Before revision:  In this paper, we consider the sliding window version of the problem: For a fixed window length d, we propose two algorithms to maintain the palindromic tree of size O(d) for every sliding window S[i.. i+d-1 ] over S, one running in O(n\log\sigma') time with O(d) space where \sigma' \leq d is the maximum number of distinct characters in the windows, and the other running in O(n + d\sigma) time with d \sigma + O(d) space.
After revision:  In this paper, we consider the sliding window version of the problem: For a sliding window of length at most d, we propose two algorithms to maintain the palindromic tree of size O(d) for every sliding window S[i.. i+d-1 ] over S, one running in O(n\log\sigma') time with O(d) space where \sigma' \leq d is the maximum number of distinct characters in the windows, and the other running in O(n + d\sigma) time with d \sigma + O(d) space.