recursion - umd
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Recursion
CMSC132,Summer2016Object-OrientedProgrammingII
AnwarMamat
Recursion
• Whenonefunctioncallsitselfdirectlyorindirectly
• amethodwherethesolutiontoaproblemdependsonsolutionstosmallerinstancesofthesameproblem.
Factorial
5!=5*4!
4=4*3!
3!=3*2!
2!=2*1!
1!=1
2*1
3 *2
4 *65 *24=120
5!=?
Definition:
GreatestCommonDivisor
• gcd:FindlargestintegerdthatevenlydividesintopandqEuclid'salgorithm.[300BCE]
GreatestCommonDivisor
• gcd:FindlargestintegerdthatevenlydividesintopandqEuclid'salgorithm.[300BCE]
TowersofHanoi
TowersofHanoi• Moveallthediscsfromtheleftmostpegtotherightmostone.– Onlyonediscmaybemovedatatime.– Adisccanbeplacedeitheronemptypegorontopofalargerdisc
TowersofHanoiLegend
• Q.Isworldgoingtoend(accordingtolegend)?– 64goldendiscson3diamondpegs.–Worldendswhencertaingroupofmonksaccomplishtask.
• Q.Willcomputeralgorithmshelp
Let’smovedisks
A B C
?
Moveonedisk
A B C
Movetwodisks
A B C
Movethreedisks
A B C
Movefourdisks
A B C
Movendisks
1.Moven-1disksfromA->B
2.MovethelargestdisktoC
3.Moven-1disksfromB->C
A B C
TowersofHanoi:RecursiveSolutionpublic class TowersOfHanoi {
public static void solve(int n, String A, String B, String C) {if (n == 1) {
System.out.println(A + " -> " + C);} else {
solve(n - 1, A, C, B);System.out.println(A + " -> " + C);solve(n - 1, B, A, C);
}}public static void main(String[] args) {
int discs = 3;solve(discs, "A", "B", "C");
}}
A B C
Remarkablepropertiesofrecursivesolution
• Takes2n stepstosolvendiscproblem.• Takes585billionyearsforn=64(atrateof1discpersecond).
• Reassuringfact:anysolutiontakesatleastthislong
FibonacciNumberFibonaccinumbers:0,1,1,2,3,5,8,13,21,34
FibonacciNumberFibonaccinumbers:0,1,1,2,3,5,8,13,21,34
InefficientRecursionFibonaccinumbers:0,1,1,2,3,5,8,13,21,34
Memoized VersionoftheFibonaccipublic Map<Integer, Integer> fibo;
public int fib(int n){
int f1=0,f2=0;
if( (n == 1)|| (n == 2)) return 1;
else{
if(fibo.containsKey(n-1)){
f1 = fibo.get(n-1);
}else{
f1 = fib(n-1);
fibo.put(n-1,f1);
}
if(fibo.containsKey(n-2)){
f2 = fibo.get(n-2);
}else{
f2 = fib(n-2);
fibo.put(n-2,f2);
}
return f2+f1;
}
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