---
_id: '309'
abstract:
- lang: eng
text: 'We present an efficient algorithm for a problem in the interface between
clustering and graph embeddings. An embedding '' : G ! M of a graph G into a 2manifold
M maps the vertices in V (G) to distinct points and the edges in E(G) to interior-disjoint
Jordan arcs between the corresponding vertices. In applications in clustering,
cartography, and visualization, nearby vertices and edges are often bundled to
a common node or arc, due to data compression or low resolution. This raises the
computational problem of deciding whether a given map '' : G ! M comes from an
embedding. A map '' : G ! M is a weak embedding if it can be perturbed into an
embedding ψ: G ! M with k'' "k < " for every " > 0. A polynomial-time
algorithm for recognizing weak embeddings was recently found by Fulek and Kyncl
[14], which reduces to solving a system of linear equations over Z2. It runs in
O(n2!) O(n4:75) time, where 2:373 is the matrix multiplication exponent and n
is the number of vertices and edges of G. We improve the running time to O(n log
n). Our algorithm is also conceptually simpler than [14]: We perform a sequence
of local operations that gradually "untangles" the image ''(G) into
an embedding (G), or reports that '' is not a weak embedding. It generalizes a
recent technique developed for the case that G is a cycle and the embedding is
a simple polygon [1], and combines local constraints on the orientation of subgraphs
directly, thereby eliminating the need for solving large systems of linear equations.'
acknowledgement: '∗Research supported in part by the NSF awards CCF-1422311 and CCF-1423615,
and the Science Without Borders program. The second author gratefully acknowledges
support from Austrian Science Fund (FWF): M2281-N35.'
article_processing_charge: No
author:
- first_name: Hugo
full_name: Akitaya, Hugo
last_name: Akitaya
- first_name: Radoslav
full_name: Fulek, Radoslav
id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87
last_name: Fulek
orcid: 0000-0001-8485-1774
- first_name: Csaba
full_name: Tóth, Csaba
last_name: Tóth
citation:
ama: 'Akitaya H, Fulek R, Tóth C. Recognizing weak embeddings of graphs. In: ACM;
2018:274-292. doi:10.1137/1.9781611975031.20'
apa: 'Akitaya, H., Fulek, R., & Tóth, C. (2018). Recognizing weak embeddings
of graphs (pp. 274–292). Presented at the SODA: Symposium on Discrete Algorithms,
New Orleans, LA, USA: ACM. https://doi.org/10.1137/1.9781611975031.20'
chicago: Akitaya, Hugo, Radoslav Fulek, and Csaba Tóth. “Recognizing Weak Embeddings
of Graphs,” 274–92. ACM, 2018. https://doi.org/10.1137/1.9781611975031.20.
ieee: 'H. Akitaya, R. Fulek, and C. Tóth, “Recognizing weak embeddings of graphs,”
presented at the SODA: Symposium on Discrete Algorithms, New Orleans, LA, USA,
2018, pp. 274–292.'
ista: 'Akitaya H, Fulek R, Tóth C. 2018. Recognizing weak embeddings of graphs.
SODA: Symposium on Discrete Algorithms, 274–292.'
mla: Akitaya, Hugo, et al. Recognizing Weak Embeddings of Graphs. ACM, 2018,
pp. 274–92, doi:10.1137/1.9781611975031.20.
short: H. Akitaya, R. Fulek, C. Tóth, in:, ACM, 2018, pp. 274–292.
conference:
end_date: 2018-01-10
location: New Orleans, LA, USA
name: 'SODA: Symposium on Discrete Algorithms'
start_date: 2018-01-07
date_created: 2018-12-11T11:45:45Z
date_published: 2018-01-01T00:00:00Z
date_updated: 2023-09-15T12:19:32Z
day: '01'
department:
- _id: UlWa
doi: 10.1137/1.9781611975031.20
external_id:
arxiv:
- '1709.09209'
isi:
- '000483921200021'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1709.09209
month: '01'
oa: 1
oa_version: Preprint
page: 274 - 292
project:
- _id: 261FA626-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: M02281
name: Eliminating intersections in drawings of graphs
publication_status: published
publisher: ACM
publist_id: '7556'
quality_controlled: '1'
related_material:
record:
- id: '6982'
relation: later_version
status: public
scopus_import: '1'
status: public
title: Recognizing weak embeddings of graphs
type: conference
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2018'
...