eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-14
16:1
16:17
10.4230/LIPIcs.ESA.2018.16
article
Near-Optimal Distance Emulator for Planar Graphs
Chang, Hsien-Chih
1
Gawrychowski, Pawel
2
Mozes, Shay
3
Weimann, Oren
4
University of Illinois at Urbana-Champaign, USA
University of Wrocław, Poland
IDC Herzliya, Israel
University of Haifa, Israel
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 O~(n) time a distance emulator of size O~(min(k^2,sqrt{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 all-pairs shortest path distances among k terminals in O~(n) time when k=O(n^{1/3}).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol112-esa2018/LIPIcs.ESA.2018.16/LIPIcs.ESA.2018.16.pdf
planar graphs
shortest paths
metric compression
distance preservers
distance emulators
distance oracles