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closestpair.pyx
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from cpython.mem cimport PyMem_Malloc, PyMem_Free
cdef extern from "math.h":
float sqrtf(float)
from libc.math cimport sqrt, fabs
from c_utils cimport read_numpy
cimport numpy as cnp
cnp.import_array()
import numpy as np
from scipy.spatial.distance import pdist
cdef struct arr_1d:
double value
size_t idx
cdef int cmp_arr(const void *a_ptr, const void *b_ptr):
cdef double a = (<double *>a_ptr)[0]
cdef double b = (<double *>b_ptr)[0]
if a < b:
return -1
elif a > b:
return 1
else:
return 0
cdef double distance(double x1, double y1, double x2, double y2):
return sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2))
cdef inline double dist_sqr(double x1, double y1, double x2, double y2):
return (x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2)
cdef inline double min_3_db(double a, double b, double c):
if a < b and a < c:
return a
if b < a and b < c:
return b
return c
cdef inline double min_2_db(double a, double b):
if a < b:
return a
else:
return b
cdef inline size_t max_2_int(size_t a, size_t b):
if a > b:
return a
else:
return b
cdef inline size_t min_2_int(size_t a, size_t b):
if a < b:
return a
else:
return b
""" ######## Using memory views ############ """
cpdef double min_dist_naive_mv(double[:, :] arr):
cdef size_t i, j, n
cdef double min, dist
n = arr.shape[0]
min = dist_sqr(arr[0, 0], arr[0, 1],
arr[1, 0], arr[1, 1])
for i in range(n - 1):
for j in range(i + 1, n):
# dist = dist_sqr(arr[i, 0], arr[i, 1], arr[j, 0], arr[j, 1])
dist = (arr[i, 0] - arr[j, 0]) * (arr[i, 0] - arr[j, 0]) + (arr[i, 1] - arr[j, 1]) * (arr[i, 1] - arr[j, 1])
if dist < min:
min = dist
return sqrt(min)
""" ########### Using direct memory access ########## """
cpdef double min_dist_naive(cnp.ndarray[double, ndim=2] arr):
cdef:
size_t size
double* data
data, size = read_numpy(arr)
return min_dist_naive_c(size, data)
cdef double min_dist_naive_c(size_t n, double *arr):
cdef size_t i, j
cdef double min, dist
min = dist_sqr(arr[0], arr[1],
arr[2], arr[3])
for i in range(n - 1):
for j in range(i + 1, n):
dist = dist_sqr(arr[2*i], arr[2*i + 1], arr[2*j], arr[2*j + 1])
# dist = (arr[2*i] - arr[2*j]) * (arr[2*i] - arr[2*j]) + (arr[2*i + 1] - arr[2*j + 1]) * (arr[2*i + 1] - arr[2*j + 1])
if dist < min:
min = dist
return sqrt(min)
""" #################### Using numpy arrays memory views ##################### """
cpdef double min_dist_mv(cnp.ndarray[cnp.float64_t, ndim=2] P):
idx = cnp.PyArray_ArgSort(P, 0, cnp.NPY_QUICKSORT)
cdef double[:, :] Px = P[idx[:, 0]]
cdef double[:, :] Py = P[idx[:, 1]]
return sqrt(_min_dist(Px, Py))
cdef double _min_dist_split(double[:, :] Sy, double delta):
cdef size_t i, j, n
n = Sy.shape[0]
for i in range(n - 1):
for j in range((i + 1), min_2_int(i + 7, n)):
delta = min_2_db(delta, dist_sqr(Sy[i, 0], Sy[i, 1], Sy[j, 0], Sy[j, 1]))
return delta
cdef cnp.ndarray[cnp.float_t, ndim=2] create_array(size_t length):
cdef cnp.npy_intp *l = [0, 2]
l[0] = length
cdef cnp.ndarray[cnp.float_t, ndim=2] buff = cnp.PyArray_SimpleNew(2, l, cnp.NPY_FLOAT64)
return buff
cdef double _min_dist(double[:, :] Px, double[:, :] Py):
cdef size_t i, j, k, n
n = Px.shape[0]
# base cases
if n == 2:
return dist_sqr(Px[0, 0], Px[0, 1], Px[1, 0], Px[1, 1])
if n == 3:
return min_3_db(dist_sqr(Px[0, 0], Px[0, 1], Px[1, 0], Px[1, 1]),
dist_sqr(Px[0, 0], Px[0, 1], Px[2, 0], Px[2, 1]),
dist_sqr(Px[1, 0], Px[1, 1], Px[2, 0], Px[2, 1]))
cdef size_t mid = n // 2
cdef double mid_x = Px[mid, 0]
cdef double[:, :] Qx, Rx
Qx, Rx = Px[:mid, :], Px[mid:, :]
# copy Rx, Ry -> Qy, Ry
cdef double[:, :] Qy = create_array(Qx.shape[0])
cdef double[:, :] Ry = create_array(Rx.shape[0])
j = 0
k = 0
# print("n=", n)
# print("i Py[i, 0], mid_x")
for i in range(n):
# print(i, Py[i, 0], " ", mid_x)
if Py[i, 0] < mid_x:
Qy[j, 0] = Py[i, 0]
Qy[j, 1] = Py[i, 1]
j += 1
else:
Ry[k, 0] = Py[i, 0]
Ry[k, 1] = Py[i, 1]
k += 1
# print("j, k")
# print(j, k)
# print("===========")
cdef double d1 = _min_dist(Qx, Qy)
cdef double d2 = _min_dist(Rx, Ry)
delta = min_2_db(d1, d2)
cdef double[:, :] Sy = create_array(n)
j = 0
for i in range(n):
if (Py[i, 0] >= mid_x - sqrt(delta)) or (Py[i, 0] <= mid_x + sqrt(delta)):
Sy[j, 0] = Py[i, 0]
Sy[j, 1] = Py[i, 1]
j += 1
if j != 0:
delta = _min_dist_split(Sy[:j, :], delta)
return delta
"""################### Using arrays in pure C ######################"""
cpdef double min_dist_c(cnp.ndarray[cnp.float64_t, ndim=2] P, cnp.NPY_SORTKIND kind = cnp.NPY_QUICKSORT):
cdef size_t i
idx = cnp.PyArray_ArgSort(P, 0, kind)
cdef double[:, :] Px_mview = P[idx[:, 0]]
cdef double[:, :] Py_mview = P[idx[:, 1]]
cdef double *Px
cdef double *Py
Px = <double *> PyMem_Malloc(Px_mview.shape[0] * 2 * sizeof(double))
Py = <double *> PyMem_Malloc(Py_mview.shape[0] * 2 * sizeof(double))
# # copy sorted arrays to Px, Py
for i in range(Px_mview.shape[0]):
Px[2*i] = Px_mview[i, 0]
Px[2*i + 1] = Px_mview[i, 1]
Py[2*i] = Py_mview[i, 0]
Py[2*i + 1] = Py_mview[i, 1]
cdef double d = _min_dist_c(Px_mview.shape[0], Px, Py)
PyMem_Free(Px)
PyMem_Free(Py)
return sqrt(d)
cdef double _min_dist_c(size_t n, double *Px, double *Py):
cdef size_t i, j, k
# base cases
if n == 2:
return dist_sqr(Px[0], Px[1], Px[2*1], Px[2*1 + 1])
if n == 3:
return min_3_db(dist_sqr(Px[0], Px[1], Px[2], Px[2 + 1]),
dist_sqr(Px[0], Px[1], Px[2*2], Px[2*2 + 1]),
dist_sqr(Px[2], Px[2 + 1], Px[2*2], Px[2*2 + 1]))
# split Px on Qx and Rx
cdef size_t mid = n // 2
cdef double mid_x = Px[2 * mid]
cdef double *Qx
cdef double *Rx
Qx = Px # Px[:mid, :], total size 2 * mid
Rx = Px + 2 * mid # Px[mid:, :], total size 2 * (n - mid)
cdef double *Qy
cdef double *Ry
Qy = <double*> PyMem_Malloc(mid * 2 * sizeof(double))
Ry = <double*> PyMem_Malloc((n - mid) * 2 * sizeof(double))
# sorting Qx, Rx by y, according to Py array
_sort_y(n, mid_x, Qy, Ry, Py)
cdef double d1 = _min_dist_c(mid, Qx, Qy)
cdef double d2 = _min_dist_c(n - mid, Rx, Ry)
cdef double delta = min_2_db(d1, d2)
PyMem_Free(Qy)
PyMem_Free(Ry)
# get points in Sy
cdef double *Sy
Sy = <double*> PyMem_Malloc(n * 2 * sizeof(double))
cdef size_t s_y_size = _get_sy(n, mid_x, delta, Py, Sy)
delta = _min_dist_split_c(s_y_size, Sy, delta)
PyMem_Free(Sy)
return delta
cdef double _min_dist_split_c(size_t n, double *Sy, double delta):
cdef size_t i, j, j_max
cdef double d, d_max
for i in range(n - 1):
# for i in prange(n - 1,=True):
# ~7% speed up compared to func call min_2_int()
if i + 7 < n:
j_max = i + 7
else:
j_max = n
for j in range((i + 1), j_max):
# ~30% speed up compared to func call dist_sqr()
# d = dist_sqr(Sy[2*i], Sy[2*i + 1], Sy[2*j], Sy[2*j + 1])
d = (Sy[2*i] - Sy[2*j]) * (Sy[2*i] - Sy[2*j]) + (Sy[2*i + 1] - Sy[2*j + 1]) * (Sy[2*i + 1] - Sy[2*j + 1])
if d < delta:
delta = d
return delta
cdef void _sort_y(size_t n, double mid_x, double *Qy, double *Ry, double *Py):
cdef size_t i, j, k
# sorting Qx, Rx by y, according to Py array
j = 0
k = 0
for i in range(n):
if Py[2*i] < mid_x:
Qy[2*j] = Py[2*i]
Qy[2*j + 1] = Py[2*i + 1]
j += 1
elif Py[2*i] >= mid_x:
Ry[2*k] = Py[2*i]
Ry[2*k + 1] = Py[2*i + 1]
k += 1
return
cdef size_t _get_sy(size_t n, double mid_x, double delta, double *Py, double *Sy):
cdef size_t i
cdef size_t j = 0
cdef double upper_bound = mid_x + sqrt(delta)
cdef double lower_bound = mid_x - sqrt(delta)
for i in range(n):
if (Py[2*i] > lower_bound) or (Py[2*i] < upper_bound):
Sy[2*j] = Py[2*i]
Sy[2*j + 1] = Py[2*i + 1]
j += 1
return j
""" ################ 32 bit version ############################## """
cdef inline float dist32(float x1, float y1, float x2, float y2):
return (x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2)
cdef inline float min32(float a, float b):
if a > b:
return b
else:
return a
cdef float min_3(float a, float b, float c):
if a < b and a < c:
return a
if b < a and b < c:
return b
return c
cpdef float min_dist32(cnp.ndarray[cnp.float32_t, ndim=2] P, cnp.NPY_SORTKIND kind = cnp.NPY_QUICKSORT):
cdef size_t i
idx = cnp.PyArray_ArgSort(P, 0, kind)
cdef float[:, :] Px_mview = P[idx[:, 0]]
cdef float[:, :] Py_mview = P[idx[:, 1]]
cdef float *Px
cdef float *Py
Px = <float *> PyMem_Malloc(Px_mview.shape[0] * 2 * sizeof(float))
Py = <float *> PyMem_Malloc(Py_mview.shape[0] * 2 * sizeof(float))
# # copy sorted arrays to Px, Py
for i in range(Px_mview.shape[0]):
Px[2*i] = Px_mview[i, 0]
Px[2*i + 1] = Px_mview[i, 1]
Py[2*i] = Py_mview[i, 0]
Py[2*i + 1] = Py_mview[i, 1]
cdef float d = _mindist32(Px_mview.shape[0], Px, Py)
PyMem_Free(Px)
PyMem_Free(Py)
return sqrtf(d)
cdef float _mindist32(size_t n, float *Px, float *Py):
cdef size_t i, j, k
# base cases
if n == 2:
return dist32(Px[0], Px[1], Px[2*1], Px[2*1 + 1])
if n == 3:
return min_3(dist32(Px[0], Px[1], Px[2], Px[2 + 1]),
dist32(Px[0], Px[1], Px[2*2], Px[2*2 + 1]),
dist32(Px[2], Px[2 + 1], Px[2*2], Px[2*2 + 1]))
# split Px on Qx and Rx
cdef size_t mid = n // 2
cdef float mid_x = Px[2 * mid]
cdef float *Qx
cdef float *Rx
Qx = Px # Px[:mid, :], total size 2 * mid
Rx = Px + 2 * mid # Px[mid:, :], total size 2 * (n - mid)
cdef float *Qy
cdef float *Ry
Qy = <float*> PyMem_Malloc(mid * 2 * sizeof(float))
Ry = <float*> PyMem_Malloc((n - mid) * 2 * sizeof(float))
# sorting Qx, Rx by y, according to Py array
_sort_y32(n, mid_x, Qy, Ry, Py)
cdef float d1 = _mindist32(mid, Qx, Qy)
cdef float d2 = _mindist32(n - mid, Rx, Ry)
cdef float delta = min32(d1, d2)
PyMem_Free(Qy)
PyMem_Free(Ry)
# get points in Sy
cdef float *Sy
Sy = <float*> PyMem_Malloc(n * 2 * sizeof(float))
cdef size_t s_y_size = _get_sy32(n, mid_x, delta, Py, Sy)
delta = _min_dist_split32(s_y_size, Sy, delta)
PyMem_Free(Sy)
return delta
cdef void _sort_y32(size_t n, float mid_x, float *Qy, float *Ry, float *Py):
cdef size_t i,j,k
# sorting Qx, Rx by y, according to Py array
j = 0
k = 0
for i in range(n):
if Py[2*i] < mid_x:
Qy[2*j] = Py[2*i]
Qy[2*j + 1] = Py[2*i + 1]
j += 1
elif Py[2*i] >= mid_x:
Ry[2*k] = Py[2*i]
Ry[2*k + 1] = Py[2*i + 1]
k += 1
return
cdef size_t _get_sy32(size_t n, float mid_x, float delta, float *Py, float *Sy):
cdef size_t i
cdef size_t j = 0
cdef float upper_bound = mid_x + sqrtf(delta)
cdef float lower_bound = mid_x - sqrtf(delta)
for i in range(n):
if (Py[2*i] > lower_bound) or (Py[2*i] < upper_bound):
Sy[2*j] = Py[2*i]
Sy[2*j + 1] = Py[2*i + 1]
j += 1
return j
cdef float _min_dist_split32(size_t n, float *Sy, float delta):
cdef size_t i, j, j_max
cdef float d
for i in range(n - 1):
# ~7% speed up compared to func call min_2_int()
if i + 7 < n:
j_max = i + 7
else:
j_max = n
for j in range((i + 1), j_max):
d = dist_sqr(Sy[2*i], Sy[2*j], Sy[2*i + 1], Sy[2*j + 1])
# d = (Sy[2*i] - Sy[2*j]) * (Sy[2*i] - Sy[2*j]) + (Sy[2*i + 1] - Sy[2*j + 1]) * (Sy[2*i + 1] - Sy[2*j + 1])
if d < delta:
delta = d
return delta
""" ################ RETURN MAX points in Sy search ###############"""
cdef size_t _get_sy_strict(size_t n, double mid_x, double delta, double *Py, double *Sy):
cdef size_t i
cdef size_t j = 0
for i in range(n):
if (Py[2*i] > mid_x - delta) or (Py[2*i] < mid_x + delta):
Sy[2*j] = Py[2*i]
Sy[2*j + 1] = Py[2*i + 1]
j += 1
return j
cpdef (size_t, double) max_points_c(cnp.ndarray[cnp.float64_t, ndim=2] P, bint strict = 0):
idx = cnp.PyArray_ArgSort(P, 0, cnp.NPY_QUICKSORT)
cdef double[:, :] Px = P[idx[:, 0]]
cdef double[:, :] Py = P[idx[:, 1]]
cdef double *Px_ptr
cdef double *Py_ptr
cdef size_t i
Px_ptr = <double *> PyMem_Malloc(Px.shape[0] * 2 * sizeof(double))
Py_ptr = <double *> PyMem_Malloc(Py.shape[0] * 2 * sizeof(double))
for i in range(Px.shape[0]):
Px_ptr[2*i] = Px[i, 0]
Px_ptr[2*i + 1] = Px[i, 1]
Py_ptr[2*i] = Py[i, 0]
Py_ptr[2*i + 1] = Py[i, 1]
cdef size_t mp = 0
cdef double delta
delta = _max_points(Px.shape[0], Px_ptr, Py_ptr, &mp, strict)
PyMem_Free(Px_ptr)
PyMem_Free(Py_ptr)
return mp, delta
cdef double _max_points_split_c(size_t n, double *Sy, double delta, size_t *mp):
cdef size_t i, j
cdef double dist
for i in range(n - 1):
for j in range((i + 1), min_2_int(i + 7, n)):
dist = distance(Sy[2*i], Sy[2*i + 1], Sy[2*j], Sy[2*j + 1])
if dist < delta:
delta = dist
mp[0] = max_2_int(mp[0], j - i)
return delta
cdef double _max_points(size_t n, double *Px, double *Py, size_t *mp, bint strict):
cdef size_t i, j, k
# base cases
if n == 2:
return distance(Px[0], Px[1], Px[2*1], Px[2*1 + 1])
if n == 3:
return min_3_db(distance(Px[0], Px[1], Px[2], Px[2 + 1]),
distance(Px[0], Px[1], Px[2*2], Px[2*2 + 1]),
distance(Px[2], Px[2 + 1], Px[2*2], Px[2*2 + 1]))
# split Px on Qx and Rx
cdef size_t mid = n // 2
cdef double mid_x = Px[2 * mid]
cdef double *Qx
cdef double *Rx
Qx = Px # Px[:mid, :], total size 2 * mid
Rx = Px + 2 * mid # Px[mid:, :], total size 2 * (n - mid)
cdef double *Qy
cdef double *Ry
Qy = <double*> PyMem_Malloc(mid * 2 * sizeof(double))
Ry = <double*> PyMem_Malloc((n - mid) * 2 * sizeof(double))
# sorting Qx, Rx by y, according to Py array
_sort_y(n, mid_x, Qy, Ry, Py)
cdef double d1
cdef size_t mp1 = 0
cdef double d2
cdef size_t mp2 = 0
d1 = _max_points(mid, Qx, Qy, &mp1, strict)
d2 = _max_points(n - mid, Rx, Ry, &mp2, strict)
cdef double delta = min_2_db(d1, d2)
mp[0] = max_2_int(mp1, mp2)
PyMem_Free(Qy)
PyMem_Free(Ry)
# get points in Sy
cdef double *Sy
Sy = <double*> PyMem_Malloc(n * 2 * sizeof(double))
cdef size_t s_y_size
if strict:
s_y_size = _get_sy_strict(n, mid_x, delta, Py, Sy)
else:
s_y_size = _get_sy(n, mid_x, delta, Py, Sy)
if s_y_size != 0:
delta = _max_points_split_c(s_y_size, Sy, delta, mp)
PyMem_Free(Sy)
return delta
""" #############################################################
###################### UNIT TESTS ###########################
#############################################################
"""
def test_min_dist_naive_c_1():
cdef double *a = [0.0, 0.0, 4.0, 3.0, 9.0, 7.0]
assert fabs(min_dist_naive_c(3, a) - 5.0) < 1e-16
def test_min_dist_c_1():
cdef double *a = [0.0, 0.0, 4.0, 3.0, 9.0, 7.0]
assert fabs(_min_dist_c(3, a, a) - 25.0) < 1e-16
def test_min_dist_c_2():
cdef double *px = [0.1, 0.2, 0.2, 0.3, 0.3, 0.4, 0.5, 0.6]
cdef double *py = [0.1, 0.2, 0.2, 0.3, 0.3, 0.4, 0.5, 0.6]
# arr = numpy.array([[0.1, 0.2], [0.2, 0.3], [0.3, 0.4], [0.5, 0.6]])
assert fabs(sqrt(_min_dist_c(4, px, px)) - min_dist_naive_c(4, px)) < 1e-16
def test_min_dist_c_3():
for i in range(100):
arr = np.random.randn(20, 2)
assert fabs(min_dist_c(arr) - pdist(arr).min()) < 1e-16
def test_mindist32_1():
cdef float *x32 = [0.1, 0.2, 0.2, 0.3, 0.3, 0.4, 0.5, 0.6]
cdef float *y32 = [0.1, 0.2, 0.2, 0.3, 0.3, 0.4, 0.5, 0.6]
cdef double *x64 = [0.1, 0.2, 0.2, 0.3, 0.3, 0.4, 0.5, 0.6]
cdef double *y64 = [0.1, 0.2, 0.2, 0.3, 0.3, 0.4, 0.5, 0.6]
# arr = numpy.array([[0.1, 0.2], [0.2, 0.3], [0.3, 0.4], [0.5, 0.6]])
assert fabs(<double>_mindist32(4, x32, y32) - _min_dist_c(4, x64, y64)) < 1e-8
assert fabs(<double>sqrt(_mindist32(4, x32, y32)) - min_dist_naive_c(4, x64)) < 1e-8