Question

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

Hence, the value of the given recurrence relation \(T(625)\) is 47001.

Step-by-step solution

Define base case

The base case has been defined as \(T(1) = 1\). This will be our starting point.

Solve for T(5)

Since \(T(n)=7 T\left(\frac{n}{5}\right)+10 n\), we can write \(T(5) = 7T(1) + 10*5 = 7*1 + 50 = 57\).

Solve for T(25)

Now we start to see the pattern. Using the same strategy, we can find \(T(25) = 7T(5) + 10 * 25 = 7*57 + 250 = 649\).

Solve for T(125)

Continuing the pattern, we find \(T(125) = 7T(25) + 10 * 125 = 7*649 + 1250 = 5793\).

Solve for T(625)

Finally, to find \(T(625)\), we calculate \(T(625) = 7T(125) + 10 * 625 = 7*5793 + 6250 = 47001\).

Key concepts

Algorithm Analysis

In the quest to understand how algorithms perform, algorithm analysis stands as a crucial tool. It involves evaluating the efficiency and scalability of an algorithm in terms of time and space. Consider the textbook exercise where we explore the recurrence relation \(T(n)=7T\left(\frac{n}{5}\right)+10n\) with a base case of \(T(1)=1\). When solving the recurrence, we're participating in a form of algorithm analysis by breaking down how the function behaves as it recursively processes inputs.

To expand our analysis, we delve into the individual components of the given relation. The term \(7T\left(\frac{n}{5}\right)\) reveals that the algorithm recursively calls itself seven times on a subproblem of size \(\frac{n}{5}\), while the term \(10n\) indicates a linear operation that's proportional to the input size. Thus, algorithm analysis here enables us to decipher the inner workings and efficiency of the recursive process—a knowledge cornerstone for developing and optimizing algorithms.

Time Complexity

Time complexity is a theoretical measure of the time an algorithm takes to run as a function of the length of the string representing the input. By applying this concept to our recurrence relation, we attempt to classify the algorithm's efficiency as the input size grows. The operation \(10n\) is linear, which might tempt us to think the overall complexity is linear, yet that's not the case due to the recurrence part.

To characterize the time complexity of our relation, we examine how the function expands. The recursive term shows that as the input becomes larger, the number of operations grows, not just linearly, but compounded by the recursive calls—a quintessential mark of non-linear growth. With each recursive step, the input is divided, and multiple instances of smaller subproblems are solved. This is indicative of a complexity that is more than linear, typically found in divide and conquer algorithms like the one in our exercise. Understanding time complexity is pivotal as it predicts running time and helps coders write more efficient programs.

Divide and Conquer

The divide and conquer strategy is elegantly exemplified in the way the given recurrence relation is broken down. This approach involves dividing a large problem into smaller subproblems, conquering each subproblem recursively, and combining the results to solve the original problem. The recursive term in our relation, \(7T\left(\frac{n}{5}\right)\), reflects the 'divide' portion, where the original problem of size 'n' is split into subproblems of size \(\frac{n}{5}\).

By observing the solution steps, we notice the process of repeated subdivision eventually leads to a base case, where the problem cannot be subdivided further, signaling the 'conquer' phase. Once the base case is reached, the algorithm combines the solved subproblems progressively until it resolves the initial large problem. The divide and conquer method is a powerful algorithmic paradigm responsible for the efficiency of many algorithms, and it is crucial for students to understand its underlying mechanics to appreciate its broad applicability in computer science.