Compressed Data Structures for Dynamic Sequences

We consider the problem of storing a dynamic string S over an alphabetΣ = { 1, . . . , σ } in compressed form. Our representation supports insertions and deletions of symbols and answers three fundamental queries: access(i, S) returns the i-th symbol in S, ranka(i, S) counts how many times a symbol a occurs among the first i positions in S, and selecta(i, S) finds the position where a symbol a occurs for the i-th time. We present the first fully-dynamic data structure for arbitrarily large alphabets that achieves optimal query times for all three operations and supports updates with worst-case time guarantees. Ours is also the first fully-dynamic data structure that needs only nHk+o(n log σ) bits, where Hk is the k-th order entropy and n is the string length. Moreover our representation supports extraction of a substring S[i..i + ] in optimal O(log n/ log log n+ / logσ n) time.