The basic k-means clustering algorithm is a simple algorithm
that separates the given data space into different clusters based on centroids
calculation using some proximity function. Using this algorithm, we first
choose the k- points as initial centroids and then each point is assigned to a
cluster with the closest centroid. The algorithm is formally described as
follows:
Input:
A data set D containing m objects (points) with n attributes in an Euclidean space
Output:
Partitioning of m objects into k-clusters C1, C2, C3,
…, Ck, i.e. Ci ⊂ D and Ci ∩ Cj = ᶲ (for 1 ≤ i, j ≤ k)
Procedure:
- Select k points as initial centroid
- Repeat
- Form k clusters by assigning each point to its closest centroid
- Re-compute the centroid of each cluster
- Until the centroids don’t change
Time Complexity of the algorithm:
To analyze the
complexity of the algorithm in terms of time required, we need to identify the
operations required in each step and in each iteration of the algorithm. Here,
for k-means algorithm, the operations for each iteration are:
- Calculation of distances: To calculate the distance from a point to the centroid, we can use the squared Euclidean proximity function. For this, we need two subtractions, one summation, two multiplications and one square - root operations, i.e., 6-operations.
- Comparisons between distances
- Calculation of centroids
So, the total number of operations for an iteration
= distance calculation operations +
comparison operations + centroids calculation operations
= 6*[k*m*n] operations +
[(k-1)*m*n] operations + [k*((m-1) + 1)*n] operations
Assume that the algorithm converges after I iterations, then
total number of operations for the algorithm
= 6*[I*k*m*n] operations +
[I*(k-1)*m*n] operations + [I*k*((m-1) + 1)*n] operations
= [6Ikmn + I(k-1)mn + Ikmn]
operations
≈ O (I*k*m*n)
Here, we’ve applied the basic assumption of each operation
requiring the same amount of time. Since, normally, k << m & n << m, the analysis shows
that the k-means is linear in m, the number of points, and is scalable and
efficient in processing large data sets.
Space Complexity of the Algorithm:
The space requirement for k-means are modest because only
the data points and centroids are needed to be stored. Specifically, the
storage required is O((m+k)n), where m is the number of objects (points) &
n is the number of attributes considering n-dimensional objects.
References
[1] Vipin Kumar, Pang-Ning Tan & Michael Steinbach.(2005). "Introduction to Data Mining." Pearson/Addison Wesley, ISBN 0-321-32136-7.
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