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APS105
Recursion
Recursion
• A function is allowed to call itself!– Example:
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• 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
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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:
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Other Examples
• Finding your way out of a maze• Fractals• Solving a sudoku
Recursive Math Functions
• Example: f(3):.
Recursive Math Functions
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Implementing Recursion in C
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Factorial Using a Loop
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Factorial Using Recursion
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Factorial Using Recursion (again)
Printing Patterns with Recursion
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Print a Row of n Stars (recursively)
.Print a Row of n Stars (attempt2)
**********
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Print a Triangle of Stars
**********
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Print an Inverted Triangle of Stars
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What Will This Print?
Recursion and Strings
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
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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
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Determining Powers Efficiently
Determining Powers
• Computing powers the easy (but slow) way:X5 = X*X*X*X*X
X20 = X*X*X*X*X*X*X*X*X*X....
• The more efficient way:.
Formula for Determining Powers
• .
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Recursively Determining Power
Ackermann’s Function and
Maze Routing
Ackermann’s Function
• .
Maze Routing
• Basic idea (see Ch8)
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Towers of Hanoi
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
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Towers of Hanoi: 1st Inner Problem
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Towers of Hanoi: Ex. base problem
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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 2n-1– for n=3: 7 moves– for n=20: 1,048,575 moves– for n=64: 1.8*1019 moves
• == 585billion years at one move per second• note: 14billion years since big bang
• The algorithm is “exponential”– roughly Xn moves where n is size of problem– exponential algorithms are impractical!
