Given a sequence of n elements, we introduce the notion of an almost-increasing subsequence in two contexts. The first notion is the longest subsequence that can be converted to an increasing subsequence by possibly adding a value, that is at most a fixed constant c, to each of the elements. We show how to optimally construct such subsequence in O(n logk) time, where k is the length of the output subsequence. As an exercise, we show how to produce in O(n 2 logk) time a special type of subsequences, that we call subsequences obeying the triangle inequality, by using as a subroutine our algorithm for the above case. The second notion is the longest subsequence where every element is at least the value of a monotonically non-decreasing function in terms of the r preceding elements (or even with respect to every r elements among those preceding it). We show how to construct such subsequence in O(n r logk) time. © 2010 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Elmasry, A. (2010). The longest almost-increasing subsequence. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6196 LNCS, pp. 338–347). https://doi.org/10.1007/978-3-642-14031-0_37
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