# Circular, Periodic, or Framed Data Clustering: Fast, Optimal,
and Reproducible

Fast, optimal, and reproducible clustering algorithms for
circular, periodic, or framed data. The algorithms introduced here
are based on a core algorithm for optimal framed clustering the authors
have developed (Debnath & Song 2021) .
The runtime of these algorithms is O(K N log^2 N), where K is the number
of clusters and N is the number of circular data points. On a desktop
computer using a single processor core, millions of data points can be
grouped into a few clusters within seconds. One can apply the algorithms
to characterize events along circular DNA molecules, circular RNA
molecules, and circular genomes of bacteria, chloroplast, and
mitochondria. One can also cluster climate data along any given
longitude or latitude. Periodic data clustering can be formulated as
circular clustering. The algorithms offer a general high-performance
solution to circular, periodic, or framed data clustering.