Ranking and unranking of B-trees

Unranking of small combinations from large sets

“…1, a group in C (8, 3) consists of (0, 2, 5), (1,3,6), (2,4,7), (0, 3,5), (1,4,6), (2,5,7), (0, 3,6), and (1,4,7). Only (0, 2, 5) belongs to R (8,3) in this group.…”

“…Since n and m are coprime in this case, we do not need to apply Lemma 1. These chosen vectors are assigned ranks 0, 8, 16, 24, 32, 40, and 48 according to their ranks in R (8,3), and the ranks of the other vectors are then decided by them. In the example of Table 1, the vector (0, 2, 5) ∈ R(8, 3) is assigned rank 16, and the vectors in the group of (0, 2, 5) are assigned ranks 17 to 23.…”

“…To the best of our knowledge, there are no known algorithms that outperform the algorithms presented in this paper. Ranking and unranking algorithms have been studied for various objects, such as permutations [19,20], trees [5,21], and B-trees [8]. The works most related to the present work among these are those about permutations of m elements chosen from an n-element set, as discussed in Mareš and Straka [19] and Myrvold and Ruskey [20].…”

Generating All and Random Instances of a Combinatorial Object

“…An algorithm for generating B-trees is described in the work by Gupta et al [16]. It is based on backtrack search, and produces B-trees with worst case delay proportional to the output size.…”

“…The order of generating B-trees becomes lexicographic if B-trees are coded as a B-tree sequence, defined in [5]. The algorithm [16] has constant expected delay in producing next B-tree, exclusive of the output, which is proven in the work by Belbaraka and Stojmenovic [5]. Using a decoding procedure, an algorithm that generates the B-tree data structure (meaning that the parent-children links are established) from given B-tree sequence can be designed, with constant average delay.…”