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

"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" I think this might be better called gridding. So first make a dict, with a set of coords as the key and put each shop in a 50km bucket near that point. then when you are calculating distances, you only look in nearby buckets, rather than iterate through each shop in the whole universe

Answer 3

Thanks everyone's help. I think I have solved this by incorporating all the suggestions.

I use numpy to import the geographic co-ordinates and then project them using "France Lambert - 93". This lets me fill scipy.spatial.cKDTree with the points and then calculate a sparse_distance_matrix by specifying a cut-off of 50km (my projected points are in metres). I then extract extract the lower-triangle to a CSV.

import numpy as np
import csv
import time
from pyproj import Proj, transform

#http://epsg.io/2154 (accuracy: 1.0m)
fr = '+proj=lcc +lat_1=49 +lat_2=44 +lat_0=46.5 +lon_0=3 \
+x_0=700000 +y_0=6600000 +ellps=GRS80 +towgs84=0,0,0,0,0,0,0 \
+units=m +no_defs'

#http://epsg.io/27700-5339 (accuracy: 1.0m)
uk = '+proj=tmerc +lat_0=49 +lon_0=-2 +k=0.9996012717 \
+x_0=400000 +y_0=-100000 +ellps=airy \
+towgs84=446.448,-125.157,542.06,0.15,0.247,0.842,-20.489 +units=m +no_defs'

path_to_csv = '.../raw_in.csv'
out_csv = '.../out.csv'

def proj_arr(points):
    inproj = Proj(init='epsg:4326')
    outproj = Proj(uk)
    # origin|destination|lon|lat
    func = lambda x: transform(inproj,outproj,x[2],x[1])
    return np.array(list(map(func, points)))

tstart = time.time()

# Import points as geographic coordinates
# ID|lat|lon
#Sample to try and replicate
#points = np.array([
#        [39007,46.585012,5.5857829],
#        [88086,48.192370,6.7296289],
#        [62627,50.309155,3.0218611],
#        [14020,49.133972,-0.15851507],
#        [1091, 42.981765,2.0104902]])
#
points = np.genfromtxt(path_to_csv,
                       delimiter=',',
                       skip_header=1)

print("Total points: %d" % len(points))
print("Triangular matrix contains: %d" % (len(points)*((len(points))-1)*0.5))
# Get projected co-ordinates
proj_pnts = proj_arr(points)

# Fill quad-tree
from scipy.spatial import cKDTree
tree = cKDTree(proj_pnts)
cut_off_metres = 1600
tree_dist = tree.sparse_distance_matrix(tree,
                                        max_distance=cut_off_metres,
                                        p=2) 

# Extract triangle
from scipy import sparse
udist = sparse.tril(tree_dist, k=-1)    # zero the main diagonal
print("Distances after quad-tree cut-off: %d " % len(udist.data))

# Export CSV
import csv
f = open(out_csv, 'w', newline='') 
w = csv.writer(f, delimiter=",", )
w.writerow(['id_a','lat_a','lon_a','id_b','lat_b','lon_b','metres'])
w.writerows(np.column_stack((points[udist.row ],
                             points[udist.col],
                             udist.data)))
f.close()

"""
Get ID labels
"""
id_to_csv = '...id.csv'
id_labels = np.genfromtxt(id_to_csv,
                       delimiter=',',
                       skip_header=1,
                       dtype='U')

"""
Try vincenty on the un-projected co-ordinates
"""
from geopy.distance import vincenty
vout_csv = '.../out_vin.csv'
test_vin = np.column_stack((points[udist.row].T[1:3].T,
                            points[udist.col].T[1:3].T))

func = lambda x: vincenty(x[0:2],x[2:4]).m
output = list(map(func,test_vin))

# Export CSV
f = open(vout_csv, 'w', newline='')
w = csv.writer(f, delimiter=",", )
w.writerow(['id_a','id_a2', 'lat_a','lon_a',
            'id_b','id_b2', 'lat_b','lon_b',
            'proj_metres','vincenty_metres'])
w.writerows(np.column_stack((list(id_labels[udist.row]),
                             points[udist.row ],
                             list(id_labels[udist.col]),
                             points[udist.col],
                             udist.data,
                             output,
                             )))

f.close()    
print("Finished in %.0f seconds" % (time.time()-tstart)

This approach took 164 seconds to generate (for 5,306,434 distances) - compared to 9 - and also around 90 seconds to save to disk.

I then compared the difference in the vincenty distance and the hypotenuse distance (on the projected co-ordinates).

The mean difference in metres was 2.7 and the mean difference/metres was 0.0073% - which looks great.

Answer 4

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!