Consider the following recursive algorithm for computing the sum of the first n cubes: S(n) = 13 + 23 + … + n3.
(a) Set up a recurrence relation for the number of multiplications made by this algorithm.
(b) Provide an initial condition for the recurrence relation you develop at the question (a).
(c) Solve the recurrence relation of the question (a) and present the time complexity as described at the question number 1.
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Consider the following recursive algorithm for computing the sum
of the first n cubes: S(n) =...
ALGORITHM RecS(n) // Input: A nonnegative integer n ifn=0 return 0 else return RecS(n+ n n...
ALGORITHM RecS(n) // Input: A nonnegative integer n ifn=0 return 0 else return RecS(n+ n n n Determine what this algorithm computes. You must justify your answer. made by this algorithm and solve it. You must justify your answer. same thing using for/while loop(s) developed in (3). You must justify your answer. 1) 2) Set up the initial condition and recurrence relation for the number of multiplications 3) Write the pseudocode for the non-recursive version of this algorithm, i.e., compute...
Consider the problem of computing the power function pow(n,x) = n^x using only multiplications. The first...
Consider the problem of computing the power function pow(n,x) = n^x using only multiplications. The first approach is to perform x multiplications ($n \cdot n \cdot n \cdot \ldots \cdot n$, x times). Find a better, recursive algorithm to solve this problem (by better, we mean one that uses fewer than $x$ multiplications). Write down the pseudocode for this new function, and then analyze the runtime of that recursive program by first writing out the recurrence relation $T(n, x)$ that...
Consider the problem of computing the power function pow(n,x) = n^x using only multiplications. The first approach is to...
Consider the problem of computing the power function pow(n,x) = n^x using only multiplications. The first approach is to perform x multiplications ($n \cdot n \cdot n \cdot \ldots \cdot n$, x times). Find a better, recursive algorithm to solve this problem (by better, we mean one that uses fewer than $x$ multiplications). Write down the pseudocode for this new function, and then analyze the runtime of that recursive program by first writing out the recurrence relation $T(n, x)$ that...
07-15 pts) Develop a recursive version of the Bubble Sort algorithm. (a) Write the pseudo code...
07-15 pts) Develop a recursive version of the Bubble Sort algorithm. (a) Write the pseudo code of the algorithm and justify that it is recursive and works correctly Write the recurrence relation for the algorithm and solve it using one of the two approaches discussed in class, as appropriate. Solve the recurrence relation and show that the time complexity of the recursive algorithm is θ(n).
given a recursive alogrithmn for computing 2^n , n greater than or equal to zero integer,...
given a recursive alogrithmn for computing 2^n , n greater than or equal to zero integer, based on the formula 2^n = 2^n-1 Set up a recurrence relation for the number of additions made by the alogrithmn, and then solve it,
For each of the following recursive methods available on the class handout, derive a worst-case recurrence...
For each of the following recursive methods available on the class handout, derive a worst-case recurrence relation along with initial condition(s) and solve the relation to analyze the time complexity of the method. The time complexity must be given in a big-O notation. 1. digitSum(int n) - summing the digits of integer: int digitSum(int n) { if (n < 10) return n; return (digitSum(n/10) + n%10); } 2. void reverseA(int l, int r) - reversing array: void...
Q4) [5 points] Consider the following two algorithms: ALGORITHM 1 Bin Rec(n) //Input: A positive decimal...
Q4) [5 points] Consider the following two algorithms: ALGORITHM 1 Bin Rec(n) //Input: A positive decimal integer n llOutput: The number of binary digits in "'s binary representation if n1 return 1 else return BinRec(ln/2)) +1 ALGORITHM 2 Binary(n) tive decimal integer nt io 's binary representation //Output: The number of binary digits in i's binary representation count ←1 while n >1 do count ← count + 1 return count a. Analyze the two algorithms and find the efficiency for...
The following algorithm (Rosen pg. 363) is a recursive version of linear search, which has access...
The following algorithm (Rosen pg. 363) is a recursive version of linear search, which has access to a global list of distinct integers a_1, a_2,..., a_n. procedure search(i, j, x : i,j, x integers, 1 < i < j < n) if a_i = x then return i else if i = j then 4. return 0 else return search(i + 1, j, x) Prove that this algorithm correctly solves the searching problem when called with parameters i = 1...
3. Recursive Program (6 points) Consider the following recursive function for n 1: Algorithm 1 int...
3. Recursive Program (6 points) Consider the following recursive function for n 1: Algorithm 1 int recurseFunc(int n) If n 0, return 1. If n 1, return 1 while i< n do while j <n do print("hi") j 1 end while i i 1 end while int a recurse Func(n/9); int b recurse Func (n/9) int c recurse Func (n/9) return a b c (1) Set up a runtime recurrence for the runtime T n) of this algorithm. (2) Solve...
write a recursive algorithm to find the sum of the first N terms of the series...
write a recursive algorithm to find the sum of the first N terms of the series 1, 1/2, 1/3, ... 1/N
ALGORITHM RecS(n) // Input: A nonnegative integer n ifn=0 return 0 else return RecS(n+ n n n Determine what this algorithm computes. You must justify your answer. made by this algorithm and solve it. You must justify your answer. same thing using for/while loop(s) developed in (3). You must justify your answer. 1) 2) Set up the initial condition and recurrence relation for the number of multiplications 3) Write the pseudocode for the non-recursive version of this algorithm, i.e., compute...
07-15 pts) Develop a recursive version of the Bubble Sort algorithm. (a) Write the pseudo code of the algorithm and justify that it is recursive and works correctly Write the recurrence relation for the algorithm and solve it using one of the two approaches discussed in class, as appropriate. Solve the recurrence relation and show that the time complexity of the recursive algorithm is θ(n).
Q4) [5 points] Consider the following two algorithms: ALGORITHM 1 Bin Rec(n) //Input: A positive decimal integer n llOutput: The number of binary digits in "'s binary representation if n1 return 1 else return BinRec(ln/2)) +1 ALGORITHM 2 Binary(n) tive decimal integer nt io 's binary representation //Output: The number of binary digits in i's binary representation count ←1 while n >1 do count ← count + 1 return count a. Analyze the two algorithms and find the efficiency for...
The following algorithm (Rosen pg. 363) is a recursive version of linear search, which has access to a global list of distinct integers a_1, a_2,..., a_n. procedure search(i, j, x : i,j, x integers, 1 < i < j < n) if a_i = x then return i else if i = j then 4. return 0 else return search(i + 1, j, x) Prove that this algorithm correctly solves the searching problem when called with parameters i = 1...
3. Recursive Program (6 points) Consider the following recursive function for n 1: Algorithm 1 int recurseFunc(int n) If n 0, return 1. If n 1, return 1 while i< n do while j <n do print("hi") j 1 end while i i 1 end while int a recurse Func(n/9); int b recurse Func (n/9) int c recurse Func (n/9) return a b c (1) Set up a runtime recurrence for the runtime T n) of this algorithm. (2) Solve...
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