  Several more and more efficient component--by--component (CBC) constructions
for suitable rank-1 lattices were developed during the last decades. On the one
hand, there exist constructions that are based on minimizing some error
functional. On the other hand, there is the possibility to construct rank-1
lattices whose corresponding cubature rule exactly integrates all elements
within a space of multivariate trigonometric polynomials.
  In this paper, we focus on the second approach, i.e., the exactness of rank-1
lattice rules. The main contribution is the analysis of a probabilistic version
of an already known algorithm that realizes a CBC construction of such rank-1
lattices. It turns out that the computational effort of the known deterministic
algorithm can be considerably improved in average by means of a simple
randomization. Moreover, we give a detailed analysis of the computational costs
with respect to a certain failure probability, which then leads to the
development of a probabilistic CBC algorithm. In particular, the presented
approach will be highly beneficial for the construction of so-called
reconstructing rank-1 lattices, that are practically relevant for function
approximation. Subsequent to the rigorous analysis of the presented CBC
algorithms, we present an algorithm that determines reconstructing rank-1
lattices of reasonable lattice sizes with high probability. We provide
estimates on the resulting lattice sizes and bounds on the occurring failure
probability. Furthermore, we discuss the computational complexity of the
presented algorithm.
  Various numerical tests illustrate the efficiency of the presented
algorithms. Among others, we demonstrate how to exploit the efficiency of our
algorithm even for the construction of exactly integrating rank-1 lattices,
provided that a certain property of the treated space of trigonometric
polynomials is known.
