Correction to: Cluster identification, selection, and description in Cluster randomized crossover trials: the PREP-IT trials

An amendment to this paper has been published and can be accessed via the original article

Delaunay graphs by divide and conquer

This document describes the LEDA program dc_delaunay.c for computing Delaunay graphs by the divide-and-conquer method. The program can be used either with exact primitives or with non-exact primitives. It handles all cases of degeneracy and is relatively robust against the use of imprecise arithmetic. We use the literate programming tool noweb by Norman Ramsey

A Separation Bound for Real Algebraic Expressions

Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, k-th root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms. We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda real.

The Reliable Algorithmic Software Challenge RASC

Algorithms are the heart of computer science. They make systems work. The theory of algorithms, i.e., their design and their analysis, is a highly developed part of theoretical computer science [7]

Rational points on circles

We solve the following problem. For a given rational circle $C$ passing through the rational points $p$, $q$, $r$ and a given angle $\alpha$, compute a rational point on $C$ whose angle at $C$ differs from $\alpha$ by a value of at most $\epsilon$. In addition, try to minimize the bit length of the computed point. This document contains the C++ program |rational_points_on_circle.c|. We use the literate programming tool |noweb| by Norman Ramsey

Fast recursive division

We present a new recursive method for division with remainder of integers. Its running time is $2K(n)+O(n \log n)$ for division of a $2n$-digit number by an $n$-digit number where $K(n)$ is the Karatsuba multiplication time. It pays in p ractice for numbers with 860 bits or more. Then we show how we can lower this bo und to $3/2 K(n)+O(n\log n)$ if we are not interested in the remainder. As an application of division with remainder we show how to speedup modular multiplication. We also give practical results of an implementation that allow u s to say that we have the fastest integer division on a SPARC architecture compa red to all other integer packages we know of

Efficient Exact Geometric Computation Made Easy

We show that the combination of the Cgal framework for geometric computation and the number type leda_real yields easy-to-write, correct and efficient geometric programs