Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal

We give new methods for maintaining a data structure that supports ray shooting and shortest path queries in a dynamically-changing connected planar subdivision S. Our approach is based on a new dynamic method for maintaining a balanced decomposition of a simple polygon via geodesic triangles. We maintain such triangulations by viewing their dual trees as balanced trees. We show that rotations in these trees can be implemented via a simple "diagonal swapping" operation performed on the corresponding geodesic triangles, and that edge insertion and deletion can be implemented on these trees using operations akin to the standard split and splice operations. We also maintain a dynamic point location structure on the geodesic triangulation, so that we may implement ray shooting queries by first locating the ray's endpoint and then walking along the ray from geodesic triangle to geodesic triangle until we hit the boundary of some region of S. The shortest path between two points in the same ...

We describe a new and simple method for constructing binary space partitions in arbitrary dimensions. We also introduce the concept of uncluttered scenes, which are scenes with a certain property that we suspect many realistic scenes exhibit, and we show that our method constructs a BSP of size O(n) for an uncluttered scene consisting of n objects. The construction time is O(n log n). Because any set of disjoint fat objects is uncluttered, our result implies an efficient method to construct a linear size BSP for fat objects. We use our BSP to develop a data structure for point location in uncluttered scenes. The query time of our structure is O(log n), and the amount of storage is O(n). This result can in turn be used to perform range queries with not-too-small ranges in scenes consisting of disjoint fat objects or, more generally, in so-called low-density scenes. 1 Introduction Many geometric problems can be solved more easily if a decomposition of the space of interest in...

A pair of points s and g on the boundary of a simple polygon P admits a walk if two guards can simultaneously walk along the two boundary chains of P from s to g such that they are always visible to each other. The walk is a counter-walk if one guard moves from s to g while the other moves from g to s in the same direction along the boundary and they are always visible to each other. The (counter-)walk is straight if no backtracking is necessary during the (counter-)walk. In this paper, we show that, given a polygon with n vertices, to test if there exists (s; g) that admits a (straight) (counter-)walk can be solved in time O(n log n) and in linear space. Also we compute all (s; g)'s that admit a (straight) walk in O(n log n) time and all vertex pairs that admit a (straight) counter-walk in O(n log n +m), where m is O(n 2 ). Keywords: Visibility, watchman routes, motion planning, simple polygon, circular-arc graph. 1. Introduction Given a simple polygon P , a pair of distinct poi...

Given a sequence of k polygons in the plane, a start point s, and a target point, t, we seek a shortest path that starts at s, visits in order each of the polygons, and ends at t. If the polygons are disjoint and convex, we give an algorithm running in time O(kn log(n/k)), where n is the total number of vertices specifying the polygons. We also extend our results to a case in which the convex polygons are arbitrarily intersecting and the subpath between any two consecutive polygons is constrained to lie within a simply connected region; the algorithm uses O(nk log n) time. Our methods are simple and allow shortest path queries from s to a query point t to be answered in time O(k log n + m), where m is the combinatorial path length. We show that for nonconvex polygons this "touring polygons" problem is NP-hard.

Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide linear-time algorithms for constructing optimal area drawings. Let T be a bounded-degree rooted tree with N nodes. Our results are summarized as follows: ffl We show that T admits a planar polyline upward grid drawing with area O(N ), and with width O(N ff ) for any prespecified constant ff such that 0 ! ff ! 1. ffl If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O(N log log N ). ffl We show that if T is ordered, it admits an O(N log N)-area planar upward grid drawing that preserves the left-to-right ordering of the children of each node. ffl We show that all of the above area bounds are asymptotically optimal in the worst case. ffl ...

Two of the fundamental questions that arise in the manufacturing industry concerning every type of manufacturing process are: 1. Given an object, can it be built using a particular process? 2. Given that an object can be built using a particular process, what is the best way to construct the object? The latter question gives rise to many different problems depending on how best is qualified. We address these problems for two complimentary categories of manufacturing processes: rapid prototyping systems and casting processes. The method we use to address these problems is to first define a geometric model of the process in question and then answer the questions on that model. In the category of rapid prototyping systems, we concentrate on stereolithography, which is emerging as one of the most popular rapid prototyping systems. We model stereolithography geometrically and then study the class of objects that admit a construction in this model. For the objects that admit a constructio...

We propose a strategy to decompose a polygon, containing zero or more holes, into “approximately convex” pieces. For many applications, the approximately convex components of this decomposition provide similar benefits as convex components, while the resulting decomposition is significantly smaller and can be computed more efficiently. Moreover, our approximate convex decomposition (ACD) provides a mechanism to focus on key structural features and ignore less significant artifacts such as wrinkles and surface texture. We propose a simple algorithm that computes an ACD of a polygon by iteratively removing (resolving) the most significant non-convex feature (notch). As a by product, it produces an elegant hierarchical representation that provides a series of ‘increasingly convex ’ decompositions. A user specified tolerance determines the degree of concavity that will be allowed in the lowest level of the hierarchy. Our algorithm computes an ACD of a simple polygon with n vertices and r notches in O(nr) time. In contrast, exact convex decomposition is NP-hard or, if the polygon has no holes, takes O(nr 2) time. Models and movies can be found on our web-pages at:

. In this paper we explore some novel aspects of visibility for stationary and moving points inside a simple polygon P . We provide a mechanism for expressing the visibility polygon from a point as the disjoint union of logarithmically many canonical pieces using a quadratic-space data structure. This allows us to report visibility polygons in time proportional to their size, but without the cubic space overhead of earlier methods. The same canonical decomposition can be used to determine visibility within a frustum, or to compute various attributes of the visibility polygon efficiently. By exploring the connection between visibility polygons and shortest path trees, we obtain a kinetic algorithm that can track the visibility polygon as the viewpoint moves along polygonal paths inside P , at a polylogarithmic cost per combinatorial change in the visibility. The combination of the static and kinetic algorithms leads to a space query-time tradeoff for the visibility from a point problem ...

We present here numerical and combinatorial methods that permit the use of exact arithmetic in the construction of unions and intersection of polygonal regions. An argument is given that, even in an exact arithmetic system, rounding of coordinates is necessary. We also argue that it is natural and useful to round to a nonuniform grid, and we give methods for calculating the nearest grid point. The main result is a shortest path rounding algorithm that restores the combinatorial consistency of a polygon after its vertices have been rounded. This algorithm runs in linear time in the number of "near" vertex-edge pairs. It is optimal in the sense that it introduces the minimum combinatorial and geometric changes. We know of no other bounded-error rounding algorithm for nonuniform grids. 1 Introduction There are many useful applications of set operations on polygons: graphics, maps, CAD/CAM, etc. There are efficient theoretical algorithms for performing the set operations: union, intersec...