Before revision:  We present efficient data structures for computing two important functions of LF and \phi^{-1} in constant time with O(r) words of space , which is a bit step forward in computation time from the previous best result of O(\log \log n)time for string length n and O(r) words of space. Finally, We present algorithms for computing queries on RLBWT by leveraging those two data structures in optimal time with O(r) words of space.
After revision:  We present efficient data structures for computing two important functions of LF and \phi^{-1} in constant time with O(r) words of space . Subsequently, we present OptBWTR (optimal-time queries on BWT-runs compressed indexes), the first string index that supports various queries including locate, count, extract queries in optimal time with O(r) words of space.