Compact directed acyclic word graphs (CDAWGs) [Blumer et al. 1987] are a fundamental data structure on strings with applications in text pattern searching, data compression, and pattern discovery. Intuitively, the CDAWG of a string T is obtained by merging isomorphic subtrees of the suffix tree [Weiner 1973] of the same string T, and thus CDAWGs are a compact indexing structure. Indeed, the CDAWG size 𝖾 can be sublinear in n for some highly repetitive strings. Of its various applications, the CDAWG allows for computing pattern occurrences, maximal exact matches (MEMs), minimal absent words (MAWs), and minimal unique substrings (MUSs) in optimal time using O(𝖾) space. For designing space-efficient data storage, it is crucial that the underlying data structure is robust against data edits and errors. As a mathematical measure for this, the notion of compression sensitivity [Akagi et al. 2023] was introduced as the maximum of the size increase in the compressed data structures after edits operations. In this paper, we investigate the sensitivity of CDAWGs when a single character edit operation is performed at an arbitrary position in the input string T. We show that the size of the CDAWG after an edit operation on T is asymptotically at most 8 times larger than the original CDAWG before the edit. This O(1) upper bound significantly improves on the only known upper bound O(n/log n) for the problem.

A suffixient set is a novel combinatorial object that captures the essential information of repetitive strings in a way that, provided with a random access mechanism, supports various forms of pattern matching. In this paper, we study the size χ of the smallest suffixient set as a repetitiveness measure. First, we study its sensitivity to various string operations. We show that χ cannot increase by more than 2 after appending or prepending a character to the string. As a consequence, we are able to give simple linear-time online algorithms to compute smallest suffixient sets. We also show that, although reversing the string can increase χ by an arbitrary O ( n ) value, it always holds χ ( T )/ χ ( T R ) ≤ 2. We also prove lower and upper bounds for the additive or multiplicative increase of χ after applying arbitrary edit operations, or rotating the text. In particular, we show that the additive increase can be as large as Ω ( n ) for all those operations. Secondly, we place χ among known repetitiveness measures. In particular, we show χ ≤ 2 r (where r is the number of runs in the Burrows-Wheeler Transform of the string), that there are string families where χ = o ( v ) (where v is the size of the smallext lexicographic parse of the string), and that χ is uncomparable to almost all reachable measures based on copy-paste mechanisms. In passing, we give precise bounds for χ for some relevant string families, for example χ ≤ σ + 2 on episturmian words over alphabets of size σ (e.g., χ ≤ 4 on Fibonacci strings, for which we precisely characterize the only two smallest suffixient sets).