REDUCING THE CLOCKWISE-ALGORITHM TO k LENGTH CLASSES

Ripà, Marco

doi:10.14710/jfma.v4i1.10106

Cited by 2 publications

(4 citation statements)

References 6 publications

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“…would solve the aforementioned problem if only we could allow self-intersecting for those 6 links of prescribed length 1 + √ 2 units. Thanks to the MΛI-algorithm, we showed that it is possible to provide acceptable covering paths, inside the MAABB, whose edges belong to the same integer (and not only irrational [3]) length class, in two, three, and more dimensions. We hope that the arising questions concerning how many valid length classes there are for any k-tuple (n 1 , n 2 , .…”

Section: Discussionmentioning

confidence: 99%

“…In Section 2, we also show how, for specific cases as n 1 = n k = 3, it is possible to shorten the link length [2] of the general solution provided by the MΛI-algorithm if we consider the bounding box k i=1 [0, n i ] (i.e., the RAABB) instead of the MAABB. Moreover, referring to the grid graphs G 3,3 = {0, 1, 2} × {0, 1, 2} and G 3,3,3 = {0, 1, 2} × {0, 1, 2} × {0, 1, 2}, we have constructively proved in [3] the existence of self-intersecting covering paths inside the MAABB (see Definition 1.3) that are formed by less than 3 k line segments, all belonging to the same irrational length class.…”

Section: E Regular Axis-aligned Bounding Box)mentioning

confidence: 94%

“…1. Visit all the points of G n The additional constraint that will be considered is to replace the RAABB with the MAABB [3]. Thus, merely P n 1 ,n 2 ,...,n k ⊆ Bn 1 ,n 2 ,...,n k , instead of P n 1 ,n 2 ,...,n k ⊂ B n 1 ,n 2 ,...,n k (as stated by the third rule above).…”

Section: E Regular Axis-aligned Bounding Box)mentioning

confidence: 99%

See 2 more Smart Citations

General Uncrossing Covering Paths Inside the Axis-Aligned Bounding Box

Ripà¹

2021

JFMA

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Given the finite set of n_1⋅n_2⋅...⋅n_k points G(n_1,n_2,...,n_k) in R^𝑘 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].

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Section: Discussionmentioning

confidence: 99%

Section: E Regular Axis-aligned Bounding Box)mentioning

confidence: 94%

See 1 more Smart Citation

General Uncrossing Covering Paths Inside the Axis-Aligned Bounding Box

Ripà¹

2021

JFMA

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“…If k = 3, the best achievable sequence of knight jumps has length 24 (see [14], page 66, Figures 4&5). In addition, Figure 3 shows another polygonal chain, P (3, 3; {(1, 1, 1), (2, 0, 0)}) := (0, 0, 0) (3,4; {(1, 1, 1, 1)})∪{(0, 2, 0, 1) → (0, 0, 0, 2)} (see Figure 4), where P c (3, 4; {(1, 1, 1, 1)}) := (0, 0, 0, 2) → (2, 1, 0, 2) → (0, 2, 0, 2) →…”

mentioning

confidence: 99%

Proving the existence of Euclidean knight’s tours on n × n × ⋯ × n chessboards for n < 4

Ripà

2024

NNTDM

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The Knight's Tour problem consists of finding a Hamiltonian path for the knight on a given set of points so that the knight can visit exactly once every vertex of the mentioned set. In the present, we provide a 5-dimensional alternative to the well-known statement that it is not ever possible for a knight to visit once every vertex of $C(3,k):=\{\underbrace{\{0,1,2\} \times \{0,1,2\}\times \cdots \times \{0,1,2\}}_\textrm{\textit{k}-times}\}$ by performing a sequence of $3^k-1$ jumps of standard length, since the most accurate answer to the original question actually depends on which mathematical assumptions we are making at the beginning of the game when we decide to extend a planar chess piece to the third dimension and above. Our counterintuitive outcome follows from the observation that we can alternatively define a 2D knight as a piece that moves from one square to another on the chessboard by covering a fixed Euclidean distance of $\sqrt{5}$ so that also the statement of Theorem 3 in [Erde, J., Golénia, B., & Golénia, S. (2012), The closed knight tour problem in higher dimensions, The Electronic Journal of Combinatorics, 19(4), #P9] does not hold anymore for such a Euclidean knight, as long as a 2 × 2 × ⋯ × 2 chessboard with at least 2<sup>7</sup> cells is given. Moreover, we construct a classical closed knight's tour on $C(3,4)-\{(1,1,1,1)\}$ whose arrival is at a distance of 2 from $(1,1,1,1)$, and then we show a closed Euclidean knight's tour on $\{\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\}\subseteq \mathbb{Z}^7$.

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REDUCING THE CLOCKWISE-ALGORITHM TO k LENGTH CLASSES

Cited by 2 publications

References 6 publications

General Uncrossing Covering Paths Inside the Axis-Aligned Bounding Box

General Uncrossing Covering Paths Inside the Axis-Aligned Bounding Box

Proving the existence of Euclidean knight’s tours on n × n × ⋯ × n chessboards for n < 4

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