Recognizing weak embeddings of graphs
Akitaya, Hugo
Fulek, Radoslav
Tóth, Csaba
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.
ACM
2018
info:eu-repo/semantics/conferenceObject
doc-type:conferenceObject
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https://research-explorer.app.ist.ac.at/record/309
Akitaya H, Fulek R, Tóth C. Recognizing weak embeddings of graphs. In: ACM; 2018:274-292. doi:<a href="https://doi.org/10.1137/1.9781611975031.20">10.1137/1.9781611975031.20</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1137/1.9781611975031.20
info:eu-repo/semantics/altIdentifier/arxiv/1709.09209
info:eu-repo/grantAgreement/FWF/M02281
info:eu-repo/semantics/closedAccess