Compressed Pattern Matching in the Annotated Streaming Model

The annotated streaming model was originally introduced by Chakrabarti, Cormode and McGregor [ICALP 09]. In this extension of the conventional streaming model, a single query on the stream has to be answered by a client with the help from an untrusted annotator that provides an annotated data stream (to be used with the input stream). Only one-way communication is allowed. We extend the model by considering multiple queries, and focus on on-going queries where a new output must be given every time an input item arrives. In this model, we show the existence of a data structure that enables us to store and recover information about past items in the stream using very little space on the client. We first use this technique to give a space-annotation trade-off for the annotated multi-indexing problem, which is a natural generalisation of the previously studied annotated indexing problem. Our main result is a space-annotation trade-off for the classic exact pattern matching problem in phrase-compressed strings. In particular, we show the existence of a O(logn) time per phrase, O(logn+m) client space solution which uses O(logn) words of annotation per phrase. If the client space is increased to O(n + m) then O(1) words of annotation and O(1) time per phrase suffices. Here n is the length of the stream and m is the length of the pattern. Our result also holds for the well-known LZ78 compression scheme which is a special case of phrase-compression. All of the problems we consider have Ω(n) randomised space lower bounds in the standard (unannotated) streaming model.