Lattice spanners of low degree

Let [Formula: see text] denote the degree [Formula: see text] dilation of a point set [Formula: see text] in the domain of plane geometric spanners. If [Formula: see text] is the infinite square lattice, it is shown that [Formula: see text] and [Formula: see text]. If [Formula: see text] is the infinite hexagonal lattice, it is shown that [Formula: see text] and [Formula: see text]. All our constructions are planar lattice tilings constrained to degree [Formula: see text] or [Formula: see text].

Lower Bounds on the Dilation of Plane Spanners

(I) We exhibit a set of 23 points in the plane that has dilation at least [Formula: see text], improving the previous best lower bound of [Formula: see text] for the worst-case dilation of plane spanners. (II) For every [Formula: see text], there exists an [Formula: see text]-element point set [Formula: see text] such that the degree [Formula: see text] dilation of [Formula: see text] equals [Formula: see text] in the domain of plane geometric spanners. In the same domain, we show that for every [Formula: see text], there exists a an [Formula: see text]-element point set [Formula: see text] such that the degree [Formula: see text] dilation of [Formula: see text] equals [Formula: see text] The previous best lower bound of [Formula: see text] holds for any degree. (III) For every [Formula: see text], there exists an [Formula: see text]-element point set [Formula: see text] such that the stretch factor of the greedy triangulation of [Formula: see text] is at least [Formula: see text].