function _ll_points_in_region($point, $data, $epsilon)
{
    $region = array();
    foreach ($data as $index => $datum) {
        if (ll_euclidian_distance($point, $datum) < $epsilon) {
            $region[$index] = $datum;
        }
    }
    return $region;
}
function _ll_distances_to_point($xs, $x)
{
    $distances = array();
    foreach ($xs as $index => $xi) {
        $distances[$index] = ll_euclidian_distance($xi, $x);
    }
    asort($distances);
    array_shift($distances);
    // has "self" as the smallest distance.
    return $distances;
}
function __ll_distance_to_centroid($x, $centroid)
{
    return ll_euclidian_distance($x, $centroid);
}