CS103L SPRING 2017 UNIT 8: RECURSION

Transcription

1 CS103L SPRING 2017 UNIT 8: RECURSION

2 RECURSION A recursion function is defined in terms of itself Applies to math, e.g. recursion relations, sequences Fibonacci: F 0 = 1, F 1 = 1, F n = F n-1 + F n-2 Applies to computer functions that call themselves Also, nerdy CS acronyms GNU = GNU is Not Unix

3 RECURSIVE FUNCTIONS To write recursive functions we break up the problem into two cases: #1 A smaller version of the same problem #2 A base case of the problem where the answer is known ahead of time, or easily computed All recursive functions *must* have a base case - or what? Factorial Example: n! = n*(n-1)*(n-1) *1 factorial(n): factorial(1) = 1 (base case) factorial(n) = n*factorial(n-1) (recursive case)

4 FACTORIAL EXAMPLE unsigned long fact(unsigned long n) { //base case if(n == 1) return 1; unsigned long r = fact(n-1); //recursive case return n * r

5 TRACING RECURSIVE CODE fact(n=3) r = fact(n-1) return 3 * 2 fact(n=2) r = fact(n-1) return 2 * 1 fact(n=1) return 1

6 TWO TYPES OF RECURSIVE FUNCTIONS Recursive functions can be classified in two type: head recursion, tail recursion Head Recursion: immediately make the recursive call then do some work Tail recursion: do some work, then make the recursive call void stopgo(int n) { if(n==1) { cout << Stop. << endl; void gostop(int n) { if(n==1) { cout << Stop. << endl; return; return; gostop(n-1); cout << Go << endl; cout << Go << endl; gostop(n-1);

7 RECURSIVE FUNCTIONS Recursive functions usually take the place of loops Example: summing up an array of integers int main() { int data[4] = {8, 6, 7, 9; int sum1 = isum_it(data, 4); int sum2 = rsum_it(data, 4); int isum_it(int data[], int len) { sum = data[0]; for(int i=1; i < len; i++){ sum += data[i]; int rsum_it(int data[], int len) { if(len == 1) return data[0]; else int sum = rsum_it(data, len-1); return sum + data[len-1];

8 TRACING RECURSIVE SUM data[4] = {8,6,7,9; rsum_it(data, 4) rsum_it(data, 3) rsum_it(data, 2) rsum_it(data, 1) int sum = rsum_it(data, 4-1) int sum = rsum_it(data, 3-1) int sum = rsum_it(data, 2-1) return data[0] sum = 21 sum = 14 sum = 8 return 21 + data[3] return 14 + data[2] return 8+data[1]

9 STACK MAKES RECURSION POSSIBLE How does this work: we call the *same* function over and over? Because of the stack! We can call the function as many times as need be, each gets its *own* stack frame: hence each has it s own stack-local variables Code for all functions Data Code for for rsum_it all functions (data=800, len=1, sum=??) and return link Data for rsum_it (data=800, len=2, sum=??) sum=8) and return link Data for rsum_it (data=800, len=3, sum=??) sum=14) and return link Data for rsum_it (data=800, len=4, sum=??) sum=21) and return link Data for main (data=800,sum2=??) and return link System stack area data[4]: int main() { int data[4] = {8, 6, 7, 9; int sum2 = rsum_it(data, 4); int rsum_it(int data[], int len) { if(len == 1) return data[0]; else int sum = rsum_it(data, len-1); return sum + data[len-1];

10 IN CLASS EXERCISE count-up count-down

11 REQUIREMENTS FOR RECURSIVE FUNCTIONS You must have at least one base case! This case must terminate the recursion, i.e it will not make a further recursive call It is possible to have more than one base case The recursive case must make progress towards the base case The problem must get smaller each time or? Recursive cases and base cases must have return statements to propagate the answer up the recursive call stack

12 LOOPS VS. RECURSION FAQ: is it better to use loops or recursion? All recursive solutions can be done with iterative solutions Recursion pros: Clean and elegant code, easy to read Usually recursive code is much simpler than iterative Sometime iterative code is *nearly impossible* to write! Power of recursion comes from making multiple recursive calls - hard to implement with iterative solutions How to choose? Iteration is faster/less memory But if implementation is difficult or inscrutable - use recursion

13 RECURSIVE EXAMPLES k = Recursive Binary Search start i end Is a given element, k, in an array? If yes, return, otherwise return -1 Initialize start, mid, end binsearch(k, list, start, end): base case: start = end return list[start] == k? start : start i end Recursive case: start i end compute mid if (list[mid] == k) return mid; else if(list[mid] > k) return binsearch(k,list,start,mid); else return binsearch(k,list,mid,end); start end i

14 RECURSIVE EXAMPLES Recursive Bubble Sort with es 0 - (n-2) Scan list and swap to bubble largest value to the right Recurse on list es 0 - (n-2), then 0 - (n - 3), etc Original After Pass After Pass After Pass After Pass After Pass 5

15 IN CLASS EXERCISE Text Fractal

16 CODING TIME! Examples to code: recursive sum version pointer version Examples to code: Recursive Insertion sort Fibonacci Recursive binary search version pointer version Recursive Bubble sort