Covering Paths and Trees for Planar Grids

Given a set of points in the plane, a covering path is a polygonal path that visits all the points. In this paper we consider covering paths of the vertices of an n x m grid. We show that the minimal number of segments of such a path is $2\min(n,m)-1$ except when we allow crossings and $n=m\ge 3$, in which case the minimal number of segments of such a path is $2\min(n,m)-2$, i.e., in this case we can save one segment. In fact we show that these are true even if we consider covering trees instead of paths. These results extend previous works on axis-aligned covering paths of n x m grids and complement the recent study of covering paths for points in general position, in which case the problem becomes significantly harder and is still open

SOLVING THE 106 YEARS OLD 3^k POINTS PROBLEM WITH THE CLOCKWISE-ALGORITHM

In this paper, we present the clockwise-algorithm that solves the extension in -dimensions of the infamous nine-dot problem, the well known two-dimensional thinking outside the box puzzle. We describe a general strategy that constructively produces minimum length covering trails, for any ∈ N−{0}, solving the NP-complete (3×3×⋯×3)-points problem inside a 3×3×⋯×3 hypercube. In particular, using our algorithm, we explicitly draw different covering trails of minimal length h() = (3^ − 1)/2, for = 3, 4, 5. Furthermore, we conjecture that, for every ≥ 1, it is possible to solve the 3^-points problem with h() lines starting from any of the 3^ nodes, except from the central one. Finally, we cover 3×3×3 points with a tree of size 12

GENERAL UNCROSSING COVERING PATHS INSIDE THE AXIS-ALIGNED BOUNDING BOX

Given the finite set of n_1⋅n_2⋅...⋅n_k points G(n_1,n_2,...,n_k) in R^ such that n_k≥...≥n_2≥n_1∈Z+, we introduce a new algorithm, called MΛI, which returns an uncrossing covering path inside the minimum axis-aligned bounding box [0,n_1−1]×[0,n_2−1]×...×[0,n_k−1], consisting of 3⋅(n_1⋅n_2⋅...⋅n_k−1)−2 links of prescribed length n_k−1 units. Thus, for any n_k≥3, the link length of the covering path provided by our MΛI-algorithm is smaller than the cardinality of the set G(n_1,n_2,...,n_k). Furthermore, assuming k>2, we present an uncrossing covering path for G(3,3,...,3), comprising only 20*3^(k−3)−2 two units long edges, which is constrained by the axis-aligned bounding box [0,4−√3]×[0,4−√3]×[0,2]×...×[0,2]

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 mini