Consider the recurrence T(n) = n²/3 . T(n²/3) +n with base case T (n) = 1 for n<2. Find an asymptotic solution using the method of recursion trees, presenting your answer as follows: (a) How deep is the recursion tree for T(n)? (b) What is the size of the subproblems at level d? (considering the root node to be at depth d = 0) (c) How many nodes does the tree have at level d? (Hint: you may find the geometric series formula helpful.) (d) What is the total “non-recursive work" (i.e., the sum of the non-recursive terms) at level d? (e) What is the asymptotic growth rate of T(n)? Give a precise answer using the O notation.

Consider the recurrence T(n) = n²/3 . T(n²/3) +n with base case T (n) = 1 for n<2. Find an asymptotic solution using the method of recursion trees, presenting your answer as follows: (a) How deep is the recursion tree for T(n)? (b) What is the size of the subproblems at level d? (considering the root node to be at depth d = 0) (c) How many nodes does the tree have at level d? (Hint: you may find the geometric series formula helpful.) (d) What is the total “non-recursive work" (i.e., the sum of the non-recursive terms) at level d? (e) What is the asymptotic growth rate of T(n)? Give a precise answer using the O notation.

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter18: Deterministic Dynamic Programming

Section18.4: Resource-allocation Problems

Problem 3P

See similar textbooks

Similar questions

Provide an example of a recursive function in which the amount of work on each activation is constant. Provide the recurrence equation and the initial condition that counts the number of operations executed. Specify which operations you are counting and why they are the critical ones to count to assess its execution time. Draw the recursion tree for that function and determine the Big-Θ by determining a formula that counts the number of nodes in the tree as a function of n.
Please explain!! Solve the recurrence: T(n)=2T(2/3 n)+n^2. first by directly adding up the work done in each iteration and then using the Master theorem. Note that this question has two parts (a) Solving IN RECURSION TREE the problem by adding up all the work done (step by step) and
For each of the following recurrences, verify the answer you get by applying the master method, by solving the recurrence algebraically OR applying the recursion tree method. T(N) = 4T(N/2) + n2logn T(N) = 5T(N/2) + n2/logn
The rooted Fibonacci trees Tn are defined recursively in the following way. T1 and T2 are both the rooted trees consisting of a single vertex, and for n = 3, 4, …, the rooted tree Tn is constructed from a root with Tn-1 as its left subtree and Tn-2 as its right subtree. 1. How many vertices, leaves, and internal vertices does the rooted Fibonacci tree Tn have, where n is a positive integer? 2. What is its height?
The function f is defined for non-negative integers a and b recursively as follows:f(a, b) ={0 if a = 0 or b = 0f(a − 1, b − 1) + 2a − 1 if a = bf(a − b, b) + f(b, b) if a > bf(a, a) + f(b − a, a) if a < b}Compute f (3, 2) by drawing a recursion tree showing all of the computationrequired and then use your tree to compute the answer.
For each of the following recurrences, verify the answer you get by applying the master method, by solving the recurrence algebraically OR applying the recursion tree method. T(N) = 2T(N-1) + 1 T(N) = 3T(N-1) + n
Provide an example of a recursive function in which the amount of work on each activation is constant. Provide the recurrence equation and the initial condition that counts the number of operations executed. Specify which operations you are counting and why they are the critical ones to count to assess its execution time. Draw the recursion tree for that function and determine the Big-© by determining a formula that counts the number of nodes in the tree as a function of n.-Use course material of CMSC 451 and additional sources. Need reference also . Note dont copy from any other sites including chat gpt also . Previous ans here was also copy from chat gpt .
Provide an example of a recursive function in which the amount of work on each activation is constant. Provide the recurrence equation and the initial condition that counts the number of operations executed. Specify which operations you are counting and why they are the critical ones to count to assess its execution time. Draw the recursion tree for that function and determine the Big-© by determining a formula that counts the number of nodes in the tree as a function of n.-Use course material of CMSC 451 and additional sources. Need reference also . Note dont copy from any other sites
Use a recursion tree to determine a good asymptotic upper bound on following recurrences. Please see Appendix of your text book for using harmonic and geometric series. a) T (n) = T(n/5) + O(n)2 b) T (n) = 10T(n/2) + O(n)2 c) T (n) = 10T(n/2) + Θ (1) d) T (n) = 2T (n/2) + n/ lg n e) T (n) = 2T (n - 1) + Θ (1)
Consider a divide-and-conquer algorithm that calculates the sum of all elements in a set of n numbers by dividing the set into two sets of n/2 numbers each, finding the sum of each of the two subsets recursively, and then adding the result. What is the recurrence relation for the number of operations required for this algorithm? Answer is f(n) = 2 f(n/2) + 1. Please show why this is the case.
Give the uppor-bound for the recurrence T(n)=2π(n/2)+n∧2, using the Recursion Tree method. You must show at least 3 levels of the tree, and give the explicit log base when using the tree height!
Draw a recursion tree for a recurrence and use the Substitution Method to prove the solution. ( make a sample question )