Design and Analysis of Algorithms Chapter 3: Recursion

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Questions and Answers

What is the definition of a recursive function?

A function that uses a loop to solve a problem.

A function that calls another function.

A function that solves only the simplest case.

A function that calls itself, either directly or indirectly. (correct)

What is the purpose of the base case in a recursive function?

To terminate the recursive function.

To simplify the problem until reaching the base case. (correct)

To make recursive calls to the function.

To solve the entire problem at once.

What is the recursion step in a recursive function?

The function returns a result to the main function.

The function simplifies the problem until reaching the base case.

The function calls itself to work on the smaller problem. (correct)

The function terminates when reaching the base case.

What is the outcome when a recursive function reaches the base case?

The function returns a result to the main function. Signup and view all the answers

What is the algebraic relationship used to define the factorial function recursively?

𝑛! = 𝑛 × (𝑛 - 1)! Signup and view all the answers

What is the recursive definition of the factorial function?

𝑓𝑎𝑐𝑡(𝑛) = 𝑛 × 𝑓𝑎𝑐𝑡(𝑛 - 1) Signup and view all the answers

What is the process of finding the factorial of a number using a recursive function?

Calling the function with a smaller input until reaching the base case. Signup and view all the answers

What happens when a recursive function terminates?

The function returns a result to the main function. Signup and view all the answers

What is the main advantage of using recursion in problem-solving?

It provides a natural way to solve problems with a recursive structure Signup and view all the answers

What happens when a recursive function reaches the base case?

The function returns a result and terminates Signup and view all the answers

What is the role of the base case in a recursive function?

To provide a trivial solution to the problem Signup and view all the answers

Why is recursion useful for solving the Towers of Hanoi puzzle?

Because it allows for a divide-and-conquer approach Signup and view all the answers

What is the outcome of the recursive calls in a recursive function?

A sequence of progressively smaller problems Signup and view all the answers

What is the purpose of the recursive call in a recursive function?

To divide the problem into smaller subproblems Signup and view all the answers

What is the result of the factorial function when n is 1?

1 Signup and view all the answers

What is the relationship between the recursive function and the problem it solves?

The function is a recursive structure that mirrors the problem Signup and view all the answers

Study Notes

Recursion

Recursion is a phenomenon where a function invokes itself, either directly or indirectly, to solve a problem.
A recursive function knows how to solve only the simplest case(s), also known as base case(s).
In a recursive function, the function calls a copy of itself to work on a smaller problem, referred to as a recursive call or the recursion step.

Characteristics of Recursion

The recursion step leads to multiple recursive calls, dividing the problem into smaller subproblems.
Recursion terminates when each call simplifies the problem until reaching the base case, forming a sequence of progressively smaller problems.
Upon reaching the base case, the function returns a result, propagating back up the call stack until the final result is returned to the main function.

Example: Recursive Factorial

A recursive definition of the factorial function is based on the algebraic relationship: 𝑛!= 𝑛 · (𝑛 – 1)!
The recursive function 𝑓𝑎𝑐𝑡(𝑛) is defined as: 𝑓𝑎𝑐𝑡(𝑛) = 𝑛 × 𝑓𝑎𝑐𝑡(𝑛 − 1), and 𝑓𝑎𝑐𝑡(1) = 1.
To find the factorial of a number, the function calls itself with a smaller input until it reaches the base case, where the result is returned and propagated back up the call stack.

Recursion

Recursion is a phenomenon where a function invokes itself, either directly or indirectly, to solve a problem.
A recursive function knows how to solve only the simplest case(s), also known as base case(s).
In a recursive function, the function calls a copy of itself to work on a smaller problem, referred to as a recursive call or the recursion step.

Characteristics of Recursion

The recursion step leads to multiple recursive calls, dividing the problem into smaller subproblems.
Recursion terminates when each call simplifies the problem until reaching the base case, forming a sequence of progressively smaller problems.
Upon reaching the base case, the function returns a result, propagating back up the call stack until the final result is returned to the main function.

Example: Recursive Factorial

A recursive definition of the factorial function is based on the algebraic relationship: 𝑛!= 𝑛 · (𝑛 – 1)!
The recursive function 𝑓𝑎𝑐𝑡(𝑛) is defined as: 𝑓𝑎𝑐𝑡(𝑛) = 𝑛 × 𝑓𝑎𝑐𝑡(𝑛 − 1), and 𝑓𝑎𝑐𝑡(1) = 1.
To find the factorial of a number, the function calls itself with a smaller input until it reaches the base case, where the result is returned and propagated back up the call stack.

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Description

This quiz covers the basics of recursion, including its importance, examples, and methods of solving recursion equations. It also includes the Towers of Hanoi puzzle and the rabbit problem.