# Planar Distance Oracles with Better Time-Space Tradeoffs

@inproceedings{Long2021PlanarDO, title={Planar Distance Oracles with Better Time-Space Tradeoffs}, author={Yaowei Long and Seth Pettie}, booktitle={SODA}, year={2021} }

In a recent breakthrough, Charalampopoulos, Gawrychowski, Mozes, and Weimann (STOC 2019) showed that exact distance queries on planar graphs could be answered in $n^{o(1)}$ time by a data structure occupying $n^{1+o(1)}$ space, i.e., up to $o(1)$ terms, optimal exponents in time (0) and space (1) can be achieved simultaneously. Their distance query algorithm is recursive: it makes successive calls to a point-location algorithm for planar Voronoi diagrams, which involves many recursive distance…

## Figures and Tables from this paper

## 11 Citations

Exact Distance Oracles for Planar Graphs with Failing Vertices

- Mathematics, Computer ScienceACM Transactions on Algorithms
- 2022

The authors' n2+o(1)/q2-size, Õ(q)-query-time oracle improves over the previously best known tradeoff of Baswana et al. 2012 and shows several tradeoffs between space, query time, and preprocessing time.

Exact Distance Oracles for Planar Graphs with Failing Vertices

- Mathematics, Computer ScienceSODA
- 2019

An oracle of size of size $\tilde{\mathcal{O}}(\frac{n^{k+3/2}}{q^{2k+1}})$ that answers queries in $tilde(q)$ time is proposed that matches, up to polylogarithmic factors, the fastest failure-free distance oracles with nearly linear space.

Truly Subquadratic Exact Distance Oracles with Constant Query Time for Planar Graphs

- Computer Science, MathematicsISAAC
- 2021

This work presents a truly subquadratic size distance oracle for reporting exact shortest-path distances between any pair of vertices of G in constant time.

An Almost Optimal Edit Distance Oracle

- Computer Science, MathematicsICALP
- 2021

This work shows that it can efficiently query for the alignment score of every pair of substrings after preprocessing the input for almost the same time it takes to compute just the alignment of S and T.

Near-Optimal Distance Oracles for Vertex-Labeled Planar Graphs

- Computer Science, MathematicsISAAC
- 2021

This is the first non-trivial exact vertex-labeled distance oracle for planar graphs and, to the authors' knowledge, for any interesting graph class other than trees.

A Simple Algorithm for Multiple-Source Shortest Paths in Planar Digraphs

- Computer ScienceSOSA
- 2022

This paper presents a data structure which can answer any query for the shortest path distance in G from u to v or from v to u in O (log n ) time, provided u is on f .

Optimal Approximate Distance Oracle for Planar Graphs

- Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

This work improves Thorup's (FOCS 2001, JACM 2004) space bound by more than a logarithmic factor while matching the query time of his structure, the first improvement for planar digraphs in two decades, both in the weighted and unweighted setting.

Single-source shortest paths and strong connectivity in dynamic planar graphs

- Computer Science, MathematicsJ. Comput. Syst. Sci.
- 2022

On the Discrete Fréchet Distance in a Graph

- Computer Science, MathematicsArXiv
- 2022

A conditional lower bound is provided showing that the Fréchet distance, or even its 1.01-approximation, between arbitrary paths in a weighted planar graph cannot be computed in O((|P | · |Q|)1−δ) time for any δ > 0 unless the Orthogonal Vector Hypothesis fails.

An efficient oracle for counting shortest paths in planar graphs

- MathematicsTheoretical Computer Science
- 2022

## References

SHOWING 1-10 OF 43 REFERENCES

Better Tradeoffs for Exact Distance Oracles in Planar Graphs

- Computer Science, MathematicsSODA
- 2018

This oracle both significantly simplifies and significantly improves the recent oracle of Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses O(n5/3)-space and answers queries in O(log n) time.

Distance Oracles for Sparse Graphs

- Mathematics, Computer Science2009 50th Annual IEEE Symposium on Foundations of Computer Science
- 2009

A new lower bound for approximate distance oracles in the cell-probe model is given, which holds even for sparse (polylog(n)-degree) graphs, and it is not an "incompressibility" bound: it is a three-way tradeoff between space, stretch, and query time.

Distance Oracles beyond the Thorup-Zwick Bound

- Computer Science, Mathematics2010 IEEE 51st Annual Symposium on Foundations of Computer Science
- 2010

It is shown that a 2-approximate distance oracle requires space $\tOmega(n^2 / \sqrt{\alpha})$ and this implies a space lower bound to achieve approximation $2d+1$.

More Compact Oracles for Approximate Distances in Undirected Planar Graphs

- Computer Science, MathematicsSODA
- 2013

The polynomial dependency on e−1 and log n is reduced, getting the first improvement in the query time--space tradeoff, and an oracle with space O(n) and query time O(e−1 is obtained.

Exact distance oracles for planar graphs

- Computer ScienceSODA
- 2012

New and improved data structures that answer exact node-to-node distance queries in planar graphs and an exact distance oracle of space O(n) such that for any pair of nodes at distance l the query time is O(min{l, √ n}).

Fast and Compact Exact Distance Oracle for Planar Graphs

- Computer Science, Mathematics2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
- 2017

An O(n 5/3)-space distance oracle which answers exact distance queries in O(log n) time for n-vertex planar edge-weighted digraphs is introduced.

Almost optimal distance oracles for planar graphs

- Computer Science, MathematicsSTOC
- 2019

We present new tradeoffs between space and query-time for exact distance oracles in directed weighted planar graphs. These tradeoffs are almost optimal in the sense that they are within…

The Space-Stretch-Time Tradeoff in Distance Oracles

- Computer Science, MathematicsESA
- 2014

New distance oracles for computing distances of stretch less than 2 on general weighted undirected graphs and for any integer k are presented, which significantly improves the state-of-the-art for each point in the space-stretch-time tradeoff space.

Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus and Minor-Free Graphs

- Mathematics, Computer ScienceICALP
- 2011

For planar graphs, bounded-genus graphs, and minor-excluded graphs, this paper gives distance-oracle constructions that require only O(n) space, and the big O hides only a fixed constant, independent of e and independent of genus or size of an excluded minor.

On-Line Algorithms for Shortest Path Problems on Planar Digraphs

- Mathematics, Computer ScienceWG
- 1996

Efficient algorithms for answering shortest path queries in digraphs with small separators and, in particular, in planarDigraphs are described, for any class ofdigraphs for which an O(√n) separator theorem holds.