Network Farthest-Point Diagrams

Consider the continuum of points along the edges of a network, i.e., an undirected graph with positive edge weights. We measure distance between these points in terms of the shortest path distance along the network, known as the network distance. Within this metric space, we study farthest points. We introduce network farthest-point diagrams, which capture how the farthest points---and the distance to them---change as we traverse the network. We preprocess a network G such that, when given a query point q on G, we can quickly determine the farthest point(s) from q in G as well as the farthest distance from q in G. Furthermore, we introduce a data structure supporting queries for the parts of the network that are farther away from q than some threshold R > 0, where R is part of the query. We also introduce the minimum eccentricity feed-link problem defined as follows. Given a network G with geometric edge weights and a point p that is not on G, connect p to a point q on G with a straight line segment pq, called a feed-link, such that the largest network distance from p to any point in the resulting network is minimized. We solve the minimum eccentricity feed-link problem using eccentricity diagrams. In addition, we provide a data structure for the query version, where the network G is fixed and a query consists of the point p.Comment: A preliminary version of this work was presented at the 24th Canadian Conference on Computational Geometr

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)

On Farthest-Point Information in Networks

Consider the continuum of points along the edges of a network, an embedded undirected graph with positive edge weights. Distance between these points can be measured as shortest path distance along the edges of the network. We introduce two new concepts to capture farthest-point information in this metric space. The first, eccentricity diagrams, are used to encode the distance towards farthest points for any point on the network compactly. With this, we can solve the minimum eccentricity feed-link problem, i.e., the problem to extend a network by one new point minimizing the largest network distance towards the new point. The second, network farthest-point diagrams, provide an implicit description of the sets of farthest points. A network farthest-point diagram is, in principle, a compressed farthest-point network Voronoi link diagram generated by the entire continuum of uncountably many points on the network at hand. We provide construction algorithms for data structures that allow for queries for the distance to farthest points as well as their location from any point on a network in optimal time. Thus, we establish first bounds on construction times and storage requirements of such data structures