Minimal Absent Words on Run-Length Encoded Strings
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A string w is called a minimal absent word (MAW) for another string T if w does not occur (as a substring) in T and any proper substring of w occurs in T. State-of-the-art data structures for reporting the set 𝖬𝖠𝖶(T) of MAWs from a given string T of length n require O(n) space, can be built in O(n) time, and can report all MAWs in O(|𝖬𝖠𝖶(T)|) time upon a query. This paper initiates the problem of computing MAWs from a compressed representation of a string. In particular, we focus on the most basic compressed representation of a string, run-length encoding (RLE), which represents each maximal run of the same characters a by a^p where p is the length of the run. Let m be the RLE-size of string T. After categorizing the MAWs into five disjoint sets ℳ_1, ℳ_2, ℳ_3, ℳ_4, ℳ_5 using RLE, we present matching upper and lower bounds for the number of MAWs in ℳ_i for i = 1,2,4,5 in terms of RLE-size m, except for ℳ_3 whose size is unbounded by m. We then present a compact O(m)-space data structure that can report all MAWs in optimal O(|𝖬𝖠𝖶(T)|) time.
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