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

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We consider the problem faced by a robot that must explore and learn an unknown room with obstacles in it. We seek algorithms that achieve a bounded ratio of the worst-case distance traversed in order to see all visible points of the environment (thus creating a map), divided by the optimum distance needed to verify the map, if we had it in the beginning. The situation is complicated by the fact that the latter off-line problem (the problem of optimally verifying a map) is NP-hard. Although we show that there is no such "competitive" algorithm for general obstacle courses, we give a competitive algorithm for the case of a polygonal room with a bounded number of obstacles in it. We restrict ourselves to the rectilinear case, where each side of the obstacles and the room is parallel to one of the coordinates, and the robot must also move either parallel or perpendicular to the sides. (In a subsequent paper, we will discuss the extension to polygons of general shapes.) We also discuss t...

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 show that two cooperating robots can learn ex-actly any strongly-connected directed graph with n in-distinguishable nodes in expected tame polynomial in n. We introduce a new type of homing sequence for two robots which helps the robots recognize certain previously-seen nodes. We then present an algorithm in which the robots learn the graph and the homing se-quence simultaneously by wandering actively through the graph. Unlike most previous learning results us-ang homing sequences, our algorithm does not require a teacher to provide counterexamples. Furthermore, the algorithm can use efficiently any additional infor-mation available that distinguishes nodes. We also present an algorithm in which the robots learn by tak-ing random walks. The rate at which a random walk converges to the stationary distribution is character-ized by the conductance of the graph. Our random-walk algorithm learns in expected time polynomial in n and in the inverse of the conductance and is more eficient than the homing-sequence algorithm for high-conductance graphs. 1

Searching for a goal is a central and extensively studied problem in computer science. In classical searching problems, the cost of a search function is simply the number of queries made to an oracle that knows the position of the goal. In many robotics problems, as well as in problems from other areas, we want to charge a cost proportional to the distance between queries (e.g., the time required to travel between two query points). With this cost function in mind, the abstract problem known as the w-lane cow-path problem was designed. There are known optimal deterministic algorithms for the cow-path problem, and we give the first randomized algorithm in this paper. We show that our algorithm is optimal for two paths (w = 2), and give evidence that it is optimal for larger values of w. Subsequent to the preliminary of version of this paper, Kao, Ma, Sipser, and Yin [10] have shown that our algorithm is indeed optimal for all w 2. Our randomized algorithm gives expected performance tha...

This paper analyzes the complexity of on-line reinforcement learning algorithms, namely asynchronous realtime versions of Q-learning and value-iteration, applied to the problem of reaching a goal state in deterministic domains. Previous work had concluded that, in many cases, tabula rasa reinforcement learning was exponential for such problems, or was tractable only if the learning algorithm was augmented. We show that, to the contrary, the algorithms are tractable with only a simple change in the task representation or initialization. We provide tight bounds on the worst-case complexity, and show how the complexity is even smaller if the reinforcement learning algorithms have initial knowledge of the topology of the state space or the domain has certain special properties. We also present a novel bidirectional Q-learning algorithm to find optimal paths from all states to a goal state and show that it is no more complex than the other algorithms.

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

A robot with k-bit 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 n-node trees.

Consider a networked environment, supporting mobile agents, where there is a black hole: a harmful host that disposes of visiting agents upon their arrival, leaving no observable trace of such a destruction. The black hole search problem is the one of assembling a team of asynchronous mobile agents, executing the same protocol and communicating by means of whiteboards, to successfully identify the location of the black hole; we are concerned with solutions that are generic (i.e., topology-independent). We establish tight bounds on the size of the team (i.e., the number of agents), and the cost (i.e., the number of moves) of a size-optimal solution protocol. These bounds depend on the a priori knowledge the agents have about the network, and on the consistency of the local labellings. In particular, we prove that: with topological ignorance ∆ + 1 agents are needed and suffice, and the cost is Θ(n 2), where ∆ is the maximal degree of a node and n is the number of nodes in the network; with topological ignorance but in presence of sense of direction only two agents suffice and the cost is Θ(n 2); and with complete topological knowledge only two agents suffice and the cost is Θ(n log n). All the upper-bound proofs are constructive.