## Abstract

In a 1989 paper titled 'shortest paths without a map', Papadimitriou and Yannakakis introduced an online model of searching in a weighted layered graph for a target node, while attempting to minimize the total length of the path traversed by the searcher. This problem, later called layered graph traversal, is parametrized by the maximum cardinality k of a layer of the input graph. It is an online setting for dynamic programming, and it is known to be a rather general and fundamental model of online computing, which includes as special cases other acclaimed models. The deterministic competitive ratio for this problem was soon discovered to be exponential in k, and it is now nearly resolved: it lies between O(2k) and O(k2k). Regarding the randomized competitive ratio, in 1993 Ramesh proved, surprisingly, that this ratio has to be at least O(k2/log1+?k) (for any constant ? > 0). In the same paper, Ramesh also gave an O(k13)-competitive randomized online algorithm. Since 1993, no progress has been reported on the randomized competitive ratio of layered graph traversal. In this work we show how to apply the mirror descent framework on a carefully selected evolving metric space, and obtain an O(k2) competitive randomized online algorithm, nearly matching the known lower bound on the randomized competitive ratio.

Original language | American English |
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Title of host publication | Proceedings - 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science, FOCS 2022 |

Publisher | IEEE Computer Society |

Pages | 1102-1113 |

Number of pages | 12 |

ISBN (Electronic) | 9781665455190 |

DOIs | |

State | Published - 2022 |

Event | 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022 - Denver, United States Duration: 31 Oct 2022 → 3 Nov 2022 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2022-October |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022 |
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Country/Territory | United States |

City | Denver |

Period | 31/10/22 → 3/11/22 |

### Bibliographical note

Funding Information:The second author s research was done while he was at Tel Aviv University, supported by the Israel Academy of Sciences and Humanities and Council for Higher Education Excellence Fellowship Program for International Postdoctoral Researchers. The third author s research was supported in part by ISF grants 3565-21 and 389-22, and by BSF grant 2018687

Funding Information:

The second author’s research was done while he was at Tel Aviv University, supported by the Israel Academy of Sciences and Humanities & Council for Higher Education Excellence Fellowship Program for International Postdoctoral Researchers. The third author’s research was supported in part by ISF grants 3565-21 and 389-22, and by BSF grant 2018687.

Publisher Copyright:

© 2022 IEEE.

## Keywords

- competitive analysis
- online computing
- shortest path