31. The given diagram shows the fow-chart for a recursive function Afn). Assume that all statementaexcept for the recursive calls, have O(1) time complexity. If the wonst case time completity of thisfunction is O(s") then the least possible valuetaccurate up to two decimal positions) of aFlew thert Sr Raive Fnction AStartARaturnSaturn

31. The given diagram shows the flow-chart for a recurtive function A(n). Assume that all statements except for the recursive calls, have O) time complexity If the worst case time complecity of this function is O(n", then the least possible valuetaccurate up to rwo decimal positiona) f a is Pwhet Raie Fn Sart AND Ratum

31. The given diagram shows the flow-chart for a recurtive function A(n). Assume that all statements except for the recursive calls, have O) time complexity If the worst case time complecity of this function is O(n", then the least possible valuetaccurate up to rwo decimal positiona) f a is Pwhet Raie Fn Sart AND Ratum

C++ Programming: From Problem Analysis to Program Design

8th Edition

ISBN:9781337102087

Author:D. S. Malik

Publisher:D. S. Malik

Chapter15: Recursion

Section: Chapter Questions

Problem 7SA

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Problem 1. Prove that the following functions are Primitive Recursive. I – 1 if x > 0, (1) mPred(x) = for x € N. if x = 0. if x > 0, (2) sgn(x) = for x E N. 0 if x = 0. if x > 0, (3) sgn(x) = for x E N. 1 if x = 0.
Find a closed form representation for the following recursively defined function. Give the run-time complexity of each recursively defined function.
1. For the function defined recursively by f(0)=5 and f(n)=4f (n-1)+3, answer the following: a. Find a closed form representation for this function. Your closed form should not include any series. b. Prove that your representation is correct using a formal inductive argument.
In programming, a recursive function calls itself. The classical example is factorial(n), which can be defined recursively as n*factorial(n-1). Nonethessless, it is important to take note that a recursive function should have a terminating condition (or base case), in the case of factorial, factorial(0)=1. Hence, the full definition is: factorial(n) = 1, for n = 0 factorial(n) = n * factorial(n-1), for all n > 1 For example, suppose n = 5: // Recursive call factorial(5) = 5 * factorial(4) factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) factorial(2) = 2 * factorial(1) factorial(1) = 1 * factorial(0) factorial(0) = 1 // Base case // Unwinding factorial(1) = 1 * 1 = 1 factorial(2) = 2 * 1 = 2 factorial(3) = 3 * 2 = 6 factorial(4) = 4 * 6 = 24 factorial(5) = 5 * 24 = 120 (DONE) Exercise (Factorial) (Recursive): Write a recursive method called factorial() to compute the factorial of the given integer. public static int factorial(int n) The recursive algorithm is:…
4. Find a closed form representation for the function defined recursively by f(0)=5 and f(n+1)=3f(n)+4. Prove that your representation is correct using an inductive argument.
s a classic example of a recursively defined function. It was originallynamed for Leonardo Fibonacci who studied it in the context of predictingthe growth of rabbit populations.
Given three sequences of length m, n, and p each, you are to design and analyze an algorithm to find the longest common subsequence (LCSS) for the three sequence. Is it possible to use dynamic programming to find the LCSS between the three sequences? If yes, provide recursive solution to the above problem. (Hint: We already designed dynamic programming algorithm for two sequences of length m and n in our class.)
A. Construct a RECURSIVE solution for following iterative solution to find a value in a circular-linked list B. Analyze the runtime complexity (correctly explain the scenario, show how much work would be done, and represent the work using asymptotic notations, i.e. big O), of the given iterative solution and your recursive solution for the best case scenario and the worse case scenario. C. prove the correctness of your recursive solution by induction /**@param value - a value to search for@return true if the value is in the list and set the current reference to itotherwise return false and not updating the current reference*/public boolean find(T value){ if(this.cur == null) return false; //get out, nothing is in here Node<T> tmp = this.cur; //start at the current position if(tmp.data == value) return true; // found it at the starting location tmp = tmp.next; //adv. to next node, if there is one while(tmp != cur) { if(tmp.data == value){ this.cur = tmp;…
QUESTION 6 Given the recursive function definition as mult(n,k) = n for k ==1 (base case) = n+mult(n.k-1) for k> 1 (recursive step) implement this recursive function in C For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac). BIUS Paragraph Arial
Consider the following problems for recursive definition/solution. Answer the following questions. [Remember that a recursive definition/solution requires base case and recursive case] We learned that its power set has 2" elements when a set has n elements. Define it in a recursive solution.
The Lucas numbers are a series of numbers where the first two Lucas numbers (i.e., at indices 0 and 1) are 2 and 1 (respectively) and the kth Lucas number L_k (where k>1) is L_(k-1) + L_(k-2). Consider the following recursive definition for a function that is supposed to find the nth element of the Lucas numbers. Select the best option that identifies the line of code that prevents this function from running recursively and provides the correct code. Question options: line 5 should be return recursive_function(n-1) + recursive_function(n-2) line 3 should be return [2,1][n-1][n-2] line 3 should be return [2,1][n-1] line 2 should be if n <= 2: line 5 should be return recursive_function(n-1 + n-2) None of these options
a. Correctness of dynamic programming algorithm: Usually, a dynamic programming algorithm can be seen as a recursion and proof by induction is one of the easiest way to show its correctness. The structure of a proof by strong induction for one variable, say n, contains three parts. First, we define the Proposition P(n) that we want to prove for the variable n. Next, we show that the proposition holds for Base case(s), such as n = 0, 1, . . . etc. Finally, in the Inductive step, we assume that P(n) holds for any value of n strictly smaller than n' , then we prove that P(n') also holds. Use the proof by strong induction properly to show that the algorithm of the Knapsack problem above is correct. b. Bounded Knapsack Problem: Let us consider a similar problem, in which each item i has ci > 0 copies (ci is an integer). Thus, xi is no longer a binary value, but a non-negative integer at most equal to ci , 0 ≤ xi ≤ ci . Modify the dynamic programming algorithm seen at class for this…