Online Graph Exploration: New Results on Old and New Algorithms

Megow, Nicole; Mehlhorn, Kurt; Schweitzer, Pascal

doi:10.1007/978-3-642-22012-8_38

Cited by 25 publications

(34 citation statements)

References 34 publications

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“…This scenario has been introduced by Kalyanasundaram and Pruhs in [18] and is known as a fixed graph scenario. While learning the endpoints of the incident edges is stronger than the typical exploration scenario, it does have justification (see [18] and [19]); it also corresponds to previously studied neighbourhood sense of direction [8].…”

Section: Introductionmentioning

confidence: 74%

Online Graph Exploration with Advice

Dobrev

Královič

Μάρκου

2012

Lecture Notes in Computer Science

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We study the problem of exploring an unknown undirected graph with non-negative edge weights. Starting at a distinguished initial vertex s, an agent must visit every vertex of the graph and return to s. Upon visiting a node, the agent learns all incident edges, their weights and endpoints. The goal is to find a tour with minimal cost of traversed edges. This variant of the exploration problem has been introduced by Kalyanasundaram and Pruhs in [18] and is known as a fixed graph scenario. There have been recent advances by Megow, Mehlhorn, and Schweitzer ([19]), however the main question whether there exists a deterministic algorithm with constant competitive ratio (w.r.t. to offline algorithm knowing the graph) working on all graphs and with arbitrary edge weights remains open. In this paper we study this problem in the context of advice complexity, investigating the tradeoff between the amount of advice available to the deterministic agent, and the quality of the solution. We show that Ω(n log n) bits of advice are necessary to achieve a competitive ratio of 1 (w.r.t. an optimal algorithm knowing the graph topology). Furthermore, we give a deterministic algorithm which uses O(n) bits of advice and achieves a constant competitive ratio on any graph with arbitrary weights. Finally, going back to the original problem, we prove a lower bound of 5/2 − for deterministic algorithms working with no advice, improving the best previous lower bound of 2− of Miyazaki, Morimoto, and Okabe from [20]. In this case, significantly more elaborate technique was needed to achieve the result.

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Section: Introductionmentioning

confidence: 74%

Online Graph Exploration with Advice

Dobrev

Královič

Μάρκου

2012

Lecture Notes in Computer Science

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“…Upon arriving at a node v, the following information is made available: all outgoing incident edges including their weight, plus the IDs (cf. [46,52]) of the corresponding nodes at the head of these edges. We call a graph explored, if a searcher starting from some node s has visited all nodes and returned to s.…”

Section: Modelmentioning

confidence: 99%

“…More closely related to the online exploration of all nodes of directed graphs is the online exploration of all nodes of undirected graphs. While a greedy algorithm achieves a competitive ratio of Θ(log n) [59], it is not known if a constant competitive ratio for general graphs is possible [52]. For cycles there is an algorithm with a sharp competitive ratio of 1+…”

Section: Related Workmentioning

confidence: 99%

“…For planar graphs a sophisticated variant of depth-first search named ShortCut by Kalyanasundaram and Pruhs achieves a competitive ratio of 16 [46]. Their result was recently extended for graphs of genus g to 16(1 + 2g) [52]. If there are just k different edge weights, there exists an algorithm with competitive ratio 2k [52].…”

Section: Related Workmentioning

confidence: 99%

“…Their result was recently extended for graphs of genus g to 16(1 + 2g) [52]. If there are just k different edge weights, there exists an algorithm with competitive ratio 2k [52]. Fleischer et al considered the problem of searching just for a node instead of a tour in [31].…”

Section: Related Workmentioning

confidence: 99%

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Lower and upper competitive bounds for online directed graph exploration

Foerster

Wattenhofer

2016

Theoretical Computer Science

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We study the problem of exploring all nodes of an unknown directed graph. A searcher has to construct a tour that visits all nodes, but only has information about the parts of the graph it already visited. Analogously to the travelling salesman problem, the goal is to minimize the cost of such a tour. In this article, we present upper and lower bounds for the competitive ratio of both the deterministic and the randomized online version of exploring all nodes of directed graphs. Our bounds are sharp or sharp up to a small constant, depending on the specific model. As it turns out, restricting the diameter, the incoming/outgoing degree, or randomly choosing a starting point does not improve lower bounds beyond a small constant factor. Even supplying the searcher in a planar euclidean graph with the nodes' coordinates does not help. Essentially, exploring a directed graph has a multiplicative overhead linear in the number of nodes. Furthermore, if one wants to search for a specific node in unweighted directed graphs, a greedy algorithm with quadratic multiplicative overhead can only be improved by a small constant factor as well.

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