A Linear Algorithm for the Random Sampling from Regular Languages

We present the first linear algorithm for the random sampling from regular languages. More precisely, for generating a uniformly random word of length n in a given regular language, our algorithm has worst-case space bit-complexity O(n) and mean time bit-complexity O(n). The previously best algorithm, due to Denise and Zimmermann (1999), has worst-case space bit-complexity O(n ²) and mean time bit-complexity O(nlog (n)). The Denise et al. algorithm was obtained by performing a floating-point optimization on the general recursive method formalized by Nijenhuis and Wilf (and further developed by Flajolet, Zimmermann and Van Cutsem). Our algorithm combines the floating-point optimization with a new divide-and-conquer scheme.