Generalised distances of sequences I: $B$-distances

Abstract

In this paper, we investigate the B-distances of infinite sequences. For this purpose we use generalized neighbourhood sequences. The general neighbourhood sequences were introduced for measuring distances in digital geometry (Z n). We extend their application to sequences, and present an algorithm which provides a shortest path between two sequences. We also present a formula to calculate the B-distance of any two sequences for a neighbourhood sequence B. We also investigate the concept of k-convergent sequences for k 2 N, that concept is generally weaker than the convergence. We will use the term k-sequence which is a kind of generalization of the concept of 0-sequence. We also show some connection between the B-distances of sequences and the properties of their difference sequences.

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

“…The Hamming-distance (H-distance) of two same-length sequences can be extended to infinite sequences over infinite alphabets (Z, or R), see [11]. Other possibilities are the supremum norm (sup-distance) (see [7]) and the inf-distance, but this latter is rarely used.…”

“…The theory of neighbourhood sequences (n-sequences, for short) comes from digital geometry, but they can also be applied for infinite sequences [11] and for (formal) languages [8]. In digital geometry, finite integer sequences are used according to the dimension of the used digital space.…”

“…The theory is, then, extended to infinite dimension in [4,9,10]. By dropping the criteria of usage of integer values in a sequence, the theory is further extended from digital spaces to (infinite) sequences [11].…”

“…By the help of an n-sequence B = (b(i)) ∞ i=1 the B-distance of sequences p, q is defined as follows [11]. The length of a shortest path from p to q is taken, such that at the i-th step, we may move from a sequence to another if and only if they are b(i)-neighbours.…”

“…In this paper, we generalize the concept of B-distances of sequences of [11] by the following reasons: Our previous theory works very well if the elements of the sequences play equal roles. However, in some cases, for instance, the tails have less important role than the first elements of the sequences.…”