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Directed Graph Exploration

Directed Graph Exploration. K laus- Tycho Förster. When in Rome …. Navigating in Zurich . Zurich: Full of one-way streets… . Formal Model. Given a strongly connected directed graph All edges have non-negative weights All nodes have a unique ID A searcher starts from some node

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Directed Graph Exploration

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  1. Directed Graph Exploration Klaus-TychoFörster

  2. When in Rome …

  3. Navigating in Zurich

  4. Zurich: Full of one-way streets…

  5. Formal Model • Given a strongly connected directed graph • All edges have non-negative weights • All nodes have a unique ID • A searcher starts from some node • With unlimited memory and computational power • Has to explore the graph • A graph is called explored, if the searcher has visited all nodes and returnedto the starting node • When the searcher arrives at a node, she knows all outgoing edges, including their cost and the ID of the node at the end of the edges cf. [Kalyanasundaram & Pruhs 1994, Megow et. al. 2011]

  6. How good is a tour, how good is a strategy? • Cost of a tour: Sum of traversed edge weights Competitive ratios for: • a tour : • deterministic algorithms: • randomized algorithms:

  7. Applications of Graph Exploration • One of the fundamental problems of robotics cf. [Burgard et. al. 2000, Fleischer & Trippen 2005] • Exploring the state space of a finite automaton cf. [Brass et. al. 2009] • A model for learning cf. [Deng & Papadimitriou 1999]

  8. Some Related Work • Offline: Asymmetric Traveling Salesman problem • Approximation ratio of [Feige & Singh 2007] Undirected graph exploration: • General case: ) [Rosenkrantz et. al. 1977] • Best known lower bound: [Dobrev et. al. 2012] • Planar graphs: [Kalyanasundaram & Pruhs1994] • Genus at most: [Megowet. al. 2011] • Unweighted: (, [Miyazaki et. al. 2009]) • Does randomization help? Directed Case factorof 4 atmost

  9. Exploring with a Greedy Algorithm • Achieves a competitive ratio of • Proof sketch: • Greedy uses paths to new nodes and then returns • The greedy path from to a not yet visited node is a shortest path • Let be an opt. Tour inducing a cyclicorderingof all nodes in with the tour consisting of segments. • The path has by definition at most the cost of the whole part of the tour , which consists of at most segments. • Therefore, the cost of each of the segments in has to be used at most times for the upper cost bound of the greedy algorithm.

  10. Exploring with a Greedy Algorithm – Unweighted Case • Achieves a competitive ratio of • Proof sketch: • The costtoreachthefirstnewnodeis, thenatmost, thenatmost, … • Ifwesumthisup, weget an upperboundof • The costof an optimal tour isat least .

  11. Lower Bounds for Deterministic Online Algorithms • No better competitive ratio than is possible. • Unweighted case: No better competitive ratio than is possible. • Both results are tight.

  12. Lower Bounds for RandomizedOnline Algorithms • No better competitive ratio than is possible. • Proof sketch: • When being at a node , with , for the first time, then the “correct” edge can be picked with a probability of at most . • Expected amount of “wrong” decisions: • The cost of an optimal tour is 1. • Unweighted case: No better competitive ratio than is possible.

  13. Variations of the Model • Randomized starting node? • Choosing best result from all starting nodes? • Possible solution: Duplicate the graphs, connect their starting nodes • No better competitive ratio possible than • (deterministic online algorithms) • (randomized online algorithms)

  14. Variations of the Model • What if the searcher also sees incoming edges? • What if the searcher does not see the IDs of the nodes at the end of outgoing edges, but knows the IDs of outgoing and incoming edges? • Greedy algorithm still works with same ratio (all nodes have been visited if all edges have been seen as incoming and outgoing edges) • Lower bound examples also still work decreaseslowerboundby a factoroflessthan 2 decreaseslowerboundby a factoroflessthan 1.5

  15. Searching for a Node • Not feasible in weighted graphs: • In unweighted graphs, lower bounds for competitive ratios: • A greedy algorithm has a competitive ratio of Randomized Deterministic

  16. Overview of our Results

  17. Thank you Klaus-Tycho Förster

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