Tighter Bounds for the Sum of Irreducible LCP Values

The suffix array is frequently augmented with the longest-common-prefix (LCP) array that stores the lengths of the longest common prefixes between lexicographically adjacent suffixes of a text. While the sum of the values in the LCP array can be \(\Omega (n^2)\) for a text of length \(n\), the sum of so-called irreducible LCP values was shown to be \(\mathcal {O}(n\lg n)\) just a few years ago. In this paper, we improve the bound to \(\mathcal {O}(n\lg r)\), where \(r\le n\) is the number of runs in the Burrows-Wheeler transform of the text. We also show that our bound is tight up to lower order terms (unlike the previous bound). Our results and the techniques used in proving them provide new insights into the combinatorics of text indexing and compression, and have immediate applications to LCP array construction algorithms.