Optimal Discovery of Subword Associations in Strings

Given a textstring x of n symbols aJld an illtcgcr constant. d, we consider the problem of finding, for any pair (H, =) of subwords of x the number of times that yand z occur in tandem (i.e., with no intermediate occurrence of either one of them) within a distance of d symbols of x. Although in principle there might be Ij4 distind subword pairs in X, we show that it suffices to consider a family of only [/2 SlLCh pairs, with the property that for any neglected pair (w', Zl), there is a corresponding pair (w, z) contained in our family and such that: (i) Wi is a prefix of wand z' is a prefix of z, and (ii) the tandem index of (w', =') equals that of (w, z). We show that an algorithm for the construction of the table of all such tandem indices can be bllilt to run in optimal O(n2 ) time and space.