In cluster analysis, single linkage, nearest neighbour or shortest distance is a method of calculating distances between clusters in hierarchical clustering. In single linkage, the distance between two clusters is computed as the distance between the two closest elements in the two clusters.

Mathematically, the linkage function – the distance D(X,Y) between clusters X and Y – is described by the expression

$D(X,Y)=\min_{x\in X, y\in Y} d(x,y),$

where X and Y are any two sets of elements considered as clusters, and d(x,y) denotes the distance between the two elements x and y.

A drawback of this method is the so-called chaining phenomenon: clusters may be forced together due to single elements being close to each other, even though many of the elements in each cluster may be very distant to each other.

## Naive Algorithm

The following algorithm is an agglomerative scheme that erases rows and columns in a proximity matrix as old clusters are merged into new ones. The $N \times N$ proximity matrix D contains all distances d(i,j). The clusterings are assigned sequence numbers 0,1,......, (n − 1) and L(k) is the level of the kth clustering. A cluster with sequence number m is denoted (m) and the proximity between clusters (r) and (s) is denoted d[(r),(s)].

The algorithm is composed of the following steps:

1. Begin with the disjoint clustering having level L(0) = 0 and sequence number m = 0.
2. Find the most similar pair of clusters in the current clustering, say pair (r), (s), according to d[(r),(s)] = min d[(i),(j)] where the minimum is over all pairs of clusters in the current clustering.
3. Increment the sequence number: m = m + 1. Merge clusters (r) and (s) into a single cluster to form the next clustering m. Set the level of this clustering to L(m) = d[(r),(s)]
4. Update the proximity matrix, D, by deleting the rows and columns corresponding to clusters (r) and (s) and adding a row and column corresponding to the newly formed cluster. The proximity between the new cluster, denoted (r,s) and old cluster (k) is defined as d[(k), (r,s)] = min d[(k),(r)], d[(k),(s)].
5. If all objects are in one cluster, stop. Else, go to step 2.

## Optimally efficient algorithm

The algorithm explained above is easy to understand but of complexity $\mathcal{O}(n^3)$. In 1973, R. Sibson proposed an optimally efficient algorithm of only complexity $\mathcal{O}(n^2)$ known as SLINK.[1]

This is essentially the same as Kruskal's algorithm for minimum spanning trees. However, in single linkage clustering, the order in which clusters are formed is important, while for minimum spanning trees what matters is the set of pairs of points that form distances chosen by the algorithm.

Alternative linkage schemes include complete linkage and Average linkage clustering - implementing a different linkage in the naive algorithm is simply a matter of using a different formula to calculate inter-cluster distances in the initial computation of the proximity matrix and in step 4 of the above algorithm. An optimally efficient algorithm is however not available for arbitrary linkages. The formula that should be adjusted has been highlighted using bold text.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Complete-linkage clustering — In cluster analysis, complete linkage or farthest neighbour is a method of calculating distances between clusters in agglomerative hierarchical clustering. In complete linkage,[1] the distance between two clusters is computed as the maximum… …   Wikipedia

• Clustering — Unter Clusteranalyse (der Begriff Ballungsanalyse wird selten verwendet) versteht man strukturentdeckende, multivariate Analyseverfahren zur Ermittlung von Gruppen (Clustern) von Objekten, deren Eigenschaften oder Eigenschaftsausprägungen… …   Deutsch Wikipedia

• Clustering-Verfahren — Unter Clusteranalyse (der Begriff Ballungsanalyse wird selten verwendet) versteht man strukturentdeckende, multivariate Analyseverfahren zur Ermittlung von Gruppen (Clustern) von Objekten, deren Eigenschaften oder Eigenschaftsausprägungen… …   Deutsch Wikipedia

• Sequence clustering — In bioinformatics, sequence clustering algorithms attempt to group sequences that are somehow related. The sequences can be either of genomic, transcriptomic (ESTs) or protein origin.For proteins, homologous sequences are typically grouped into… …   Wikipedia

• Consensus clustering — Clustering is the assignment of objects into groups (called clusters) so that objects from the same cluster are more similar to each other than objects from different clusters. Often similarity is assessed according to a distance measure.… …   Wikipedia

• Document clustering — (also referred to as Text clustering) is closely related to the concept of data clustering. Document clustering is a more specific technique for unsupervised document organization, automatic topic extraction and fast information retrieval or… …   Wikipedia

• Cluster analysis — The result of a cluster analysis shown as the coloring of the squares into three clusters. Cluster analysis or clustering is the task of assigning a set of objects into groups (called clusters) so that the objects in the same cluster are more… …   Wikipedia

• Nearest-neighbor chain algorithm — In the theory of cluster analysis, the nearest neighbor chain algorithm is a method that can be used to perform several types of agglomerative hierarchical clustering, using an amount of memory that is linear in the number of points to be… …   Wikipedia

• Ballungsanalyse — Unter Clusteranalyse (der Begriff Ballungsanalyse wird selten verwendet) versteht man strukturentdeckende, multivariate Analyseverfahren zur Ermittlung von Gruppen (Clustern) von Objekten, deren Eigenschaften oder Eigenschaftsausprägungen… …   Deutsch Wikipedia

• Cluster-Analyse — Unter Clusteranalyse (der Begriff Ballungsanalyse wird selten verwendet) versteht man strukturentdeckende, multivariate Analyseverfahren zur Ermittlung von Gruppen (Clustern) von Objekten, deren Eigenschaften oder Eigenschaftsausprägungen… …   Deutsch Wikipedia