Approximation algorithms for hitting objects with straight lines
Discrete Applied Mathematics
Angle-restricted tours in the plane
Computational Geometry: Theory and Applications
Improved lower bounds for the link length of rectilinear spanning paths in grids
Information Processing Letters
Information Processing Letters
Optimal Covering Tours with Turn Costs
SIAM Journal on Computing
Approximating minimum bending energy path in a simple corridor
Computational Geometry: Theory and Applications
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Given a finite set of points S in Rd, consider visiting thepoints in S with a polygonal path that makes a minimum number ofturns, or equivalently, has the the minimum number of segments(links). We call this minimization problem the minimum linkspanning path problem. This natural problem has appeared severaltimes in the literature under different variants. The simplest oneis where the allowed paths are axis-aligned. Let L(S) be theminimum number of links of an axis-aligned path for S denote by Gdn the d-dimensional grid of size n. Kranakis, Krizanc andMeertens (Ars Combinatoria, vol. 38, pp. 177--192, 1994)showed that in 2-dimensions L(G2n)=2n-1 and in three dimensions 4/3 n2-O(n)3n) 2+O(n). Kranakiset al. conjectured that, for all d ≥ 3, L(Gdn)= d/d-1 nd-1 ± O(nd-2). We prove theconjecture for d=3 by showing that L(G3n) ≥ 3/2 n2 -O(n). For d=4, we prove that 4/3 n3 -O(n2) ≤ L(G4n) ≤ 4/3 n3 +O(n5/2).For general d, we give new estimates on L(Gdn), that bring usvery close to the conjectured value. The new lower bound of (1+ 1/d)nd-1-O(nd-2) improves previous result byCollins and Moret (Information Processing Letters, vol. 68,pp. 317--319, 1998), while the new upper bound of (1+ 1/d-1)nd-1+O(nd-3/2) differs from the conjecturedvalue only in the lower order terms. For arbitrary point sets, we give an exact bound on the minimumnumber of links needed in an axis-aligned path traversing any planar n-point set. We obtain similar tight estimates (within 1) in anynumber of dimensions d. For the general problem of traversing anarbitrary set of points in Rd with an axis-aligned spanning pathhaving a minimum number of links, we present a constant ratio(depending on the dimension d) approximation algorithm.