X. Deng and C. Papadimitriou. Exploring an unknown graph. In Proc. 31st IEEE FOCS, pages 355--361, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....assumed, e.g. unknown terrain with convex obstacles [10] or room with polygonal [14] or rectangular [5] obstacles. Another approach is to model the environment as a graph, assuming that the robot may only move along its edges. The graph setting can be further speci ed in two di erent ways. In [1, 7, 8, 15] the robot explores strongly connected directed graphs and it can move only in the direction from head to tail of an edge, not vice versa. In [3, 11, 16, 24] the explored graph is undirected and the robot can traverse edges in both directions. In the graph setting it is often required that apart ....

....scenarios considered in the literature di er in an important way: it is either assumed that nodes of the graph have unique labels which the robot can recognize, or it is assumed that nodes are anonymous. Exploration of directed graphs assuming the existence of labels was investigated in [1, 15]. In this case no restrictions on the robot moves were imposed, other than by directions of edges, and fast exploration and mapping algorithms were sought. Exploration of undirected labeled graphs was considered in [3, 11, 16, 24] Since in this case a simple exploration based on depth rst ....

X. Deng and C. H. Papadimitriou, Exploring an unknown graph, Journal of Graph Theory 32 (1999), 265-297.

....a given problem P to be solved. Example of this type of investigations are the studies on topology reconstruction: E is typically a single entity (sometimes two) G is unknown to the entity, and P is the construction of a map of the graph (e.g. see [2, 5, 6] Other examples are graph exploration [7, 8, 11], wake up [1, 15] black hole search [9, 10] searching for a mobile intruder [17] etc. In this paper, we focus on a fundamental problem in distributed mobile computing: election, that is the process by which a group of autonomous asynchronous mobile entities initially in the same state and ....

X. Deng and C. H. Papadimitriou. Exploring an unknown graph. Journal of Graph Theory, 32:265-297, 1999.

....dout(v) Communicated by S. Khuller: submitted January 2002; revised June 2002 Work by S. Even supported by the Fund for the Promotion of Research at the Technion. 1 Introduction and Model The question of how best to traverse an unknown maze, given only local knowledge, has been long studied. [8, 3, 5, 7, 6, 1, 4, 2]. Without the ability to store some knowledge about the maze the searcher can easily become trapped in cycles. In this paper we consider the case of directed Eulerian mazes graphs, and show that a finite state automaton, with access to some local memory stored in the vertices, can find an ....

....a method to e#ectively backtrack on directed edges. Their method uses O( V E log V ) edge traversals, and requires that vertices are labeled with distinct names. We shall later say more about the comparison of our results with those of Afek and Gafni. Deng and Papadimitriou [4] also use a more powerful model than we do. Their purpose is to explore the graph; i.e. to discover its structure. They assume that vertices have distinct identities and that the robot is a general purpose computer (Turing equivalent) They describe a recursive algorithm which finds an Eulerian ....

X. Deng and C.H. Papadimitriou, Exploring an Unknown Graph, J. of Graph Th., Vol. 32, No. 3, 1999, pp. 265-297.

....investigation is part of an ongoing research e ort on understanding the algorithmic limits of computing with mobile agents. Among the problems being attacked, network exploration in all its variants (e.g. search, map drawing, map veri cation, etc. has de nitely played a central role (e.g. see [1, 4, 7, 9, 10, 11, 21]) As for the election problem, its study in environments other than the traditional distributed ones is currently being carried out. Such is the case for radio networks, ad hoc networks, etc. e.g. see [14, 18, 19] However, they all assume comparable labels. 2. A NECESSARY CONDITION FOR ....

X. Deng and C. Papadimitriou, Exploring an unknown graph. Journal of Graph Theory, 32:265-297, 1999.

....they visit all vertices of the graph. One can just stop the exploration when one hits a goal state. The following algorithm has been used earlier as part of proofs on Eulerian tours, for example by Hierholzer[2] Recently it was re considered by several researchers as a graph learning algorithm [1, 4]: Building a Eulerian Tour Algorithm (BETA) Traverse unexplored edges whenever possible (ties can be broken arbitrarily) If all edges emanating from the current vertex have been explored, execute the initial sequence of edge traversals again, this time stopping at all vertices that have ....

....procedures. BETA is similar to DepthFirst Search in some sense, but instead of backtracking its latest moves when it gets stuck it repeats the initial walk, because backtracking is not always possible on arbitrary Eulerian graphs. BETA executes every action at most twice (see, for example [1]) which implies the following theorem: Theorem 1 BETA explores any Eulerian unknown graph with the complexity of at most two weights of the graph. Since BETA can be applied to the treasure hunt problem too, it solves the problem with the complexity that is linear on the weight of the graph. ....

[Article contains additional citation context not shown here]

X. DENG AND C. PAPADIMITRIOU, Exploring an unknown graph, in Proceedings of the Symposium on Foundations of Computer Science, 1990, pp. 355--361.

....we call Greedy Mapping. Then we explain how to model it as a graph search problem and use D Lite to solve it. 3 Greedy Mapping Mapping is an important task for mobile robots and a large number of mapping methods have been developed for them, both in robotics and in theoretical computer science [9, 20, 13, 18, 3, 8, 12, 21, 23, 29, 2, 10, 11, 19, 30, 27, 6, 24, 25]. A good overview is given in [31] In this paper, we study Greedy Mapping, a simple sensor based planning method that discretizes terrain into cells and then always moves the robot from its current cell to the closest cell with unknown blockage status, that is, to the closest unobserved cell, ....

X. Deng and C. Papadimitriou. Exploring an unknown graph. In Proceedings of the Symposium on Foundations of Computer Science, pages 355--361, 1990.

....and learn about the environment as they progress. The complexity measure associated with such an algorithm is the ratio of the total distance traversed by the algorithm to the length of the shortest source target path. Related work on exploring graphs with incomplete information is considered in [DP]. This measure is closely related to the concept of competitive analysis, introduced by Sleator and Tarjan [ST] which gives a worst case complexity measure for on line algorithms. An on line algorithm is an algorithm which must deal with a sequence of events, responding to events in real time ....

X. Deng and C.H. Papadimitriou. Exploring an Unknown Graph. In Proc. 31st IEEE FOCS, pages 355--361, October 1990.

....learning theory. The model of the environment is a deterministic finite automaton (DFA) Rivest and Schapire s algorithms infer the structure of DFAs correctly with high probability. Related work on inferring the structure of DFAs is done by Basye, Dean, and Kaelbling [12] Deng and Papadimitriou [40] and Betke [15] address the problem of exploring unknown directed graphs that are characterized by their deficiency, i.e. the distance from being Eulerian graphs. Directed graphs may model cities with one way streets. 18 The difficulty in learning directed graphs lies in the fact that the ....

....competitive algorithm. For polygons of arbitrary shape and with a bounded number of obstacles they show that there exists a competitive algorithm. However, the only ratio that they can prove is huge. Model Properties Results Authors directed deficiency measures distance competitive algorithm [40] graphs from being Eulerian graphs, of ratio 4 [15] distinguishable nodes 2 cooperating learners, algorithm with expected [14] indistinguishable nodes time polynomial in number of nodes undirected piecemeal learning graphs grid graphs with linear algorithms [19] rectangular obstacles ....

Xiaotie Deng and Christos H. Papadimitriou. Exploring an unknown graph. In Proceedings of the 31st Symposium on Foundations of Computer Science, volume I, pages 355--361, 1990.

....state has some probability of being incorrect. They give an algorithm for learning finite automata, assuming that the robot has access to a distinguishing sequence. Freund et al. 43] give algorithms for learning typical deterministic finite automata from random walks. Deng and Papadimitriou [35] and Betke [16] model the robot s environment as a directed graph, with distinct and recognizable vertices and edges. They give a learning algorithm with a constant competitive ratio when the graph is Eulerian or when the deficiency of the graph is 1. For general graphs, they give a competitive ....

Xiaotie Deng and Christos H. Papadimitriou. Exploring an unknown graph. In Proceedings of the 31st Symposium on Foundations of Computer Science, volume I, pages 355--361, 1990.

....restaurant finding problem, routing under faults corresponds to the problem where the exact address of the restaurant is known and one wishes avoid roadblocks or traffic jams. Other related lines of research including searching for a fugitive in a graph [4, 12, 15] or exploring an unknown graph [2, 7, 16]. In neither case do the nodes contain information of the type considered here that might direct the search. In the restaurant analogy, there are no informed policeman. We feel our work is closer in spirit to that of computing with uncertainty or with noisy computing elements [9, 10, 13, 17] An ....

X. Deng and C. Papadimitriou, "Exploring an unknown graph," FOCS 90, 356-361.

....[9] give a k trip shortest path algorithm. The exploration can also be modeled with a nite state automata as shown by Rivest and Schapire [18] or as a graph. In the later case the graph can be directed or undirected. The exploration is modeled by a directed graph by Deng and Papadimitriou [13] and Albers and Henzinger [1] Bender and Slonim [6] consider the problem with two cooperating robots. Randomized robot navigation was addressed by Berman et al. 7] In situations corresponding to physical settings the exploration is modeled by an undirected graph. Without the piecemeal ....

X. Deng and C. H. Papadimitriou. Exploring an unknown graph. In Proceedings of the 31st Symposium on Foundations of Computer Science (FOCS), pages 355-361,

....tables under network failures (surviving graph [23] In many cases it is not possible to de ne a well structured topology or even the topology may 3 be unknown or unstable like in the internet. Theoretical results and heuristic algorithms can also be obtained for generic or even unknown graphs [22]. We have focused on some important parameters to be considered in order to choose a network topology. There are, however, many other aspects we have not considered here and may also be important in some cases, like, for instance, the ecient layout of the network in the case of VLSI systems ....

X. Deng and C.H. Papadimitriou. Exploring an unknown graph. Journal of Graph Theory, 32:265-297, 1999.

.... indegree is greater than its outdegree; its deficiency d(v) is d(v) 8 : indeg(v) Gamma outdeg(v) if v is deficient 0 if v is not deficient The minimum number of edges needed to add to G to make it Eulerian is called the deficiency of G (see Kutten [4] or Deng and Papadimitriou [2], denoted by d. Clearly, d = X v2V d(v) A node is exhausted if we have explored all outgoing edges. If we reach an exhausted node v then we are stuck at v. Theorem 2 ( 6] DFS traverses at most minfmn; d 1)n(n Gamma 1) mg number of edges on a graph with n nodes, m edges, and deficiency ....

....at most dn times. Each time we get stuck we must backtrack at a cost of at most n Gamma 1. So the total number of edge traversals is at most (d 1)n(n Gamma 1) m. ut For dense graphs with d = Omega (n 2 ) DFS needs at most O(dm) edge traversals, so it is d competitive. It is conjectured [2] that there exists an exploration algorithm which is poly(d) competitive on all graphs with deficiency d. So far, only a d O(d) competitive [2] and a d O(log d) competitive [1] algorithm are known (and both algorithms are really complicated) We now turn to lower bounds. Theorem 3 ( 2] In ....

[Article contains additional citation context not shown here]

X. Deng and C. H. Papadimitriou. Exploring an unknown graph. Journal of Graph Theory, 32:265--297,

.... [Ark90, EM92] Kortenkamp and Weymouth considered the use of multiple sensing modalities to instantiate the nodes of a graph like (topological) representation [KW94] Other work has also considered the theoretical issues involved in fully covering an unknown graph using topological exploration [DP90]. In previous work it has been observed that while topological mapping with ambiguous signatures with absolute certainty is infeasible, the use of a single movable marker allows e#cient mapping. With one marker, a mapping algorithm is possible that uses a number of robot steps which is a ....

Xiaotie Deng and Christos Papadimitriou. Exploring an unknown graph. In Annual Symposium on the Foundations of Computer Science, pages 335--361, 1990.

....For example, the learner might be a security guard robot, a taxi driver, or a trail guide. Exploration of unknown environments has been addressed by many previous authors, such as Papadimitriouand Yanakakis [10] Blum, Raghavan, and Schieber [4] Rivest and Schapire [12] Deng and Papadimitriou [7], Betke [3] Deng, Kameda, and Papadimitriou [6] Rao, Kareti, Shi and Iyengar [11] and Bar Eli, Berman, Fiat, and Yan [2] This paper considers a new constraint: for some reason learning must be done piecemeal that is, a little at a time. For example, a rookie taxi driver might learn a city ....

Deng, Xiaotie and Christos H. Papadimitriou. "Exploring an Unknown Graph,"Proceedings of the 31st Symposium on Foundations of Computer Science, 1990, pp. 355--361.

....Many researchers have studied problems in environment learning and robot motion planning. Papadimitriou and Yanakakis [19] developed one of the first formal models for exploring unknown environments. They show how to find a shortest path in an unknown, undirected graph. Deng and Papadimitriou [13] and Betke [6] address the problem of learning an unknown directed graph. Bender and Slonim [5] show how two cooperating robots can learn a directed graph. Rivest and Schapire [21] model the robot s unknown environment by a deterministic finite automaton. They describe algorithms that efficiently ....

Xiaotie Deng and Christos H. Papadimitriou. Exploring an unknown graph. In Proceedings of the 31st Symposium on Foundations of Computer Science, volume I, pages 355--361, 1990.

....is visited. They address the problem of devising a strategy which guarantees a given ratio to the shortest path. This problem is shown to be PSPACE complete, which indicates that this problem is computationally very hard (see [2] for details on PSPACE complete problems) Deng and Papadimitriou [21] consider the following problem of exploring a directed and strongly connected graph. At each point we have a map of all nodes and edges that have been visited, and these nodes and edges can be recognized if they are visited again. We know how many unexplored edges emanate form each node we have ....

....of the total number of edges traversed divided by the optimum number of edge traversals, had we known the graph. A graph is Eulerian if there exists a path that visits each edge precisely once. For Eulerian graphs the ratio cannot be better 45 than two and can be achieved by a simple algorithm [21]: a) take unexplored edges whenever possible; b) if stuck, consider the closed walk of unexplored edges just completed, and retrace it, stopping at nodes that have unexplored edges, and apply this algorithm recursively from each such node. The deficiency of a graph is the number of edges to be ....

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X. Deng and C. H. Papadimitriou. Exploring an unknown graph. In Proc. 31st Ann. Symp. on Foundations of Computer Science, pages 355--361, October 1990. 49

....of prior knowledge about parts of the terrain (if available) and can be used by several robots cooperatively. 1 Introduction Mapping is an important task for mobile robots and a large number of mapping methods have been developed for them, both in robotics and in theoretical computer science [7, 18, 11, 15, 3, 6, 10, 19, 20, 24, 2, 8, 9, 16, 25, 23, 4, 21, 22]. A good overview is given in [26] In this paper, we show that greedy mapping methods are easy to implement and easy to integrate into complete robot architectures. At the same time, planning is efficient and results in short travel distances of the robot. We study Greedy Mapping, a simple ....

X. Deng and C. Papadimitriou. Exploring an unknown graph. In Proceedings of the Symposium on Foundations of Computer Science, pages 355--361, 1990.

....changes, or where certain paths become blocked or unblocked with some regularity. It would be useful to discover and make use of such regularities in navigation and exploration. 3 Related Work Graph based world models have long been used in theoretical work on a priori map learning, such as [27, 15, 25, 1]. These models have also been extended to include sensor and effector noise [4, 13] The main problem that is addressed in such work is that of figuring out how to distinguish different nodes that look the same to the agent. This is typically done by discovering distinguishing sequences which ....

X. Deng and C. H. Papadimitriou. Exploring an unknown graph. In IEEE, editor, Proceedings of the 31st Annual Symposium on Foundations of Computer Science, pages 355--361. IEEE Computer Society Press, Oct. 1990.

....mapping an unknown environment is a fundamental problem with applications ranging from robot navigation to searching the World Wide Web. As such, a large body of work has focused on nding ecient solutions to variants of the problem, with restrictive assumptions on the form of the environment (cf. [16, 15, 22, 31, 17, 35, 10, 6, 2]) In this paper, we consider a model that makes very limited assumptions about the environment, and give ecient algorithms to solve the mapping problem in this general setting. A natural way to model the problem is by a robot exploring a graph G = V; E) The case in which the graph has both ....

....a city (having one way streets) or the Internet, it contains directed edges. This alone does not make the problem substantially more dicult, since the problem with directed edges and labeled vertices can be solved by a greedy search algorithm in time O(jVj jEj) More sophisticated techniques [22, 2] yield improved running times. Regardless of whether there are directed edges, a more daunting diculty arises if vertices are not uniquely labeled. This situation could arise in applications from the limited sensory capabilities of a robot or from the changing appearance of vertices. If no ....

[Article contains additional citation context not shown here]

X. Deng and C. H. Papadimitriou. Exploring an unknown graph. In Proceedings of the Thirty First Annual Symposium on Foundations of Computer Science, pages 356-361, 1990.

....[27, 1] We focus on line graphs and general metric spaces, but in future work, we plan to expand our consideration of metric spaces to consider rotational latency as well. Finally, on line navigation and on line searching problems such as the cow path problem 9 bear some relation to our problem [26, 10, 19, 22]. However, the goal function and the requirements on the optimal algorithm are quite di erent for these problems than for our problem. In most on line searching problems, the typical goal is to nd a speci c unknown target. The o ine algorithm which knows the location of this target can move ....

X. Deng and C.H. Papadimitriou. Exploring an unknown graph. In Proc. 31st IEEE Symp. on Foundations of Computer Science, pages 355-361, 1990.

....edges, and apply this algorithm recursively from each such node. This algorithm is similar to depth first search, with the following difference: Since chronological backtracking is not always possible in directed graphs, BETA repeats its 3 The exact origin of the algorithm is unclear. (Deng Papadimitriou 1990) and (Korach, Kutten, Moran 1990) stated it explicitly as a search algorithm, but it has been used earlier as part of proofs about Eulerian tours (Hierholzer 1873) first actions when it gets stuck instead of backtracking its latest actions. BETA fits our real time search skeleton if we ....

Deng, X., and Papadimitriou, C. 1990. Exploring an unknown graph. In Proceedings of the Symposium on Foundations of Computer Science, 355--361.

....is tight. Thus, the average case complexity of a random walk in 1 step invertible state spaces is polynomial (and no longer exponential) in n. For 1 step invertible state spaces, however, there are tabula rasa on line algorithms that have a smaller big O worst case complexity than Q learning [ Deng and Papadimitriou, 1990 ] Determining Optimal Policies We now consider the problem of finding shortest paths from all states to a goal state. We present a novel extension of the Q learning algorithm that determines the goal distance of every state and has the same big O worst case complexity as the algorithm for ....

Deng, X. and Papadimitriou, C.H. 1990. Exploring an unknown graph. In Proceedings of the FOCS.

....but it is more complicated than our fastest piecemeal learning algorithm. Previous work One of the first models for exploring unknown environments was developed by Papadimitriou and Yanakakis [20] They show how to find a shortest path in an unknown, undirected graph. Deng and Papadimitriou [13] and Betke [6] address the problem of exploring an unknown directed graph. Blum, Raghavan, and Schieber [10] consider a robot navigating in an unknown two dimensional geometric terrain with convex obstacles. They compare the distance that a robot covers from one point to another in the scene with ....

Xiaotie Deng and Christos H. Papadimitriou. Exploring an unknown graph. In Proceedings of the 31st Symposium on Foundations of Computer Science, volume I, pages 355--361, 1990.

....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 Omega Gamma d 2 m) and Deng and Papadimitriou [12] showed an upper bound of d O(d) m, where m is the number of 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(log d) m. We ....

....a path that is as short as possible. In many situations it is convenient to model the environment in which the robot operates by a graph. This allows to neglect geometric features of the environment and to concentrate on combinatorial aspects of the exploration problem. Deng and Papadimitriou [12] formulated thus the following exploration problem. A robot has to explore all nodes and edges of an unknown, strongly connected directed graph. The robot visits an edge when it traverses the edge. A node or edge is explored when it is visited for the first time. The goal is to determine a map, ....

[Article contains additional citation context not shown here]

X. Deng and C. H. Papadimitriou, Exploring an unknown graph, in Proc. 31st Symp. on Foundations of Computer Science, 1990, pp. 356--361.

....of some single walk on G. If we could have access to G in order to get more information of G, it may be easier to reconstruct G. Such a situation has been dealt with in the problem of learning an unknown graph, and some of important results under various conditions were published (e.g. DP90, BS94, BRS95] Broadly speaking, in the problem, we must reconstruct an unknown graph G by wandering actively through G and gathering partial information of G we can see there. Bender and Slonim [BS94] showed that two cooperating robots can learn any strongly connected graph with n ....

....a graph) by a natural search method, called the piecemeal search. The piecemeal constraint requires that each of the learner s exploration phases must be of limited duration. They devised linear time piecemeal search learning algorithms on some kinds of graphs. Deng and Papadim 7 itriou [DP90] devised algorithms to explore all edges of an unknown directed, strongly connected graph. They evaluated the algorithm performance by means of the ratio of the total number of edges traversed divided by the optimal number of traversals. 1.2 Contributions In this section, we briefly view our ....

X. Deng and C. H. Papadimitriou. Exploring an unknown graph. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science, pages 355--361, 1990.

....and partially by DARPA grant DABT63 96 C 0018. Laboratory for Computer Science, MIT, 545 Technology Square, Cambridge, MA 02139. Email: salil math.mit.edu. Supported by a DOD NDSEG doctoral fellowship and partially by DARPA grant DABT63 96 C 0018. tions on the form of the environment (cf. [13, 12, 16, 23, 14, 27, 7, 4, 1]. In this paper, we consider a model that makes very limited assumptions about the environment, and give efficient algorithms to solve the mapping problem in this general setting. A natural way to model the problem is by a robot exploring a graph G = V;E) The case in which the graph has both ....

....(having one way streets) or the Internet, it contains directed edges. This alone does not make the problem substantially more difficult, since the problem with directed edges and labeled vertices can be solved by a greedy search algorithm in time O(jVj Delta jEj) More sophisticated techniques [16, 1] yield improved running times. Regardless of whether there are directed edges, a more daunting difficulty arises if vertices are not uniquely labeled. This situation could arise from the limited sensory capabilities of a robot or from the changing appearance of vertices. If no assumptions are made ....

[Article contains additional citation context not shown here]

X. Deng and C. H. Papadimtriou. Exploring an unknown graph. In Proceedings of the Thirty First Annual Symposium on Foundations of Computer Science, pages 356--361, 1990.

....of goal states and consequently do not use any prior knowledge to guide the search towards them. They can be used for goal directed exploration, because they visit all states during their exploration, including the goal states. The following algorithm, whose exact origin is unclear, is an example. (Deng Papadimitriou 1990) and (Korach, Kutten, Moran 1990) stated it explicitly as a search algorithm, but it has been used earlier as part of proofs about Eulerian tours, for example in (Hierholzer 1873) BETA ( Building a Eulerian Tour Algorithm) Take unexplored actions whenever possible (ties can be broken ....

....(ties can be broken arbitrarily) If all actions in the current state have been explored, execute the initial sequence of actions again, this time stopping at all states that have unexplored actions and apply the algorithm recursively from each such state. BETA executes every action at most twice (Deng Papadimitriou 1990). This implies the following theorem: Theorem 1 BETA solves any goal directed exploration problem with a cost of Theta(1) Theta weight (to be precise: with a cost of at most 2 Theta weight) BETA does not make use of any prior knowledge to guide the search towards a goal state, although such ....

Deng, X., and Papadimitriou, C. 1990. Exploring an unknown graph. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), 355--361.

....graph. A one processor per node implementation of the heuristic updating phase of LCM could be viewed as an extension of their message passing algorithm where the single source is the dummy goal node of the augmented local graph. A similar algorithm is also presented in [1] Deng and Papadimitriou [6] have considered the problem of exploring an unknown graph. Although their work is related, they only consider the task of exploring edges rather than nodes, and they concentrate on directed rather than undirected graphs. In addition, their work is only concerned with finding all edges in a graph, ....

Deng, X. and C. H. Papadimitriou, Exploring an Unknown Graph, in: Proceedings of the Thirty-First Annual Symposium on Foundations of Computer Science (FOCS-90), St. Louis, (1990) 355-361.

....by a similar modification of the construction, i.e. replacing the line segment obstacles with long, thin diamonds. Notice that non rectilinear obstacles must be used. 2 The present paper can be considered as a geometric continuation of our previous work on exploring an unknown directed graph [5]; in fact, it addresses and solves the main open problem posed in that paper. 1 In [5] it is shown that general directed graphs are not learnable, while Eulerian or almost Eulerian directed graphs are learnable. There are several new difficulties in the present geometric problem, when ....

....with long, thin diamonds. Notice that non rectilinear obstacles must be used. 2 The present paper can be considered as a geometric continuation of our previous work on exploring an unknown directed graph [5] in fact, it addresses and solves the main open problem posed in that paper. 1 In [5] it is shown that general directed graphs are not learnable, while Eulerian or almost Eulerian directed graphs are learnable. There are several new difficulties in the present geometric problem, when compared with the graphtheoretic problem studied in [5] First, the off line problem in [5] the ....

[Article contains additional citation context not shown here]

X. Deng and C.H. Papadimitriou, "Exploring an unknown graph," Proc. 31st Annual IEEE Symp. on Foundations of Computer Science, 1990, 355-361.

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X. Deng, and C. H. Papadimitriou, "Exploring an Unknown Graph," FOCS 1990, pp.355361.

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X. Deng, and C. H. Papadimitriou, "Exploring an Unknown Graph," FOCS 1990, pp.355361.

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X. Deng and C. Papadimitriou. Exploring an unknown graph. In Proc. 31st IEEE FOCS, pages 355--361, 1990.

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Deng, X., and Papadimitriou, C. H. (1990), Exploring an unknown graph, in "Proceedings, 31st IEEE Symposium on Foundations of Computer Science," pp. 355--361.

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X. Deng and C. H. Papadimitriou, Exploring an unknown graph, J. of Graph Th., 32, (1999), 265-297.

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X. Deng and C.H. Papadimitriou. Exploring an unknown graph. In Proc. 31st IEEE Symp. on Foundations of Computer Science, pages 355--361, 1990.

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X. Deng and C.H. Papadimitriou. Exploring an unknown graph. In Proc. 31st IEEE Symp. on Foundations of Computer Science, pages 355-361, 1990.

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Xiaotie Deng and Christos H. Papadimitriou. Exploring an unknown graph. Journal of Graph Theory, 32:265--297, 1999.

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X. Deng and C. H. Papadimitriou. Exploring an unknown graph. Journal of Graph Theory, 32:265--297, 1999.

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X. Deng and C. H. Papadimitriou, Exploring an unknown graph, Journal of Graph Theory 32 (1999), 265-297.

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X. Deng and C. H. Papadimitriou. Exploring an unknown graph. In Proc. 31st Annual Symposium on Foundations of Computer Science, pages 356-361, 1990.

No context found.

X. Deng and C. Papadimitriou, Exploring an unknown graph. Journal of Graph Theory, 32:265-297, 1999.

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X. Deng and C. H. Papadimitriou. Exploring an unknown graph. In Proceedings of the Thirty First Annual Symposium on Foundations of Computer Science, pages 356--361, 1990.

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X. Deng and C. H. Papadimitriou, Exploring an unknown graph, Journal of Graph Theory 32 (1999), 265297.

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X. Deng and C. Papadimitriou, Exploring an unknown graph, Proc 31st Symposium on Foundations of Computer Science, 1990, pp. 356--361.

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X. Deng and C. H. Papadimitriou. Exploring an unknown graph. Revised version of [12].

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Deng, X. and Papadimitriou, C.H. 1990. Exploring an unknown graph. In Proceedings of the Symposium on Foundations of Computer Science. 355--361.

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X. Deng and C. H. Papadimitriou. Exploring an unknown graph. Revised version of [9].

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Deng, X., and Papadimitriou, C. H. (1990), Exploring an unknown graph, in "Proceedings, 31st IEEE Symposium on Foundations of Computer Science," pp. 355--361.

No context found.

X. Deng and C. H. Papadimitriou. "Exploring an Unknown Graph." FOCS, pp. 355--361, 1990.

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