Traversing a set of points with a minimum number of turns. (English summary) Discrete Comput. Geom. 41 (2009), no. 4, 513–532.1432-0444 For a finite set of points S ⊆ Rd, let L(S) denote the minimum number of line segments of an axis-aligned polygonal path spanning S. The authors show the following bounds when S = Gd n = [1, n] × · · · × [1, n] is a d-dimensional grid of size n: • L(G3 n) ≥ 3 2n2 − O(n), thus proving the case d = 3 of a conjecture of E. Kranakis, D. Krizanc and L. G. L. T. Meertens [Ars Combin. 38 (1994), 177–192; MR1310418 (95m:05135)]. They also obtain almost tight lower and upper bounds for the more general case of an a × b × c grid. • 4 3n3 − O(n2) ≤ L(G4 n) ≤
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