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Doubt about Hough Circle Transform

In the book "Learning OpenCv" I can read

The Hough gradient method works as follows. First the image is passed through an edge detection phase (in this case, cvCanny() ). Next, for every nonzero point in the edge image, the local gradient is considered (the gradient is computed by first computing the first- order Sobel x - and y-derivatives via cvSobel()). Using this gradient, every point along the line indicated by this slope—from a specified minimum to a specified maximum distance—is incremented in the accumulator. At the same time, the location of every one of these nonzero pixels in the edge image is noted. The candidate centers are then selected from those points in this (two-dimensional) accumulator that are both above some given threshold and larger than all of their immediate neighbors. These candidate centers are sorted in descending order of their accumulator values, so that the centers with the most supporting pixels appear first. Next, for each center, all of the nonzero pixels (recall that this list was built earlier) are considered. These pixels are sorted according to their distance from the center. Working out from the smallest distances to the maximum radius, a single radius is selected that is best supported by the nonzero pixels.

Now I am not sure about the "these pixels are sorted according to their distance from the center". Why do we do that? In order to speed up radius support checking? I wasn't able to find any explanation. Thanks a lot for the help