Continuous amortization and extensions: With applications to bisection-based root isolation

Abstract

Subdivision-based algorithms recursively subdivide an input region until the smaller subregions can be processed. It is a challenge to analyze the complexity of such algorithms because the work they perform is not uniform across the input region. Continuous amortization was introduced in Burr et al. (2009) as a way to bound the complexity of subdivision-based algorithms. The main features of this new technique are that (1) the technique can be applied, uniformly, to a variety of subdivision-based algorithms, (2) the technique considers a function directly related to the subdivision-based algorithm under consideration, and (3) the output of the technique is often explicitly expressed in terms of the intrinsic complexity of the problem instance.In this paper, the theory of continuous amortization is generalized and applied in several directions. The theory is generalized (1) to allow the domain to be higher dimensional or an abstract measure space, (2) to allow more general subdivisions than bisections, and (3) to bound the value of general functions on the regions of the final partition. The theory is applied to seven examples of subdivision-based algorithms. These applications include (1) bounding the number of subdivisions performed by algorithms for isolating real and complex roots of polynomials, (2) bounding the bit-complexity of subdivision-based algorithms for isolating the real roots of polynomials, and (3) bounding the expected run-time of an algorithm for approximating a biased coin. In each of these applications, by using continuous amortization, we achieve or improve the best-known complexity bounds.