Question

I am looking at the k-means++ initialization algorithm. The following two steps of the algorithm give rise to non-uniform probabilities:

For each data point x, compute D(x), the distance between x and the
nearest center that has already been chosen.

Choose one new data point at random as a new center, using a weighted
probability distribution where a point x is chosen with probability
proportional to D(x)^2.

How can I select with this stated weighted probability distribution in C++?

Answer 1

With a finite set of individual data points X, this calls for a discrete probability distribution.

The easiest way to do this is to enumerate the points X in order, and calculate an array representing their cumulative probability distribution function: (pseudocode follows)

/* 
 * xset is an array of points X,
 * cdf is a preallocated array of the same size
 */
function prepare_cdf(X[] xset, float[] cdf)
{
   float S = 0;
   int N = sizeof(xset);
   for i = 0:N-1
   {
      float weight = /* calculate D(xset[i])^2 here */
      // create cumulative sums and write to the element in cdf array
      S += weight;
      cdf[i] = S;
   }

   // now normalize so the CDF runs from 0 to 1
   for i = 0:N-1
   {
      cdf[i] /= S;
   }
}

function select_point(X[] xset, float[] cdf, Randomizer r)
{
   // generate a random floating point number from a 
   // uniform distribution from 0 to 1
   float p = r.nextFloatUniformPDF();
   int i = binarySearch(cdf, p);
   // find the lowest index i such that p < cdf[i]

   return xset[i];
}

You call prepare_cdf once, and then call select_point as many times as you need to generate random points.