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

Exploring and mapping an unknown environment is a fundamental problem, which is studied in various contexts. Many works have focused on finding efficient solutions to restricted versions of the problem. In this paper, we consider a model that makes very limited assumptions on the environment and solve the mapping problem in this general setting. We model

We consider exploration problems where a robot has to construct a complete map of an unknown environment. We assume that the environment is modeled by a directed, strongly connected graph. The robot's task is to visit all nodes and edges of the graph using the minimum number R of edge traversals. Koutsoupias [16] gave a lower bound for R of #(d 2 m), and Deng and Papadimitriou [12] showed an upper bound of d O(d) m, where m is the number edges in the graph and d is the minimum number of edges that have to be added to make the graph Eulerian. We give the first sub-exponential algorithm for this exploration problem, which achieves an upper bound of d O(logd) m. We also show a matching lower bound of d #(logd) m for our algorithm. Additionally, we give lower bounds of 2 #(d) m, resp. d #(logd) m for various other natural exploration algorithms. 1 Introduction Suppose that a robot has to construct a complete map of an unknown environment using a path that is as sho...

We study scheduling problems in battery-operated computing devices, aiming at schedules with low total energy consumption. While most of the previous work has focused on finding feasible schedules in deadline-based settings, in this article we are interested in schedules that guarantee good response times. More specifically, our goal is to schedule a sequence of jobs on a variable-speed processor so as to minimize the total cost consisting of the energy consumption and the total flow time of all jobs. We first show that when the amount of work, for any job, may take an arbitrary value, then no online algorithm can achieve a constant competitive ratio. Therefore, most of the article is concerned with unit-size jobs. We devise a deterministic constant competitive online algorithm and show that

We study exploration problems where a robot has to construct a complete map of an unknown environment using a path that is as short as possible. In the first problem setting we consider, a robot has to explore n rectangles. We show that no deterministic or randomized online algorithm can be better than\Omega\Gamma p n)-competitive, solving an open problem by Deng, Kameda and Papadimitriou [5]. We also generalize this bound to the problem of exploring three-dimensional rectilinear polyhedra without obstacles. In the second problem setting we study, a robot has to explore a grid graph with obstacles in a piecemeal fashion. The piecemeal constraint was defined by Betke, Rivest and Singh [3] and implies that the robot has to return a start node every so often. Betke et al. gave an efficient algorithm for exploring grids with rectangular obstacles. We present an efficient strategy for piecemeal exploration of grids with arbitrary obstacles. 1 Introduction In robot exploration problems, a...

We consider online routing algorithms for finding paths between the vertices of plane graphs.

We present a competitive strategy for walking into the kernel of an initially unknown star-shaped polygon. From an arbitrary start point, s, within the polygon, our strategy finds a path to the closest kernel point, k, whose length does not exceed 5.3331 ...times the distance from s to k. This is complemented by a general lower bound of # 2. Our analysis relies on a result about a new and interesting class of curves which are self-approaching in the following sense.

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We present an on-line strategy that enables a mobile robot with vision to explore an unknown simple polygon. We prove that the resulting tour is less than 26.5 times as long as the shortest watchman tour that could be computed o#-line.

We consider the problem of a robot which has to find a target in an unknown simple polygon, based only on what it has seen so far. A street is a polygon for which the two boundary chains from start to target are mutually weakly visible. A target inside a street can be found by walking a path that is at most a constant times longer than the shortest path in the street from start to target. We define a strictly larger class of polygons, called generalized streets or G-streets, which are characterized by the property that every point on the boundary of a G-street is visible from a point on a horizontal line segment connecting the two boundary chains. We present an on-line strategy for a robot to find the target in an unknown rectilinear G-street; the length of its path is at most 9 times the length of the shortest path in the L 1 metric, and 9.06 times the length of the L 2 -shortest path. These bounds are optimal. Key words: Simple polygon, street, searching, doubling, competitive...