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.
After revision:  We present efficient data structures for computing two important functions of LF and \phi^{-1} so that they can be computed in a 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.