## Abstract

Given a graph G and a set of terminals T, a distance emulator of G is another graph H (not necessarily a subgraph of G) containing T, such that all the pairwise distances in G between vertices of T are preserved in H. An important open question is to find the smallest possible distance emulator. We prove that, given any subset of k terminals in an n-vertex undirected unweighted planar graph, we can construct in Õ(n) time a distance emulator of size Õ(min(k^{2}, √k · n)). This is optimal up to logarithmic factors. The existence of such distance emulator provides a straightforward framework to solve distance-related problems on planar graphs: Replace the input graph with the distance emulator, and apply whatever algorithm available to the resulting emulator. In particular, our result implies that, on any unweighted undirected planar graph, one can compute ll-pairs shortest path distances among k terminals in Õ(n) time when k = O(n^{1/3}).

Original language | English |
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Title of host publication | 26th European Symposium on Algorithms, ESA 2018 |

Editors | Hannah Bast, Grzegorz Herman, Yossi Azar |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Print) | 9783959770811 |

DOIs | |

State | Published - 1 Aug 2018 |

Event | 26th European Symposium on Algorithms, ESA 2018 - Helsinki, Finland Duration: 20 Aug 2018 → 22 Aug 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 112 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 26th European Symposium on Algorithms, ESA 2018 |
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Country/Territory | Finland |

City | Helsinki |

Period | 20/08/18 → 22/08/18 |

### Bibliographical note

Funding Information:Supported partially by NSF grant CCF-1408763. 2 Supported partially by ISF grants 794/13 and 592/17 3 Supported partially by ISF grants 794/13 and 592/17

Funding Information:

Supported partially by NSF grant CCF-1408763. Supported partially by ISF grants 794/13 and 592/17

Publisher Copyright:

© Steven Chaplick, Minati De, Alexander Ravsky, and Joachim Spoerhase.

## Keywords

- Distance emulators
- Distance oracles
- Distance preservers
- Metric compression
- Planar graphs
- Shortest paths

## ASJC Scopus subject areas

- Software