Recursion: Recursion: repeated curses

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Lucas Robbins
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1 Recursion: Programming can be viewed as the process of breaking a whole into parts, properly. This can be done in many ways. Recursion, is a special process of programming, where A whole is broken into some parts, but also where the parts are similar to the whole. Recursion is a way to organize both commands and data. It is a declarative way of coding; it describes WHAT is to be done, It differs from the previous imperative way, of commanding HOW it's to be done. Recursion, in its general form is abstract: it needs no computers, it needs no languages, it needs no mathematics. Views of recursion in programming have two differing forms: a. Structure, or form, where looping is done without whiles or fors. b. Definition, or behavior, where something is defined in terms of itself. 1 Recursion: repeated curses Recursion is the process where a method (typed or void) calls itself. For example, the following void method recurse has two slots, a curse string, and an integer many. Within the body, if many is greater than 0 then it outputs a curse and calls itself one fewer time or (many - 1) times. So, ultimately many becomes a value of 0 and the cursing process stops after an output of many curses. Public static void recurse (String curse, int many ) { // Does repeat a curse many times if (many > 0) { System.out.println (curse); recurse (curse, many-1); }//end static void method recurse Public static void main (String[] args) { String curse ; recurse (curse, 6); }//end Trace of this recursion is shown in the following diagram, where the string curse is abbreviated to c, and the integer counter many is m, initially 3. Notice that the many slot m decreases slowly and ultimately reaches a final value of 0. main caller String c; c = "#!*&"; recurse(c,3); recurse (c,3) 3 m recurse(c,2); recurse (c,2) recurse (c,1) recurse (c,0) 2 m recurse(c,1); 1 m recurse(c,0); 0 m recurse(c,2); 2

2 Recursive Functions (typed methods) Typed methods can be defined recursively, similar to the previous void method. For example, the following factorial function has one slot, num, and calls itself with an ever decreasing value of num. Public static int factorial (int num) { int result; // Does compute factorial recursively if (num == 0) { num * factorial (num - 1) }//end int function factorial Trace of this recursion is shown in the following diagram, which is similar to the previous one involving a routine. Notice that the num slot n decreases slowly and ultimately reaches a final value of 0. main caller factorial(3) 3 n factorial(2) 2 n factorial(1) 1 n factorial(0) 0 n Int x; x = 3; y=factorial(3); outint (y); 3*factorial(2); 6 result 2*factorial(1); 2 result 1*factorial(0); 1 result 0*factorial(?); 1 result 3 Recursive Needs Recursion, of either structure or definition requires two situations, specified by some condition. First, it needs a halting situation, so that it doesn't continue forever. This condition usually involves a value approaching 0 or 1, or some other such condition. Secondly, recursion needs a continuing situation, so that a value keeps approaching the halting situation. This usually takes the form of some function fun(n) defined in terms of that function fun(n-1), with some difference in the slot (parameter or argument). 4

3 Recursive Examples power (double x, int n){ // Does the power recursively // Need the int power (n >= 0) 1.0; x * power (x, n-1); }//end double method power tearint (int i) { // Does tear apart int i and print digits if (i > 0) { tearint (i / 10); System.out.println (i % 10); }//end void method tearint public static int fib (int n) { int result; // Does compute fibonacci recursively // Note: fib is called twice within fib }else if (n == 1) { fib(n - 1) + fib(n-2); }//endif }//end int method fib0nacci main (String [ ] args) { int i; // Does test the recursive methods i = 5; System.out.println (factorial (i)); recurse ("#@!!%", i); System.out.println (fib (i)); System.out.println (power (2.0, i)); tearint (fib(i)); }//end void method main 5 Defining power (of a positive integer): many ways power1 (double x, int n) { // Does the power recursively // Need the int power (n >= 0) 1.0; x * power1 (x, n-1); }//end double method power1 power2 (double x, int n) { // Does the positive power recursively 1.0; }else if (n == 1) { x; x * x * power2 (x, n - 2); }//end double method power2 power3 ( double x, int n) { // Does the positive power recursively 1.0; }else if ((n % 2) == 1) { // odd x * power3 (x, n-1); // n is even power3 (x, n/2)* power3 (x, n/2); }//end double method power3 power4 ( double x, int n) { double temp; // Does the positive power efficiently 1.0; }else if ((n % 2) == 1) { // odd x * power4 (x, n-1); // n is even temp = power4 (x, n/2); return temp * temp; }//end double method power4 6

4 Some Recursive Patterns (many ways!) 1. Split off one end 2. Split off both ends 3. Split into two halves 4. Split into two halves and a mid 7 Defining things recursively: Example sq: find the square of an integer. sq(n) = n*n = n + n + n + n To put sq(n) in terms of sq(n-1) try: sq(n-1) = (n - 1)*(n - 1) = n*n - 2*n + 1 = sq(n) - n - n + 1 sq(n) = sq(n-1) + n + n - 1 for (n >= 0) Sq(n) = 0 for (n == 0) // end condition For example: Sq(3) = sq(2) = sq(1) (3+3-1) = sq(0) (2+2-1)+(3+3-1) = = 9 public static int sq (int n) { int result; // Does compute the square recursively 0; sq(n-1) + n + n - 1; }//end int method sq 8

5 Recursion, and arrays The sum of all items of an aray a having n entries (from 1 to n) can be represented as sum(a,n) and determined recursively as: sum (a,n) = if (n == 0) and = sum(a,n-1) + a[n] a = n Repeated applications of this definitions unfolds as: sum (a,n) = a[n] + sum(a,n-1) = a[n] + a[n-1] + sum(a,n-2) = a[n] + a[n-1] + a[n-2] + sum(a,n-3) = a[n] + a[n-1] + a[n-2] a[2] + a[1] 9 Output of Arrays, many ways: with recursion A = n main (String args[]) { // Does output array recursively, many ways int A[ ] = {999, 8,5,4,9,1,7,6,3,2 }; int n = A.length - 1; // Display the arrays, many ways show1 ( A, n); System.out.println(); show2 (A,1,n); System.out.println(); show3 (A,1,n); System.out.println(); }// end void method main show2 (int B[], int low, int high) { // Does output array breaking into two parts if (low == high) { System.out.print (B[low] + " "); }else { int mid = (low + high)/2; show2 ( B, low, mid); show2 ( B, mid+1, high ); } // end if }// end routine show2 array show1 (int B[], int n) { // Does output array beginning at far end if (n > 0) { show1 (B, n-1); System.out.print (B[n] + " "); }// end routine show1 array show3 (int B[], int low, int high ) { // Does output array by breaking into mid and two parts if (low <= high) { int mid = (low + high)/2; show3 (B, low, mid - 1); System.out.print (B[mid] + " "); show3 (B, mid+1, high ); } // end if }// end routine show3 array 10

6 Recursive Reverses: of Arrays and Strings revarray (int[] a, int i, int j) { // Does reverse array a recursively if (i <= j) { swap (a, i, j); // can break out revarray (a, i+1, j-1); }//end void method revarray public static String srev(string str) { // Does reverse a string str recursively int n = str.length(); if(n > 0) { String last = str.substring (n-1); // rest or first is str.substring (0, n-1) return last + srev (str.substring (0, n-1)); return ""; }//end function SRev 11 Problems on Recursion Indicate which of the following defintions are recursive; (or fix them if possible to be recursive) a. A human is a creature whose mother is a human. b. A chicken is an egg's way of creaing more eggs. c. An ancestor of a person is that person's parents or the ancestors of that person's parents. d. A multiple of 10 is 10 or any multiple of 10 multiplied by 10 Trace the following recursive functions: Trace the following pseudocodes; try many ways a. Trace f(x) when x is 4: if (x==0) f(x) = 3 else f(x) = 2*f(x-1) b. Trace g(7): g(x) = 1 if x is 0 or 1 g(x) = g(x-1) - g(x-2) if x is positive c. Trace h(7): if (x==0) return h(0) = 0; else if (x==1) return h(1) = 1 else return h(x) = 2*h(x-1) - h(x-2) 12

7 Define the following recursively, as methods in Java a. rem (a,b), the remainder when positive int a is divided by int b b. iseven (i), determines if a positive int I is even c. product (a,b), multiplication of ints a, b using addition (2 ways) d. quotient (a,b), when int a is divided by int b e. fibonacci (n), another way f. combinations (m,n), m things taken n at a time g. compounded (a,n,r), growth of amount a after n durations at rate r h. maximum maxa (A,n) of an array A having n values i. reversea (A,n) of an array A having n values j. SortA (A,n) of an array A having n values 13 Some Recursive Problems solutions are in the Schaums Outline page 88) 4.1 sumsquares (int num) // does sum the first num squares 4.2 sumpow (double b, int n) // Does sum the first n powers of real value b 4.4 maxarray (arr, num) // Does find the maximum value of first num items in array arr 4.8 lg (num) //Does find the integer binary logarithem of num 4.9 ispal (str) // Does tell whether str is a palendrome (or reversable string) 4.10 binary (num) // Does convert a decimal num returning the binary form 14

Programming & Data Structure Laboratory. Day 2, July 24, 2014

Programming & Data Structure Laboratory Day 2, July 24, 2014 Loops Pre and post test loops for while do-while switch-case Pre-test loop and post-test loop Condition checking True Loop Body False Loop Body