1. RecursionCIS 1057 Fall 20131 Recursion CIS 1057 Computer Programming in C Fall 2013 (Many slides based on/borrowed from Professor Hugh C. Lauer. Slides include materials from The C Programming Language, 2 nd edition, by Kernighan and Ritchie and from C: How to Program, 5 th and 6 th editions, by Deitel and Deitel)

2. Recursion and Function Implementation CIS 1057 Fall 20132 Definition Recursive Function:– a function that calls itself Directly or indirectly Each recursive call is made with a new, independent set of arguments Previous calls are suspended Allows very simple programs for very complex problems

3. Recursion and Function Implementation CIS 1057 Fall 20133 Simplest Example int factorial(int x) { if (x <= 1) return 1; else return x * factorial (x-1); }// factorial

4. Recursion and Function Implementation CIS 1057 Fall 20134 Simplest Example (continued) int factorial(int x) { if (x <= 1) return 1; else return x * factorial (x-1); }// factorial This puts the current execution of factorial “on hold” and starts a new one With a new argument!

5. Recursion and Function Implementation CIS 1057 Fall 20135 Simplest Example (continued) int factorial(int x) { if (x <= 1) return 1; else return x * factorial (x-1); }// factorial When factorial(x-1 ) returns, its result it multiplied by x Mathematically:– x! = x  (x-1)!

6. Recursion and Function Implementation CIS 1057 Fall 20136 Recursion (continued) There is an obvious circularity here –factorial calls factorial calls factorial, etc. But each one is called with a smaller value for argument! –Eventually, argument becomes ≤ 1 –Invokes if clause to terminate recursion

7. Recursion and Function Implementation CIS 1057 Fall 20137 Simplest Example int factorial(int x) { if (x <= 1) return 1; else return x * factorial (x-1); }// factorial This if statement “breaks” the recursion

8. Recursion and Function Implementation CIS 1057 Fall 20138 Questions?

9. Recursion and Function Implementation CIS 1057 Fall 20139 More Interesting Example Towers of Hanoi Move stack of disks from one peg to another Move one disk at a time Larger disk may never be on top of smaller disk

10. Recursion and Function Implementation CIS 1057 Fall 201310 Tower of Hanoi Program #include void move (int disks, int a, int c, int b); int main() { int n; printf ("How many disks?"); scanf ("%d", &n); printf ("\n"); move (n, 1, 3, 2); return 0; }// main /* PRE: n >= 0; a, b, and c represent some order of the distinct integers 1, 2, 3 POST: the function displays the individual moves necessary to move n disks from needle a to needle c, using needle b as a temporary storage needle */ void move (int disks, int a, int c, int b) { if (disks > 0) { move (disks-1, a, c, b); printf ("Move one disk from %d to %d\n", a, c); move (disks-1, b, a, c); }// if (disks > 0 return; }//move

11. Recursion and Function Implementation CIS 1057 Fall 201311 Tower of Hanoi Program #include void move (int disks, int a, int c, int b); int main() { int n; printf ("How many disks?"); scanf ("%d", &n); printf ("\n"); move (n, 1, 3, 2); return 0; }// main /* PRE: n >= 0; a, b, and c represent some order of the distinct integers 1, 2, 3 POST: the function displays the individual moves necessary to move n disks from needle a to needle c, using needle b as a temporary storage needle */ void move (int disks, int a, int c, int b) { if (disks > 0){ move (disks-1, a, c, b); printf ("Move one disk from %d to %d\n", a, c); move (disks-1, b, a, c); }// if (disks > 0 return; }//move The function main – gets number of disks and invokes function move

12. Recursion and Function Implementation CIS 1057 Fall 201312 Tower of Hanoi Program #include void move (int disks, int a, int c, int b); int main() { int n; printf ("How many disks?"); scanf ("%d", &n); printf ("\n"); move (n, 1, 3, 2); return 0; }// main /* PRE: n >= 0; a, b, and c represent some order of the distinct integers 1, 2, 3 POST: the function displays the individual moves necessary to move n disks from needle a to needle c, using needle b as a temporary storage needle */ void move (int disks, int a, int c, int b) { if (disks > 0){ move (disks-1, a, c, b); printf ("Move one disk " "from %d to %d\n",a,c); move (disks-1, b, a, c); }// if (disks > 0 return; }//move The function move – where the action is

13. Recursion and Function Implementation CIS 1057 Fall 201313 Tower of Hanoi Program #include void move (int disks, int a, int c, int b); int main() { int n; printf ("How many disks?"); scanf ("%d", &n); printf ("\n"); move (n, 1, 3, 2); return 0; }// main /* PRE: n >= 0; a, b, and c represent some order of the distinct integers 1, 2, 3 POST: the function displays the individual moves necessary to move n disks from needle a to needle c, using needle b as a temporary storage needle */ void move (int disks, int a, int c, int b) { if (disks > 0){ move (disks-1, a, c, b); printf ("Move one disk " "from %d to %d\n",a,c); move (disks-1, b, a, c); }// if (disks > 0 return; }//move First move all but one of the disks to temporary peg

14. Recursion and Function Implementation CIS 1057 Fall 201314 Tower of Hanoi Program #include void move (int disks, int a, int c, int b); int main() { int n; printf ("How many disks?"); scanf ("%d", &n); printf ("\n"); move (n, 1, 3, 2); return 0; }// main /* PRE: n >= 0; a, b, and c represent some order of the distinct integers 1, 2, 3 POST: the function displays the individual moves necessary to move n disks from needle a to needle c, using needle b as a temporary storage needle */ void move (int disks, int a, int c, int b) { if (disks > 0){ move (disks-1, a, c, b); printf ("Move one disk " "from %d to %d\n",a,c); move (disks-1, b, a, c); }// if (disks > 0 return; }//move Next, move the remaining disk to the destination peg

15. Recursion and Function Implementation CIS 1057 Fall 201315 Tower of Hanoi Program #include void move (int disks, int a, int c, int b); int main() { int n; printf ("How many disks?"); scanf ("%d", &n); printf ("\n"); move (n, 1, 3, 2); return 0; }// main /* PRE: n >= 0; a, b, and c represent some order of the distinct integers 1, 2, 3 POST: the function displays the individual moves necessary to move n disks from needle a to needle c, using needle b as a temporary storage needle */ void move (int disks, int a, int c, int b) { if (disks > 0){ move (disks-1, a, c, b); printf ("Move one disk " "from %d to %d\n",a,c); move (disks-1, b, a, c); }// if (disks > 0 return; }//move Finally, move disks from temporary to destination peg

16. Recursion and Function Implementation CIS 1057 Fall 201316 Tower of Hanoi Program #include void move (int disks, int a, int c, int b); int main() { int n; printf ("How many disks?"); scanf ("%d", &n); printf ("\n"); move (n, 1, 3, 2); return 0; }// main /* PRE: n >= 0; a, b, and c represent some order of the distinct integers 1, 2, 3 POST: the function displays the individual moves necessary to move n disks from needle a to needle c, using needle b as a temporary storage needle */ void move (int disks, int a, int c, int b) { if (disks > 0){ move (disks-1, a, c, b); printf ("Move one disk " "from %d to %d\n",a,c); move (disks-1, b, a, c); }// if (disks > 0 return; }//move Notice that move calls itself twice, but with one fewer disks each time

17. Recursion and Function Implementation CIS 1057 Fall 201317 Why Recursion? Not often used by programmers with ordinary skills in some areas, but … … some problems are too hard to solve without recursion –Most notably, the compiler! –Tower of Hanoi problem –Most problems involving linked lists and trees (Later in the course)

18. Recursion and Function Implementation CIS 1057 Fall 201318 Recursion vs. Iteration Some simple recursive problems can be “unwound” into loops But code becomes less compact, harder to follow! Hard problems cannot easily be expressed in non-recursive code Tower of Hanoi Robots or avatars that “learn” Advanced games

19. Recursion and Function Implementation CIS 1057 Fall 201319 Recursion is so important … … that all modern computer architectures specifically support it Stack register Instructions for manipulating The Stack … most modern programming languages allow it But not Fortran and not Cobol

20. Recursion and Function Implementation CIS 1057 Fall 201320 Personal Observation From my own experience, programming languages and environments that do not support recursion … … are usually not rich enough to support a diverse portfolio of programs I.e., a wide variety of applications in many different disciplines

21. Recursion and Function Implementation CIS 1057 Fall 201321 Data Storage in Memory Variables may be automatic or static Automatic variables may only be declared within functions and compound statements (blocks) Storage allocated when function or block is entered Storage is released when function returns or block exits Parameters and result are (somewhat) like automatic variables Storage is allocated and initialized by caller of function Storage is released after function returns to caller.

22. Recursion and Function Implementation CIS 1057 Fall 201322 Static variables may be declared within or outside of functions Storage allocated when program is initialized Storage is released when program exits Static variables outside of functions may be visible to linker Compiler sets aside storage for all static variables at compiler or link time Values retained across function calls Initialization must evaluate to compile-time constant Static Data

23. Recursion and Function Implementation CIS 1057 Fall 201323 Static Variable Examples int j;//static, visible to linker & all functs static float f; // not visible to linker, visible to // to all functions in this program int fcn (float a, int b) { // nothing inside of {} is visible to linker int i = b;//automatic double g;//automatic static double h;//static, not visible to // linker; value retained from call to call body – may access j, f, a, b, i, g, h }//int fcn( … )

24. Recursion and Function Implementation CIS 1057 Fall 201324 Static Variable Examples (continued) int j;//static, visible to linker & all functs static float f; // not visible to linker, visible to // to all functions in this program int fcn (float a, int b) { // nothing inside of {} is visible to linker int i = b;//automatic double g;//automatic static double h;//static, not visible to // linker; value retained from call to call body – may access j, f, a, b, i, g, h }//int fcn( … ) Declaration outside any function:– always static static storage class:– not visible to linker

25. Recursion and Function Implementation CIS 1057 Fall 201325 Static Variable Examples (continued) int j;//static, visible to linker & all functs static float f; // not visible to linker, visible to // to all functions in this program int fcn (float a, int b) { // nothing inside of {} is visible to linker int i = b;//automatic double g;//automatic static double h;//static, not visible to // linker; value retained from call to call body – may access j, f, a, b, i, g, h }//int fcn( … ) Inside function:– default is automatic static storage class:– not visible to linker

26. Recursion and Function Implementation CIS 1057 Fall 201326 Static Variable Examples (continued) int j;//static, visible to linker & all functs static float f; // not visible to linker, visible to // to all functions in this program int fcn (float a, int b) { // nothing inside of {} is visible to linker int i = b;//automatic double g;//automatic static double h;//static, not visible to // linker; value retained from call to call body – may access j, f, a, b, i, g, h }//int fcn( … ) Note: value of h is retained from one call to next Value of h is also retained across recursions

27. Recursion and Function Implementation CIS 1057 Fall 201327 Extern Variables int j;//static, visible to linker & all functs static float f; // not visible to linker, visible to // to all functions in this program extern float p; // static, defined in another program int fcn (float a, int b) { // nothing inside of {} is visible to linker int i = b;//automatic double g;//automatic static double h;//static, not visible to // linker; value retained from call to call body – may access j, f, a, b, i, g, h, p }//int fcn( … ) extern storage class:– a static variable defined in another C program

28. Recursion and Function Implementation CIS 1057 Fall 201328 Automatic Variables, Arguments, & Results Allocated on The Stack Definition – The Stack –A last-in, first-out data structure provided by the operating system for each running program –For temporary storage of automatic variables, arguments, function results, and other stuff Used by all modern programming languages –Even assembly languages for modern processors –… but not Fortran or Cobol!

29. Recursion and Function Implementation CIS 1057 Fall 201329 Automatic Variables Allocated when function or compound statement is entered Released when function or compound statement is exited Values not retained from execution to next

30. Recursion and Function Implementation CIS 1057 Fall 201330 Arguments and Results Arguments are values calculated by caller of function Placed on The Stack by caller for use by function Function may assign new value to argument, but … …caller never looks at argument values again! Result is storage allocated by caller On The Stack Assigned by return statement of function For use by caller Arguments & result are removed by caller After function returns, after caller has absorbed return value

31. Recursion and Function Implementation CIS 1057 Fall 201331 Typical Implementation of The Stack Linear region of memory Stack Pointer “growing” downward Each time some information is pushed onto The Stack, pointer moves downward Each time info is popped off of The Stack, pointer moves back upward

32. Recursion and Function Implementation CIS 1057 Fall 201332 Typical Memory for Running Program (Windows & Linux) 0x00000000 0xFFFFFFFF address space program code (text) static data heap (dynamically allocated) stack (dynamically allocated) PC SP

33. Recursion and Function Implementation CIS 1057 Fall 201333 Typical Memory for Running Program (Windows & Linux) 0x00000000 0xFFFFFFFF address space program code (text) static data heap (dynamically allocated) stack (dynamically allocated) PC SP Heap to be discussed later in course

34. Recursion and Function Implementation CIS 1057 Fall 201334 How it works Imagine the following program:– int factorial(int n){ … /* body of factorial function */ … }// factorial Imagine also the caller:– int x = factorial(100); What does compiled code look like?

35. Recursion and Function Implementation CIS 1057 Fall 201335 Compiled code: the caller int x = factorial(100); Put the value “100” somewhere that factorial function can find Put the current program counter somewhere so that factorial function can return to the right place in calling function Provide a place to put the result, so that calling function can find it

36. Recursion and Function Implementation CIS 1057 Fall 201336 Compiled code: factorial function Save the caller’s registers somewhere Get the argument n from the agreed-upon place Set aside some memory for local variables and intermediate results Do whatever factorial was programmed to do Put the result where the caller can find it Restore the caller’s registers Transfer back to the program counter saved by the caller

37. Recursion and Function Implementation CIS 1057 Fall 201337 Question: Where is “somewhere”? So that caller can provide as many arguments as needed (within reason)? So that called routine can decide at run-time how much temporary space is needed? So that called routine can call any other routine, potentially recursively?

38. Recursion and Function Implementation CIS 1057 Fall 201338 Answer: The Stack Calling function Push space for result, and arguments onto stack Push return address and Jump to called function Called function Push registers and automatic storage space onto stack Do work of the routine, store result in space pushed above Pop registers and automatic storage off stack Jump to return address left by calling routine

39. Recursion and Function Implementation CIS 1057 Fall 201339 Typical Address Space (Windows & Linux) 0x00000000 0xFFFFFFFF Memory address space program code (text) static data heap (dynamically allocated) stack (dynamically allocated) PC SP

40. Recursion and Function Implementation CIS 1057 Fall 201340 Note Through the magic of operating systems, each running program has its own memory –Complete with stack & everything else Called a process Windows, Linux, Unix, etc.

41. Recursion and Function Implementation CIS 1057 Fall 201341 Note (continued) Not necessarily true in small, embedded systems Real-time & control systems Mobile phone & PDA Remote sensors, instrumentation, etc. Multiple running programs share a memory Each in own partition with own stack Barriers to prevent each from corrupting others

42. Recursion and Function Implementation CIS 1057 Fall 201342 Shared Physical Memory OS Kernel stack Process 1 stack Process 2 0x00000000 0x0000FFFF Physical memory stack Process 3

43. Recursion and Function Implementation CIS 1057 Fall 201343 Why a Stack? Engineering reason – Computer architectures without stacks are not “rich” enough to support demands of modern computing Multiple running programs at the same time Complex interactions among running programs Modern computer languages Mathematical reason – To support recursive programs Not often used by programmers with ordinary skills, but … Some problems are too hard to solve without recursion Most notably, the compiler!

44. Recursion and Function Implementation CIS 1057 Fall 201344 Why a Stack? Engineering reason – Computer architectures without stacks are not “rich” enough to support demands of modern computing Multiple running programs at the same time Complex interactions among running programs Modern computer languages Mathematical reason – To support recursive programs Not often used by programmers with ordinary skills, but … Some problems are too hard to solve without it Also, advanced games and robotics with “intelligent” objects

45. Recursion and Function Implementation CIS 1057 Fall 201345 Tower of Hanoi Program & Use of Stack #include void move (int disks, int a, int c, int b); int main() { int n; printf ("How many disks?"); scanf ("%d", &n); printf ("\n"); move (n, 1, 3, 2); return 0; }// main /* PRE: n >= 0; a, b, and c represent some order of the distinct integers 1, 2, 3 POST: the function displays the individual moves necessary to move n disks from needle a to needle c, using needle b as a temporary storage needle */ void move (int disks, int a, int c, int b) { if (disks > 0){ move (disks-1, a, c, b); printf ("Move one disk " "from %d to %d\n",a,c); move (disks-1, b, a, c); }// if (disks > 0 return; }//move

46. Recursion and Function Implementation CIS 1057 Fall 201346 Implementation on The Stack n SP Stack entry for main …

47. Recursion and Function Implementation CIS 1057 Fall 201347 Implementation on The Stack (continued) n SP Stack entry for main … return address a = 1 c = 3 b = 2 disks = n Args for call to move

48. Recursion and Function Implementation CIS 1057 Fall 201348 Implementation on The Stack (continued) n SP Stack entry for main … return address a = 1 c = 3 b = 2 disks = n Args for call to move return address a = 1 c = 2 b = 3 disks = n-1 Args for 2 nd call to move

49. Recursion and Function Implementation CIS 1057 Fall 201349 Implementation on The Stack (continued) n SP Stack entry for main … return address a = 1 c = 3 b = 2 disks = n Args for call to move return address a = 1 c = 2 b = 3 disks = n-1 Args for 2 nd call to move Args for subsequent calls to move go here

50. Recursion and Function Implementation CIS 1057 Fall 201350 Implementation on The Stack (continued) n SP Stack entry for main … return address a = 1 c = 3 b = 2 disks = n Args for call to move return address a = 1 c = 2 b = 3 disks = n-1 Args for 2 nd call to move Eventually, disks = 0, so no more calls to move All existing calls return

51. Recursion and Function Implementation CIS 1057 Fall 201351 Implementation on The Stack (continued) n SP Stack entry for main … return address a = 1 c = 3 b = 2 disks = n Args for call to move Leaving stack like this again

52. Recursion and Function Implementation CIS 1057 Fall 201352 Tower of Hanoi Program #include void move (int disks, int a, int c, int b); int main() { int n; printf ("How many disks?"); scanf ("%d", &n); printf ("\n"); move (n, 1, 3, 2); return 0; }// main /* PRE: n >= 0; a, b, and c represent some order of the distinct integers 1, 2, 3 POST: the function displays the individual moves necessary to move n disks from needle a to needle c, using needle b as a temporary storage needle */ void move (int disks, int a, int c, int b) { if (disks > 0){ move (disks-1, a, c, b); printf ("Move one disk " "from %d to %d\n",a,c); move (disks-1, b, a, c); }// if (disks > 0 return; }//move

53. Recursion and Function Implementation CIS 1057 Fall 201353 Implementation on The Stack (continued) n SP Stack entry for main … return address a = 1 c = 3 b = 2 disks = n Args for call to move move now moves one disk from peg1 to peg3, and then calls move again!

54. Recursion and Function Implementation CIS 1057 Fall 201354 Implementation on The Stack (continued) n SP Stack entry for main … return address a = 1 c = 3 b = 2 disks = n Args for call to move return address a = 3 c = 2 b = 1 disks = n-1 Args for call to move disks back from peg3 to peg2

55. Recursion and Function Implementation CIS 1057 Fall 201355 Implementation on The Stack (continued) n SP Stack entry for main … return address a = 1 c = 3 b = 2 disks = n Args for call to move return address a = 3 c = 2 b = 1 disks = n-1 Args for call to move back Args for subsequent calls to move back go here

56. Recursion and Function Implementation CIS 1057 Fall 201356 Implementation on The Stack (continued) n SP Stack entry for main … return address a = 1 c = 3 b = 2 disks = n Args for call to move return address a = 3 c = 2 b = 1 disks = n-1 Args for 2 nd call to move Eventually, disks = 0, so no more calls to move All existing calls return

57. Recursion and Function Implementation CIS 1057 Fall 201357 Implementation on The Stack (continued) n SP Stack entry for main … return address a = 1 c = 3 b = 2 disks = n Args for call to move Leaving stack like this again, but with all disks moved to peg 3

58. Recursion and Function Implementation CIS 1057 Fall 201358 Implementation on The Stack n SP Stack entry for main … move returns back to main

59. Recursion and Function Implementation CIS 1057 Fall 201359 Fundamental Principle Stack provides a simple mechanism for implementing recursive functions Arbitrary numbers of intermediate functions Eventually, something must break the circularity of recursion Or stack will grow until it overflows memory!

60. Recursion and Function Implementation CIS 1057 Fall 201360 Note on Recursive Functions Some simple recursive functions can be “unwound” into loops – e.g., int factorial(int n) { return (n <= 1) ? 1 : (n * factorial(n-1)); }//factorial is equivalent to int factorial(int n) { int product = 1; for (int k=1; k <= n; k++) product *= k; return product } //factorial

61. Recursion and Function Implementation CIS 1057 Fall 201361 Note on Recursive Functions (continued) Other recursive functions are much too difficult to understand if unwound E.g., Tower of Hanoi Keeping track of which disk is on which peg requires superhuman intellect, very complex code!

62. Recursion and Function Implementation CIS 1057 Fall 201362 Questions?