def _small_order_select(B, p, r, i):
n = r - p + 1
m = n // 2
if i >= m:
return _select_with_cascaded_swaps(B, n, p, r, i)
for j in between(0, m - 1):
if B[p + j] < B[p + m + j]:
_cascaded_swap(B, n, p + j, p + m + j)
if n % 2 == 1:
_cascaded_align(B, n, p + m)
m += 1
n += 1
r = B.length
p = r - n + 1
idx = _small_order_select(B, p + m, r, i)
r = B.length
q = idx - i + 1
m = r - q + 1
n = 2 * m
p = r - n + 1
for j in between(0, i - 1):
_cascaded_swap(B, n, p + i + j, p + m + j)
_select_with_cascaded_swaps(B, n, p, p + 2 * i - 1, i)
return p + i - 1
def tasks_independent_bruteforce(deadlines):
n = deadlines.length
N = Array([len([d for d in deadlines if d <= t]) for t in between(1, n)])
for t in between(1, n):
if N[t] > t:
return False
return True
def young_sort(A):
n = int(math.sqrt(A.length))
Y = Matrix([[math.inf] * n] * n)
for i in between(1, n ** 2):
young_insert(Y, n, n, A[i])
for i in between(1, n ** 2):
A[i] = young_extract_min(Y, n, n, 1, 1)
def memoized_matrix_chain(p):
n = p.length - 1
m = Array([Array.indexed(1, n) for _ in between(1, n)])
for i in between(1, n):
for j in between(i, n):
m[i, j] = math.inf
return lookup_chain(p, m, 1, n)
def young_sort(A):
n = int(math.sqrt(A.length))
Y = Matrix([[math.inf] * n] * n)
for i in between(1, n**2):
young_insert(Y, n, n, A[i])
for i in between(1, n**2):
A[i] = young_extract_min(Y, n, n, 1, 1)
def segments_intersect(a, b):
a_vertical = is_verticle(a)
b_vertical = is_verticle(b)
if(a_vertical == b_vertical):
# Lines are parallel
if a_vertical and a[0][0] == b[0][0]:
# Both verticle and in same column
return (
util.between(a[0][1], b[0][1], b[1][1])
or util.between(a[1][1], b[0][1], b[1][1])
or util.between(b[0][1], a[0][1], a[1][1])
or util.between(b[1][1], a[0][1], a[1][1])
)
if not a_vertical and a[0][1] == b[0][1]:
# Both horiztonal and in same row
return (
util.between(a[0][0], b[0][0], b[1][0])
or util.between(a[1][0], b[0][0], b[1][0])
or util.between(b[0][0], a[0][0], a[1][0])
or util.between(b[1][0], a[0][0], a[1][0])
)
else:
# Mixed vertical and horizontal
vert, hori = (a, b) if a_vertical else (b, a)
return util.between(vert[0][0], hori[0][0], hori[1][0]) and util.between(hori[0][1], vert[0][1], vert[1][1])
return False
def test_checkerboard(self):
n = random.randint(1, 8)
# profit[i, j] contains a triple (a, b, c), where the profit of moving from square of coords (i, j):
# to square of coords (i+1, j-1) is a, to square of coords (i+1, j) is b, to square of coords (i+1, j+1) is c,
# where (i, j) means i-th row from the bottom and j-th column from the left
profit = Array([Array.indexed(1, n) for _ in between(1, n - 1)])
for i in between(1, n - 1):
profit[i,
1] = (None, random.randint(-100,
100), random.randint(-100, 100))
for j in between(2, n - 1):
profit[i, j] = (random.randint(-100,
100), random.randint(-100, 100),
random.randint(-100, 100))
profit[i, n] = (random.randint(-100,
100), random.randint(-100,
100), None)
captured_output = io.StringIO()
actual_maximum_profit, squares, last_square = checkerboard(
n, lambda x, y: checkerboard_profit(profit, x, y))
with redirect_stdout(captured_output):
print_moves(squares, n, last_square)
expected_maximum_profit = \
get_optimal_checkerboard_path_bruteforce(n, lambda x, y: checkerboard_profit(profit, x, y))
assert_that(actual_maximum_profit,
is_(equal_to(expected_maximum_profit)))
assert_squares_path(n,
captured_output.getvalue().splitlines(), profit,
expected_maximum_profit)
def merge_(A, p, q, r):
n1 = q - p + 1
n2 = r - q
L = Array.indexed(1, n1)
R = Array.indexed(1, n2)
for i in between(1, n1):
L[i] = A[p + i - 1]
for j in between(1, n2):
R[j] = A[q + j]
i = j = 1
k = p
while i <= n1 and j <= n2:
if L[i] <= R[j]:
A[k] = L[i]
i = i + 1
else:
A[k] = R[j]
j = j + 1
k = k + 1
while i <= n1:
A[k] = L[i]
i = i + 1
k = k + 1
while j <= n2:
A[k] = R[j]
j = j + 1
k = k + 1
def memoized_matrix_chain(p):
n = p.length - 1
m = Array([Array.indexed(1, n) for _ in between(1, n)])
for i in between(1, n):
for j in between(i, n):
m[i, j] = math.inf
return lookup_chain(p, m, 1, n)
def bucket_sort(A):
n = A.length
B = Array([[] for _ in range(n)], start=0)
for i in between(1, n):
B[math.floor(n * A[i])].append(A[i])
for i in between(0, n - 1):
_insertion_sort_list(B[i])
_concatenate_lists(B, A)
def selection_sort(A):
n = A.length
for j in between(1, n - 1):
min = j
for i in between(j + 1, n):
if A[i] < A[min]:
min = i
A[min], A[j] = A[j], A[min]
def activity_selector_bruteforce(s, f):
n = s.length - 2
max_size = 0
for m in between(1, n):
for activities_ids in itertools.combinations(between(1, n), m):
if activities_compatible(activities_ids, s, f):
max_size = max(max_size, m)
return max_size
def knapsack_bruteforce(w, v, W):
max_value = 0
n = w.length
for m in between(1, n):
for item_ids in itertools.combinations(between(1, n), m):
if items_total_weight(item_ids, w) <= W:
max_value = max(max_value, items_total_value(item_ids, v))
return max_value
def memoized_lcs_length(X, Y):
m = X.length
n = Y.length
c = Array([Array.indexed(0, n) for _ in between(0, m)], start=0)
for i in between(0, m):
for j in between(0, n):
c[i, j] = math.inf
return lookup_lcs(c, X, Y, m, n)
def knapsack_bruteforce(w, v, W):
max_value = 0
n = w.length
for m in between(1, n):
for item_ids in itertools.combinations(between(1, n), m):
if items_total_weight(item_ids, w) <= W:
max_value = max(max_value, items_total_value(item_ids, v))
return max_value
def get_shortest_bitonic_path_length_bruteforce(points):
n = points.length
min_length = math.inf
for k in between(0, n - 2):
for right_path in itertools.combinations(between(2, n - 1), k):
left_path = [x for x in rbetween(n - 1, 2) if x not in right_path]
path_length = get_path_length(points, [1] + list(right_path) + [n] + left_path + [1])
min_length = min(min_length, path_length)
return min_length
def get_optimal_schedule_bruteforce(times, profits, deadlines):
n = times.length
max_profit = 0
for m in between(1, n):
for schedule in itertools.permutations(between(1, n), m):
profit = get_schedule_total_profit(schedule, times, profits,
deadlines)
max_profit = max(max_profit, profit)
return max_profit
def refueling_bruteforce(stations, n):
min_stops = math.inf
m = stations.length
for nstops in between(0, m):
for stops in itertools.combinations(between(1, m), nstops):
if stops_valid(stops, stations, n):
min_stops = min(min_stops, nstops)
break
return min_stops
def refueling_bruteforce(stations, n):
min_stops = math.inf
m = stations.length
for nstops in between(0, m):
for stops in itertools.combinations(between(1, m), nstops):
if stops_valid(stops, stations, n):
min_stops = min(min_stops, nstops)
break
return min_stops
def polynomial_evaluate(a, x):
y = 0.0
n = a.length - 1
for i in between(0, n):
s = a[i]
for j in between(1, i):
s = s * x
y = y + s
return y
def binary_add(A, B):
n = A.length
C = Array.indexed(1, n + 1)
for i in between(1, n + 1):
C[i] = 0
for i in between(1, n):
sum = A[i] + B[i] + C[i]
C[i] = sum % 2
C[i + 1] = math.floor(sum / 2)
return C
def unit_circle_sort(A):
n = A.length
B = Array([[] for _ in range(n)], start=0)
for i in between(1, n):
d = math.sqrt(A[i].x**2 + A[i].y**2)
B[math.ceil(d**2 * n) - 1].append(
(A[i], d)) # store distances with points; we'll sort by them later
for i in between(0, n - 1):
_insertion_sort_list_by_distance(B[i])
_concatenate_lists_of_points(B, A)
def max_overlapping_activities(s, f):
n = s.length
max_overlaps = 0
for i in between(1, n):
overlaps = 0
for j in between(1, n):
if s[j] <= s[i] < f[j]:
overlaps += 1
max_overlaps = max(max_overlaps, overlaps)
return max_overlaps
def max_overlapping_activities(s, f):
n = s.length
max_overlaps = 0
for i in between(1, n):
overlaps = 0
for j in between(1, n):
if s[j] <= s[i] < f[j]:
overlaps += 1
max_overlaps = max(max_overlaps, overlaps)
return max_overlaps
def get_shortest_bitonic_path_length_bruteforce(points):
n = points.length
min_length = math.inf
for k in between(0, n - 2):
for right_path in itertools.combinations(between(2, n - 1), k):
left_path = [x for x in rbetween(n - 1, 2) if x not in right_path]
path_length = get_path_length(points, [1] + list(right_path) +
[n] + left_path + [1])
min_length = min(min_length, path_length)
return min_length
def matrix_multiply(A, B):
if A.columns != B.rows:
raise RuntimeError('incompatible dimensions')
else:
C = Matrix.of_dimensions(A.rows, B.columns)
for i in between(1, A.rows):
for j in between(1, B.columns):
C[i, j] = 0
for k in between(1, A.columns):
C[i, j] = C[i, j] + A[i, k] * B[k, j]
return C
def get_probabilities_for_optimal_bst():
n = random.randint(1, 10)
p, _ = get_random_array(min_size=n, max_size=n)
q, _ = get_random_array(min_size=n + 1, max_size=n + 1)
q.start = 0
total = sum([x for x in p.elements + q.elements])
for i in between(1, n):
p[i] /= total
for i in between(0, n):
q[i] /= total
return p, q
def counting_sort(A, B, k):
C = Array.indexed(0, k)
for i in between(0, k):
C[i] = 0
for j in between(1, A.length):
C[A[j]] = C[A[j]] + 1
for i in between(1, k):
C[i] = C[i] + C[i - 1]
for j in rbetween(A.length, 1):
B[C[A[j]]] = A[j]
C[A[j]] = C[A[j]] - 1
def make_change(n, d):
c = Array.indexed(0, n)
denom = Array.indexed(1, n)
c[0] = 0
for j in between(1, n):
c[j] = math.inf
for i in between(1, d.length):
if j >= d[i] and 1 + c[j - d[i]] < c[j]:
c[j] = 1 + c[j - d[i]]
denom[j] = d[i]
return c, denom
def knapsack(w, v, W):
n = w.length
K = Array([Array.indexed(0, W) for _ in between(0, n)], start=0)
for j in between(0, W):
K[0, j] = 0
for i in between(1, n):
for j in between(0, W):
K[i, j] = K[i - 1, j]
if w[i] <= j and K[i - 1, j - w[i]] + v[i] > K[i, j]:
K[i, j] = K[i - 1, j - w[i]] + v[i]
return K
def get_probabilities_for_optimal_bst():
n = random.randint(1, 10)
p, _ = get_random_array(min_size=n, max_size=n)
q, _ = get_random_array(min_size=n + 1, max_size=n + 1)
q.start = 0
total = sum([x for x in p.elements + q.elements])
for i in between(1, n):
p[i] /= total
for i in between(0, n):
q[i] /= total
return p, q
def tasks_independent(A, k):
C = Array.indexed(0, k)
for i in between(0, k):
C[i] = 0
for j in between(1, A.length):
C[A[j]] = C[A[j]] + 1
for i in between(1, k):
C[i] = C[i] + C[i - 1]
for t in between(1, k):
if C[t] > t:
return False
return True
def _sort_by_character(A, p, r, position):
k = ord('z') - ord('a')
C = Array([0] * (k + 1), start=0)
for j in between(p, r):
x = ord(A[j][position - 1]) - ord('a')
C[x] += 1
for i in between(1, k):
C[i] += C[i - 1]
B = Array.indexed(1, r - p + 1)
for j in rbetween(r, p):
x = ord(A[j][position - 1]) - ord('a')
B[C[x]] = A[j]
C[x] -= 1
A.elements[p - 1:r] = B.elements
def test_print_optimal_parens(self):
n = random.randint(1, 10)
s = Array([Array.indexed(1, n) for _ in between(1, n)])
for i in between(1, n - 1):
for j in between(i + 1, n):
s[i, j] = random.randint(i, j - 1)
captured_output = io.StringIO()
with redirect_stdout(captured_output):
print_optimal_parens(s, 1, n)
actual_output = captured_output.getvalue().splitlines()[0]
expected_output = get_optimal_parens_bruteforce(s, 1, n)
assert_that(actual_output, is_(equal_to(expected_output)))
def josephus_simulate(n, m):
L = List()
singly_linked_list_insert(L, SNode(n))
x = L.head
for i in rbetween(n - 1, 1):
singly_linked_list_insert(L, SNode(i))
x.next = L.head
for i in between(1, n):
for j in between(1, m):
x = x.next
print(x.next.key)
if L.head is x.next:
L.head = x.next.next
x.next = x.next.next
def median_nearest(A, k):
n = A.length
x = select(A, 1, n, math.floor((n + 1) / 2))
dist = Array.indexed(1, n)
for i in between(1, n):
dist[i] = abs(A[i] - x)
y = select(dist, 1, n, k)
N = set()
for i in between(1, n):
if abs(A[i] - x) <= y:
N.add(A[i])
if len(N) == k + 1:
N.remove(x + y)
return N
def _sort_by_character(A, p, r, position):
k = ord('z') - ord('a')
C = Array([0] * (k + 1), start=0)
for j in between(p, r):
x = ord(A[j][position - 1]) - ord('a')
C[x] += 1
for i in between(1, k):
C[i] += C[i - 1]
B = Array.indexed(1, r - p + 1)
for j in rbetween(r, p):
x = ord(A[j][position - 1]) - ord('a')
B[C[x]] = A[j]
C[x] -= 1
A.elements[p - 1:r] = B.elements
def test_print_optimal_parens(self):
n = random.randint(1, 10)
s = Array([Array.indexed(1, n) for _ in between(1, n)])
for i in between(1, n - 1):
for j in between(i + 1, n):
s[i, j] = random.randint(i, j - 1)
captured_output = io.StringIO()
with redirect_stdout(captured_output):
print_optimal_parens(s, 1, n)
actual_output = captured_output.getvalue().splitlines()[0]
expected_output = get_optimal_parens_bruteforce(s, 1, n)
assert_that(actual_output, is_(equal_to(expected_output)))
def dynamic_activity_selector(s, f):
n = s.length - 2
c = Array([Array.indexed(0, n + 1) for _ in between(0, n + 1)])
A = Array([Array.indexed(0, n + 1) for _ in between(0, n + 1)])
for l in between(2, n + 2):
for i in between(0, n - l + 2):
j = i + l - 1
c[i, j] = 0
A[i, j] = set()
for k in between(i + 1, j - 1):
q = c[i, k] + c[k, j] + 1
if f[i] <= s[k] < f[k] <= s[j] and q > c[i, j]:
c[i, j] = q
A[i, j] = A[i, k] | {'a' + str(k)} | A[k, j]
return A[0, n + 1]
def jugs_group(R, B):
n = R.length
for i in between(1, n - 1):
j = i
while R[i] != B[j]:
j = j + 1
B[i], B[j] = B[j], B[i]
def monge_minimums(A):
m = A.rows
minimums_indices = _monge_minimums_indices(A)
minimums = Array.indexed(1, m)
for i in between(1, m):
minimums[i] = A[i, minimums_indices[i]]
return minimums
def get_maximum_lcs_length_bruteforce(sequence1, sequence2):
max_length = 0
for i in between(1, min(sequence1.length, sequence2.length)):
for subsequence in itertools.combinations(sequence1, i):
if is_subsequence_of(subsequence, sequence2):
max_length = len(subsequence)
return max_length
def randomize_in_place_(A):
n = A.length
j = random(1, n)
A[1], A[j] = A[j], A[1]
for i in between(2, n):
j = random(i, n)
A[i], A[j] = A[j], A[i]
def _move_strings_with_exact_length_to_front(A, p, r, position):
q = p
for j in between(p, r):
if len(A[j]) == position - 1:
A[q], A[j] = A[j], A[q]
q += 1
return q
def permute_by_sorting(A):
n = A.length
P = Array.indexed(1, n)
for i in between(1, n):
P[i] = random(1, n ** 3)
_sort_using_priorities(A, P)
return A
def sum_search(S, x):
n = S.length
merge_sort(S, 1, n)
for i in between(1, n - 1):
if recursive_binary_search(S, x - S[i], i + 1, n) is not None:
return True
return False
def get_maximum_lis_length_bruteforce(sequence):
max_length = 0
for i in between(1, sequence.length):
for subsequence in itertools.combinations(sequence, i):
if is_monotonically_increasing(subsequence):
max_length = len(subsequence)
return max_length