arXiv subjects
Publications and source records attributed to Mohammad Moharrami.
There is a constant c > 0 such that for every $ε\in (0,1)$ and $n \geq 1/ε^2$, the following holds. Any mapping from the $n$-point star metric into $\ell_1^d$ with bi-Lipschitz distortion $1+ε$ requires dimension $$d \geq {c\log n\over ε^2\log (1/ε)}.$$
The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal c > 0, if the node cut conditions are satisfied, then one can simultaneously route a c-fraction of all the demands. This answers an open question of Chekuri and Kawarabayashi. More generally, we show that this holds in the setting of multi-commodity polymatroid networks introduced by Chekuri, et. al. Our approach employs a new type of random metric embedding in order to round the convex programs corresponding to these more general flow problems.
For every e>0, any subset of R^n with Hausdorff dimension larger than (1-e)n must have ultrametric distortion larger than 1/(4e).
We show that every n-point tree metric admits a (1+eps)-embedding into a C(eps) log n-dimensional L_1 space, for every eps > 0, where C(eps) = O((1/eps)^4 log(1/eps)). This matches the natural volume lower bound up to a factor depending only on eps. Previously, it was unknown whether even complete binary trees on n nodes could be embedded in O(log n) dimensions with O(1) distortion. For complete d-ary trees, our construction achieves C(eps) = O(1/eps^2).
We show that for every $α> 0$, there exist $n$-point metric spaces (X,d) where every "scale" admits a Euclidean embedding with distortion at most $α$, but the whole space requires distortion at least $Ω(\sqrt{α\log n})$. This shows that the scale-gluing lemma [Lee, SODA 2005] is tight, and disproves a conjecture stated there. This matching upper bound was known to be tight at both endpoints, i.e. when $α= Θ(1)$ and $α= Θ(\log n)$, but nowhere in between. More specifically, we exhibit $n$-point spaces with doubling constant $λ$ requiring Euclidean distortion $Ω(\sqrt{\log λ\log n})$, which also shows that the technique of "measured descent" [Krauthgamer, et. al., Geometric and Functional Analysis] is optimal. We extend this to obtain a similar tight result for $L_p$ spaces with $p > 1$.