Abstract
We show that there are asymptotically \(\gamma n\) LMS-factors in a random word of length \(n\), for some explicit \(\gamma \) that depends on the model of randomness under consideration. Our results hold for uniform distributions, memoryless sources and Markovian sources. From this analysis, we give new insight on the typical behavior of the IS-algorithm [9], which is one of the most efficient algorithms available for computing the suffix array.
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Notes
- 1.
In their proof, they compute the mean length of an LMS-factor. The types of such a factor form a word of \(SS^{*}LL^{*}S\). For the considered model, the mean length of an element of \(S^{*}\) (and of \(L^{*}\)) is one. Hence, the average length of an LMS-factor is 5 (and not the announced 4).
- 2.
The formulas below hold when the extended letters are in \(\underline{\mathrm {LS}}(A)\) only. For instance, \(\alpha L=a_{1}L\) is not part of the definition, since it is not in \(\underline{\mathrm {LS}}(A)\).
- 3.
This may be a consequence of the well-known fact that in a vertebrate genome, a C is very rarely followed by a G. This property is well captured by a Markov chain of order 1, but invisible to a memoryless model.
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Nicaud, C. (2015). A Probabilistic Analysis of the Reduction Ratio in the Suffix-Array IS-Algorithm. In: Cicalese, F., Porat, E., Vaccaro, U. (eds) Combinatorial Pattern Matching. CPM 2015. Lecture Notes in Computer Science(), vol 9133. Springer, Cham. https://doi.org/10.1007/978-3-319-19929-0_32
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