KDTree exampleΒΆ

'Note: there is an implementation of a kdtree in scipy: http://docs.scipy.org/scipy/docs/scipy.spatial.kdtree.KDTree/ It is recommended to use that instead of the below.'

This is an example of how to construct and search a kd-tree in Pythonwith NumPy. kd-trees are e.g. used to search for neighbouring data points in multidimensional space. Searching the kd-tree for the nearest neighbour of all n points has O(n log n) complexity with respect to sample size.

Building a kd-tree¶

In [ ]:
#!python numbers=disable

# Copyleft 2008 Sturla Molden
# University of Oslo

#import psyco

import numpy

def kdtree( data, leafsize=10 ):
    build a kd-tree for O(n log n) nearest neighbour search

        data:       2D ndarray, shape =(ndim,ndata), preferentially C order
        leafsize:   max. number of data points to leave in a leaf

        kd-tree:    list of tuples

    ndim = data.shape[0]
    ndata = data.shape[1]

    # find bounding hyper-rectangle
    hrect = numpy.zeros((2,data.shape[0]))
    hrect[0,:] = data.min(axis=1)
    hrect[1,:] = data.max(axis=1)

    # create root of kd-tree
    idx = numpy.argsort(data[0,:], kind='mergesort')
    data[:,:] = data[:,idx]
    splitval = data[0,ndata/2]

    left_hrect = hrect.copy()
    right_hrect = hrect.copy()
    left_hrect[1, 0] = splitval
    right_hrect[0, 0] = splitval

    tree = [(None, None, left_hrect, right_hrect, None, None)]

    stack = [(data[:,:ndata/2], idx[:ndata/2], 1, 0, True),
             (data[:,ndata/2:], idx[ndata/2:], 1, 0, False)]

    # recursively split data in halves using hyper-rectangles:
    while stack:

        # pop data off stack
        data, didx, depth, parent, leftbranch = stack.pop()
        ndata = data.shape[1]
        nodeptr = len(tree)

        # update parent node

        _didx, _data, _left_hrect, _right_hrect, left, right = tree[parent]

        tree[parent] = (_didx, _data, _left_hrect, _right_hrect, nodeptr, right) if leftbranch \
            else (_didx, _data, _left_hrect, _right_hrect, left, nodeptr)

        # insert node in kd-tree

        # leaf node?
        if ndata <= leafsize:
            _didx = didx.copy()
            _data = data.copy()
            leaf = (_didx, _data, None, None, 0, 0)

        # not a leaf, split the data in two      
            splitdim = depth % ndim
            idx = numpy.argsort(data[splitdim,:], kind='mergesort')
            data[:,:] = data[:,idx]
            didx = didx[idx]
            nodeptr = len(tree)
            stack.append((data[:,:ndata/2], didx[:ndata/2], depth+1, nodeptr, True))
            stack.append((data[:,ndata/2:], didx[ndata/2:], depth+1, nodeptr, False))
            splitval = data[splitdim,ndata/2]
            if leftbranch:
                left_hrect = _left_hrect.copy()
                right_hrect = _left_hrect.copy()
                left_hrect = _right_hrect.copy()
                right_hrect = _right_hrect.copy()
            left_hrect[1, splitdim] = splitval
            right_hrect[0, splitdim] = splitval
            # append node to tree
            tree.append((None, None, left_hrect, right_hrect, None, None))

    return tree

Searching a kd-tree¶

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#!python numbers=disable

def intersect(hrect, r2, centroid):
    checks if the hyperrectangle hrect intersects with the
    hypersphere defined by centroid and r2
    maxval = hrect[1,:]
    minval = hrect[0,:]
    p = centroid.copy()
    idx = p < minval
    p[idx] = minval[idx]
    idx = p > maxval
    p[idx] = maxval[idx]
    return ((p-centroid)**2).sum() < r2

def quadratic_knn_search(data, lidx, ldata, K):
    """ find K nearest neighbours of data among ldata """
    ndata = ldata.shape[1]
    param = ldata.shape[0]
    K = K if K < ndata else ndata
    retval = []
    sqd = ((ldata - data[:,:ndata])**2).sum(axis=0) # data.reshape((param,1)).repeat(ndata, axis=1);
    idx = numpy.argsort(sqd, kind='mergesort')
    idx = idx[:K]
    return zip(sqd[idx], lidx[idx])

def search_kdtree(tree, datapoint, K):
    """ find the k nearest neighbours of datapoint in a kdtree """
    stack = [tree[0]]
    knn = [(numpy.inf, None)]*K
    _datapt = datapoint[:,0]
    while stack:

        leaf_idx, leaf_data, left_hrect, \
                  right_hrect, left, right = stack.pop()

        # leaf
        if leaf_idx is not None:
            _knn = quadratic_knn_search(datapoint, leaf_idx, leaf_data, K)
            if _knn[0][0] < knn[-1][0]:
                knn = sorted(knn + _knn)[:K]

        # not a leaf

            # check left branch
            if intersect(left_hrect, knn[-1][0], _datapt):

            # chech right branch
            if intersect(right_hrect, knn[-1][0], _datapt):
    return knn

def knn_search( data, K, leafsize=2048 ):

    """ find the K nearest neighbours for data points in data,
        using an O(n log n) kd-tree """

    ndata = data.shape[1]
    param = data.shape[0]

    # build kdtree
    tree = kdtree(data.copy(), leafsize=leafsize)

    # search kdtree
    knn = []
    for i in numpy.arange(ndata):
        _data = data[:,i].reshape((param,1)).repeat(leafsize, axis=1);
        _knn = search_kdtree(tree, _data, K+1)

    return knn

def radius_search(tree, datapoint, radius):
    """ find all points within radius of datapoint """
    stack = [tree[0]]
    inside = []
    while stack:

        leaf_idx, leaf_data, left_hrect, \
                  right_hrect, left, right = stack.pop()

        # leaf
        if leaf_idx is not None:
            distance = numpy.sqrt(((leaf_data - datapoint.reshape((param,1)))**2).sum(axis=0))
            near = numpy.where(distance<=radius)
            if len(near[0]):
                idx = leaf_idx[near]
                distance = distance[near]
                inside += (zip(distance, idx))


            if intersect(left_hrect, radius, datapoint):

            if intersect(right_hrect, radius, datapoint):

    return inside

Quadratic search for small data sets¶

In contrast to the kd-tree, straight forward exhaustive search has quadratic complexity with respect to sample size. It can be faster than using a kd-tree when the sample size is very small. On my computer that is approximately 500 samples or less.

In [ ]:
#!python numbers=disable

def knn_search( data, K ):
    """ find the K nearest neighbours for data points in data,
        using O(n**2) search """
    ndata = data.shape[1]
    knn = []
    idx = numpy.arange(ndata)
    for i in numpy.arange(ndata):
        _knn = quadratic_knn_search(data[:,i], idx, data, K+1) # see above
        knn.append( _knn[1:] )
    return knn

Parallel search for large data sets¶

While creating a kd-tree is very fast, searching it can be time consuming. Due to Python's dreaded "Global Interpreter Lock" (GIL), threads cannot be used to conduct multiple searches in parallel. That is, Python threads can be used for asynchrony but not concurrency. However, we can use multiple processes (multiple interpreters). The pyprocessing package makes this easy. It has an API similar to Python's threading and Queue standard modules, but work with processes instead of threads. Beginning with Python 2.6, pyprocessing is already included in Python's standard library as the "multiprocessing" module. There is a small overhead of using multiple processes, including process creation, process startup, IPC, and process termination. However, because processes run in separate address spaces, no memory contention is incurred. In the following example, the overhead of using multiple processes is very small compared to the computation, giving a speed-up close to the number of CPUs on the computer.

In [ ]:
#!python numbers=disable

    import multiprocessing as processing
    import processing

import ctypes, os

def __num_processors():
    if os.name == 'nt': # Windows
        return int(os.getenv('NUMBER_OF_PROCESSORS'))
    else: # glibc (Linux, *BSD, Apple)
        get_nprocs = ctypes.cdll.libc.get_nprocs
        get_nprocs.restype = ctypes.c_int
        get_nprocs.argtypes = []
        return get_nprocs()

def __search_kdtree(tree, data, K, leafsize):
    knn = []
    param = data.shape[0]
    ndata = data.shape[1]
    for i in numpy.arange(ndata):
        _data = data[:,i].reshape((param,1)).repeat(leafsize, axis=1);
        _knn = search_kdtree(tree, _data, K+1)
    return knn

def __remote_process(rank, qin, qout, tree, K, leafsize):
    while 1:
        # read input queue (block until data arrives)
        nc, data = qin.get()
        # process data
        knn = __search_kdtree(tree, data, K, leafsize)
        # write to output queue

def knn_search_parallel(data, K, leafsize=2048):

    """ find the K nearest neighbours for data points in data,
        using an O(n log n) kd-tree, exploiting all logical
        processors on the computer """

    ndata = data.shape[1]
    param = data.shape[0]
    nproc = __num_processors()
    # build kdtree
    tree = kdtree(data.copy(), leafsize=leafsize)
    # compute chunk size
    chunk_size = data.shape[1] / (4*nproc)
    chunk_size = 100 if chunk_size < 100 else chunk_size
    # set up a pool of processes
    qin = processing.Queue(maxsize=ndata/chunk_size)
    qout = processing.Queue(maxsize=ndata/chunk_size)
    pool = [processing.Process(target=__remote_process,
                args=(rank, qin, qout, tree, K, leafsize))
                    for rank in range(nproc)]
    for p in pool: p.start()
    # put data chunks in input queue
    cur, nc = 0, 0
    while 1:
        _data = data[:,cur:cur+chunk_size]
        if _data.shape[1] == 0: break
        cur += chunk_size
        nc += 1
    # read output queue
    knn = []
    while len(knn) < nc:
        knn += [qout.get()]
    # avoid race condition
    _knn = [n for i,n in sorted(knn)]
    knn = []
    for tmp in _knn:
        knn += tmp
    # terminate workers
    for p in pool: p.terminate()
    return knn

Running the code¶

The following shows how to run the example code (including how input data should be formatted):

In [ ]:
#!python numbers=disable

from time import clock

def test():
    K = 11
    ndata = 10000
    ndim = 12
    data =  10 * numpy.random.rand(ndata*ndim).reshape((ndim,ndata) )
    knn_search(data, K)

if __name__ == '__main__':
    t0 = clock()
    t1 = clock()
    print "Elapsed time %.2f seconds" % t1-t0

    #import profile          # using Python's profiler is not useful if you are
    #profile.run('test()')   # running the parallel search.
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