K-Nearest Neighbors in pure Python
Assumption: k &lt;&lt; N, where N = len(points)

There is no need to sort the entire list of points!

Instead, take the first k points, and determine their distance values, and sort them.  Then, for each success point:


determine its distance,
if it is smaller than the maximum,


discard the maximum, and insert the new point in the sorted list.



Sorting N points by distance is O(N log N); creating and maintaining a sorted list of k smallest elements is only O(N log k), which should be considerably faster.

I'm not sure if heapq.nsmallest() is built into CPython or not ...

k_neighbours = heapq.nsmallest(k, points, key=distance)
counter = Counter(x.classif for x in k_neighbours)




Well, I'm disappointed to see heapq.nsmallest() performed up to 40% worse that sorted on CPython, but I'm happy to see PyPy validates my assertion that you don't need to sort the entire list.

Continuing with that thought, bisect.insort() may be used to maintain a list of the k-nearest neighbours so far:

    neighbours = [(float('inf'), None)] * k
    for pnt in points:
        dist = distance(pnt)
        if dist &lt; neighbours[-1][0]:
            neighbours.pop()
            bisect.insort(neighbours, (dist, pnt))

    counter = Counter(pnt.classif for dist, pnt in neighbours)


This gave me 4% speedup over sorted()[:k] with your gist sample set.

Significant, but not impressive.  Still, it was enough encouragement to press on an look for other inefficiencies.

How about the distance() code.  It gets called a lot; can we speed it up?  Sure!

def predict(target: Coordinates, points: Sequence[KNNPoint], k: int, *,
            sum=sum, zip=zip) -&gt; str:

    def distance(p: KNNPoint) -&gt; float:
        return sum((a - b) ** 2 for a, b in zip(target, p.coords))

    # ... 


Instead of searching the global scope for the sum and zip functions, they are saved as variables sum, zip in the local scope, along with target, for use in distance().  Total improvement: 6%.

Applying the same sum=sum, zip=zip change to the original code, without the bisect.insort() change, also speeds it up by 2%.

Further, adding insort=bisect.insort to the function declaration, and using insort(neighbours, (dist, pnt)) in the function body also provides a minor improvement.

Finally, I was concerned about neighbours[-1][0].  Looking up the first tuple of the last element in the array seemed inefficient.  We could keep track of this in a local threshold variable.  Final total speedup: 7.7%.

neighbours = [(float('inf'), None)] * k
threshold = neighbours[-1][0]
for pnt in points:
   dist = distance(pnt)
   if dist &lt; threshold:
      neighbours.pop()
      insort(neighbours, (dist, pnt))
      threshold = neighbours[-1][0]


YMMV