V-Order: New combinatorial properties & a simple comparison algorithm

Computation of the suffix array, Burrows-Wheeler transform and FM-index in V-order

“…By Lemma 8, a comparison of two strings in V -order can ignore equal prefixes (or suffixes). Therefore, on this basis, Algorithm COMPARE(x,y), first presented in [8] and later upgraded in [9], ignores common prefixes of strings x and y while comparing them. COMPARE also makes use of the following:…”

“…Lemma 10. [3,4,5,7,8,9] V -order comparison of given strings x and y requires linear time and constant space.…”

“…Analogous to lexicographic comparison, the central problem of efficient Vordering of strings was considered in [3,4,5,6,7], culminating in a remarkably simple, linear time, constant space comparison algorithm [8], further improved in [9]. In work closely related to ideas in this paper, [3,4] also described efficient Lyndon-like factorization of a string x = x [1..n] into V -words (Definition 11).…”

Reconstructing a string from its Lyndon arrays

“…If in Figure 6 only rotations I-III are considered, then the selection x[2] = c rather than d would be made; if rotations II-IV were used, no determination could be made for x [1]; and if rotations I, III, IV were used, column i = 3 would become an instance of Case 2(b), and procedure ConstructString (λ, σ, n; x) i ← 1; cover ← σ n while i < n do maxlen ← 0 -Length of maximum λ [1..σ, i] for j ← 1 to σ do Treat λ as a two-dimensional array; find final letter jmax & frequency freqmax for maxlen.…”

“…Also, linear-time algorithms for computing Lyndon border and Lyndon suffix arrays have recently been proposed [2]. There is scope to eztend the results of this paper to the UMFF based on V-order [1], then further to UMFFs in their full generality.…”