Compressed indexes for dynamic text collections

Abstract

Let T be a string with n characters over an alphabet of constant size. A recent breakthrough on compressed indexing allows us to build an index for T in optimal space (i.e., O ( n ) bits), while supporting very efficient pattern matching [Ferragina and Manzini 2000; Grossi and Vitter 2000]. Yet the compressed nature of such indexes also makes them difficult to update dynamically. This article extends the work on optimal-space indexing to a dynamic collection of texts. Our first result is a compressed solution to the library management problem, where we show an index of O ( n ) bits for a text collection L of total length n , which can be updated in O (| T | log n ) time when a text T is inserted or deleted from L ; also, the index supports searching the occurrences of any pattern P in all texts in L in O (| P | log n + occ log 2 n ) time, where occ is the number of occurrences. Our second result is a compressed solution to the dictionary matching problem, where we show an index of O ( d ) bits for a pattern collection D of total length d , which can be updated in O (| P | log 2 d ) time when a pattern P is inserted or deleted from D ; also, the index supports searching the occurrences of all patterns of D in any text T in O ((| T | + occ )log 2 d ) time. When compared with the O ( d log d )-bit suffix-tree-based solution of Amir et al. [1995], the compact solution increases the query time by roughly a factor of log d only. The solution to the dictionary matching problem is based on a new compressed representation of a suffix tree. Precisely, we give an O ( n )-bit representation of a suffix tree for a dynamic collection of texts whose total length is n , which supports insertion and deletion of a text T in O (| T | log 2 n ) time, as well as all suffix tree traversal operations, including forward and backward suffix links. This work can be regarded as a generalization of the compressed representation of static texts. In the study of the aforementioned result, we also derive the first O ( n )-bit representation for maintaining n pairs of balanced parentheses in O (log n /log log n ) time per operation, matching the time complexity of the previous O ( n log n )-bit solution.