Fast algorithms for finding disjoint subsequences with extremal densities

We derive fast algorithms for the problem of finding, on the real line, a prescribed number of intervals of maximum total length that contain at most some prescribed number of points from a given point set. Basically this is a typicaldynamic programming problem, however, for input sizes much bigger than the two parameters we can improve the obvious time bound by selecting a restricted set of candidate intervals that are sufficient to build some optimal solution. As a byproduct, the same idea improves an algorithm for a similar subsequence problem recently brought up by Chen, Lu and Tang at IWBRA 2005. The problems are motivated by the search for significant patterns in certain biological data. While the algorithmic idea for the asymptotic worst-case bound is rather evident, we also consider further heuristics to save even more time in typical instances. One of them, described in this paper, leads to an apparently open problem of computational geometry flavour (where we are seeking a subquadratic algorithm) which might be interesting in itself

Fast algorithms for finding disjoint subsequences with extremal densities

We derive fast algorithms for the following problem: Given a set of n points on the real line and two parameters s and p, find s disjoint intervals of maximum total length that contain at most p of the given points. Our main contribution consists of algorithms whose time bounds improve upon a straightforward dynamic programming algorithm, in the relevant case that input size n is much bigger than parameters s and p. These results are achievedby selecting a few candidate intervals that are provably sufficient for building an optimal solution via dynamic programming. As a byproduct of this idea we improve an algorithm for a similar subsequence problem of Chen, Lu andTang (2005). The problems are motivated by the search for significant patterns in biological data. Finally we propose several heuristics that further reduce the time complexity in typical instances. One of them leads to an apparently open subsequence sum problem of independent interest

Fast algorithms for finding disjoint subsequences with extremal densities

We derive fast algorithms for the following problem: Given a set of n points on the real line and two parameters s and p, find s disjoint intervals of maximum total length that contain at most p of the given points. Our main contribution consists of algorithms whose time bounds improve upon a straightforward dynamic programming algorithm, in the relevant case that input size n is much bigger than parameters s and p. These results are achievedby selecting a few candidate intervals that are provably sufficient for building an optimal solution via dynamic programming. As a byproduct of this idea we improve an algorithm for a similar subsequence problem of Chen, Lu andTang (2005). The problems are motivated by the search for significant patterns in biological data. Finally we propose several heuristics that further reduce the time complexity in typical instances. One of them leads to an apparently open subsequence sum problem of independent interest