def testPartition():
"""Test partition function"""
print "Test"
arr = [4, 3, 2, 1]
partition(arr, 0, len(arr) - 1)
pivoted = [1, 3, 2, 4]
assert isEqual(arr, pivoted)
def note(self):
'''
Summary
====
Print chapter7.1 note
Example
====
>>> Chapter7_1().note()
'''
print('chapter7.1 note as follow')
print('第7章 快速排序')
print('快速排序是一种排序算法，对包含n个数的输入数组进行排序，最坏情况的运行时间为Θ(n^2)')
print('虽然这个最坏情况运行时间比较差，但是快速排序通常是用于排序最佳的实用选择，这是因为其平均性能相当好')
print('快速排序期望的运行时间为Θ(nlgn),且Θ(nlgn)记号中隐含的常数因子很小')
print('快速排序能够进行就地排序，在虚存坏境中也能很好地工作')
print('7.1 快速排序的描述')
print('像合并排序一样，快速排序也是基于分治模式的')
print(' 1.分解:数组A[p..r]被划分成两个(可能为空的)子数组A[p..q-1]和A[q+1..r]')
print(' 使得A[p..q-1]中的每个元素都小于等于A(q),而且，小于等于A[q+1..r]')
print(' 下标q也在这个划分过程中进行计算')
print(' 2.解决:通过递归调用快速排序，对子数组A[p..q-1]和A[q+1..r]排序')
print(' 3.合并:因为这两个子数组是就地排序的(不开辟新的数组),将他们合并不需要任何操作，整个数组A[p..r]已经排好序')
print('子数组快速排序伪代码')
print('QUICKSORT(A,p,r)')
print(' 1. if q < r')
print(' 2. q <- PARTITION(A,p,r)')
print(' 3. QUICKSORT(A,p,q-1)')
print(' 3. QUICKSORT(A,q+1,r)')
print('排序完整的数组A，调用QUICKSORT(A,0,len(A))即可')
print('快速排序算法的关键是PARTITION过程，它对子数组A[q..r]进行就地重排')
print('PARTITION(A,p,r)')
print(' 1. x <- A[r]')
print(' 2. i <- p-1')
print(' 3. for j <- p to r-1')
print(' 4. if A[j] <= x')
print(' 5. i <- i+1')
print(' 6. exchange A[i] <-> A[j]')
print(' 7. exchange A[i+1] <-> A[r]')
print(' 8. return i + 1')
A = [8, 9, 6, 7, 4, 5, 2, 3, 1]
print('数组A', _deepcopy(A), '的快速排序过程为:', quicksort.quicksort(A))
A = [13, 19, 9, 5, 12, 8, 7, 4, 11, 2, 6, 21]
print('练习7.1-1 数组A', _deepcopy(A), '的一步partition过程得到middle索引为：',
quicksort.partition(A, 0,
len(A) - 1))
A = [11, 11, 11, 11, 11]
print('练习7.1-2 数组A', _deepcopy(A), '的一步partition过程得到middle索引为：',
quicksort.partition(A, 0,
len(A) - 1))
print('练习7.1-3 就一个长度为n的for循环，且一定会执行，所以时间复杂度为Θ(n)，然后用确界的夹逼定义证明')
print('练习7.1-4 不等号方向改变即可')
def quickselect(arr, low, high, k):
if low < high:
p = partition(arr, low, high)
if p < k:
quickselect(arr, p + 1, high, k)
elif p > k:
quickselect(arr, low, p - 1, k)
def randomized_selection(A, i, first=None, last=None):
'''Returns the `i`th smallest element (indexed from 0) in `A[first:last]`.
Requires `len(A) > i`.
`A` must not contain duplicate elements.
Recursively partitions `A`, as in randomized quicksort, where the pivot is
chosen uniformly at random.
>>> randomized_selection([3, 5, 7, 1, 9, 4, 6, 8], 4)
6
'''
first = first if first is not None else 0
last = last if last is not None else len(A)
if last - first == 1:
return A[first]
pivot_index, _ = quicksort.partition(A, first, last,
quicksort.random_pivot)
normalized_pivot_index = pivot_index - first
if normalized_pivot_index == i:
return A[pivot_index]
elif normalized_pivot_index > i:
return randomized_selection(A, i, first, pivot_index)
else:
return randomized_selection(A, i - normalized_pivot_index - 1,
pivot_index + 1, last)
def test_partition():
"""Test the partition returns a modified list and value we expect."""
from quicksort import partition
test_list = [0, 1]
lo = 0
hi = len(test_list) - 1
pivot_point, a_list = partition(test_list, lo, hi)
assert pivot_point == 1
assert a_list == [0, 1]
def selection(L, p, r, i):
if p == r:
return L[p]
q = partition(L, p, r)
k = q - p + 1
if i == k:
return L[q]
elif i < k:
return selection(L, p, q - 1, i)
else:
return selection(L, q + 1, r, i - k)
def tail_recursive_quicksort(A, p, r):
while p < r:
q = partition(A, p, r)
l_size = q - p
r_size = r - q
if l_size < r_size:
tail_recursive_quicksort(A, p, q - 1)
p = q + 1 # larger is from A[q+1...r], hence p = q+1
else:
tail_recursive_quicksort(A, q + 1, r)
r = q - 1 # larger is from A[p...q-1], hence r = q-1
def test_partition(seq: list, l_i: int, r_i: int, p_i: int,
p_i_expected: int) -> None:
"""Verify partition step works."""
pivot = seq[p_i]
seq_actual, p_i_actual = quicksort.partition(seq, l_i, r_i, p_i)
assert p_i_actual == p_i_expected
assert seq_actual[p_i_expected] == pivot
assert all(x < seq_actual[p_i_expected] for x in seq_actual[:p_i_expected])
assert all(x > seq_actual[p_i_expected]
for x in seq_actual[p_i_expected + 1:])
def order_select(l, p, r, i):
# no randomization implemented here
if p == r:
return l[p]
index = i - 1
pivot_index = partition(l, p, r)
if index == pivot_index:
return l[pivot_index]
if index > pivot_index:
return order_select(l, pivot_index + 1, r, index - pivot_index)
else:
return order_select(l, p, pivot_index - 1, i)
def introSort(data, start, end, depth):
if start < end:
if depth == 0:
# Switch from quicksort to heapsort
heapSort(data, start, end)
Plot(max(data[start:end + 1]), data)
return
# Return the pivot index
p = partition(data, start, end)
# Sort all the elements to the left and to the right of the pivot
introSort(data, start, p - 1, depth - 1)
introSort(data, p + 1, end, depth - 1)
def kthtop(array,k):
shuffle(array)
low = 0
high = len(array) - 1
while(high > low):
j = partition(array, low, high)
if j < k: low = j + 1
elif j > k: high = j - 1
else: return array[k]
return array[k]
def RSelection(A, low, n, order_statistic):
if low == n:
return A[low]
if order_statistic == 0:
return -1
if n > low:
pivot = random.randrange(low, n + 1)
pivot_index = quicksort.partition(A, 0, n, pivot)
i = pivot_index - low + 1
if i == order_statistic:
return A[pivot_index]
elif i > order_statistic:
return RSelection(A, low, pivot_index - 1, order_statistic)
else:
return RSelection(A, pivot_index + 1, n, order_statistic - i)
def RSelect(arr, l, r, k):
if len(arr) == 1:
return arr[0]
if l > r:
i = l
else:
i = randint(l, r)
print(f"i : {i}")
arr[l], arr[i] = arr[i], arr[l]
j = partition(arr, l, r)
if j + 1 == k:
print(arr)
return arr[j]
elif j > k:
return RSelect(arr, l, j - 1, k)
else:
return RSelect(arr, j + 1, r, k)
def select_statistic_random(lst, start, end, i):
""" select i-statistic algorithm use random element everytime """
if start >= end:
return lst[start]
# select random pivot element
pivot = random.randint(start, end)
# swap pivot and last element into list
lst[-1], lst[pivot] = lst[-1], lst[pivot]
# regrouping the list
new_pivot = partition(lst, start, end)
k = new_pivot - start + 1
if i == k:
return lst[new_pivot]
elif i < k:
# run recursive quick sort for left half
return select_statistic_random(lst, start, new_pivot - 1, i)
else:
# run for right half
return select_statistic_random(lst, new_pivot + 1, end, i - k)
def top(i, array, start=None, end=None):
try:
if i > len(array):
raise UnboundLocalError('Error: ith position greater that the length of the array')
quicksort.randomize_pivot(array, end, start)
pivot = quicksort.partition(array, start, end)
ith_pos = i-1
if ith_pos == pivot:
return array[pivot]
if ith_pos < pivot:
return top(i, array, 0, pivot-1)
else:
return top(i, array, pivot + 1, end)
except UnboundLocalError as e:
raise e
def select(data, p, r, i):
""" Selects the ith smallest elements of self.data from p to r. Guarantees a good split by choosing the median
of medians of blocks of size 5.
Attributes:
p -- a starting index
r -- an end index
i -- an index of the smallest element
"""
if p == r:
return data[p]
# divide the group of elements into n/5 groups of 5 elements each
medians = [] # a list of medians from each group
last = r
j = p
while j < last + 1:
end = j + 5
if end > last + 1:
end = last + 1
insertion_sort(data, j, end)
mid = (j + end - 1) / 2
medians.append(data.pop(mid))
last -= 1
j += 4
# put medians at the head of the list
data[p:p] = medians
# use select recursively to find the median of medians
median = select(data, p, p + len(medians) - 1, (len(medians) + 1) / 2)
data[data.index(median)], data[r] = data[r], data[data.index(median)]
q = quicksort.partition(data, p, r)
k = q - p + 1
if i == k:
return data[q]
elif i < k:
return select(data, p, q - 1, i)
else:
return select(data, q + 1, r, i - k)
def solution(A):
m = 1
left = 0
right = len(A) - 1
s = [(0, 0)]
while left < right:
i, j = partition(A, left, right)
_jl = j - left
_ri = right - i
if _jl < m and _ri < m:
left, right = s.pop()
elif _jl < m and _ri >= m:
left = i
elif _jl >= m and _ri < m:
right = j
elif _jl >= m and _ri >= m:
if _jl >= _ri:
s.append((left, j))
left = i
else:
s.append((i, right))
right = j
return A
def test_kind_of_unsorted_list_4(self):
array = [1, 3, 2, 4]
pivot = partition(array, 2)
self.assertEqual(pivot, 1)
self.assertPivot(array, pivot)
def test_sorted_non_contiguous_list(self):
array = [1, 2, 4]
pivot = partition(array, 2)
self.assertEqual(pivot, 2)
self.assertPivot(array, pivot)
def test_unsorted_list(self):
array = [3, 1, 2]
pivot = partition(array, 2)
self.assertEqual(pivot, 1)
self.assertPivot(array, pivot)
def test_reversed_list(self):
array = [3, 2, 1]
pivot = partition(array, 1)
self.assertEqual(pivot, 1)
self.assertPivot(array, pivot)
def randomizedpartition(A, p, r):
i = random.randrange(p, r)
quicksort.exchange(A, i, r)
return quicksort.partition(A, p, r)
def test_rotated_list__5(self):
array = [4, 1, 5, 2, 3]
pivot = partition(array, 3)
self.assertEqual(pivot, 1)
self.assertPivot(array, pivot)
def randomized_partition(A, p, r):
i = random.randint(p, r)
A[i], A[r] = A[r], A[i]
return partition(A, p, r)
def test_partition(self):
a = [2, 1, 3]
q = qs.partition(a, 0, len(a) - 1)
self.assertEqual(q, 2)
def random_partition(arr, left, right):
p = random.randint(left, right)
arr[right], arr[p] = arr[p], arr[right]
mid = partition(arr, left, right)
return mid
def partition_random(A, p, r):
i = random.randint(p, r)
A[r], A[i] = A[i], A[r]
return quicksort.partition(A, p, r)
def test_partition_2(self):
array = [3, 8, 2, 5, 1, 4, 7, 6]
self.assertEquals(partition(array, 0, 7), (0, 4, 6, 7))
def test_yet_another_unsorted_list_4(self):
array = [3, 1, 2, 4]
pivot = partition(array, 2)
self.assertEqual(pivot, 1)
self.assertPivot(array, pivot)
def test_partition():
# l&i r
arr = [4, 8, 7, 0, 2, 6, 5, 3, 1]
i = 0 # index of pivot element
l = 0 # index of starting element
r = len(arr) - 1 # index of ending element
assert i == 0, 'index of pivot element is set to first element'
assert arr[i] == 4
i = partition(arr, l, r, i)
assert i == 4
assert arr[i] == 4
assert all(x < 4 for x in arr[l:i])
assert all(x > 4 for x in arr[i+1:r])
# l i r
arr = [4, 8, 7, 0, 2, 6, 5, 3, 1]
i = 1 # index of pivot element
l = 0 # index of starting element
r = len(arr) - 1 # index of ending element
assert i == 1
assert arr[i] == 8
i = partition(arr, l, r, i)
assert i == 8
assert arr[i] == 8
assert all(x < 8 for x in arr[l:i])
assert all(x > 8 for x in arr[i+1:r])
# l i&r
arr = [4, 8, 7, 0, 2, 6, 5, 3, 1]
i = 8 # index of pivot element
l = 0 # index of starting element
r = len(arr) - 1 # index of ending element
assert i == 8
assert arr[i] == 1
i = partition(arr, l, r, i)
assert i == 1
assert arr[i] == 1
assert all(x < 1 for x in arr[l:i])
assert all(x > 1 for x in arr[i+1:r])
# l i r
arr = [4, 8, 7, 0, 2, 6, 5, 3, 1]
# [ 7, 0, 2, 6, 5 ]
# [ 0, 2, 7, 6, 5 ]
# 0 1 2 3
i = 4 # index of pivot element
l = 2 # index of starting element
r = 6 # index of ending element
assert i == 4
assert arr[i] == 2
i = partition(arr, l, r, i)
assert i == 3
assert arr[i] == 2
assert all(x < 2 for x in arr[l:i])
assert all(x > 2 for x in arr[i+1:r])
# l&i&r
arr = [1]
i = 0 # index of pivot element
l = 0 # index of starting element
r = 0 # index of ending element
comparisons = Counter()
assert i == 0
assert arr[i] == 1
i = partition(arr, l, r, i, count=comparisons)
assert i == 0
assert arr[i] == 1
assert comparisons.total == 0
assert all(x < 1 for x in arr[l:i])
assert all(x > 1 for x in arr[i+1:r])
# l i&r
arr = [1, 2]
i = 1 # index of pivot element
l = 0 # index of starting element
r = 1 # index of ending element
comparisons = Counter()
assert i == 1
assert arr[i] == 2
i = partition(arr, l, r, i, count=comparisons)
assert i == 1
assert arr[i] == 2
assert comparisons.total == 1
assert all(x < 2 for x in arr[l:i])
assert all(x > 2 for x in arr[i+1:r])
# l&i r
arr = [1, 2]
i = 0 # index of pivot element
l = 0 # index of starting element
r = 1 # index of ending element
comparisons = Counter()
assert i == 0
assert arr[i] == 1
i = partition(arr, l, r, i, count=comparisons)
assert i == 0
assert arr[i] == 1
assert comparisons.total == 1
assert all(x < 1 for x in arr[l:i])
assert all(x > 1 for x in arr[i+1:r])
def test_sorted_list(self):
array = [1, 2, 3]
pivot = partition(array, 1)
self.assertEqual(pivot, 1)
self.assertPivot(array, pivot)
def test_quicksort():
eq_(quicksort.partition([4, 5, 3, 7, 2]), [3, 2, 4, 5, 7])
def test_partition_1(self):
array = [3, 5, 1, 4, 2, 6]
self.assertEquals(partition(array, 0, 5), (0, 4, 6, 5))
def test_partition(self):
a = [2, 8, 7, 1, 3, 5, 6, 4]
partition(a, 0, 7)
self.assertEquals(a, [2, 1, 3, 4, 7, 5, 6, 8])
def randomquicksort(A: list, begin: int, end: int):
if end > begin:
randomize(A, begin, end)
compare = quicksort.partition(A, begin, end)
randomquicksort(A, begin, compare - 1)
randomquicksort(A, compare + 1, end)
def test_partition_1(self):
array = [3, 5, 1, 4, 2, 6]
self.assertEquals(partition(array, 0, 5), (0, 4, 6, 5))
def randomised_partition(A, p, r):
random_index = random.randint(p, r)
A[r], A[random_index] = A[random_index], A[r]
return partition(A, p, r)
def test_partition_2(self):
array = [3, 8, 2, 5, 1, 4, 7, 6]
self.assertEquals(partition(array, 0, 7), (0, 4, 6, 7))
