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Recursion • A method of defining a concept which refers to the concept itself • A method of solving a problem by reducing it to smaller versions of itself • A powerful, elegant, efficient way to solve certain classes of programming problems • A recursive function calls itself 5/24/2017 cosc237/recursion 1 Recursive definitions Relative: x is a relative of y iff (if and only • • • • if) x is x is x is x is 5/24/2017 y’s parent y’s child y’s spouse a relative of a relative of y cosc237/recursion 2 Recursive definitions cont’d Natural numbers • 1 is a natural number • if i is an natural number, then i+1 is a natural number • Definition in which a problem is expressed in terms of a smaller version of itself • Has one or more base cases 5/24/2017 cosc237/recursion 3 Example 1: void message() //infinite recursive function { System.out.println("This is a recursive function.)"; message(); } 5/24/2017 cosc237/recursion 4 Example 2: void message(int times) { if (times > 0) { System.out.println("This is a recursive function.)"; message(times -1); } } 5/24/2017 cosc237/recursion 5 message (4); //Each call creates an activation record or frame. 1st call times: 4 2nd call times: 3 3rd call times: 2 5/24/2017 cosc237/recursion 4th call times: 1 5th call times: 0 6 Recursion requires 1. base case 2. general case 5/24/2017 cosc237/recursion 7 General format for Many Recursive Functions if (some easily-solved condition) // base case solution statement else // general case recursive function call 8 • Base part – states one or more cases that satisfy the definition – No base->infinite recursion • recursive part – states how to apply the definition to satisfy other cases – with repeated applications, reduces to base • implicitly: nothing else satisfies the definition 5/24/2017 cosc237/recursion 9 Example #3: • Find the sum of all int numbers between 1 and n. • Sum(5) = 1 + 2 + 3 + 4 + 5 = 15 • Base case: Sum (1) = 1 • General Case: Sum (2) = 1 + 2 = Sum(1) + 2 = Sum (n-1) + n 5/24/2017 cosc237/recursion 10 Finding the Sum of the Numbers from 1 to n int Summation ( int n ) { if ( n == 1) return 1 ; else // base case // general case return ( n + Summation ( n - 1 ) ) ; } 1111 Exercises • Write a function using recursion to print numbers from n to 0. • Write a function using recursion to print numbers from 0 to n. (You just need to shift one line in the program above) 5/24/2017 cosc237/recursion 12 Solution: //n to 0 void Print(int n) { System.out.println(n ); if (n <= 0) return; Print(n-1); } 5/24/2017 //0 to n void Print(int n) { if (n > 0) Print(n-1); System.out.println(n ); } cosc237/recursion 13 Example #4: • • • • Write a recursive function to find n factorial. n! = n * (n-1) * (n-2)... 3 * 2 * 1 Factorial(5) = 120 = 5 * 4 * 3 * 2 * 1 Base case: if (n == 1) return 1; • General case: n! = n * (n-1) * (n-2)... 3 * 2 * 1 = n * [(n-1) * (n-2)... 3 * 2 * 1] = n * (n-1)! OR: Factorial(n) = n * Factorial(n-1) 5/24/2017 cosc237/recursion 14 Factorial n! = n x (n-1)! int factorial(int num) { if (num == 0) return num; else return num * Factorial(num – 1); } 5/24/2017 cosc237/recursion 15 Recursive Factorial Method (continued) 16 Example #5: • Write a recursive function to find xn . • Base case: 20 = 1 • General Case: 2n = 2 * 2n-1 5/24/2017 cosc237/recursion 17 power function int Power(int x, int n) // n >0 { if (n == 1) return x; // base case else return x * Power(x,n-1); } 5/24/2017 cosc237/recursion 18 Power, n<0 • Note that 2-3 = 1/23 = 1/8 • In general, xn = 1/ x-n for non-zero x, and integer n<0. 5/24/2017 cosc237/recursion 19 Power float Power (float x, int n) { if (n == 0) // base case return 1; else if (n > 0) // first general case return (x * Power(x , n - 1)); else // second general case return (1.0 / Power(x , - n)); } 5/24/2017 cosc237/recursion 20 Use the box method to trace • Activation record • Represent each call to the function during the course of the execution by a new box • Local environment 5/24/2017 cosc237/recursion 21 Tracing a Recursive Method • After completing a recursive call: – Control goes back to the calling environment – Recursive call must execute completely before control goes back to previous call – Execution in previous call begins from point immediately following recursive call 22 Exercises void mystery(char param1, char param2) { if (param1 != param2) { param1++; param2--; Mystery(param1, param2); System.out.println(param1); System.out.println(param2); } } • Show all output resulting from the following invocation:mystery('A','G'); • Under what circumstances does mystery result in 5/24/2017 infinite recursion? cosc237/recursion 23 int TermUnknown (int someInt) // This is a classic number theory problem for which // termination for all input is uncertain. { if (someInt == 1) return 0; if (someInt % 2 == 1) return 1 + TermUnknown(3 * someInt +1); return 100 + TermUnknown(someInt/2); } What a. b. c. 5/24/2017 value does each of the following invocations return? TermUnknow(1); TermUnknown(4); TermUnknown(3); cosc237/recursion 24 Largest Value in Array public static int largest(int[] list, int lowerIndex, int upperIndex) { int max; if (lowerIndex == upperIndex) return list[lowerIndex]; else { max = largest(list, lowerIndex + 1, upperIndex); if (list[lowerIndex] >= max) return list[lowerIndex]; else return max; } } 25 Largest Value in Array (continued) 26 Recursive Functions • Recursive function - invokes itself directly or indirectly – direct recursion - function is invoked by a statement in its own body – indirect recursion - one function initiates a sequence of function invocations that eventually invokes the original • A->B->C->A 5/24/2017 cosc237/recursion 27 • When a function is invoked - activated • activation frame - collection of information maintained by the run-time system(environment) – current contents of local automatic variables – current contents of all formal parameters – return address 5/24/2017 – pushed on stackcosc237/recursion 28 Linear Search • • • • array a two subscripts, low and high Key Return -1 if key is not found in the elements a[low...high]. • Otherwise, return the subscript where key is found 29 // Recursive definition int LinearSearch ( /* in */ /* in */ /* in */ /* in */ { if ( a [ low ] == key ) const int a[ ] , int low , int high, int key ) // base case return low ; else if ( low == high) // second base case return -1 ; else ; // general case return LinearSearch( a, low + 1, high, key ) } 30 30 Function BinarySearch( ) BinarySearch that takes sorted array a, and two subscripts, low and high, and a key as arguments. It returns -1 if key is not found in the elements a[low...high], otherwise, it returns the subscript where key is found BinarySearch can be written using iteration or using recursion 31 // Iterative definition int BinarySearch (int a[ ] , int low , int high,int key ) { int mid; while ( low <= high ) { // more to examine mid = (low + high) / 2 ; if ( a [ mid ] == key ) return mid ; else if ( key < a [ mid ] ) high = mid - 1 ; else low = mid + 1 ; } return -1 ; // found at mid // search in lower half // search in upper half // key was not found } 32 32 // Recursive definition int BinarySearch (int a[ ], int low, int high, int key) { int mid = (low + high) / 2 ; if ( low > high ) // base case -- not found return -1; if ( a [ mid ] < key ) // search in upper half return BinarySearch( a, mid + 1, high, key ) ; else if ( key < a [ mid ] ) // search in lower half return BinarySearch ( a, low, mid - 1, key ); else return mid; } 33 33 "One Small Step/Complete the Solution" • Each invocation of a recursive function performs one small step towards the entire solution and makes a recursive call to complete the solution. 5/24/2017 cosc237/recursion 34 Recursive Functions • Non-value-returning – invoked as separate stand-alone statement • value-returning – invoked in an expression – often embedded within return statement 5/24/2017 cosc237/recursion 35 Recursion or Iteration? • Two ways to solve particular problem: – Iteration or Recursion • Every loop can be replaced by recursion • not every recursion can be replaced by a loop • Iterative control structures use looping to repeat a set of statements • Tradeoffs between two options: – Sometimes recursive solution is easier – Recursive solution is 36 often slower Factorial(loop version) int LoopFactorial (int n) { int i; int fact = 1; for (i = 1; i <= n; i++) fact = fact * i; return fact; } 5/24/2017 cosc237/recursion 37 Factorial - table lookup • int factorial[8] = {1,1,2,6,24,120,720,5040}; • faster, can simplify algorithms • more memory space 5/24/2017 cosc237/recursion 38 Programming Example: Decimal to Binary public static void decToBin(int num, int base) { if (num > 0) { decToBin(num / base, base); System.out.print(num % base); } } Java Programming: Program Design Including Data Structures 39 Programming Example: Decimal to Binary Java Programming: Program Design 40 When to Use Recursion • If the problem is stated recursively • if recursive algorithm is less complex than recursive algorithm • if recursive algorithm and nonrecursive algorithm are of similar complexity nonrecursive is probably more efficient • consider table-driven code 5/24/2017 cosc237/recursion 41 • References: [1] Programming and Problem Solving with C++, third edition, by Neal Dale, Chip Weems, and Mark Headington, Jones and Bartlett Publishers, 2002. [2] Data Abstraction and Structures Using C++, by Mark R. Headington, and David D. Riley, Jones and Bartlett Publishers, 1997. [3] Starting Out with C++, by Tony Gaddis, Scott Jones Publishers, 2004. Java Programming: • [4] Program Design Including Data Structures, D.S. Malik 5/24/2017 cosc237/recursion 42 •