1.7k views

Let $Q$ denote a queue containing sixteen numbers and $S$ be an empty stack. Head(Q) returns the element at the head of the queue $Q$ without removing it from $Q$. Similarly Top (S) returns the element at the top of $S$ without removing it from $S$. Consider the algorithm given below.

while Q is not Empty do
if S is Empty OR Top(S) ≤ Head (Q) then
x:= Dequeue (Q);
Push (S, x);
else
x:= Pop(S);
Enqueue (Q, x);
end
end

The maximum possible number of iterations of the while loop in the algorithm is _______.

edited | 1.7k views

256 when 16,15,14,...,1 are present in queue

alternately 15 dequeue & push,15 pop & enqueue followed by 1 dequeue & push i.e. 31 iterations

brings it to the state 16,15,...,2 in queue and 1 in stack

29 iterations to get to 16,15,...,3 in queue and 2,1 in stack and so on

31+29+27+...+1=16^2=256

selected
not clear can someone explain more clearly?

@madhab it creates little bit confusion bcoz of 16 elements lets solve for 3 elements
for worst case take 3,2,1
step1.   initially queue contains 3 elements so after 5 while loop iteration  queue contains 3,2 and stack contains 1
step 2. now after 3 more while loop iterations queue contains 3 and stack contains 1,2{top=2}
step3. now after 1 more while loop iteration just push 3 in stack so queue is empty  and stack contains 1,2,3{top=3}
so total no of iteration =5+3+1 =9.
now u can easily generalize the result for n elements it is
[2n-1]+[2(n-1)-1]+[2(n-2)-1]+[2(n-3)-1]+ ............,+3+1  //it is an a.p. with n terms and common difference=2
so sum= n/2*(1+2n-1)=n2
so now for 16 elements ans =16=256

It's an eminent explanation saurabh rai sir,

#Boss

Thanks

its ok brother just chillllll...
btw i m nt sir i m a fellow aspirant like u  :)
While loop will run for the maximum number of times when the Queue elements are sorted in descending order.

Let's suppose that initially, Queue elements are 16, 15, 14, 13...............2, 1

Now, 16 will be first pushed into the stack, So, now stack top is 16 and Head(Q) is 15, So 16 will be popped out of the stack ( since, "if S is Empty OR Top(S) ≤ Head (Q) "returns false, hence else part will be executed) and enqueued into the Queue.

So, after two iterations Queue elements will be -> 15, 14, 13, .........................2, 1, 16 and stack will be empty.

Similarly, each of the elements 15,14,13...........2 will be pushed into the stack and immediately popped out of the stack(when popped out of the stack then also enqueue into the queue).So after 30 iterations stack will be empty and Queue contains will be like==> 1, 16, 15, 14, .....................2.

Now 1 will be Dequeued and pushed into the stack.Once 1 is pushed into the stack, it will never be popped(or we can say never be enqueued into the Queue again) because in order to Pop 1, there should be an element into the Queue which is less than 1 and that element comes at the front of the queue, since there is no element currently present in the Queue which is less than 1, there is no way to pop 1.

So, after 31 iterations Queue is==> 16, 15, 14, ........................2 and stack contains 1.

Now, the problem boils down to Queue with 15 elements.

Using the similar logic we can say after another 29 iterations (Total =31+29 )Queue will be like==> 16, 15, 14, ............3 and stack contains 1,2  (stack top is 2) and then 2 will remain there in the stack forever.

Similar way if we go on then, after 31 + 29 + 27 + 25 + ......+1 iterations Queue will be empty.

This is in A.P series with d=2. Sum = (16 *(1+31))/2= 16*32/2= 256
Queue Q contains 16,15,14,13,.....,1 where front pointing to 16 and rear is pointing to 1, Let denote Top(S) as T(s) and Head(Q) as H(q)
Stack S is empty now.
Now See While loop condition, As T(s) is empty,we do Dequeue 16 and push 16 to S, So one DEQUEUE and one PUSH, now T(s)=16 and H(q)=15
...........................................................................................................................
As T(S)>H(Q) we do POP 16 and then do Enqueue 16, So one POP and one ENQUEUE , now T(s)=empty and H(q)=15
..........................................................................................................................
As T(s) is empty,we do Dequeue 15 and push 15 to S, So one DEQUEUE and one PUSH, now T(s)=15 and H(q)=14
.......................................................................................................................
As T(S)>H(Q) we do POP 15 and then do Enqueue 15, So one POP and one ENQUEUE , now T(s)=empty and H(q)=14
---------------------------------------------------------------------------------------
..........
...........
For each number upto 2, we dequeue and push once, then POP and enqueue once, for last number that is 1 we dequeue and push once but after that T(S)<H(Q) as then H(Q)=16 and T(S)=1
so it takes 15*2+1=31 iterations to place 1 in stack and make Q(16 to 2), then 29 iterations to place 1,2 in stack and make Q(16 to 3)..then 27 iterations to place 1,2,3 in stack and make Q(16 to 4)..so on....
so total iterations=31+29+27+....+1=256
+1 vote
i m not getting ??? how it came ??

Maximum no of iterations of the while loop is n2 (in worst case , when (n-1) are in ascending order and nth element is in min of all the element)

I try for n=3 (2 3 1) got 9 iteration of while loop

for base case , it will be n iteration ( no are in ascending order)

and for descending order it will infinite time

edited
there can be 3 cases

case 1:

all elements in queue are arranged in ascending order,suppose 1,2,3......16

then it will take 16 iterations are there

but we are asked to compute max possible iterations so consider next case

case 2:

all elements are same ,  16 loops are possible in this case

case 3:

it will take 31+29+27+25.......+1=256

which can be calculated as

15             15

⅀(2n+1)=2  ⅀n+15=2*120+15=256

n=0           n=0

Ans: 256

Worst case input: 16, 15, 14, .... 1

First 16 will get into stack(since S is Empty) then popped out to the end of the queue (since Top(S)>Head(Q)). This happens for all the 15 elements (from 16 to 2). i.e, 2 operations per item (one push to stack and one pop from stack since top(S)>Head(Q)).

After this element 1 gets pushed to stack.

Therefore, in total = 2 * 15 + 1 = 31 operations to insert just one element into stack.

Now the problem gets reduced to handling 15 numbers in the queue (16, 15, 14, .... 2).

Total number of operations  = 31+29+27+...+1

an =31, a1=1.

an=a1+(n-1)*d   ==> n=16

Total = n/2 * (a1+an) = 16/2*(1+31) = 256