[FIXED] Python calculate lots of distances quickly

Issue

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?

Solution

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

Answered By - ali_m