Lecture 24 Tao Wang 1

Transcription

1 Lecture 24 Tao Wang 1

2 Objectives Introduction of recursion How recursion works How recursion ends Infinite recursion Recursion vs. Iteration Recursion that Returns a Value Edition 2

3 Introduction If we put a while statement inside another while statement, it sort of exhibits a circular fashion. In mathematics, this kind of circular definition is called a recursive definition. More precisely, a function definition may contain a call to itself. We call such function, recursive function. Edition 3

4 Recursion in C++ In the form of a function that calls itself void recurse() { recurse(); //Function calls itself } int main() { recurse(); //Sets off the recursion } Edition 4

5 Recursive Functions for Tasks When writing function, one basic design technique is to break the task into subtasks. Sometimes one of the subtasks is a smaller example of the same task. Example: Print digits of a number vertically Write a function that writes numbers to the screen with the digits written vertically. Example: 1984 would be written as Edition 5

6 Algorithm Design For a simple case, If n < 10, then write the number n to the screen. For a general case, suppose you want to write the number 1234 vertically. Break the task into two subtasks: Output all the digits except the last digit like so: Output the last digit, which in this example is 4. Notice that subtask 1 is a smaller version of the original task, so we can implement this task with a recursive call. And subtask 2 is just the simple case listed above. Edition 6

7 Sample Program #include <iostream> using namespace std; void write_vertical(int n) { Tracing the Call: if (n<10) cout<<n<<endl; write vertical(123); else // n is two or more digits long write vertical(12); { write vertical(1); write_vertical(n/10); cout<<(n%10)<<endl; } } int main() { cout<<"write_vertical(123):"<<endl; write_vertical(123); return 0; } Edition 7

8 How recursion works When a function is called, the computer plugs in the arguments to the parameter(s) and begins to execute the code. If it encounter a recursive call, then it temporarily stops its computation. Because it must know the result of the recursive call before it continues. It saves all the necessary information to continue the computation, and proceed to evaluate the recursive call. Most computers use a structure called stack to keep track of such information. Edition 8

9 How recursion ends C++ places no restrictions on how recursive calls are used. In order to be useful, any recursive call must ultimately terminate with some piece of code that does not depend on recursion. General outline of a successful recursive function definition is as follows: One or more cases in which the function accomplishes its task by using recursive calls to accomplish one or more smaller versions of the task. One or more cases in which the function accomplishes its task without the use of any recursive calls. These cases without any recursive calls are called base cases or stopping cases. Edition 9

10 Infinite Recursion If we don t have a stopping case, then every recursive call produces another recursive call, then a call to the function will, in theory, run forever. This is called infinite recursion In practice, such a function will typically run until the computer runs out of resources and the program crashes. Example: void new_write_vertical(int n) { new_write_vertical(n/10); cout<<(n%10)<<endl; } It s syntactically correct, but will loop infinitely when n reaches 0. Edition 10

11 Recursion vs. Iteration Recursion is not absolutely necessary. Any task that can be accomplished using recursion can also be done in some other way without recursion. This is referred to as an iterative version. Recursion will usually run slower and use more storage than an equivalent iterative version (because it has to keep track of the stack). However, since the computer is keeping track of the stack automatically for you, using recursion can sometimes make your job as a programmer easier, and produce code that is easier to understand. Edition 11

12 Recursion that Returns a Value Recursion is not limited to void functions. A recursive function can return a value of any type. Designing recursive functions is the same as before: One or more recursive cases. One or more base or stopping cases. Edition 12

13 Power function pow(2.0, 3.0) returns We can implement our own power function using recursion. Formula: x n is equal to x n 1 * n Translating this into C++ says that the value returned by power(x, n) is power(x, n-1) * x Edition 13

14 Sample Program - Power int power(int x, int n) { } if (n < 0) { cout<<"illegal argument to power.\n"; exit(1); } Trace: if (n > 0) 1) power(2,0) return (power(x, n-1) * x); 2) power(2,1) else // n == 0 3) power(2,3) power(2,3) is power(2, 2) * 2 return 1; power(2,2) is power(2, 1) * 2 power(2,1) is power(2, 0) * 2 power(2,0) is 1 (stopping case) Edition 14

15 More on Recursive Design Techniques When designing a recursive function, you need not trace out the entire sequence of recursive calls for that instance of function. Make sure the following three properties are satisfied: There is no infinite recursion. Each stopping case returns the correct value for that case. For recursive cases: if all recursive calls return the correct value, then the final value returned by the function is the correct value. The above technique of checking is also known as mathematical induction. Edition 15

16 Example Tower of Hanoi It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape. The objective of the puzzle is to move the entire stack to another rod, obeying the following rules: Only one disk may be moved at a time. Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod. No disk may be placed on top of a smaller disk Edition 16

17 Example Tower of Hanoi Demo 17

18 Recursive Solution Break the problem down into a collection of smaller problems and further breaking those problems down into even smaller problems until a solution is reached. label the pegs A, B, C these labels may move at different steps let n be the total number of discs number the discs from 1 (smallest, topmost) to n (largest, bottommost) To move n discs from peg A to peg C: move n 1 discs from A to B. This leaves disc n alone on peg A move disc n from A to C move n 1 discs from B to C so they sit on disc n Edition 18

19 Common Programming Errors Forgot the stopping case or poorly written stopping case. Don t know how to break into subtasks. Infinite recursion. Edition 19

20 Summary If a problem can be reduced to a smaller task, then a recursive solution is likely to be easy to implement. Two kind of cases: one is with a stopping case and one is without. When defining a recursive function, use the three properties given in this slide to check that the function is correct. Edition 20