Parallel computation of the correlation dimension from a time series

A parallel algorithm is presented to compute the Correlation Dimension from a time series generated by a dynamical system. Three versions are described: the first computes all distances between points in the phase space, whereas the second and third compute only distances less than a threshold eps; the third version in particular is very powerful since it employs a box-assisted approach and linked lists for a fast search of neighboring points. The parallelization is designed for coarse-grained multiprocessor systems with distributed memory and is accomplished using a message passing model and partitioning points evenly among processors. Uniform implementation and computational analysis allow a clear comparison of the three versions. The algorithms, tested on the Transtech PARAstation multiprocessor, are well balanced, give a linear speed-up and show a good scalability. The third version is particularly suitable for fast processing of very long time series and allows the estimation of D_2 even for medium- and high-dimensional systems, where an extremely large number of points is needed. The algorithms can be adapted with few modifications to the computation of the generalized dimensions D_q, and they can also be useful in other applications involving the efficient computation of distances between points in a large set. More generally, the computational framework can be used in similar problems involving long-range interactions.