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

In this paper, we show that the universal covering space of a surface can be used to unify previous results on computing paths in a simple polygon. We optimize a given path among obstacles in the plane under the Euclidean and link metrics and under polygonal convex distance functions. Besides revealing connections between the minimum paths under these three distance functions, the framework provided by the universal cover leads to simplified linear-time algorithms for shortest path trees, for minimum-link paths in simple polygons, and for paths restricted to c given orientations. 1 Introduction If a wire, a pipe, or a robot must traverse a path among obstacles in the plane, then one might ask what is the best route to take. For the wire, perhaps the shortest distance is best; for the pipe, perhaps the fewest straight-line segments. For the robot, either might be best depending on the relative costs of turning and moving. In this paper, we find shortest paths and shortest closed curve...

We give algorithms to find shortest paths homotopic to given disjoint paths that wind amongst n point obstacles in the plane. Our deterministic algorithm runs in time O(k log n+n √n), and the randomized version in time O(k log n+n(log n)1(1+ε)), where k is the input plus output sizes of the paths.

Traditionally, graph drawing algorithms represent vertices as circles and edges as curves connecting the vertices. We introduce the problem of drawing with “fat ” edges, i.e., with edges of variable thickness. The thickness of an edge is often used as a visualization cue, to indicate importance, or to convey some additional information. We present a model for drawing with fat edges and a corresponding polynomial time algorithm that uses the model. We focus on a restricted class of graphs that occur in VLSI wire routing and show how to extend the algorithm to general planar graphs. We show how to convert an arbitrary wire routing into a homotopically equivalent routing that maximizes the distance between any two wires. Among such, we obtain the routing with minimum total wire length. A homotopically equivalent routing that maximizes the distance between any two wires yields a graph drawing which maximizes edge thickness. Finally, our algorithm also allows for different edge weights, that is, the requirement for unit wire thickness can be removed.

INTRODUCTION We present an algorithm for growing fat graphs. Traditionally, graph drawing algorithms represent vertices as circles and edges as closed curves connecting the vertices. The thickness of an edge is often used as a visualization cue, to indicate importance, or to convey some additional information. We show how to grow fat graphs with edges of variable thickness. For the purpose of the demonstration we focus on a restricted class of graphs that occur in VLSI wire routing. This class corresponds to planar, max-degree-1 graphs. The underlying algorithm also extends to general planar graphs as shown in [2]. In VLSI wire routing it is often desirable to maximize the distance between di#erent wires. Maximizing the distance between wires is equivalent to finding the drawing in which the edges are drawn as thick as possible, i.e., allowing the graph to grow as fat as possible. The continuous homotopic routing problem [1, 3, 5] is a classic VLSI problem. The input is an initial sk

We study the problem of finding shortest non-crossing thick paths in a polygonal domain, where a thick path is the Minkowski sum of a usual (zero-thickness, or thin) path and a disk. Given K pairs of terminals on the boundary of a simple n-gon, we compute in O(n + K) time a representation of the set of K shortest non-crossing thick paths joining the terminal pairs; using the representation, any particular path can be output in time proportional to its complexity. We compute K shortest thick non-crossing paths in a polygon with h holes in O ` (K + 1) h h! poly(n, K) ´ time, using an efficient method to compute any one of the K thick paths if the “threadings ” of all paths amidst the holes are specified. We show that if h is not constant, the problem is NP-hard; we also show the hardness of approximation. We give a pseudopolynomial-time algorithm for some rectilinear versions of the problem. We apply our thick paths algorithms to obtain the first algorithmic results for the minimum-cost continuous flow problem — an extension of the standard discrete minimumcost network flow problem to continuous domains. The results are based on showing a continuous analog of the Network

Path planning, a topic of much interest in military planning, is largely treated as the task of finding the best path with respect to some criterion such as length, travel time, etc, for which efficient algorithms are already available. Military planning requires understanding enemy intentions and devising unexpected plans to fox the enemy which calls for not a best path, but a number of alternative paths. We study the problem of computing multiple paths with different properties, such as all paths with at most L loops, in free space among polygonal obstacles using a framework of Voronoi diagram. The complexity of the algorithms have been analyzed. We show that the Voronoi diagram, though widely used, is inadequate to represent certain important properties of representative paths in free space. Further, we show how this framework might be applied in three different military problems – entity reidentification, ambush analysis, and rapid re-routing in urban operations. 1.

Let P be a path between two points s and t in a polygonal subdivision T with obstacles and weighted regions. Given a relative error tolerance ε ∈ (0,1), we present the first algorithm to compute a path between s and t that can be deformed to P without passing over any obstacle and the path cost is within a factor 1+ε of the optimum. The running time is O ( h3 ε2 knpolylog(k,n, 1)), where k is the number of segments in P and ε h and n are the numbers of obstacles and vertices in T, respectively. The constant in the running time of our algorithm depends on some geometric parameters and the ratio of the maximum region weight to the minimum region weight.

We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimum-weight cycles in a given homotopy or homology class, and computing minimum-weight cycle bases for the fundamental group or various homology groups.