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A cluster is a collection of data objects that are similar to one another within the same cluster and are dissimilar to the objects in other clusters. Moreover, there will exist more or less similarities among these large amounts of initial cluster results in real life data set. Accordingly, analyzer may have difficulty to implement further analysis if they know nothing about these similarities. Therefore, it is very valuable to analyze these similarities and construct the hierarchy structures of the initial clusters. The traditional cluster methods are unfit for this cluster post-processing problem for their favor of finding the convex cluster result, impractical hypothesis and multiple scans of the data set. Based on multifractal theory, this paper proposes the FCHO (fractal-based cluster hierarchy optimization) algorithm, which integrates the cluster similarity with cluster shape and cluster distribution to construct the cluster hierarchy tree from the disjoint initial clusters. The elementary time-space complexity of the FCHO algorithm is presented. Several comparative experiments using synthetic and real life data set show the

Finding Natural Cluster Hierarchies Based on MultiFractal