Inductive Definition to Recursive Function

Transcription

1 PDS: CS Computer Sc & Egg: IIT Kharagpur 1 Iductive Defiitio to Recursive Fuctio

2 PDS: CS Computer Sc & Egg: IIT Kharagpur 2 Factorial Fuctio Cosider the followig recursive defiitio of the factorial fuctio.! = 1, if = 0, ( 1)!, if > 0. The fuctio is used to defie itself. The defiitio is a equatio with a computatioal couterpart.

3 PDS: CS Computer Sc & Egg: IIT Kharagpur 3 The Equatio The factorial fuctio satisfies the fuctioal equatio a. F() = 1, if = 0, F( 1), if > 0. a The factorial is the fixed-poit of this equatio

4 PDS: CS Computer Sc & Egg: IIT Kharagpur 4 Computatio of 4! 4! = 4 3! = 4 (3 2!) = 4 (3 (2 1!)) = 4 (3 (2 (1 0!))) = 4 (3 (2 (1 1))) = 4 (3 (2 1)) = 4 (3 2) = 4 6 = 24

5 PDS: CS Computer Sc & Egg: IIT Kharagpur 5 Note There is o value computatio i the first four steps. The fuctio is beig ufolded. The value computatio starts oly after the basis of the defiitio is reached. Last four steps computes the values.

6 PDS: CS Computer Sc & Egg: IIT Kharagpur 6 A fuctio i C Laguage may call itself A fuctio that calls itself directly or idirectly is called a recursive fuctio. Ufoldig ad delayed computatio ca be simulated by such a fuctio.

7 PDS: CS Computer Sc & Egg: IIT Kharagpur 7 Recursive Call If a fuctio calls itself, the obvious questio is about the termiatio of the process. A() call A A() call A A() call A A()

8 PDS: CS Computer Sc & Egg: IIT Kharagpur 8 Recursive Call Naturally the call caot be ucoditioal. The basis of a iductive (recursive) defiitio provides the coditio for termiatio. The fuctio calls itself to reach the termiatio coditio, the basis.

9 PDS: CS Computer Sc & Egg: IIT Kharagpur 9 Useless for Computatio The factorial fuctio also satisfies the followig equatio, F() = 1, if = 0, F(+1) +1, if > 0. but caot be used for computatio as a call to itself does ot reach the basis.

10 PDS: CS Computer Sc & Egg: IIT Kharagpur 10 Recursive factorial Fuctio it factorial(it ) { if ( == 0) retur 1 ; else retur *factorial( - 1) ; } // factorialfr1.c The fuctio takes a actual parameter p (a o-egative iteger) ad returs the value of p!.

11 PDS: CS Computer Sc & Egg: IIT Kharagpur 11 Differet Calls ad Icaratios of Actual Parameter is 4

12 PDS: CS Computer Sc & Egg: IIT Kharagpur d Call : 3 1 3rd Call : th Call :!= 0 * th Call :!= 0 * 0 Retur 1 First Call : 4 1 3!= 0 * Retur 2 4!= 0 Retur * 6 Retur 24 1 Retur 1

13 PDS: CS Computer Sc & Egg: IIT Kharagpur 13 Same Code Differet Data Same code is used for every recursive call. Data (local) is differet i every recursive call.

14 PDS: CS Computer Sc & Egg: IIT Kharagpur 14 Code for factorial computatio mai() stack frame factorial() factorial() factorial() factorial() factorial() stack frams High address p retur address retur address retur address retur address retur address Text/Code Regio of Memory Low address Stack Regio of Memory

15 PDS: CS Computer Sc & Egg: IIT Kharagpur 15 Note The first phase does ot compute the value but ufolds the recursio upto the base case. The value computatio starts from the base case.

16 PDS: CS Computer Sc & Egg: IIT Kharagpur 16 Note For every call there is ew icaratio of all the formal parameters ad the local variables (that are ot static). The variable ames get bid to differet memory locatios. Variables of oe ivocatio are ot visible from aother ivocatio.

17 PDS: CS Computer Sc & Egg: IIT Kharagpur 17 Note Oce a retur statemet is executed, all the variables of the correspodig ivocatio die. The last icaratio of a variable ame dies first - last i first out (LIFO). The last call is retured first.

18 PDS: CS Computer Sc & Egg: IIT Kharagpur 18 Stack Frame or Activatio Record mai() stack frame factorial(4) stack frame 4 x retur address mai() stack frame factorial(4) factorial(3) 4 3 x retur address retur address Stack Regio Stack Regio

19 PDS: CS Computer Sc & Egg: IIT Kharagpur 19 mai() stack frame factorial(4) factorial(3) factorial(2) factorial(1) factorial(0) retur addr 4 x retur address retur address retur address retur address Retur 1 mai() stack frame factorial(4) factorial(3) factorial(2) factorial(1) 4 x retur address retur address retur address retur address Stack Regio Stack Regio

20 PDS: CS Computer Sc & Egg: IIT Kharagpur 20 mai() stack frame factorial(4) factorial(3) factorial(2) factorial(1) retur addr 4 x retur address retur address retur address mai() stack frame factorial(4) factorial(3) factorial(2) Retur 1 4 x 3 2 retur address retur address retur address Stack Regio Stack Regio

21 PDS: CS Computer Sc & Egg: IIT Kharagpur 21 mai() stack frame x mai() stack frame x factorial(4) retur addr 4 Retur 24 Stack Regio Stack Regio

22 PDS: CS Computer Sc & Egg: IIT Kharagpur 22 Note The recursive factorial fuctio uses more memory tha its o-recursive couter part. The o-recursive fuctio uses fixed amout of memory for a it data, whereas the memory usage by the recursive fuctio is proportioal to the value of data. Moreover a fuctio call ad retur takes some amout of extra time.

23 PDS: CS Computer Sc & Egg: IIT Kharagpur 23 Recursive Fuctio with Iterative Dyamics We have see that the value computatio i our factorial fuctio starts after ufoldig the recursio. But this dyamics of computatio i a recursive fuctio ca be made differet. The fuctio may start the computatio from the begiig.

24 PDS: CS Computer Sc & Egg: IIT Kharagpur 24 it factiter(it, it acc) { if ( == 0) retur acc ; else retur factiter(-1, *acc) ; } // factorialfr2.c This fuctio is called as factiter(, 1) to calculate the value of!. The secod parameter is the value of the basis.

25 PDS: CS Computer Sc & Egg: IIT Kharagpur 25 Computatio of factiter(4,1)

26 PDS: CS Computer Sc & Egg: IIT Kharagpur 26 4th Call : 5th Call : First Call : 4 1 4*1 2d Call : acc 3 4!= *4 3rd Call : acc 2 1 2* acc!= != 0 Retur *24 acc 0 24 Retur acc 4 1 Retur 24!= 0 Retur 24 Retur 24

27 PDS: CS Computer Sc & Egg: IIT Kharagpur 27 mai() stack frame factiter() stack frams factiter() factiter() factiter() High address p acc retur address acc retur address 12 2 acc retur address 1 24 acc retur address Low address Stack Regio

28 PDS: CS Computer Sc & Egg: IIT Kharagpur 28 Note The computatio of! i this recursive fuctio is very similar to the computatio i a for-loop. it factiter(it ) { it acc = 1, i ; } for(i = ; i > 0 ; --i) acc *= i ; retur acc ;

29 PDS: CS Computer Sc & Egg: IIT Kharagpur 29 Iductive Defiitio: gcd(m, ) gcd(m,) = if m = 0, gcd( mod m,m) if m > 0.

30 PDS: CS Computer Sc & Egg: IIT Kharagpur 30 Recursive Fuctio gcd(m,) it gcd(it s, it l){ if(s == 0) retur l; retur gcd(l%s, s); } // gcdfr.c

31 PDS: CS Computer Sc & Egg: IIT Kharagpur 31 Differet Calls to gcd() gcd(0,0) retur 0 gcd(0,5) retur 5 gcd(5,0) gcd(0,5) retur 5 gcd(18, 12) gcd(12, 18) gcd(6,12) gcd(0,6) retur 6