String Comparison in $V$-Order: New Lexicographic Properties & On-line Applications

Abstract

V -order is a global order on strings related to Unique Maximal Factorization Families (UMFFs) [6,7], which are themselves generalizations of Lyndon words [14]. V -order has recently been proposed as an alternative to lexicographical order in the computation of suffix arrays and in the suffix-sorting induced by the Burrows-Wheeler transform.Efficient V -ordering of strings thus becomes a matter of considerable interest. In this paper we present new and surprising results on V -order in strings, then go on to explore the algorithmic consequences.

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

“…The groups G 0 , G 2 , G 4 , G 7 , G 8 are empty and not shown in the figure. EVSA = (2,9,5,12,1,4,11,3,10,8,7,6), describing the ordering of the strings in M V w , while L = V -BWT w = 153302299629. Note that although w contains no letter repetitions, in contrast V -BWT w does, as is typical with this transformation scheme.…”

“…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).…”