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Chapter 2: Problem 14

Given the recurrence relation \\[\begin{array}{l} T(n)=7 T\left(\frac{n}{5}\right)+10 n \quad \text { for } n>1 \\\T(1)=1\end{array}\\] find \(T(625)\)

Short Answer

Expert verified
The solution to the recurrence relation \(T(625) = T(7)T\left(\frac{n}{5}\right) + 10 n\) given \(T(1) = 1\) is \(T(625) = 65151\).

Step by step solution

01

Start with the given \(n\)

We are asked to find \(T(625)\). To calculate this, we will first substitute \(n=625\) into the recurrence relation.
02

Apply Recurrence Relation

Substitute \(n=625\) into the recurrence relation to get \(T(625) = 7T(125) + 6250\). Now, we need to calculate \(T(125)\). Repeat the same process i.e., substitute \(n=125\) into the recurrence relation to get \(T(125) = 7T(25) + 1250\). Continue this iterative process until we reach \(T(1)\).
03

Final Calculation

Here we will repeatedly substitute \(n\) in the relation till we reach \(T(1)\): \(T(625) = 7T(125) + 6250 = 7[7T(25) + 1250] + 6250 = 49T(25) + 13750 = 49[7T(5) + 250] + 13750 = 343T(5) + 24350 = 343[7T(1) + 50] + 24350 = 2401T(1) + 41150 = 2401*1 + 41150 = 65151\).

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Most popular questions from this chapter

Suppose that there are \(n=2^{k}\) teams in an elimination tournament, in which there are \(n / 2\) games in the first round, with the \(n / 2=2^{k-1}\) winners playing in the second round, and so on. a. Develop a recurrence equation for the number of rounds in the tournament. b. (b) How many rounds are there in the tournament when there are 64 teams? c. Solve the recurrence equation of part (a).

Modify Algorithm 2.9 (Large Integer Multiplication) so that it divides each \(n\) digit integer into a. three smaller integers of \(n / 3\) digits (you may assume that \(n=3^{k}\) ). b. four smaller integers of \(n / 4\) digits (you may assume that \(n=4^{k}\) ). Analyze your algorithms, and show their time complexities in order notation.

Write algorithms that perform the operations \(u \times 10^{m} ; u\) divide \(10^{m} ; u\) rem \(10^{m}\) where \(u\) represents a large integer, \(m\) is a nonnegative integer, divide returns the quotient in integer division, and rem returns the remainder. Analyze your algorithms, and show that these operations can be done in linear time.

Use the divide-and-conquer approach to write an algorithm that finds the largest item in a list of \(n\) items. Analyze your algorithm, and show the results in order notation.

Verify the following identity \(\sum_{p=1}^{n}[A(p-1)+A(n-p)]=2 \sum_{p=1}^{n} A(p-1)\) This result is used in the discussion of the average-case time complexity analysis of Algorithm 2.6 (Quicksort).
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