APS105 Recursion

A function is allowed to call itself! –Example:. A function calling itself is called “recursion” Such a function is called a “recursive” function –Why would you want to do this?

Recursion: Motivation Sometimes a prob has self-similar sub-probs –E.g., problems-within-a-problem –“inner” problems have same description as outer but inner problems are smaller in some way –inner problems themselves have inner problems Recursion continues until: –some small indivisible problem reached –or some other stopping condition is reached –called the “base case” or “terminating case” Note: –sometimes recursion can be expressed as a loop –or vice-versa

Recursion: Walking Across the Room Loop-based solution basic idea: –while not at the wall take another step Recursive solution basic idea:.

Recursion: Cake Cutting Cut a jelly-roll into equal parts < 100g each.

Reading/Evaluating Expressions Recall: BEDMAS: order of evaluation –Brackets –Exponents –Division & Multiplication –Addition & Subtraction Apply BEDMAS to every expression –and to every sub-expression within the expression

Example: Evaluating Expressions BEDMAS Example:.

Other Examples Finding your way out of a maze Fractals Solving a sudoku

Recursive Math Functions

Example: f(3):. Recursive Math Functions

. Implementing Recursion in C

. Factorial Using a Loop

. Factorial Using Recursion

. Factorial Using Recursion (again)

Printing Patterns with Recursion

. Print a Row of n Stars (recursively)

. Print a Row of n Stars (attempt2)

**** *** ** *. Print a Triangle of Stars

* ** *** ****. Print an Inverted Triangle of Stars

. What Will This Print?

Recursion and Strings

Can think of strings using a recursive definition –Example: one of these recursive definitions: –a string is a character followed by a string –a string is a character preceded by a string –a string is two characters with a string in the middle

Palindromes Palindrome: –a string that is the same backwards and forwards –Examples: racecar, level, civic, madam, noon Write a function to determine if a palindrome

. Function to test Palindrome

. Palindrome tester with a Helper

Greatest Common Divisor (GCD)

GCD Algorithm GCD of two numbers is –if the numbers are the same: the GCD is either number –if the numbers are different: the GCD of the smaller of the two and the difference between the two Example: GCD(20,8) = GCD(8,12) = GCD(8,4) = GCD(4,4) The GCD is 4

Formalized GCD Alg., and Code..

Determining Powers Efficiently

Determining Powers Computing powers the easy (but slow) way: X 5 = X*X*X*X*X X 20 = X*X*X*X*X*X*X*X*X*X.... The more efficient way:.

Formula for Determining Powers.

. Recursively Determining Power

Ackermann’s Function and Maze Routing

Ackermann’s Function.

Maze Routing Basic idea (see Ch8).

Towers of Hanoi

Move all disks to another rod, but: –Only one disk may be moved at a time. –No disk may be placed on a smaller disk. Initial:Goal:

Towers of Hanoi: Outer Problem =

Towers of Hanoi: 1 st Inner Problem =

Towers of Hanoi: Ex. base problem =

Towers of Hanoi Write a program that prints a solution –assume rods 1,2,3 –assume some number of discs given by height

. Towers of Hanoi

Towers Algorithm Legend: –Monks found 64 disk tower-game –Universe will end when they finish the game number of moves is 2 n -1 –for n=3: 7 moves –for n=20: 1,048,575 moves –for n=64: 1.8*10 19 moves == 585billion years at one move per second note: 14billion years since big bang The algorithm is “exponential” –roughly X n moves where n is size of problem –exponential algorithms are impractical!