Algorithmic graph embeddings

Algorithmic embeddings

We present several computationally efficient algorithms, and complexity results on low distortion mappings between metric spaces. An embedding between two metric spaces is a mapping between the two metric spcaes and the distortion of the embedding is the factor by which the distances change. We have pioneered theoretical work on relative (or approximation) version of this problem. In this setti...

Crossing Minimization within Graph Embeddings Crossing Minimization within Graph Embeddings

We propose a novel optimization-based approach to embedding heterogeneous high-dimensional data characterized by a graph. The goal is to create a two-dimensional visualization of the graph structure such that edge-crossings are minimized while preserving proximity relations between nodes. This paper provides a fundamentally new approach for addressing the crossing minimization criteria that exp...

Directed Graph Embeddings

Definitions of graph embeddings and graph minors for directed graphs are introduced. For the class of 2-terminal directed acyclic graphs (TDAGs) the definitions coincide, and the class is closed under both operations. The usefulness of the directed embedding operation is demonstrated be characterizing all TDAGs with parallel-width at most $k$, which generalizes earlier characterizations of seri...

Simultaneous Geometric Graph Embeddings

We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straight-line drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two graphs the problem is NP-hard. We also...

Algorithmic Applications of Low-Distortion Geometric Embeddings