- We Are Given Three Subsets Of Numbers A B C C 0 N Design An Algorithm That Runs In Worst Case Time N Log N T 1 (55.3 KiB) Viewed 41 times
We are given three subsets of numbers A, B, C C {0,...,n). Design an algorithm that runs in worst case time (n log(n)) that checks if there exists numbers ₁, ₂ in A, B, respectively and number n3 in C such that n₁ + m₂ = n3. Hint: Convert the set A = {no, n₁,...,nk} into the polynomial P₁(x) : x + x² + ... + x²k. Suppose P₁(x), PB(x) are polynomials obtained from the sets A, B respectively, interpret what the product P₁(x) X PB(x) signifies. Use this to complete an algorithm for the problem at hand that runs in n log(n) time. # inputs sets a, b, c # return True if there exist n1 in a, n2 in B such that n1+n2 in C # return False otherwise # number n which signifies the maximum number in a, b, c #here is a useful reference to set data structure in python #
https://docs.python.org/3/tutorial/data ... .html#sets def check_sum_exists(a, b, c, n): a_coeffs = [0] *n b_coeffs = [0] *n # convert sets a, b into polynomials as provided in the hint #a_coeffs and b_coeffs should contain the result # your code here # multiply them together print ("a_coeffs. ", a_coeffs) print ("b_coeffs ", b_coeffs) c_coeffs = polynomial_multiply(a_coeffs, b_coeffs) print("Co-efficients of the testcase are") coeffs_copy = [] for num in c_coeffs: if(abs (num-0) < abs(num-1)): coeffs_copy.append(0) elif (abs (num-1) < abs(num-2)): coeffs_copy.append (1) else: coeffs_copy.append(2) print (coeffs_copy) # use the result to solve the problem at hand # your code here # return True/False
print('-- Test 1 --') a = set([1, 2, 10, 11]) b = set([2, 5, 8, 10]) c = set([1, 2, 5, 8]) assert not check_sum_exists(a, b, c, 12), f'Failed Test 1: your code returned true when the expected answer is false' print('Passed') print('-- Test 2 --') a = set([1, 2, 10, 11]) b = set ([2, 5, 8, 10]) c = set([1, 2, 5, 8, 11]) assert check_sum_exists(a, b, c, 12), f'Failed Test 2: your code returns false but note that 1 in a + 10 in b = 11 in c' print('Passed') print('-- Test 3 --') a={1, 4, 5, 7, 11, 13, 14, 15, 17, 19, 22, 23, 24, 28, 34, 35, 37, 39, 42, 44} b={0, 1, 4, 9, 10, 11, 12, 15, 18, 20, 25, 31, 34, 36, 38, 40, 43, 44, 47, 49} c={3, 4, 5, 7, 8, 10, 19, 20, 21, 24, 31, 35, 36, 37, 38, 39, 42, 44, 46, 49} assert check_sum_exists(a, b, c, 50), f'Failed Test 3: your code returns False whereas the correct answer is true eg., 4 + 0 print('-- Test 4 --') a-{98, 2, 99, 40, 77, 79, 87, 88, 89, 27} b-{64, 66, 35, 69, 70, 40, 76, 45, 12, 60} c={36, 70, 10, 44, 15, 16, 83, 20, 84, 55} assert not check_sum_exists(a, b, c, 100), f'Failed Test 4: your code returns True whereas the correct answer is False' print('All Tests Passed (15 points)!')