Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital Rays

Optimal Bounds for Weak Consistent Digital Rays in 2D

“…To state our results in the context of the work of Chiu et al [2], recall they showed that for all segments with length N and k inner leaves between diagonals N/2 and N has error Ω( N log N N +k ). This implies that Ω(N log N ) inner leaves are required to obtain O(1) error.…”

“…Certainly s satisfies both properties (1) and ( 2) above, but it may not satisfy (3). But we will show that we do not have to go "too many" diagonals before D(s) before we must find a point p ∈ R(s) that is close to the center of its cone and still satisfies (1) and (2). In order to show this last part, we need D(s) to be sufficiently large.…”

“…We will do so by first showing there exists a diagonal d * that is the smallest diagonal for which the cone width of Subtree(s) is greater than 2 − δ. We will then show that there must be a diagonal greater than d * and less than or equal to D(s) for which ℓ(t) crosses at most 2 3 δ below a grid point. Let us begin by bounding for which diagonals the cone width of Subtree(s) must be greater than 2 − δ.…”

“…Chiu et al [2] considered 2D WCDRs and considered their impact on (non-weak) CDRs in higher dimensions. In particular, they consider the tradeoff between the number of inner leaves a 2D WCDR can have and the error of the system.…”

“…Next we consider the case when p ↓ is the parent of p. We will show that p ↓ x −t d−1 x > 1.5−δ. Let t ′ be the point that is 2 3 δ below p ↖ . Note that t d will always be strictly above t ′ .…”