We show that every n-point tree metric admits a (1 + Îµ)-embedding into ` C(Îµ) logn 1 , for every Îµ > 0, where C(Îµ) â‰¤ O ( ( 1Îµ ) 4 log 1Îµ ) ) . This matches the natural volume lower bound up to aâ€¦ (More)

The classical theorem of FÃ¡ry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of FÃ¡ryâ€™sâ€¦ (More)

Given a graph G cellularly embedded on a surface Î£ of genus g, a cut graph is a subgraph of G such that cutting Î£ along G yields a topological disk. We provide a fixed parameter tractableâ€¦ (More)

We prove new upper and lower bounds on the number of homotopy moves required to tighten a closed curve on a compact orientable surface (with or without boundary) as much as possible. First, we proveâ€¦ (More)

In this paper, we give a conditional lower bound of n on running time for the classic k-median and k-means clustering objectives (where n is the size of the input), even in lowdimensional Euclideanâ€¦ (More)

Normal surface theory, a tool to represent surfaces in a triangulated 3-manifold combinatorially, is ubiquitous in computational 3-manifold theory. In this paper, we investigate a relaxed notion ofâ€¦ (More)