Python calculate lots of distances quickly

Question

I have an input of 36,742 points which means if I wanted to calculate the lower triangle of a distance matrix (using the vincenty approximation) I would need to generate 36,742*36,741*0.5 = 1,349,974,563 distances.

I want to keep the pair combinations which are within 50km of each other. My current set-up is as follows

shops= [[id,lat,lon]...]

def lower_triangle_mat(points):
    for i in range(len(shops)-1):
        for j in range(i+1,len(shops)):
            yield [shops[i],shops[j]]

def return_stores_cutoff(points,cutoff_km=0):
    below_cut = []
    counter = 0
    for x in lower_triangle_mat(points):
        dist_km = vincenty(x[0][1:3],x[1][1:3]).km
        counter += 1
        if counter % 1000000 == 0:
            print("%d out of %d" % (counter,(len(shops)*len(shops)-1*0.5)))
        if dist_km <= cutoff_km:
            below_cut.append([x[0][0],x[1][0],dist_km])
    return below_cut

start = time.clock()
stores = return_stores_cutoff(points=shops,cutoff_km=50)
print(time.clock() - start)

This will obviously take hours and hours. Some possibilities I was thinking of:

• Use numpy to vectorise these calculations rather than looping through
• Use some kind of hashing to get a quick rough-cut off (all stores within 100km) and then only calculate accurate distances between those stores
• Instead of storing the points in a list use something like a quad-tree but I think that only helps with the ranking of close points rather than actual distance -> so I guess some kind of geodatabase
• I can obviously try the haversine or project and use euclidean distances, however I am interested in using the most accurate measure possible
• Make use of parallel processing (however I was having a bit of difficulty coming up how to cut the list to still get all the relevant pairs).

Edit: I think geohashing is definitely needed here - an example from:

from geoindex import GeoGridIndex, GeoPoint

geo_index = GeoGridIndex()
for _ in range(10000):
    lat = random.random()*180 - 90
    lng = random.random()*360 - 180
    index.add_point(GeoPoint(lat, lng))

center_point = GeoPoint(37.7772448, -122.3955118)
for distance, point in index.get_nearest_points(center_point, 10, 'km'):
    print("We found {0} in {1} km".format(point, distance))

However, I would also like to vectorise (instead of loop) the distance calculations for the stores returned by the geo-hash.

Edit2: Pouria Hadjibagheri - I tried using lambda and map:

# [B]: Mapping approach           
lwr_tr_mat = ((shops[i],shops[j]) for i in range(len(shops)-1) for j in range(i+1,len(shops)))

func = lambda x: (x[0][0],x[1][0],vincenty(x[0],x[1]).km)
# Trying to see if conditional statements slow this down
func_cond = lambda x: (x[0][0],x[1][0],vincenty(x[0],x[1]).km) if vincenty(x[0],x[1]).km <= 50 else None

start = time.clock()
out_dist = list(map(func,lwr_tr_mat))
print(time.clock() - start)

start = time.clock()
out_dist = list(map(func_cond,lwr_tr_mat))
print(time.clock() - start)

And they were all around 61 seconds (I restricted number of stores to 2000 from 32,000). Perhaps I used map incorrectly?

Answer 1

This sounds like a classic use case for k-D trees.

If you first transform your points into Euclidean space then you can use the query_pairs method of scipy.spatial.cKDTree:

from scipy.spatial import cKDTree

tree = cKDTree(data)
# where data is (nshops, ndim) containing the Euclidean coordinates of each shop
# in units of km

pairs = tree.query_pairs(50, p=2)   # 50km radius, L2 (Euclidean) norm

pairs will be a set of (i, j) tuples corresponding to the row indices of pairs of shops that are ≤50km from each other.

---

The output of tree.sparse_distance_matrix is a scipy.sparse.dok_matrix. Since the matrix will be symmetric and you're only interested in unique row/column pairs, you could use scipy.sparse.tril to zero out the upper triangle, giving you a scipy.sparse.coo_matrix. From there you can access the nonzero row and column indices and their corresponding distance values via the .row, .col and .data attributes:

from scipy import sparse

tree_dist = tree.sparse_distance_matrix(tree, max_distance=10000, p=2)
udist = sparse.tril(tree_dist, k=-1)    # zero the main diagonal
ridx = udist.row    # row indices
cidx = udist.col    # column indices
dist = udist.data   # distance values

Answer 2

Have you tried mapping entire arrays and functions instead of iterating through them? An example would be as follows:

from numpy.random import rand

my_array = rand(int(5e7), 1)  # An array of 50,000,000 random numbers in double.

Now what is normally done is:

squared_list_iter = [value**2 for value in my_array]

Which of course works, but is optimally invalid.

The alternative would be to map the array with a function. This is done as follows:

func = lambda x: x**2  # Here is what I want to do on my array.

squared_list_map = map(func, test)  # Here I am doing it!

Now, one might ask, how is this any different, or even better for that matter? Since now we have added a call to a function, too! Here is your answer:

For the former solution (via iteration):

1 loop: 1.11 minutes.

Compared to the latter solution (mapping):

500 loop, on average 560 ns. 

Simultaneous conversion of a map() to list by list(map(my_list)) would increase the time by a factor of 10 to approximately 500 ms.

You choose!