f(*). Let M\u27 be a bichromatic matching. The best known upper bound for f(M\u27) is the same as for F(M\u27), which is essentially O(n^3). We prove that f(M\u27)<=slant n-2 for points in convex position, and f(M\u27)= O(n^2) for semi-collinear points. The flip operation can also be defined on spanning trees. For a spanning tree T on a convex point set we show that f(T)=O(n log n)

Efikasni algoritmi za probleme iz diskretne geometrije

The first class of problem we study deals with geometric matchings. Given a set of points in the plane, we study perfect matchings of those points by straight line segments so that the segments do not cross. Bottleneck matching is such a matching that minimizes the length of the longest segment. We are interested in finding a bottleneck matching of points in convex position. In the monochromatic case, where any two points are allowed to be matched, we give an O(n 2 )-time algorithm for finding a bottleneck matching, improving upon previously best known algorithm of O(n 3 ) time complexity. We also study a bichromatic version of this problem, where each point is colored either red or blue, and only points of different color can be matched. We develop a range of tools, for dealing with bichromatic non-crossing matchings of points in convex position. Combining that set of tools with a geometric analysis enable us to solve the problem of finding a bottleneck matching in O(n 2 ) time. We also design an O(n)-time algorithm for the case where the given points lie on a circle. Previously best known results were O(n 3 ) for points in convex position, and O(n log n) for points on a circle. The second class of problems we study deals with dilation of geometric networks. Given a polygon representing a network, and a point p in the same plane, we aim to extend the network by inserting a line segment, called a feed-link, which connects p to the boundary of the polygon. Once a feed link is fixed, the geometric dilation of some point q on the boundary is the ratio between the length of the shortest path from p to q through the extended network, and their Euclidean distance. The utility of a feed-link is inversely proportional to the maximal dilation over all boundary points. We give a linear time algorithm for computing the feed-link with the minimum overall dilation, thus improving upon the previously known algorithm of complexity that is roughly O(n log n).Prva klasa problema koju proučavamo tičee se geometrijskih mečinga. Za dat skup tačaaka u ravni, posmatramo savršene mečinge tih tačaka spajajućii ih  dužima koje   se ne smeju sećui. Bottleneck mečing je takav mečing koji minimizuje dužinu najduže duži. Naš cilj je da nađemo bottleneck mečiing tačaka u konveksnom položaju.Za monohromatski slučaj, u kom je dozvoljeno upariti svaki par tačaka, dajemo algoritam vremenske složenosti O(n 2) za nalaženje bottleneck mečinga. Ovo  je bolje od prethodno najbolji poznatog algoritam, čiija je složenost O(n 3 ). Takođe proučavamo bihromatsku verziju ovog problema, u kojoj je svaka tačka  obojena ili u crveno ili u plavo, i dozvoljeno je upariti samo tačke različite boje. Razvijamo niz alata za rad sa bihromatskim nepresecajućim mečinzima tačaka u konveksnom položaju. Kombinovanje ovih alata sa geometrijskom analizom omogućava nam da rešimo problem nalaženja bottleneck mečinga u O(n 2 ) vremenu. Takođe, konstruišemo algoritam vremenske složenosti O(n) za slučaj kada  sve date tačkke leže na krugu. Prethodno najbolji poznati algoritmi su imali složenosti  O(n 3 ) za tačkeke u konveksnom položaju i O(n log n) za tačke na krugu. Druga klasa problema koju proučaavamo tiče se dilacije u geometrijskim mrežama. Za datu mrežu predstavljenu poligonom, i tačku p u istoj ravni, želimo proširiti mrežu  dodavanjem duži zvane feed-link koja povezuje p sa obodom poligona. Kada je feed- link fiksiran, definišemo geometrijsku dilaciju neke tačke q na obodu kao odnos izme  đu  dužine najkraćeg puta od p do q kroz proširenu mrežu i njihovog Euklidskog rastojanja. Korisnost feed-linka je obrnuto proporcionalna najvećoj dilaciji od svih ta čaka na obodu poligona. Konstruišemo algoritam linearne vremenske složenosti koji nalazi feed-link sa najmanom sveukupnom dilacijom. Ovim postižemo bolji rezultat od prethodno najboljeg poznatog algoritma složenosti približno O(n log n)