Akitaya, Hugo ; Fulek, RadoslavIST Austria ; 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 , 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 : 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 , and combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.
∗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.
274 - 292
SODA: Symposium on Discrete Algorithms
New Orleans, LA, USA
2018-01-07 – 2018-01-10
Akitaya H, Fulek R, Tóth C. Recognizing weak embeddings of graphs. In: ACM; 2018:274-292. doi:10.1137/1.9781611975031.20
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
Akitaya, Hugo, Radoslav Fulek, and Csaba Tóth. “Recognizing Weak Embeddings of Graphs,” 274–92. ACM, 2018. https://doi.org/10.1137/1.9781611975031.20.
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.
Akitaya H, Fulek R, Tóth C. 2018. Recognizing weak embeddings of graphs. SODA: Symposium on Discrete Algorithms 274–292.
Akitaya, Hugo, et al. Recognizing Weak Embeddings of Graphs. ACM, 2018, pp. 274–92, doi:10.1137/1.9781611975031.20.
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