A data structure for a sequence of string accesses in external memory

We introduce a new paradigm for querying strings in external memory, suited to the execution of sequences of operations. Formally, given a dictionary of n strings S1, . . ., Sn, we aim at supporting a search sequence for m not necessarily distinct strings T1, T2, . . ., Tm, as well as inserting and deleting individual strings. The dictionary is stored on disk, where each access to a disk page fetches B items, the cost of an operation is the number of pages accessed (I/Os), and efficiency must be attained on entire sequences of string operations rather than on individual ones. Our approach relies on a novel and conceptually simple self-adjusting data structure (SASL) based on skip lists, that is also interesting per se. The search for the whole sequence T1, T2, . . ., Tm can be done in an expected number of I/Os: O (∑ ^m _j=1 |T_j|/B + ∑ⁿ_i=1 (ni log _B m/ni)), where each Tj may or may not be present in the dictionary, and ni is the number of times Si is queried (i.e., the number of Tj s equal to Si ). Moreover, inserting or deleting a string Si takes an expected amortized number O( |Si |/ B + log_B n) of I/Os. The term ∑^m _{j =1} |Tj |/ B in the search formula is a lower bound for reading the input, and the term ∑ⁿ_i=1 ni log_B m/ni (entropy of the query sequence) is a standard information-theoretic lower bound.We regard this result as the static optimality theorem for external-memory string access, as compared to Sleator and Tarjan's classical theorem for numerical dictionaries [Sleator and Tarjan 1985]. Finally,we reformulate the search bound if a cache is available, taking advantage of common prefixes among the strings examined in the search.