Computing a rectilinear shortest path amid splinegons in plane

We reduce the problem of computing a rectilinear shortest path between two given points s and t in the splinegonal domain \calS to the problem of computing a rectilinear shortest path between two points in the polygonal domain. As part of this, we define a polygonal domain \calP from \calS and transform a rectilinear shortest path computed in \calP to a path between s and t amid splinegon obstacles in \calS. When \calS comprises of h pairwise disjoint splinegons with a total of n vertices, excluding the time to compute a rectilinear shortest path amid polygons in \calP, our reduction algorithm takes O(n + h \lg{n}) time. For the special case of \calS comprising of concave-in splinegons, we have devised another algorithm in which the reduction procedure does not rely on the structures used in the algorithm to compute a rectilinear shortest path in polygonal domain. As part of these, we have characterized few of the properties of rectilinear shortest paths amid splinegons which could be of independent interest

Bicriteria Rectilinear Shortest Paths among Rectilinear Obstacles in the Plane

Given a rectilinear domain P of h pairwise-disjoint rectilinear obstacles with a total of n vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points s and t in P. Three types of bicriteria rectilinear paths are considered: minimum-link shortest paths, shortest minimum-link paths, and minimum-cost paths where the cost of a path is a non-decreasing function of both the number of edges and the length of the path. The one-point and two-point path queries are also considered. Algorithms for these problems have been given previously. Our contributions are threefold. First, we find a critical error in all previous algorithms. Second, we correct the error in a not-so-trivial way. Third, we further improve the algorithms so that they are even faster than the previous (incorrect) algorithms when h is relatively small. For example, for computing a minimum-link shortest s-t path, the previous algorithm runs in O(n log^{3/2} n) time while the time of our new algorithm is O(n + h log^{3/2} h)

Planar rectilinear shortest path computation using corridors

AbstractThe rectilinear shortest path problem can be stated as follows: given a set of m non-intersecting simple polygonal obstacles in the plane, find a shortest L1-metric (rectilinear) path from a point s to a point t that avoids all the obstacles. The path can touch an obstacle but does not cross it. This paper presents an algorithm with time complexity O(n+m(lgn)3/2), which is close to the known lower bound of Ω(n+mlgm) for finding such a path. Here, n is the number of vertices of all the obstacles together

L_1 Shortest Path Queries among Polygonal Obstacles in the Plane

Given a point s and a set of h pairwise disjoint polygonal obstacles with a total of n vertices in the plane, after the free space is triangulated, we present an O(n+h log h) time and O(n) space algorithm for building a data structure (called shortest path map) of size O(n) such that for any query point t, the length of the L_1 shortest obstacle-avoiding path from s to t can be reported in O(log n) time and the actual path can be found in additional time proportional to the number of edges of the path. Previously, the best algorithm computes such a shortest path map in O(n log n) time and O(n) space. In addition, our techniques also yield an improved algorithm for computing the L_1 geodesic Voronoi diagram of m point sites among the obstacles

Shortest Paths and Steiner Trees in VLSI Routing

Routing is one of the major steps in very-large-scale integration (VLSI) design. Its task is to find disjoint wire connections between sets of points on a chip, subject to numerous constraints. This problem is solved in a two-stage approach, which consists of so-called global and detailed routing steps. For each set of metal components to be connected, global routing reduces the search space by computing corridors in which detailed routing sequentially determines the desired connections as shortest paths. In this thesis, we present new theoretical results on Steiner trees and shortest paths, the two main mathematical concepts in routing. In the practical part, we give computational results of BonnRoute, a VLSI routing tool developed at the Research Institute for Discrete Mathematics at the University of Bonn. Interconnect signal delays are becoming increasingly important in modern chip designs. Therefore, the length of paths or direct delay measures should be taken into account when constructing rectilinear Steiner trees. We consider the problem of finding a rectilinear Steiner minimum tree (RSMT) that --- as a secondary objective --- minimizes a signal delay related objective. Given a source we derive some structural properties of RSMTs for which the weighted sum of path lengths from the source to the other terminals is minimized. Also, we present an exact algorithm for constructing RSMTs with weighted sum of path lengths as secondary objective, and a heuristic for various secondary objectives. Computational results for industrial designs are presented. We further consider the problem of finding a shortest rectilinear Steiner tree in the plane in the presence of rectilinear obstacles. The Steiner tree is allowed to run over obstacles; however, if it intersects an obstacle, then no connected component of the induced subtree must be longer than a given fixed length. This kind of length restriction is motivated by its application in VLSI routing where a large Steiner tree requires the insertion of repeaters which must not be placed on top of obstacles. We show that there are optimal length-restricted Steiner trees with a special structure. In particular, we prove that a certain graph (called augmented Hanan grid) always contains an optimal solution. Based on this structural result, we give an approximation scheme for the special case that all obstacles are of rectangular shape or are represented by at most a constant number of edges. Turning to the shortest paths problem, we present a new generic framework for Dijkstra's algorithm for finding shortest paths in digraphs with non-negative integral edge lengths. Instead of labeling individual vertices, we label subgraphs which partition the given graph. Much better running times can be achieved if the number of involved subgraphs is small compared to the order of the original graph and the shortest path problems restricted to these subgraphs is computationally easy. As an application we consider the VLSI routing problem, where we need to find millions of shortest paths in partial grid graphs with billions of vertices. Here, the algorithm can be applied twice, once in a coarse abstraction (where the labeled subgraphs are rectangles), and once in a detailed model (where the labeled subgraphs are intervals). Using the result of the first algorithm to speed up the second one via goal-oriented techniques leads to considerably reduced running time. We illustrate this with the routing program BonnRoute on leading-edge industrial chips. Finally, we present computational results of BonnRoute obtained on real-world VLSI chips. BonnRoute fulfills all requirements of modern VLSI routing and has been used by IBM and its customers over many years to produce more than one thousand different chips. To demonstrate the strength of BonnRoute as a state-of-the-art industrial routing tool, we show that it performs excellently on all traditional quality measures such as wire length and number of vias, but also on further criteria of equal importance in the every-day work of the designer

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

Guarding and Searching Polyhedra

Guarding and searching problems have been of fundamental interest since the early years of Computational Geometry. Both are well-developed areas of research and have been thoroughly studied in planar polygonal settings. In this thesis we tackle the Art Gallery Problem and the Searchlight Scheduling Problem in 3-dimensional polyhedral environments, putting special emphasis on edge guards and orthogonal polyhedra. We solve the Art Gallery Problem with reflex edge guards in orthogonal polyhedra having reflex edges in just two directions: generalizing a classic theorem by O'Rourke, we prove that r/2 + 1 reflex edge guards are sufficient and occasionally necessary, where r is the number of reflex edges. We also show how to compute guard locations in O(n log n) time. Then we investigate the Art Gallery Problem with mutually parallel edge guards in orthogonal polyhedra with e edges, showing that 11e/72 edge guards are always sufficient and can be found in linear time, improving upon the previous state of the art, which was e/6. We also give tight inequalities relating e with the number of reflex edges r, obtaining an upper bound on the guard number of 7r/12 + 1. We further study the Art Gallery Problem with edge guards in polyhedra having faces oriented in just four directions, obtaining a lower bound of e/6 - 1 edge guards and an upper bound of (e+r)/6 edge guards. All the previously mentioned results hold for polyhedra of any genus. Additionally, several guard types and guarding modes are discussed, namely open and closed edge guards, and orthogonal and non-orthogonal guarding. Next, we model the Searchlight Scheduling Problem, the problem of searching a given polyhedron by suitably turning some half-planes around their axes, in order to catch an evasive intruder. After discussing several generalizations of classic theorems, we study the problem of efficiently placing guards in a given polyhedron, in order to make it searchable. For general polyhedra, we give an upper bound of r^2 on the number of guards, which reduces to r for orthogonal polyhedra. Then we prove that it is strongly NP-hard to decide if a given polyhedron is entirely searchable by a given set of guards. We further prove that, even under the assumption that an orthogonal polyhedron is searchable, approximating the minimum search time within a small-enough constant factor to the optimum is still strongly NP-hard. Finally, we show that deciding if a specific region of an orthogonal polyhedron is searchable is strongly PSPACE-hard. By further improving our construction, we show that the same problem is strongly PSPACE-complete even for planar orthogonal polygons. Our last results are especially meaningful because no similar hardness theorems for 2-dimensional scenarios were previously known

Attempted reconstruction of design procedures and concepts during the reign of Sultan Qaytbay (872/1468-901/1496) in Jerusalem and Cairo: with special reference to the Madrasa Al-Ashrafiyya and the Minbar in the Khanaqah of Farag Ibn Barquq

Scholars in all areas of Islamic study are fascinated by and are often at pains to underline the unity inherent in Islam. This can be manifested in a multitude of ways, it can be demonstrated against the theological background established by the ,ur'an, or, it may be seen as transcending the barriers enclosing the various arts and sciences of Islam.If it is seen in terms of art and architecture, it can be explained on the grand scale by trading connections built up in times of peace and stability or by forced population movements in the face of conflict, both forces that can transfer from one geographical location to another the artistic traditions, expertise, and techniques previously reserved to the former location but which through their transference are unified with other Islamic traditions.I can see that the nature of unity can be visually expressed by, and is found within, the three broad categories noted by Grabar; but there are others in the field of Islamic art who maintain that a meaning can be attached to a specific design and that this meaning can be verbalised_. I find that I cannot accept that a verbalised meaning can be conveyed by each and every decorative composition, but I find I cannot deny the likeliehood of a visualised one. In fact my hypothesis is that in the notable examples of high cuality iriamluk architecture a visualised meaning does exist. But, just as the expression cf the nuances and form of a piece of music have a greater clarity when in an expos use is main of a musical instrument to illustrate them, rather than an excessive reliance on words to provide the meaning, I believe that architectural experiences are often best expressed in visual terms. For example, later in this Dissertation the stone mi.nbar presented by Sultan tb.y to the Khanaq h of Farag b. Barqúq is analysed in great detail, to my eyes it offers a statement not only on its own symmetry, which is normal for a minbar, but much more it concerns itself with a statement about the symmetry of the Iihtinagah; orally to present my findings and the reasons behind them takes tens of pages compared to a graphical presentation of only five figures.In one sense I admit to this being a final statement, but in another sense it should be the beginning of Islamic architectural appreciation and investigation leading into areas where it may be possible theoretically to reconstitute Islamic works of art from criteria deduced from other known and existing works of art. Those of us who are fortunate enough to work on the evidence at first hand as a job, or, those who have time from other interests for such diversions, must strive to understand and show precisely what was in the minds of the craftsmen and how they detailed their architectural contributions to make a statement. An efficacious route is to choose a high period in the architectural arts sustained either by one bowerful patron or by one atelier over a reasonable ueriod of years. Having thoroughly investigated the period it would then be advantageous to use this fund of knowledge by applying it to the periods immediately before and after the chosen one. However, in my case the first stage has still years of basic cataloguing to do and so I have not embarked on the second stage, nor have I had the opportunity to trace the antecedents of the many decorative elements current at the time of Sultan jytbay.The methods I have used to reach my conclusions are, for the architecture accurate surveys drawn-up at a scale not smaller than 1:50, and for the architectural decoration paper squeezes. (See Appendix A). But I have also consciously developed my own natural visual sequence into a chronology of 'Appreciation Levels'.The method using Appreciation Level is n attempt at recording the precise order in which an initiate assimilates the numerous elements of a design +. design with continual reference back to previous visual exrperiences and mental stimuli. I suggest that in the elements a natural order can be found which, in general terms, assists the efforts of the uninitiated as well as those initiates who can apperceive the visual compatibilities and conflicts reflected in the desin. iith these activities in mind I originally chose the term '?ecognition Levels' before discarding it on the grounds that it implied the item is 'known again' or 'identified as known before', thus I saw the term to be restrictive. On the other hand the word 'Appreciation' is apposite, it may be defined as: estimation; judgement; percebtion; critique. o value; adequate recognition; rise in o each of the words chosen to qualify and define the word 'Appreciation' can be ascribed a function to be attempted by a person when confronted by an object for the first time. This, therefore, is a more accurate term to apply to the ordering of elements.There is, I think, a general thought process connected to :appreciation Levels regardless of the size and type of artifact. The first level is the recognition of the material(s) used in the artifact's construction, assuming that there does exist between the observer and the observed a satisfactory proximity. The second level in this chronology can involve an appreciation of size, of scale and of the architectonic qualities and purposes of the object, leading on perhaps to an appreciation of the over -all shape, and then on to the substructures or repetitious elements which may then lead back to a fuller appreciation of the over -all shape by emphasising it. At this stage in the chronological ordering of the Appreciation Levels a slight change can occur in the type of information perceived, the object's main visual elements come into focus allowing the component parts to be determined and classified, e.g. a geometric skeleton with nodes or the softer curving lines of vegetal patterns.This last Appreciation Level is likely to be the final level attained by the observer and thus it is here that the judgements concerning the beauty and the interest of the artifact are made based on the design content seen in this and the preceding levels. I like to think that as the medieval craftsmen realised the limited visual abilities of the average casual observer so he strived to present the major design elements with the strength, clarity, and meaning which might be com- prehended without excessive mental exertions. _'he prizes await those with more developed powers of observation and perce_,tion in the guise of further appreciation Levels hidden in the subtle End complex relationships of the smaller repetitive elements and whose exploration creates new rythms and harmonies

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop