Fluctuated Fitting Under the $\ell _1$-metric

Abstract

We consider the problem of fitting a given sequence of integers by an $(\alpha ,\beta )$-fluctuated one. For a sequence of numbers, those elements which are larger than their direct precursors are called ascends, those elements which are smaller than their direct precursors are called descends. A sequence is said to be $(\alpha ,\beta )$-fluctuated if there is a descend between any $\alpha +1$ ascends and an ascend between any $\beta +1$ descends; or equivalently, if it has at most $\alpha$ consecutive ascends and at most $\beta$ consecutive descends, when adjacent equal values are ignored.

Given a sequence of integers $\mathbf {a}=(a_1,\ldots ,a_n)$ and two parameters $\alpha ,\beta$ in [1, n], we compute (1) a sequence $\mathbf {b}=(b_1,\ldots ,b_n)$ of integers that is $(\alpha ,\beta )$-fluctuated and is closest to $\mathbf {a}$ among all such sequences; (2) a sequence $\mathbf {b}'=(b'_1,\ldots ,b'_n)$ of integers that is $(\alpha ,\beta )$-fluctuated and is bounded by $\mathbf {a}$ (i.e. $b'_i\le a_i$ for all i) and is closest to $\mathbf {a}$ among all such sequences. We measure the distance between sequences under $\ell _1$ metric.

Our algorithm runs in $O((\alpha +\beta )\cdot n)$ time, which is linear when $\alpha ,\beta$ are considered as constants. We also show that a variation of our problem can be solved in the same time complexity. We achieve our result mainly by exploiting and utilizing the property of the closest sequence.