Efficient Algorithms for Longest Closed Factor Array

We consider a family of strings called closed strings and a related array of Longest Closed Factors LCF. We show that the reconstruction of a string from its LCF array is easier than the construction and verification of this array. Moreover, the reconstructed string is unique. We improve also the time of construction/verification, reducing it from $$\mathcal {O}n \log n/\log \log n$$ the best previously known to $$\mathcal {O}n\sqrt{\log n}$$. We use connections between the LCF array and the longest previous/next factor arrays.