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Consider the following function $foo()$

void foo(int n){
  if(n<=0) printf("Bye");

Let $P(n)$ represent recurrence relation, indicating number of time print statement executed. What will best recurrence for $P(n)?$

$A)P(n)=P(n-1)+P(n-3)+2,$ for $n>0; $ $1$ for $n=0$

$A)P(n)=P(n-1)+P(n-3)+2,$ for $n>0; $ $2$ for $n=0$

$A)P(n)=P(n-1)+P(n-3)+2,$ for $n>0; $ $0$ for $n=0$

$A)P(n)=P(n-1)+P(n-3)+1,$ for $n>0; $ $2$ for $n=0$

The options are confusing to me. Can someone explain the options well. Moreover , what will be constant added  $1$ or $2?$

in Programming by Veteran (119k points) | 160 views

2 Answers

+2 votes
Best answer

 If we give n=0 then printf will execute only 1 time (else part will not execute) so option a should be the answer by eliminating the answer.

$\rightarrow$ The constants 1 are 2 are used since the extra printf will be there if the loop does not go in the recursive conditions.

$\rightarrow$ For eg n=1.

printf("hi"); //extra print.

foo(n-3) //foo(-2)= print("Bye");  p(n-3)

printf("hi"); //extra print

foo(n-1) //foo(0)=print("bye"); p(n-1)

$\rightarrow$ so the 2 constant used is for the extra printf which i have shown in the above code.

$A)P(n)=P(n−1)+P(n−3)+2, for n>0; 1 for n=0$ is the correct answer


Why 2 and not 1 ?

$\rightarrow$ Its simple, suppose the code is

    void foo(int n){
      if(n<=0) printf("Bye");

$\rightarrow$ Now  the recurrence relation is

$\rightarrow$ P(n)= 2, for n>0; 1 for n=0

$\rightarrow$ if n=0 we are executing print statement once as we are not going in the else part.

$\rightarrow$ when n>0 we are going to the else part in which we are using 2 print statements.

$\rightarrow$ so P(n)=2 for n>0.

$\rightarrow$ Now, in the actual code we are just adding two recursive parts in the else section of the above code.

$\rightarrow$ so the recursion becomes

P(n)=P(n−1)+P(n−3)+2,for n>0;1 for n=0

    void foo(int n){
      if(n<=0) printf("Bye"); //p(n) =1 if n=0.
      else{                   // else
          printf("Hi");       // 1
          foo(n-3);           // p(n-3)
          printf("Hi");       // 1
          foo(n-1);           // p(n-1)
      }                      // i.e. p(n)= 1+p(n-3)+1+p(n-1) = P(n−1)+P(n−3)+2

NOTE :- we are not doing O(1) + O(1) = O(1) since we have to count the print statements (not to calculate the time complexity.)

by Boss (23.8k points)
selected by
I havenot got it, why $2$ and not $1?$

Please check my answer @srestha


and what is meaning of 

1 for n=0??


2 for n=0 ??


for 1 time print statement will be executed if n=0


for 2 time print statement will be executed if n=0


ok, thanks :)
First will be correct as evident for n = 0  only 1 print is there
0 votes
Option A is correct as for n=0;the time complexity will be O(1) because the if statement will be executed and function will terminate after printing "Bye"
by (303 points)

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