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When they are recursiond6d9dd69 Recursive void Function Example  Consider task: Search list for a value Subtask 1: search 1st half of list Subtask 2: search 2nd half of list Subtasks are smaller versions of original task! When this occurs, recursive function can be used. Usually results in  elegant solution 'dHdbd'd'"b' *Recursive void Function: Vertical Numbers+ uTask: display digits of number vertically, one per line Example call: writeVertical(1234); Produces output: 1 2 3 48vdG/*G   " 'Vertical Numbers: Recursive Definition( Break problem into two cases Simple/base case: if n<10 Simply write number n to screen Recursive case: if n>=10, two subtasks: 1- Output all digits except last digit 2- Output last digit Example: argument 1234: 1st subtask displays 1, 2, 3 vertically 2nd subtask displays 4 `8d d)d<dd?dd8 )<  &    !writeVertical Function Definition  Given previous cases: void writeVertical(int n) { if (n < 10) //Base case cout << n << endl; else { //Recursive step writeVertical(n/10); cout << (n%10) << endl; } }Jd$     #    "          writeVertical Trace  BExample call: writeVertical(123); writeVertical(12); (123/10) writeVertical(1); (12/10) cout << 1 << endl; cout << 2 << endl; cout << 3 << endl; Arrows indicate task function performs Notice 1st two calls call again (recursive) Last call (1) displays and  ends |"d/C                       } 2Recursion  A Closer Look :Computer tracks recursive calls Stops current function Must know results of new recursive call before proceeding Saves all information needed for current call To be used later Proceeds with evaluation of new recursive call When THAT call is complete, returns to  outer computation  dddnd i Recursion Big Picture Outline of successful recursive function: One or more cases where function accomplishes it s task by: Making one or more recursive calls to solve smaller versions of original task Called  recursive case(s) One or more cases where function accomplishes it s task without recursive calls Called  base case(s) or stopping case(s)*d<didPd*d*<iP* I Infinite Recursion Base case MUST eventually be entered If it doesn t infinite recursion Recursive calls never end! Recall writeVertical example: Base case happened when down to 1-digit number That s when recursion stoppedHdddMdHM*j   W Infinite Recursion Example Consider alternate function definition: void newWriteVertical(int n) { newWriteVertical(n/10); cout << (n%10) << endl; } Seems  reasonable enough Missing  base case ! Recursion never stopsHd(TE-            J Stacks for Recursion A stack Specialized memory structure Like stack of paper Place new on top Remove when needed from top Called  last-in/first-out memory structure Recursion uses stacks Each recursive call placed on stack When one completes, last call is removed from stackd1d-d,ddXd1-,X Stack Overflow Size of stack limited Memory is finite Long chain of recursive calls continually adds to stack All are added before base case causes removals If stack attempts to grow beyond limit: Stack overflow error Infinite recursion always causes thisdd8d/d(dd&d8/(&  Recursion Versus Iteration Recursion not always  necessary Not even allowed in some languages Any task accomplished with recursion can also be done without it Nonrecursive: called iterative, using loops Recursive: Runs slower, uses more storage Elegant solution; less codingd-d d=d- =*   i !'Recursive Functions that Return a Value( Recursion not limited to void functions Can return value of any type Same technique, outline: 1- One+ cases where value returned is computed by recursive calls Should be  smaller sub-problems 2- One+ cases where value returned computed without recursive calls Base case^dBd!dDd d^B!D  ")Return a Value Recursion Example: Powers* jRecall predefined function pow(): result = pow(2.0,3.0); Returns 2 raised to power 3 (8.0) Takes two double arguments Returns double value Let s write recursively For simple example9dRddd9RF      #Function Definition for power()  int power(int x, int n) { if (n<0) { cout <<  Illegal argument ; exit(1); } if (n>0) return (power(x, n-1)*x); else return (1); }(dp       a $Calling Function power() 0Example calls: power(2, 0); returns 1 power(2, 1); returns (power(2, 0) * 2); returns 1 Value 1 multiplied by 2 & returned to original callted4dV3 %Calling Function power() .Larger example: power(2,3); power(2,2)*2 power(2,1)*2 power(2,0)*2 1 Reaches base case Recursion stops Values  returned back up stackbUdCdEC &Tracing Function power()  'Thinking Recursively Ignore details Forget how stack works Forget the suspended computations Yes, this is an  abstraction principle! And encapsulation principle! Let computer do  bookkeeping Programmer just think  big picture ddd$d$ ,Thinking Recursively: power BConsider power() again Recursive definition of power: power(x, n) returns: power(x, n  1) * x Just ensure  formula correct And ensure base case will be met`cd?d6-? (Recursive Design Techniques RDon t trace entire recursive sequence! Just check 3 properties: 1- No infinite recursion 2- Stopping cases return correct values 3- Recursive cases return correct valuesN@djd@j ) Recursive Design Check: power()! Check power() against 3 properties: 1- No infinite recursion: 2nd argument decreases by 1 each call Eventually must get to base case of 1 2- Stopping case returns correct value: power(x,0) is base case Returns 1, which is correct for x0 3- Recursive calls correct: For n>1, power(x,n) returns power(x,n-1)*x Plug in values correctb$ddLd(d;ddDd$I(9 DM * Binary Search Recursive function to search array Determines IF item is in list, and if so: Where in list it is Assumes array is sorted Breaks list in half Determines if item in 1st or 2nd half Then searches again just that half Recursively (of course)!#d>d,dIdd#>,)  +Pseudocode for Binary Search    .Checking the Recursion fCheck binary search against criteria: 1- No infinite recursion: Each call increases first or decreases last Eventually first will be greater than last 2- Stopping cases perform correct action: If first > last no elements between them, so key can t be there! IF key == a[mid] correctly found! 3- Recursive calls perform correct action If key < a[mid] key in 1st half  correct call If key > a[mid] key in 2nd half  correct callb&ddWd*dgd*dbd&W*g* / -Execution of Binary Search  /Efficiency of Binary Search  Extremely fast Compared with sequential search Half of array eliminated at start! Then a quarter, then 1/8, etc. Essentially eliminate half with each call Example: Array of 100 elements: Binary search never needs more than 7 compares! Logarithmic efficiency (log n)d d#dId d0dd #I 0   0Recursive Solutions Notice binary search algorithm actually solves  more general problem Original goal: design function to search an entire array Our function: allows search of any interval of array By specifying bounds first and last Very common when designing recursive functionsFdpd$d/dFp/   s Summary 1 6  8Reduce problem into smaller instances of same problem -> recursive solution Recursive algorithm has two cases: Base/stopping case Recursive case Ensure no infinite recursion Use criteria to determine recursion correct Three essential properties Typically solves  more general problemod"dIdd(do"I( /T[R  ` 33PP` 3333` ___MMM` 13` 333fpKNāvI` j@v۩ῑ΂H` Q_{>?" dd@(3?n<d@ `7 `2@`7``2 r?" dd@  @@ ``PR    @ ` ` p>> ph  (     C vA^C:\WINDOWS\Desktop\Savitch_jpg\Savitchleft.jpg"  6d0 "0`P`  X Click to edit Master title style!! j  6e0 "0<$D `   R*Copyright 2002 Pearson Education, Inc.**B ) ,  6f0 "Y <$D `  tSlide * X   `h0 ??"   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