Post on 12-Jan-2016
transcript
Recursion
Functions – reminder
A function can call other functions. Return causes the execution of the function to
terminate and returns a value to the calling function.
The type of the value returned must be the same as the return-type defined for the function.
return-type name(argType1 arg1, argType2 arg2, …) {function body;return value;
}
A recursive definition
C functions can also call themselves! However, usually not with the same
parameters (why?) Some functions can be defined using
smaller occurrences of themselves.Such a definition is called a “recursive
definition” of a function.
Recursive calling
Example:void func(int n){
putchar(`*`);func(n);
}Infinite series of *What is the problem ?Look for other problem …
Factorial
By definition : n! = 1*2*3*… *(n-1)*n
Thus, we can also define factorial the following way: 0! = 1 n! = n*(n-1)! for n>0
(n-1)!
*n
Example - factorial
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);
}
int factorial(int n){ int fact = 1; while (n >= 1) {
fact *=n; n--;
} return fact;}
Conclusions for Recursive calling
Every recursive function has a “boundary condition”. The function stops calling itself when it is satisfied.
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…4*…
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
FactRec(3)n
3
Returns…
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
FactRec(3)n
3
Returns…
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
FactRec(3)n
3
Returns…3*…
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
FactRec(3)n
3
Returns…
FactRec(2)n
2
Returns…
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
FactRec(3)n
3
Returns…
FactRec(2)n
2
Returns…
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
FactRec(3)n
3
Returns…
FactRec(2)n
2
Returns…2*…
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
FactRec(3)n
3
Returns…
FactRec(2)n
2
Returns…
FactRec(1)n
1
Returns…
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
FactRec(3)n
3
Returns…
FactRec(2)n
2
Returns…
FactRec(1)n
1
Returns…
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
FactRec(3)n
3
Returns…
FactRec(2)n
2
Returns…
FactRec(1)n
1
Returns…1
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
FactRec(3)n
3
Returns…
FactRec(2)n
2
Returns…2*1
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…
FactRec(3)n
3
Returns…3*2
Recursive factorial – step by step
int factRec(int n){ if (n==0 || n==1) return 1;
return n*factRec(n-1);}
FactRec(4)n
4
Returns…4*6
1הדפסת מספרים- דוגמא
#include <stdio.h>void print1(int n){
if (n>=0){printf("%d
",n);print1(n-1);
{{void main(){
int i = 3;print1(i);putchar('\n');
{
3 2 1 0
2הדפסת מספרים- דוגמא
#include <stdio.h>void print2(int n){
if (n>=0){print2(n-1);printf("%d
",n);{
{void main(){
int i = 3;print2(i);putchar('\n');
{
0 1 2 3
3הדפסת מספרים- דוגמא
#include <stdio.h>void print3(int n){
if (n>=0){printf("%d ",n);print3(n-1);printf("%d ",n);
{{void main(){
int i = 3;print3(i);putchar('\n');
{
3 2 1 0 0 1 2 3
4הדפסת מספרים- דוגמא
#include <stdio.h>void print4(int n){
if (n>=0){print4(n-1);printf("%d ",n);print4(n-1);
{{void main(){
int i = 3;print4(i);putchar('\n');
{
0 1 0 2 0 1 0 3 0 1 0 2 0 1 0
Another example - power
Xy = x*x*…*x
Recursive definitions (assume non-negative y):
Base: x0=11. Xy = X*(Xy-1)
2. Xy = (Xy/2)2 (for even y’s only)
y times
Fibonacci Series
Fibonacci definition: n0 = 0
n1 = 1
nn = nn-1 + nn-2
0 1 1 2 3 5 8 13 21 34 55 …
Fibonacci Iterative
void fibonacci(int n) { int Fn, Fn1, Fn2, ind; Fn2 = 0 ;
Fn1 = 1 ; Fn = 0 ;
if ( n == 1 )Fn = 1 ;
for (ind=2 ; ind <= n ; ind++){ Fn = Fn1 + Fn2; Fn2 = Fn1;
Fn1 = Fn;}
printf("F(%d) = %d \n", n, Fn);}
Fibonacci Recursive
fibonacci(1)
fibonacci(5)
fibonacci(4) fibonacci(3)
fibonacci(3) fibonacci(2) fibonacci(2)
fibonacci(1)
fibonacci(2) fibonacci(1) fibonacci(1)
fibonacci(0)
fibonacci(0) fibonacci(1) fibonacci(0)
int fibonacci(int n) { if (n==0)
return 0; if (n==1)
return 1; return fibonacci(n-1) + fibonacci(n-2);}
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
2*…
y
5
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
y
4
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
y
4
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
y
4
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
y
2
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
y
2
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
y
2
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
square(…)
y
2
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
square(…)
y
2
rec_pow(2, 1)x
2
Returns…
y
1
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
square(…)
y
2
rec_pow(2, 1)x
2
Returns…
y
1
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
square(…)
y
2
rec_pow(2, 1)x
2
Returns…
y
1
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
square(…)
y
2
rec_pow(2, 1)x
2
Returns…
y
1
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
square(…)
y
2
rec_pow(2, 1)x
2
Returns…
2*…
y
1
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
square(…)
y
2
rec_pow(2, 1)x
2
Returns…
2*…
y
1
rec_pow(2, 0)x
2
Returns…
y
0
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
square(…)
y
2
rec_pow(2, 1)x
2
Returns…
2*…
y
1
rec_pow(2, 0)x
2
Returns…
y
0
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
square(…)
y
2
rec_pow(2, 1)x
2
Returns…
2*…
y
1
rec_pow(2, 0)x
2
Returns…
1
y
0
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
square(…)
y
2
rec_pow(2, 1)x
2
Returns…
2*1
y
1
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(…)
y
4
rec_pow(2, 2)x
2
Returns…
square(2)
y
2
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
y
5
rec_pow(2, 4)x
2
Returns…
square(4)
y
4
rec_pow – step by step
int rec_pow(int x, int y){if (y == 0)
return 1;if (y%2 == 0)
return square(rec_pow(x,y/2));else
return x*rec_pow(x,y-1);}
rec_pow(2, 5)x
2
Returns…
2*16
y
5
Exercise
Write a program that receives two non-negative integers and computes their product recursively.
Not * multiple operator !!
Hint: Notice that the product a*b is actually a+a+…+a (b times).
Solution
int recMult( int x, int y ) {if( x == 0)
return 0;return y + recMult( x-1,y);
{
Exercise
Given the following iterative version of sum-of-digits calculation
Find the recursive definition of this function
(don’t forget the base case!)
int sum_digits(int n){int sum = 0;while (n > 0) {
sum += n%10;n = n/10;}return sum;
}
Solution
int sumOfDigits( int x ) {if( x < 0)
x *= -1;if( x == 0 )
return 0;else
return x % 10 + sumOfDigits( x / 10 );
}
More uses
Recursion is a general approach to programming functions.
Its uses are not confined to calculating mathematical expressions!
For example : write a function that finds the max member in an array of integer.
Solution
int rec_max(int arr[ ], int size){int rest;if (size == 1)
return arr[0];else {
rest = rec_max(arr+1, size-1); if (arr[0] > rest) return arr[0];else return rest;
}}
חיפוש בינארי
int BinarySearch(int arr[],int x, int left, int right) { int middle;if(left>right) return -1; else { middle=(left+right)/2); if(arr[middle]==x)
return middle; else if(x < arr[middle])
return BinarySearch(arr,x,left,middle-1); else return
BinarySearch(arr,x,middle+1,right); }
}
Towers of Hanoi
The Towers of Hanoi problem 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.
Towers of Hanoi
Recursive Solution
To move n disks from peg A to peg C: move n−1 disks from A to B. This leaves disk
#n alone on peg A move disk #n from A to C move n−1 disks from B to C so they sit on
disk #n
Recursive Solution - Function
void hanoi(int x, char from, char to, char aux){ if(x==1){
printf("Move Disk From %c to %c\n", from, to);
}else {
hanoi(x-1,from,aux,to);printf("Move Disk From %c to %c\n",
from, to);hanoi(x-1,aux,to,from);
}}
Scope of variables - reminder
A variable declared within a function is unrelated to variables declared elsewhere.
A function cannot access variables that are declared in other functions.