Author(s):

Akitaya, Hugo A; Fulek, Radoslav; Tóth, Csaba D

Title: 
Recognizing weak embeddings of graphs

Affiliation 
IST Austria 
Abstract: 
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 interiordisjoint 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 polynomialtime 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.

Keywords: 
Polynomial approximation; Polynomialtime algorithms; Graph theory; MAtrix multiplication; Computational problem; Clustering algorithms; Data visualization; Linear equations; Maps; Corresponding vertices; Graph embeddings; Local constraints; Local operations; System of linear equations

Conference Title:

29th Annual ACM SIAM Symposium on Discrete Algorithms SODA 2018

Conference Dates:

January 7  10, 2018

Conference Location:

New Orleans, LA, USA

ISBN:

9781611975031

Publisher:

ACM

Date Published:

20180101

Start Page: 
274

End Page:

292

DOI: 
10.1137/1.9781611975031.20

Notes: 
∗Research supported in part by the NSF awards CCF1422311 and CCF1423615, and the Science Without Borders program. The second author gratefully acknowledges support from Austrian Science Fund (FWF): M2281N35.

Open access: 
no 