Sorting strings and constructing digital search trees in parallel

We describe two simple optimal-work parallel algorithms for sorting a list L=(X₁,X₂,...,X_m) of m strings over an arbitrary alphabet Σ, where Σ_i=1^m|X_i|=n. The first algorithm is a deterministic algorithm that runs in O((log² m)/(log log m)) time and the second is a randomized algorithm that runs in O(log m) time. Both algorithms use O(m log(m)+n) operations. Compared to the best known parallel algorithms for sorting strings, the algorithms offer the following improvements: the total number of operations used by the algorithms is optimal while all previous parallel algorithms use a non-optimal number of operations; we make no assumption about the alphabet while the previous algorithms assume that the alphabet is restricted to {1,2,..., n^O(1)}; the computation model assumed by the algorithms is the Common CRCW PRAM unlike the known algorithms that assume the Arbitrary CRCW PRAM; and the presented algorithms use O(m log m+n) space, while previous parallel algorithms use O(n^1+ε ) space, where ε is a positive constant. We also present optimal-work parallel algorithms to construct a digital search tree for a given set of strings and to search for a string in a sorted list of strings. We use the parallel sorting algorithms to solve the problem of determining a minimal starting point of a circular string with respect to lexicographic ordering