The dendrogram of DIANA can be constructed by letting the splinter group <math>C_\textrm{new}</math> be a child of the hollowed-out cluster <math>C_*</math> each time. This constructs a tree with <math>C_0</math> as its root and <math>n</math> unique single-object clusters as its leaves.
== Greedy Nature of the Algorithm ==
Hierarchical clustering is often described as a greedy algorithm because it makes a series of locally optimal choices without reconsidering previous steps. At each iteration, it merges the two clusters that are closest together based on a selected distance metric, always choosing the best immediate option available. This approach is "greedy" because it seeks to optimize the current decision rather than planning for the best possible overall clustering <ref name=":4" />. Once two clusters are merged, the decision is final and irreversible, without the possibility of backtracking, which can lead to suboptimal results if earlier choices were not ideal. Despite this, the greedy nature of hierarchical clustering makes it computationally efficient and simple to implement, though it may not always capture the true underlying structure of complex datasets <ref name=":5" />.
== Limitations ==
Hierarchical clustering, particularly in its standard agglomerative form, presents several notable limitations: (a) Time Complexity: Hierarchical clustering, especially in its basic agglomerative form, has a high time complexity of O(n³). This becomes a significant bottleneck for large datasets, limiting its scalability <ref name="CLINK2">{{cite journal |author=D. Defays |year=1977 |title=An efficient algorithm for a complete-link method |journal=The Computer Journal |publisher=British Computer Society |volume=20 |issue=4 |pages=364–6 |doi=10.1093/comjnl/20.4.364 |doi-access=}}</ref>. (b) Scalability: Due to the time and space complexity, hierarchical clustering algorithms struggle to handle very large datasets efficiently <ref name=DE/> (c) Sensitivity to Noise and Outliers: Hierarchical clustering methods can be sensitive to noise and outliers in the data, which can lead to the formation of inaccurate or misleading cluster hierarchies <ref name="SLINK2">{{cite journal |author=R. Sibson |year=1973 |title=SLINK: an optimally efficient algorithm for the single-link cluster method |url=http://www.cs.gsu.edu/~wkim/index_files/papers/sibson.pdf |journal=The Computer Journal |publisher=British Computer Society |volume=16 |issue=1 |pages=30–34 |doi=10.1093/comjnl/16.1.30 |doi-access=free}}</ref>. (d) Difficulty with High-Dimensional Data: In high-dimensional spaces, hierarchical clustering can face challenges due to the curse of dimensionality, where data points become sparse, and distance measures become less meaningful. This can result in poorly defined clusters<ref name=":6" /><ref name=":3">{{Cite conference |last1=Herr |first1=Dominik |last2=Han |first2=Qi |last3=Lohmann |first3=Steffen |last4=Ertl |first4=Thomas |date=2016 |title=Visual Clutter Reduction through Hierarchy-based Projection of High-dimensional Labeled Data |url=https://graphicsinterface.org/wp-content/uploads/gi2016-14.pdf |conference=Graphics Interface |language=en-CA |doi=10.20380/gi2016.14 |access-date=2022-11-04 |website=Graphics Interface}}</ref>. (e) Inability to Handle Non-Convex Shapes and Varying Densities: Traditional hierarchical clustering methods, like many other clustering algorithms, often assume that clusters are convex and have similar densities. They may struggle to accurately identify clusters with non-convex shapes or varying densities <ref name=":5">{{Cite journal |last=Wani |first=Aasim Ayaz |date=2024-08-29 |title=Comprehensive analysis of clustering algorithms: exploring limitations and innovative solutions |journal=PeerJ Computer Science |language=en |volume=10 |pages=e2286 |doi=10.7717/peerj-cs.2286 |issn=2376-5992 |pmc=11419652 |pmid=39314716 |doi-access=free}}</ref>.
