Abstract
Given a finite set of points S in ℝd, 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 d n be an n×…×n grid in ℤd. Kranakis et al. (Ars Comb. 38:177–192, 1994) showed that L(G 2 n )=2n−1 and \(\frac{4}{3}n^{2}-O(n)\le L(G^{3}_{n})\le \frac{3}{2}n^{2}+O(n)\) and conjectured that, for all d≥3, \(L(G^{d}_{n})=\frac{d}{d-1}n^{d-1}\pm O(n^{d-2}).\) We prove the conjecture for d=3 by showing the lower bound for L(G 3 n ). For d=4, we prove that \(L(G^{4}_{n})=\frac{4}{3}n^{3}\pm O(n^{5/2}).\)
For general d, we give new estimates on L(G d n ) that are very close to the conjectured value. The new lower bound of \((1+\frac{1}{d})n^{d-1}-O(n^{d-2})\) improves previous result by Collins and Moret (Inf. Process. Lett. 68:317–319, 1998), while the new upper bound of \((1+\frac{1}{d-1})n^{d-1}+O(n^{d-3/2})\) 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 ℝd with an axis-aligned spanning path having a minimum number of links, we present a constant ratio (depending on the dimension d) approximation algorithm.
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Arkin, E.M., Mitchell, J.S.B., Piatko, C.D.: Minimum-link watchman tours. Inf. Process. Lett. 86, 203–207 (2003)
Arkin, E.M., Bender, M.A., Demaine, E.D., Fekete, S.P., Mitchell, J.S.B., Sethia, S.: Optimal covering tours with turn costs. SIAM J. Comput. 35(3), 531–566 (2005)
Braß, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)
Collins, M.J.: Covering a set of points with a minimum number of turns. Int. J. Comput. Geom. Appl. 14(1–2), 105–114 (2004)
Collins, M.J., Moret, M.E.: Improved lower bounds for the link length of rectilinear spanning paths in grids. Inf. Process. Lett. 68, 317–319 (1998)
Fekete, S.P., Woeginger, G.J.: Angle-restricted tours in the plane. Comput. Geom.: Theory Appl. 8(4), 195–218 (1997)
Gaur, D.R., Bhattacharya, B.: Covering points by axis parallel lines. In: Proc. 23rd European Workshop on Computational Geometry, pp. 42–45 (2007)
Gavril, F.: Some NP-complete problems on graphs. In: Proc. 11th Conference on Information Sciences and Systems, pp. 91–95 (1977)
Hassin, R., Megiddo, N.: Approximation algorithms for hitting objects with straight lines. Discrete Appl. Math. 30(1), 29–42 (1991)
Kranakis, E., Krizanc, D., Meertens, L.: Link length of rectilinear Hamiltonian tours in grids. Ars Comb. 38, 177–192 (1994)
Maheshwari, A., Sack, J.-R., Djidjev, H.N.: Link distance problems. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 519–558. Elsevier, Amsterdam (2000). Chap. 12
Megiddo, N., Tamir, A.: On the complexity of locating linear facilities in the plane. Oper. Res. Lett. 1(5), 194–197 (1982)
Stein, C., Wagner, D.P.: Approximation algorithms for the minimum bends traveling salesman problem. In: Proc. 8th Internat. Conf. on Integer Programming and Combinatorial Optimization. LNCS, vol. 2081, pp. 406–421. Springer, Berlin (2001)
Wagner, D.P.: Path planning algorithms under the link-distance metric. Ph.D. thesis, Dartmouth College (2006)
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Work by A. Dumitrescu was partially supported by NSF CAREER grant CCF-0444188.
Work by F. Hurtado was partially supported by projects MECMTM2006-01267 and Gen. Cat. 2005SGR00692.
Work by P. Valtr was partially supported by the project 1M0545 of the Ministry of Education of the Czech Republic.
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Bereg, S., Bose, P., Dumitrescu, A. et al. Traversing a Set of Points with a Minimum Number of Turns. Discrete Comput Geom 41, 513–532 (2009). https://doi.org/10.1007/s00454-008-9127-1
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DOI: https://doi.org/10.1007/s00454-008-9127-1