C OMP 110/401 R ECURSION Instructor: Prasun Dewan P REREQUISITES - - PowerPoint PPT Presentation

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C OMP 110/401 R ECURSION Instructor: Prasun Dewan P REREQUISITES - - PowerPoint PPT Presentation

Nov 06, 2023 416 likes •757 views

C OMP 110/401 R ECURSION Instructor: Prasun Dewan P REREQUISITES Conditionals 2 R ECURSION English definition Return (Oxford/Webster) procedure repeating itself indefinitely or until condition met, such as grammar rule (Webster)

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COMP 110/401 RECURSION

Instructor: Prasun Dewan

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PREREQUISITES

 Conditionals

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RECURSION

 English definition

 Return (Oxford/Webster)  procedure repeating itself indefinitely or until condition

met, such as grammar rule (Webster)

 English examples

 adequate: satisfactory  satisfactory: adequate

 My definition: recursion: recursion  Mathematics

 expression giving successive terms of a series (Oxford)

 Programming

 Method calling itself.  On a smaller problem.  Alternative to loops.

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FACTORIAL(N)

public static int factorial(int n) { int product = 1; while (n > 0) { product *= n; n -= 1; } return product; } public static void main (String[] args) { while (true) { // loop condition never false int n = Console.readInt(); if (n < 0) break; System.out.println("factorial = " + factorial(n)); } } 1*2*3*…*n

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DEFINING FACTORIAL(N)

Product of the first n numbers 1*2*3*…*n factorial(0) = 1 factorial(1) = 1 = 1*factorial(0) factorial(2) = 2*1 = 2*factorial(1) factorial(3) = 3*2*1 = 3*factorial(2) factorial(4) = 4*3*2*1 = 4*factorial(3) factorial(n) = n*n-1*…*1 = n*factorial(n-1)

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DEFINING FACTORIAL(N)

factorial(n) = 1 if n == 0 factorial(n) = n*factorial(n-1) if n > 0

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IMPLEMENTING FACTORIAL(N) (EDIT)

factorial(n) = 1 if n == 0 factorial(n) = n*factorial(n-1) if n > 0 public static int factorial(int n) { … }

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IMPLEMENTING FACTORIAL(N)

factorial(n) = 1 if n == 0 factorial(n) = n*factorial(n-1) if n > 0 public static int factorial (int n) { if (n == 0) return 1; if (n > 0) return n*factorial(n-1); } n < 0 ? Function must return something for all cases Multiple return Early return

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IMPLEMENTING FACTORIAL(N)

factorial(n) = 1 if n == 0 factorial(n) = n*factorial(n-1) if n > 0 public static int factorial(int n) { if (n == 0) return 1; else if (n < 0) return factorial(-n); else return n*factorial(n-1); } Base (terminating) case Recursive Reduction Steps factorial(n) = factorial(-n) if n < 0

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RECURSIVE METHODS

 Should have base case(s)  Recurse on smaller problem(s)

 recursive calls should converge to base case(s)

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GENERAL FORM OF A RECURSIVE METHOD

if (base case 1 ) return solution for base case 1 else if (base case 2) return solution for base case 2 …. else if (base case n) return solution for base case n else if (recursive case 1) do some preprocessing recurse on reduced problem do some postprocessing … else if (recursive case m) do some preprocessing recurse on reduced problem do some postprocessing

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RECURSION VS. LOOPS (ITERATION)

public static int factorial(int n) { int product = 1; while (n > 0) { product *= n; n -= 1; } return product; } public static int factorial(int n) { if (n == 0) return 1; else if (n < 0) return factorial(-n); else return n*factorial(n-1); } Implementation follows from definition

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TRACING RECURSIVE CALLS

Call Stack Locals

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TRACING RECURSIVE CALLS

Invocation n return value factorial(1) 1 ? factorial(2) 2 ? Invocation n return value factorial(0) ? factorial(1) 1 ? factorial(2) 2 ?

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TRACING RECURSIVE CALLS

Call Stack Locals

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TRACING RECURSIVE CALLS

Invocation n return value factorial(0) 1 factorial(1) 1 ? factorial(2) 2 ? Invocation n return value factorial(0) 1 factorial(1) 1 1 factorial(2) 2 ? Invocation n return value factorial(0) 1 factorial(1) 1 1 factorial(2) 2 2

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RECURSION PITFALLS

public static int factorial (int n) { return n*factorial(n-1); } factorial(2) 2*factorial(1) 1*factorial(0) 0*factorial(-1)

  • 1*factorial(-2)

… Infinite recursion! (stack overflow) No base case

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RECURSION PITFALLS

public static int factorial (int n) { if (n == 0) return 1; else if (n < 0) return factorial(-n); else return factorial(n+1)/n; } factorial(2) factorial(3)/3 factorial(4)/4 factorial(5)/5 factorial(6)/6 … Infinite recursion! Recurses on bigger problem

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RECURSIVE FUNCTIONS WITH MULTIPLE PARAMETERS

power(base, exponent) baseexponent base*base*base*…*base Exponent # of times power(0, exponent) = 0 power(1, exponent) = 1 power(2, exponent) = 2*2*…*2 (exponent times) power(3, exponent) = 3*3*…*3 (exponent times) No pattern!

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RECURSIVE FUNCTIONS WITH MULTIPLE PARAMETERS (EDIT)

power(base, exponent) baseexponent base*base*base*…*base Exponent # of times

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RECURSIVE FUNCTIONS WITH MULTIPLE PARAMETERS (EDIT)

power(base, exponent) baseexponent base*base*base*…*base Exponent # of times power(base, 0) = 1 power(base, 1) = base*1 = base*power(base, 0) power(base, 2) = base*base*1 = base*power(base, 1) power(base, exponent) = base*power(base, exponent-1)

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DEFINING POWER(BASE, EXPONENT)

power(base, exponent) = 1 if exponent <= 0 if exponent > 0 power(base, exponent) = base*power(base, exponent-1) public static int power(int base, int exponent) { if (n <= 0) return 1; else return base*power(base, exponent-1); }

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RECURSIVE PROCEDURES: GREET(N)

greet(0) greet(1) print “hello” greet(2) print “hello” print “hello” greet(n) print “hello” print “hello” print “hello” n times

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DEFINING GREET(N) (EDIT)

greet(0) greet(1) print “hello” greet(2) print “hello” print “hello” greet(n) print “hello” print “hello” print “hello” n times

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DEFINING GREET(N)

greet(0) greet(1) print “hello” greet(2) print “hello” print “hello” greet(n) print “hello” print “hello” print “hello” n times do nothing; greet(0); print “hello”; greet(1); print “hello”; greet(n-1); print “hello”;

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DEFINING GREET(N)

greet(n) do nothing; if n <= 0 greet(n) greet(n-1); print “hello”; if n > 0

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IMPLEMENTING GREET(N) (EDIT)

greet(n) do nothing; if n <= 0 greet(n) greet(n-1); print “hello”; if n > 0

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IMPLEMENTING GREET(N)

greet(n) do nothing; if n <= 0 greet(n) greet(n-1); print “hello”; if n > 0 public static void greet (int n) { if (n > 0) { greet(n-1); System.out.println(“hello”); } }

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LIST-BASED RECURSION: MULTIPLYLIST()

multiplyList() = 1 if remaining input is: -1 multiplyList() = 2 if remaining input is: 2, -1 multiplyList() = 12 if remaining input is: 2, 6, -1 multiplyList() = readNextVal() * multiplyList() if nextVal > 0 multiplyList() = readNextVal() if nextVal < 0

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LIST-BASED RECURSION: MULTIPLYLIST()

multiplyList() = 1 if nextVal < 0 multiplyList() = readNextVal() * multiplyList() if nextVal > 0 public static int multiplyList () { int nextVal = Console.readInt(); if (nextVal < 0) return 1; else return nextVal*multiplyList(); }

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TRACING MULTIPLYLIST()

public static int multiplyList () { int nextVal = Console.readInt(); if (nextVal < 0) return 1; else return nextVal*multiplyList(); } Invocation Remaining input Return value multiplyList() 2, 30, -1 ? Invocation Remaining input Return value multiplyList() 30, -1 ? multiplyList() 2, 30, -1 ? Invocation Remaining input Return value multiplyList()

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? multiplyList() 30, -1 ? multiplyList() 2, 30, -1 ? Invocation Remaining input Return value multiplyList()

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1 multiplyList() 30, -1 ? multiplyList() 2, 30, -1 ? Invocation Remaining input Return value multiplyList()

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1 multiplyList() 30, -1 30 multiplyList() 2, 30, -1 ? Invocation Remaining input Return value multiplyList()

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1 multiplyList() 30, -1 30 multiplyList() 2, 30, -1 60

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SUMMARY

 Recursive Functions  Recursive Procedures  Number-based Recursion  List-based Recursion  See sections on trees, graphs, DAGs and visitor for other kinds

  • f recursion