How is centroid clustering performed with only pair wise distances

Question

I am new to clustering and am a bit confused on how the following code can compute centroid information without the original euclidean vectors.

import numpy as np
from scipy.cluster import hierarchy
import matplotlib.pyplot as plt

# Create an array of distances
x = np.array([100., 200., 300., 400., 500., 250.,
              450., 280., 450., 750.])
 
# compute linkage
temp = hierarchy.linkage(x, method="centroid")

This code runs without error and is using 'centroid' as the link method. I am confused as to how this is even possible. My guess is there might be a formula which maps distance pair values to the centroid distance value.

I tried looking at various sources, including scipy documentation, but they all mention taking on average of the original data. Is there some reference which discusses how one can compute centroid distances without the original euclidean data?

Answer 1

Thanks to @Henry for pointing me to the end of the documentation which points to this paper:

Modern hierarchical, agglomerative clustering algorithms

In the paper a general clustering algorithm is described in Fig 1. The author specifies that centroid, and many other clustering strategies, can be thought of as a distance update algorithm. Thus, the need for the original euclidean data is not needed to calculate the centroids. The author does not derive the formulas but this was enough to satisfy my curiosity.