Generalised distances of sequences II: B-distances with weight sequences

Abstract

In this paper, we investigate the weighted B-distances of infinite sequences. The general neighbourhood sequences were introduced for measuring distances in digital geometry (Z n), and the theory was recently extended for application to sequences. By assigning various weights to the elements of the sequences the concept is further generalized. An algorithm is presented which provides a shortest path between two sequences. Formula is also provided to calculate the weighted B-distance of any two sequences with a neighbourhood sequence B and a weight sequence. There are several neighbourhood sequences, which do not generate metrics. We prove a necessary and sufficient condition for a B-distance to define a generalized metric space above the sequences. Moreover, our results can be applied if the elements of the sequences used with various weights. In case of weight functions used B-distances we present also the metric conditions.

Generalised distances of sequences I: $B$-distances

“…(There is no problem with the symmetry by Lemma 4) Based on the proof of the case Z 1 using the properties of uids sequences the next theorem can be established. (This theorem is proven in a more general form in the second part of the paper, see [12].) By Lemma 7 one can decide, whether an neighbourhood sequence give a generalized metric space or not: Now we study the structure of the set of neighbourhood sequences based on another kind of partial ordering (see [7]).…”

“…Their properties are independent of their using conditions, therefore we do not give more details on them here. Instead of it, we are going to show another type of extensions of these distances in [12] as a continuation of this paper.…”

“…Hence calculating the L-distances we have some difficulties. (In [12] we return to this problem.) For these reasons we recommend the B-distances which have more pleasant properties in this point of view.…”

“…However, in geometry those distances are the most useful (e.g., have more practical interest), which have this property. In the next part of this section we give a necessary and sufficient condition for a distance based on a neighbourhood sequence to be a (generalized) metric (and we continue this analysis in [12], too).…”

On the number of shortest paths by neighborhood sequences on the square grid

“…A greedy algorithm that provides a shortest path between any two points by any neighborhood sequence B can be found in [10], while a general formula to compute B-distances can be found in [12]. We should note here that digital distances, and especially B-distances, can be applied in various image processing tasks [7,17] and also in communication networks [13], their extension for computing distances of sequences was also investigated [14,15]. In digital geometry, similarly as in graph theory, there is usually more than one minimal (i.e., shortest) path between two points, and it is an interesting combinatorial problem to find their numbers.…”