On fast and simple algorithms for finding Maximal subarrays and applications in learning theory

Abstract

Consider the following problem SUB(k, n). Given an array A = (a ₁,..., a_n) of real elements a _i and a natural number k, find (at most) k disjoint subarrays A ₁,...,A_k in A such that the sum of the elements contained in the subarrays is maximum.

In this paper, we present a simple algorithm, based on Dynamic Programming, solving SUB(k, n) in time O(kn). Extracting the main idea of the dynamic programming scheme, we are able to extend the algorithm such that it is applicable for a wider class of related optimization problems. We show efficient applications of this algorithm in the area of agnostic learning by means of minimum disagreement, such as learning k intervals or identifying objects in a pixel matrix. In particular, the algorithm enables us to generate optimal functions inside a special class of piecewise constant functions. In restricted settings, our algorithm is better by a factor of sample-size-order compared to solutions induced by the dynamic scheme presented in [5].

Furthermore, we develop a generalization of a tree data structure introduced in [1]. Using this tree structure, we can solve corresponding online learning tasks efficiently.