10.1137/1.9781611975031.20
Akitaya, Hugo
Hugo
Akitaya
Fulek, Radoslav
Radoslav
Fulek0000-0001-8485-1774
Tóth, Csaba
Csaba
Tóth
Recognizing weak embeddings of graphs
ACM
2018
2018-12-11T11:45:45Z
2020-01-21T13:21:55Z
conference
https://research-explorer.app.ist.ac.at/record/309
https://research-explorer.app.ist.ac.at/record/309.json
1709.09209
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.