Combinatorial theorems about embedding trees on the real line

Amit Chakrabarti, Subhash Khot
2011 Journal of Graph Theory  
We consider the combinatorial problem of embedding the metric defined by an unweighted graph into the real line, so as to minimize the distortion of the embedding. This problem is inspired by connections to Banach space theory and to computer science. After establishing a framework in which to study line embeddings, we focus on metrics defined by three specific families of trees: complete binary trees, fans, and combs. We construct asymptotically optimal (i.e., distortion-minimizing) line
more » ... ings for these metrics and prove their optimality via suitable lower bound arguments. We show that even such specialized metrics require nontrivial constructions and proofs of optimality require sophisticated combinatorial arguments. Our results about these metrics show that the local density of a graph -an a priori reasonable lower bound on the optimum distortion -might in fact be arbitrarily smaller than the true optimum, even for tree metrics. They also show that the optimum distortion for a general tree can be arbitrarily low or high, even when it has bounded degree. The combinatorial techniques from our work might prove useful in further algorithmic research on low distortion metric embeddings.
doi:10.1002/jgt.20608 fatcat:vtkcxehxerggrnp2jf4lgajotq