Christos Papadimitriou and Mihalis Yannakakis. Shortest Paths without a Map. Theoretical Computer Science, 84:127--150, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of machine learning. It is about the design and analysis of efficient learning algorithms for confident explorers. Similar formal approaches to environment learning have been studied previously in the context of graph theory, computational geometry, on line algorithms, and automata theory (e.g. [89, 21, 93]) In this related work, formal models are defined that try to capture the difficulties that lie in the confident explorer s limited knowledge of its environment. In our work, we assume that the unknown environment can be described by a simple discrete model, an undirected graph G = V; E) We ....

....give an optimal randomized algorithm with competitive ratio of 4:5911. The cow path problem for the case of two paths is similar to the problem of exploring layered graphs of width 2. Layered graphs of width 2, however, allow shortcuts . For searching such a graph, Papadimitriou and Yannakakis [89] give an optimal algorithm with competitive ratio of 9. Baeza Yates, Culberson, and Rawlings [8] describe an algorithm for searching a goal t at distance n from s in a grid graph. If the distance n is known in advance, the algorithm takes 9n Gamma 2 steps. Otherwise it takes 2n 5n 2 ....

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Christos H. Papadimitriou and Mihalis Yanakakis. Shortest paths without a map. Theoretical Computer Science, 84:127--150, 1991. 126

....1) m Gamma1 ) Kao, Reif and Tate [49] give a randomized algorithm for this problem that has better expected performance than any deterministic algorithm. Kao, Ma, Sipser and Yin [48] give an optimal deterministic search strategy for the case of multiple robots. Papadimitriou and Yanakakis [62] consider the problem of a robot with vision moving around in a plane filled with obstacles. The robot does not know its environment, but knows its exact absolute location at all times, as well as its start position and its goal position. The robot s goal is to travel from the start position to ....

....rectangles whose opposing sides have the same number of edges. A 1 Theta 1 face might correspond to a standard city block; larger faces might correspond to obstacles (parks or shopping malls) Figure 3.3 gives an example. City block graphs are also studied by Papadimitriou and Yanakakis [62], Blum, Raghavan, and Schieber [22] and Bar Eli, Berman, Fiat and Yan [10] An m Theta n city block graph with no obstacles has exactly mn vertices (at points (i; j) for 1 i m, 1 j n) and 2mn Gamma (m n) edges (between points at distance 1 from each other) Obstacles, if present, ....

Christos H. Papadimitriou and Mihalis Yanakakis. Shortest paths without a map. Theoretical Computer Science, 84:127--150, 1991.

....of edges and vertices within radius Delta = ffi T o(ffi T ) from s. Our final treasure hunting algorithm is also a solution to the piecemeal learning problem. Related work 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 ....

Christos H. Papadimitriou and Mihalis Yanakakis. Shortest paths without a map. Theoretical Computer Science, 84:127--150, 1991.

....There is a Most of this research was done during a visit of F. Fomin at the F. SchillerUniversit at Jena which was supported by a fellowship of the DAAD (Kennziffer: A 99 09594) Preprint submitted to Elsevier Preprint 31 March 2000 strong resemblance of graph searching to certain pebble games [17] that model sequential computation. Other applications of graph searching can be found in the VLSI theory: the game theoretic approach to some important parameters of graph layouts such as the cutwidth [24] the topological bandwidth [23] the bandwidth [12] the profile [13] and the vertex ....

....used in this program is max i2f0; 2m Gamma1g jD i j. 5 The smallest number of searchers that are necessary to be used in a winning node search program on a graph G is denoted by ns(G) and is said to be the node search number of G. Using results of La Paugh [21] Kirousis and Papadimitriou [17] obtained the following fundamental result about node searching (see also [4] for an alternative proof) Theorem 2 (Kirousis Papadimitriou) For any graph G there exists a monotone search program using ns(G) searchers, i.e. G can be cleared by ns(G) searchers without recontamination of previously ....

L. M. Kirousis and C. H. Papadimitriou, Searching and pebbling, Theoretical Computer Science, 47 (1986), pp. 205--218.

.... VLSI layout, processor management, node searching and vertex separation (see the excellent survey of Mohring on pathwidth and the relations among these problems [33] PATHWIDTH is NP complete on arbitrary graphs [28, 2] and even for chordal graphs [24] and (using the equivalence to node search [29]) for planar graphs with vertex degrees at most three [35] On the other hand, it is polynomial when k is fixed. The results of Robertson and Seymour [39] imply non constructively that an O(n 2 ) algorithm exists for fixed k [14, 15] The recent results of Bodlaender [3] yield a linear algorithm ....

L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theoretical Computer Science, 47:205--218, 1986.

....well known that the path width of a graph is an important parameter that can significantly influence the time complexity of algorithmic graph problems. It is also well know that path width is identical to vertex separation [Kin92] and that vertex separation is identical to (node search number 1) [KP86]. Computing this parameter is, in general, an NP complete problem. In [EST94] a linear time algorithm was described for the computation of the vertex separation of trees. That paper suggested an O(n log n) time algorithm for the computation of an actual layout with minimum vertex separation but ....

L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theoretical Computer Science, 47(2):205--218, 1986.

.... showed that progressive black white pebble game (important to compiler theory) and vertex separation are polynomially reducible to each other [Len81] Node search number, a variant of search number [Par76] was shown equivalent to the vertex separation plus one by Kirousis and Papadimitriou [KP86]. From [EST94] the search number is informally defined in terms of pebbeling to be the minimum number of searchers needed to capture a fugitive who is allowed to move with arbitrary speed about the edges of the graph. For node search number, a searcher blocks all neighboring nodes without the ....

L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theoretical Computer Science, 47:205--216, 1986.

.... algorithms are useful: there are working systems such as meal delivery robots in hospitals [15] and vehicles which navigate autonomously on highways [4] More formal approaches to mobile robot navigation and environment learning have been studied extensively in theoretical computer science (e.g. [20, 10, 22, 5]) In the area of mobile robot environment learning, we study the problem of piecemeal learning of an unknown environment [8] The robot s goal is to learn a complete map of its environment while satisfying the piecemeal constraint that exploration must be done a piece at a time, with a return ....

....final algorithm we give for treasure hunting also gives a solution to the piecemeal learning problem, 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 ....

Christos H. Papadimitriou and M. Yanakakis. Shortest paths without a map. Theoretical Computer Science, 84:127--150, 1991.

....the pattern of interactions between P and E, we can record this pattern during the computation of the result and output it as a part of the result. On line algorithms for finding a point on a line or a location in a Manhattan graph require the on line agent with its environment to be closed [PY, We3]. Supplying an environment for an on line algorithm is a special case of supplying an argument to an algorithm and may be compared to supplying an initial tape to a TM. Abstractly, we close an open system and observe its algorithmic behavior at an inner interface between two subsystems [We3] The ....

....a message to an object is a form of closure (the data structure created when a procedure is invoked is actually called a closure) Taking the fixed point of a recursive function or taking the fixed point of a class in establishing its run time context is a form of closure. On line algorithms [PY] require closure by providing a context or environment to which the on line algorithm will be applied (see section 7) Alternative forms of closure that arise in empirical computer science are the subject of a future paper. Logic provides a framework for expressing the relation between syntactic ....

Christos Papadimitriou and Mihalis Yannakakis, Shortest Paths Without a Map, Theoretical Computer Science 84-1, July 1991.

....a layout problem, the Colored Vertex Separation. Both problems were shown identical in [Din96] The non colored version of the vertex separation problem is an NP Complete problem with several applications in Computer Science (VLSI design, compiler construction, etc. and it has been well studied [Len81, EST79, KP86, Kin92]. In section 2 we introduce the basic definitions and the formal statement for the problems. We show that the ICG problem for caterpillars with hairs of length at most 1 is in the class NC. In section 3 we show that the ICG problem is NP Complete for caterpillars with hairs of length at most 2. ....

L.M. Kirousis and C.H. Papadimitriou. Searching and pebbling. Theoretical Computer Science, 47:205--216, 1986.

....learn about or navigate in its environment. Such algorithms are now useful in practice: there are working meal delivery robots in hospitals [15] and vehicles that navigate autonomously on highways [4] More formal theoretical approaches to these problems have also been studied extensively (e.g. [20, 10, 22, 5]) We study the problem of piecemeal learning of an unknown environment [8] The robot s goal is to learn a complete map of its environment while satisfying the piecemeal constraint that learning must be done a piece at a time, with the robot returning to the starting point s after each ....

....s than the unvisited vertex nearest to s) then it may traverse up to O(E 2 ) edges. Our final treasure hunting algorithm is also a solution to the piecemeal learning problem, but it is more complicated than our fastest piecemeal learning algorithm. Previous work Papadimitriou and Yanakakis [20] developed one of the first 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 ....

Christos H. Papadimitriou and M. Yanakakis. Shortest paths without a map. Theoretical Computer Science, 84:127--150, 1991.

.... a constant k, is there an interval supergraph of G whose clique size is at most k Equivalently, is the pathwidth of G at most k Gamma 1 Discussion: The Pathwidth problem is NP complete on arbitrary graphs [29, 2] and even for chordal graphs [25] and (using the equivalence to node search [30]) for planar graphs with vertex degrees at most three [41] On the other hand, it is solvable in linear time when k is fixed [31, chapter 11] We shall prove that under the restriction of the problem to bipartite graphs it remains NP complete. D. Handling probe clone errorless data under ....

L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theoretical Computer Science, 47:205--218, 1986. - 19 -

....on line algorithms. An on line process P with the closed system property may be viewed as the state transition mechanism of a Turing machine whose environment E is a finite tape. It performs algorithmic TM computations. On line algorithms for exploring graphs or maps have been extensively studied [PY]. Their complexity is measured by interaction cost rather than instruction execution cost. The lower bound on interaction cost of finding a point on a line is the basis for complexity results of a family of related problems, such as finding a line in a plane or an intersection in a Manhattan ....

....cost. The lower bound on interaction cost of finding a point on a line is the basis for complexity results of a family of related problems, such as finding a line in a plane or an intersection in a Manhattan graph. D5 (complexity) Interactive complexity = number of interactive steps (see [PY] for definition) Agents for exploring static graphs have the closed system property and are algorithms. This closed system condition can be relaxed, so that the interactive behavior of agents that interact with systems evolving according to predictable algorithmic interactions can be described by ....

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Christos Papadimitriou and Mihalis Yannakakis, Shortest Paths Without a Map, Theoretical Computer Science 84-1, July 1991.

....localization strategies, this ratio is called the competitive ratio of the strategy. If a dynamic navigation strategy always reaches the target t by traveling a distance no more than k times the length of the shortest path, then the strategy is called k competitive. Papadimitriou and Yannakakis [PY91] gave a strategy for navigating between two points, where all obstacles are unit squares, that achieves a competitive ratio of 1.5, which they show is optimal. For squares of arbitrary size they gave a strategy achieving a ratio of p 26=3. They also showed, along with Eades, Lin and Wormald ....

Christos Papadimitriou and Mihalis Yannakakis. Shortest Paths without a Map. Theoretical Computer Science, 84:127--150, 1991.

No context found.

Christos Papadimitriou and Mihalis Yannakakis. Shortest Paths without a Map. Theoretical Computer Science, 84:127--150, 1991.

No context found.

L. Kirousis and C. Papadimitriou, Searching and pebbling, Theoretical Computer Science 47, 205-218, 1986.

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