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28
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied 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 ..."
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Cited by 187 (15 self)
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Introduction A natural and wellstudied 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
The Power of a Pebble: Exploring and Mapping Directed Graphs
 A PRELIMINARY VERSION OF THIS WORK APPEARED IN STOC `98
, 1998
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Exploring Unknown Environments
 SIAM Journal on Computing
, 1997
"... 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 trave ..."
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Cited by 95 (2 self)
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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 subexponential 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...
Tree Exploration with Little Memory
 SODA'02
, 2002
"... A robot with kbit memory has to explore a tree whose nodes are unlabeled and edge ports are locally labeled at each node. The robot has no a priori knowledge of the topology of the tree or of its size, and its aim is to traverse all the edges. While O(log ) bits of memory suce to explore any tre ..."
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Cited by 58 (21 self)
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A robot with kbit memory has to explore a tree whose nodes are unlabeled and edge ports are locally labeled at each node. The robot has no a priori knowledge of the topology of the tree or of its size, and its aim is to traverse all the edges. While O(log ) bits of memory suce to explore any tree of maximum degree if stopping is not required, we show that bounded memory is not sucient to explore with stop all trees of bounded degree (indeed nde log log n) bits of memory are needed for some such trees of size n). For the more demanding task requiring to stop at the starting node after completing exploration, we show a sharper lower bound nd n) on required memory size, and present an algorithm to accomplish this task with O(log n)bit memory, for all nnode trees.
Localizing a Robot with Minimum Travel
, 1995
"... We consider the problem of localizing a robot in a known environment modeled by a simple polygon P . We assume that the robot has a map of P but is placed at an unknown location inside P . From its initial location, the robot sees a set of points called the visibility polygon V of its location. I ..."
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Cited by 48 (3 self)
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We consider the problem of localizing a robot in a known environment modeled by a simple polygon P . We assume that the robot has a map of P but is placed at an unknown location inside P . From its initial location, the robot sees a set of points called the visibility polygon V of its location. In general, sensing at a single point will not suffice to uniquely localize the robot, since the set H of points in P with visibility polygon V may have more than one element. Hence, the robot must move around and use range sensing and a compass to determine its position (i.e.
Collective Tree Exploration
 In: Proc. LATIN 2004. Volume
, 2004
"... An nnode tree has to be explored by k mobile agents (robots), starting in its root. Every edge of the tree must be traversed by at least one robot, and exploration must be completed as fast as possible. Even when the tree is known in advance, scheduling optimal collective exploration turns out t ..."
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Cited by 41 (6 self)
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An nnode tree has to be explored by k mobile agents (robots), starting in its root. Every edge of the tree must be traversed by at least one robot, and exploration must be completed as fast as possible. Even when the tree is known in advance, scheduling optimal collective exploration turns out to be NPhard. We investigate the problem of distributed collective exploration of unknown trees.
Optimal Constrained Graph Exploration
"... We address the problem of constrained exploration of an unknown graph G = (V, E) from a given start node s with either a tethered robot or a robot with a fuel tank of limited capacity, the former being a tighter constraint. In both variations of the problem, the robot can only move along the edges o ..."
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Cited by 25 (1 self)
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We address the problem of constrained exploration of an unknown graph G = (V, E) from a given start node s with either a tethered robot or a robot with a fuel tank of limited capacity, the former being a tighter constraint. In both variations of the problem, the robot can only move along the edges of the graph, i.e, it cannot jump between nonadjacent vertices. In the tethered robot case, if the tether (rope) has length l, then the robot must remain within distance l from the start node s. In the second variation, a fuel tank of limited capacity forces the robot to return to s after traversing C edges. The efficiency of algorithms for both variations of the problem is measured by the number of edges traversed during the exploration. We present an algorithm for a tethered robot which explores the graph in Θ(E) edge traversals. The problem of exploration using a robot with a limited fuel tank capacity can be solved with a simple reduction from the tethered robot case and also yields a Θ(E) algorithm. This improves on the previous best known bound of O(E  + V  log 2 V ) in [4]. Since the lower bound for the graph exploration problems is E, our algorithm is optimal within a constant factor, thus answering the open problem of Awerbuch, Betke, Rivest, and Singh [3].
Going Home Through an Unknown Street
, 1998
"... We present a new strategy for searching for a goal in a street. The strategy works in two phases. First it follows an angular bisector, then it uses circular arcs based only on one side of the street. A competitive factor of 1.514 is achieved which is remarkably close to the lower bound of # 2. ..."
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Cited by 19 (13 self)
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We present a new strategy for searching for a goal in a street. The strategy works in two phases. First it follows an angular bisector, then it uses circular arcs based only on one side of the street. A competitive factor of 1.514 is achieved which is remarkably close to the lower bound of # 2. Secondly, we assume that the location of the goal is known to the robot. We prove a lower bound of # 2 on the competitive ratio of any deterministic strategy for searching in streets with known destination.
Exploring an unknown graph efficiently
 In Proc. 13th Annu. European Sympos. Algorithms
, 2005
"... Abstract. We study the problem of exploring an unknown, strongly connected directed graph. Starting at some node of the graph, we must visit every edge and every node at least once. The goal is to minimize the number of edge traversals. It is known that the competitive ratio of online algorithms for ..."
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Cited by 15 (1 self)
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Abstract. We study the problem of exploring an unknown, strongly connected directed graph. Starting at some node of the graph, we must visit every edge and every node at least once. The goal is to minimize the number of edge traversals. It is known that the competitive ratio of online algorithms for this problem depends on the deficiency d of the graph, which is the minimum number of edges that must be added to make the graph Eulerian. We present the first deterministic online exploration algorithm whose competitive ratio is polynomial in d (it is O(d 8)). 1
Competitive complexity of mobile robot on line motion planning problems
 In WAFR
, 2004
"... This paper is concerned with online problems where a mobile robot of size D has to achieve a task in an unknown planar environment whose geometry is acquired by the robot during task execution. The critical parameter in such problems is physical motion time which corresponds to length or cost of th ..."
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Cited by 10 (1 self)
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This paper is concerned with online problems where a mobile robot of size D has to achieve a task in an unknown planar environment whose geometry is acquired by the robot during task execution. The critical parameter in such problems is physical motion time which corresponds to length or cost of the path traveled by the robot. The competitiveness of an online algorithm measures its performance relative to the optimal o ® line solution to the problem. While competitiveness usually means constant relative performance, this paper generalizes competitiveness to any functional relationship between online performance and optimal o®line solution. Given an online task, its competitive complexity class is a pair of lower and upper bounds on the competitive performance of all online algorithms for the task, such that the two bounds satisfy the same functional relationship. We classify some common online motion planning problems into competitive classes. In particular, it is shown that navigation to a target whose position is either apriori known or recognized only upon arrival belongs to a quadratic competitive class. The hardest online problems belong to exponential and even nonboundable competitive classes. We present examples of such problems, which involve navigation in unknown variable traversibility environments. 1