Near-Optimal Distance Emulator for Planar Graphs
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}).
planar graphs
shortest paths
metric compression
distance preservers
distance emulators
distance oracles
Theory of computation~Design and analysis of algorithms
16:1-16:17
Regular Paper
https://arxiv.org/abs/1807.01478
Hsien-Chih
Chang
Hsien-Chih Chang
University of Illinois at Urbana-Champaign, USA
Supported partially by NSF grant CCF-1408763.
Pawel
Gawrychowski
Pawel Gawrychowski
University of Wrocław, Poland
Shay
Mozes
Shay Mozes
IDC Herzliya, Israel
Supported partially by ISF grants 794/13 and 592/17
Oren
Weimann
Oren Weimann
University of Haifa, Israel
Supported partially by ISF grants 794/13 and 592/17
10.4230/LIPIcs.ESA.2018.16
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Hsien-Chih Chang, Paweł Gawrychowski, Shay Mozes, and Oren Weimann
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