 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.
<meaning-changed>  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.
 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.