Design and Implement Special Stack Data Structure  Added Space Optimized Version
Question: Design a Data Structure SpecialStack that supports all the stack operations like push(), pop(), isEmpty(), isFull() and an additional operation getMin() which should return minimum element from the SpecialStack. All these operations of SpecialStack must be O(1). To implement SpecialStack, you should only use standard Stack data structure and no other data structure like arrays, list, .. etc.
Example:
Consider the following SpecialStack 16 > TOP 15 29 19 18 When getMin() is called it should return 15, which is the minimum element in the current stack. If we do pop two times on stack, the stack becomes 29 > TOP 19 18 When getMin() is called, it should return 18 which is the minimum in the current stack.
Solution: Use two stacks: one to store actual stack elements and other as an auxiliary stack to store minimum values. The idea is to do push() and pop() operations in such a way that the top of auxiliary stack is always the minimum. Let us see how push() and pop() operations work.
Push(int x) // inserts an element x to Special Stack
1) push x to the first stack (the stack with actual elements)
2) compare x with the top element of the second stack (the auxiliary stack). Let the top element be y.
…..a) If x is smaller than y then push x to the auxiliary stack.
…..b) If x is greater than y then push y to the auxiliary stack.
int Pop() // removes an element from Special Stack and return the removed element
1) pop the top element from the auxiliary stack.
2) pop the top element from the actual stack and return it.
The step 1 is necessary to make sure that the auxiliary stack is also updated for future operations.
int getMin() // returns the minimum element from Special Stack
1) Return the top element of auxiliary stack.
We can see that all above operations are O(1).
Let us see an example. Let us assume that both stacks are initially empty and 18, 19, 29, 15 and 16 are inserted to the SpecialStack.
When we insert 18, both stacks change to following. Actual Stack 18 < top Auxiliary Stack 18 < top When 19 is inserted, both stacks change to following. Actual Stack 19 < top 18 Auxiliary Stack 18 < top 18 When 29 is inserted, both stacks change to following. Actual Stack 29 < top 19 18 Auxiliary Stack 18 < top 18 18 When 15 is inserted, both stacks change to following. Actual Stack 15 < top 29 19 18 Auxiliary Stack 15 < top 18 18 18 When 16 is inserted, both stacks change to following. Actual Stack 16 < top 15 29 19 18 Auxiliary Stack 15 < top 15 18 18 18
Following is implementation for SpecialStack class. In the below implementation, SpecialStack inherits from Stack and has one Stack object min which work as auxiliary stack.
 C++

#include<iostream>
#include<stdlib.h>
using
namespace
std;
/* A simple stack class with basic stack funtionalities */
class
Stack
{
private
:
static
const
int
max = 100;
int
arr[max];
int
top;
public
:
Stack() { top = 1; }
bool
isEmpty();
bool
isFull();
int
pop();
void
push(
int
x);
};
/* Stack's member method to check if the stack is iempty */
bool
Stack::isEmpty()
{
if
(top == 1)
return
true
;
return
false
;
}
/* Stack's member method to check if the stack is full */
bool
Stack::isFull()
{
if
(top == max  1)
return
true
;
return
false
;
}
/* Stack's member method to remove an element from it */
int
Stack::pop()
{
if
(isEmpty())
{
cout<<
"Stack Underflow"
;
abort
();
}
int
x = arr[top];
top;
return
x;
}
/* Stack's member method to insert an element to it */
void
Stack::push(
int
x)
{
if
(isFull())
{
cout<<
"Stack Overflow"
;
abort
();
}
top++;
arr[top] = x;
}
/* A class that supports all the stack operations and one additional
operation getMin() that returns the minimum element from stack at
any time. This class inherits from the stack class and uses an
auxiliarry stack that holds minimum elements */
class
SpecialStack:
public
Stack
{
Stack min;
public
:
int
pop();
void
push(
int
x);
int
getMin();
};
/* SpecialStack's member method to insert an element to it. This method
makes sure that the min stack is also updated with appropriate minimum
values */
void
SpecialStack::push(
int
x)
{
if
(isEmpty()==
true
)
{
Stack::push(x);
min.push(x);
}
else
{
Stack::push(x);
int
y = min.pop();
min.push(y);
if
( x < y )
min.push(x);
else
min.push(y);
}
}
/* SpecialStack's member method to remove an element from it. This method
removes top element from min stack also. */
int
SpecialStack::pop()
{
int
x = Stack::pop();
min.pop();
return
x;
}
/* SpecialStack's member method to get minimum element from it. */
int
SpecialStack::getMin()
{
int
x = min.pop();
min.push(x);
return
x;
}
/* Driver program to test SpecialStack methods */
int
main()
{
SpecialStack s;
s.push(10);
s.push(20);
s.push(30);
cout<<s.getMin()<<endl;
s.push(5);
cout<<s.getMin();
return
0;
}
Output:
10
5
Space Optimized Version
The above approach can be optimized. We can limit the number of elements in auxiliary stack. We can push only when the incoming element of main stack is smaller than or equal to top of auxiliary stack. Similarly during pop, if the pop off element equal to top of auxiliary stack, remove the top element of auxiliary stack. Following is modified implementation of push() and pop().
 C++

/* SpecialStack's member method to insert an element to it. This method
makes sure that the min stack is also updated with appropriate minimum
values */
void
SpecialStack::push(
int
x)
{
if
(isEmpty()==
true
)
{
Stack::push(x);
min.push(x);
}
else
{
Stack::push(x);
int
y = min.pop();
min.push(y);
/* push only when the incoming element of main stack is smaller
than or equal to top of auxiliary stack */
if
( x <= y )
min.push(x);
}
}
/* SpecialStack's member method to remove an element from it. This method
removes top element from min stack also. */
int
SpecialStack::pop()
{
int
x = Stack::pop();
int
y = min.pop();
/* Push the popped element y back only if it is not equal to x */
if
( y != x )
min.push(y);
return
x;
}
Please write comments if you find the above code incorrect, or find other ways to solve the same problem.
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