Traversing a Set of Points with a Minimum Number of Turns

Given a finite set of points S in R-d , consider visiting the points in S with a polygonal path which makes a minimum number of turns, or equivalently, has the minimum number of segments (links). We call this minimization problem the minimum link spanning path problem. This natural problem has appeared several times in the literature under different variants. The simplest one is that in which the allowed paths are axis-aligned. Let L(S) be the minimum number of links of an axis-aligned path for S, and let G (n) (d) be an n x ... x n grid in sign Z(d) . Kranakis et al. (Ars Comb. 38:177-192, 1994) showed that L(G(n)(2) ) = 2n - 1 and 4/3 n (2) - O (n) L (G(n)(3)) <= 3/2 n(2) + O (n) and conjectured that, for all d >= 3, L (G(n)(d)) = d/d-1n(d-1) +/- O (n(d-2)). We prove the conjecture for d = 3 by showing the lower bound for L (G(n)(3)). For d = 4, we prove that L (G(n)(4)) + 4/3n(3) +/- O (n(5/2)). For general d, we give new estimates on L (G(n)(d)) that are very close to the conjectured value. The new lwer bound of (1 + 1/d) n(d-1) - O (n(d-2)) improves previous result by Collins and Moret (Inf. Process. Lett. 68:317-319, 1998), while the new upper bound of (1 + 1/d-1)n(d-1) + O (n(d-3/2)) differs from the conjectured value only in the lower order terms. For arbitrary point sets, we include an exact bound on the minimum number of links needed in an axis-aligned path traversing any planar n-point set. We obtain similar tight estimates (within 1) in any number of dimensions d. For the general problem of traversing an arbitrary set of points in R-d with an axis-aligned spanning path having a minimum number of links, we present a constant ratio (depending on the dimension d) approximation algorithm.