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4.11.3
P4.11.3 Prove the claim at the end of the section about the Euclidean Algorithm and...
1. Consider the following claim. Claim: For two integers a and b, if a + b...
1. Consider the following claim. Claim: For two integers a and b, if a + b is odd then a is odd or b is odd. (a) If we consider the claim as the implication P =⇒ Q, which statement is P and which is Q? (b) Write the negations ¬P and ¬Q. (c) (1 point) Write the contrapositive of the claim. (d) Prove the contrapositive of the claim. 2. Use contraposition (proof by contrapositive )to prove the following claim....
Recall from class that the Fibonacci numbers are defined as follows: fo = 0,fi-1 and for all n fn...
Recall from class that the Fibonacci numbers are defined as follows: fo = 0,fi-1 and for all n fn-n-1+fn-2- 2, (a) Let nEN,n 24. Prove that when we divide In by f-1, the quotient is 1 and the remainder is fn-2 (b) Prove by induction/recursion that the Euclidean Algorithm takes n-2 iterations to calculate gcd(fn,fn-1) for n 2 3. Check your answer for Question 1 against this.
Recall from class that the Fibonacci numbers are defined as follows: fo =...
We discuss the Euclidean algorithm that finds the greatest common divisor of 2 numbers u and...
We discuss the Euclidean algorithm that finds the greatest common divisor of 2 numbers u and v. We want to extend and compute the gcd of n integers gcd(u1,u2,….un). One way to do it is to assume all numbers are non-negative, so if only one of if uj≠0 it is the gcd. Otherwise replace uk by uk mod uj for all k≠j where uj is the minimum of the non-zero elements (u’s). The algorithm can be made significantly faster if one...
We discuss the Euclidean algorithm that finds the greatest common divisor of 2 numbers u and...
We discuss the Euclidean algorithm that finds the greatest common divisor of 2 numbers u and v. We want to extend and compute the gcd of n integers gcd(u_1,u_2,….u_n). One way to do it is to assume all numbers are non-negative, so if only one of if u_j≠0 it is the gcd. Otherwise replace u_k by u_k mod u_j for all k≠j where u_j is the minimum of the non-zero elements (u’s). The algorithm can be made significantly faster if...
2,3,4,5,6 please 2. Use the Euclidean algorithm to find the following: a gcd(100, 101) b. ged(2482,...
2,3,4,5,6 please
2. Use the Euclidean algorithm to find the following: a gcd(100, 101) b. ged(2482, 7633) 3. Prove that if a = bq+r, then ged(a, b) = ged(b,r). such that sa tb ged(a,b) for the following pairs 4. Use Bézout's theorem to find 8 and a. 33, 44 b. 101, 203 c. 10001, 13422 5. Prove by induction that if p is prime and plaja... An, then pla, for at least one Q. (Hint: use n = 2 as...
2) In class we showed a Matrix algorithm for Fibonacci numbers: 1 112 1 0 n+1...
2) In class we showed a Matrix algorithm for Fibonacci numbers: 1 112 1 0 n+1 F (Note: No credit for an induction proof that this is true. I'm not asking that.) a) What is the running time for this algorithm? (3 pts.) b) Prove it. (9 pts.)
answer all parts please 1. (12 points) Prove that if n is an integer, then na...
answer all parts please
1. (12 points) Prove that if n is an integer, then na +n + 1 is odd. 2. (12 points) Prove that if a, b, c are integers, c divides a +b, and ged(a,b) -1, then god (ac) - 1. 3. (a) (6 points) Use the Euclidean Algorithm to find ged(270, 105). Be sure to show all the steps of the Euclidean algorithm and, once you have finished the Euclidean Algorithm, to finish the problem by...
Exercise 7.3.5: Worst-case time complexity - mystery algorithm. The algorithm below makes some changes to an...
Exercise 7.3.5: Worst-case time complexity - mystery algorithm. The algorithm below makes some changes to an input sequence of numbers. MysteryAlgorithm Input: a1, a2....,an n, the length of the sequence. p, a number Output: ?? i != 1 j:=n While (i < j) While (i <j and a < p) i:= i + 1 End-while While (i <j and a 2 p) j:=j-1 End-while If (i < j), swap a, and a End-while Return( aj, a2,...,an) (a) Describe in English...
Need help with 5.25. I have attached the definition of the i-th binomial tree. fundamentals of...
Need help with
5.25. I have attached the definition of the i-th binomial
tree.
fundamentals of algorithmics.pdf (page 200 of 530) Q Search Protiom 8.24. For heapsort, what are the best and the worst initial arrangements of the elements to be sorted, as far as the execution time of the algorithm is concerned? Justify your answer Problem 5.25. Prove that the binomial tree B defined in Section 5.8 contains 2 nodes, of which (i) are at depth k, 0 s...
SECTION B. This part is about using KNN algorithms. In this section we will consider the...
SECTION B. This part is about using KNN algorithms. In this section we will consider the following points and we will use the Euclidean distance. D = (2). D;= 0) (1.D = (). 0,= (*) Y3 = x, Y4 = x. And their labels are yı = 0, y2 = 0, Now we are given a test point T: 1 = () B1. What is the label of T if we use 1NN algorithm to C? (a) 50% o and...
Recall from class that the Fibonacci numbers are defined as follows: fo = 0,fi-1 and for all n fn-n-1+fn-2- 2, (a) Let nEN,n 24. Prove that when we divide In by f-1, the quotient is 1 and the remainder is fn-2 (b) Prove by induction/recursion that the Euclidean Algorithm takes n-2 iterations to calculate gcd(fn,fn-1) for n 2 3. Check your answer for Question 1 against this.
Recall from class that the Fibonacci numbers are defined as follows: fo =...
2,3,4,5,6 please
2. Use the Euclidean algorithm to find the following: a gcd(100, 101) b. ged(2482, 7633) 3. Prove that if a = bq+r, then ged(a, b) = ged(b,r). such that sa tb ged(a,b) for the following pairs 4. Use Bézout's theorem to find 8 and a. 33, 44 b. 101, 203 c. 10001, 13422 5. Prove by induction that if p is prime and plaja... An, then pla, for at least one Q. (Hint: use n = 2 as...
2) In class we showed a Matrix algorithm for Fibonacci numbers: 1 112 1 0 n+1 F (Note: No credit for an induction proof that this is true. I'm not asking that.) a) What is the running time for this algorithm? (3 pts.) b) Prove it. (9 pts.)
answer all parts please
1. (12 points) Prove that if n is an integer, then na +n + 1 is odd. 2. (12 points) Prove that if a, b, c are integers, c divides a +b, and ged(a,b) -1, then god (ac) - 1. 3. (a) (6 points) Use the Euclidean Algorithm to find ged(270, 105). Be sure to show all the steps of the Euclidean algorithm and, once you have finished the Euclidean Algorithm, to finish the problem by...
Exercise 7.3.5: Worst-case time complexity - mystery algorithm. The algorithm below makes some changes to an input sequence of numbers. MysteryAlgorithm Input: a1, a2....,an n, the length of the sequence. p, a number Output: ?? i != 1 j:=n While (i < j) While (i <j and a < p) i:= i + 1 End-while While (i <j and a 2 p) j:=j-1 End-while If (i < j), swap a, and a End-while Return( aj, a2,...,an) (a) Describe in English...
Need help with
5.25. I have attached the definition of the i-th binomial
tree.
fundamentals of algorithmics.pdf (page 200 of 530) Q Search Protiom 8.24. For heapsort, what are the best and the worst initial arrangements of the elements to be sorted, as far as the execution time of the algorithm is concerned? Justify your answer Problem 5.25. Prove that the binomial tree B defined in Section 5.8 contains 2 nodes, of which (i) are at depth k, 0 s...
SECTION B. This part is about using KNN algorithms. In this section we will consider the following points and we will use the Euclidean distance. D = (2). D;= 0) (1.D = (). 0,= (*) Y3 = x, Y4 = x. And their labels are yı = 0, y2 = 0, Now we are given a test point T: 1 = () B1. What is the label of T if we use 1NN algorithm to C? (a) 50% o and...
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