Akra–Bazzi method

Abstract: This paper presents new theorems to analyze divide-and-conquer recurrences, which improve other similar ones in several aspects. In particular, these theorems provide more information, free us almost completely from technicalities like floors and ceilings, and cover a wider set of toll functions and weight distributions, stochastic recurrences included.

Abstract: We consider the following problem: Given n points in a unit square in the Euclidean plane, find a matching of the points such that the cost (i.e., the sum of the lengths of the edges between matched points) is minimum. In particular, we present a class of linear time heuristic algorithms for this problem and analyze their worst case performance. The worst case performance of an algorithm is defined as the greatest possible cost, as a function of n, of the matching produced by the algorithm on a set of n points. Each of the algorithms studied here divides the unit square into a few smaller regions, and then is applied recursively to the points in each of these regions.

Abstract: Divide and Conquer (DC) is conceptually well suited to high-dimensional optimization by decomposing a problem into multiple small-scale sub-problems. However, appealing performance can be seldom observed when the sub-problems are interdependent. This paper suggests that the major difficulty of tackling interdependent sub-problems lies in the precise evaluation of a partial solution (to a sub-problem), which can be overwhelmingly costly and thus makes sub-problems non-trivial to conquer. Thus, we propose an approximation approach, named Divide and Approximate Conquer (DAC), which reduces the cost of partial solution evaluation from exponential time to polynomial time. Meanwhile, the convergence to the global optimum (of the original problem) is still guaranteed. The effectiveness of DAC is demonstrated empirically on two sets of non-separable high-dimensional problems.