Our task is to 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 performed with time complexity 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 Special-Stack 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.
An approach that uses O(1) time and O(1) extra space is discussed here. However, in the previous article the original elements are not recovered. Only the minimum element is returned at any given point of time.
In this article, the previous approach is modified so that original elements can also be retrieved during a pop() operation.
Approach:
Consider a variable minimum in which we store the minimum element in the stack. Now, what if we pop the minimum element from the stack? How do we update the minimum variable to the next minimum value? One solution is to maintain another stack in sorted order so that the smallest element is always on the top. However, this is an O(n) space approach.
To achieve this in O(1) space, we need a way to store the current value of an element and the next minimum value in the same node. This can be done by applying simple mathematics:
new_value = 2*current_value - minimum
We push this new_value into the stack instead of current_value. To retrieve current_value and next minimum from new_value:
current_value = (new_value + minimum)/2 minimum = new_value - 2*current
When the operation Push(x) is done, we follow the given below algorithm:
- If stack is empty
- insert x into the stack
- make minimum equal to x.
- If stack is not empty
- if x is less than minimum
- set temp equal to 2*x-minimum
- set minimum equal to x
- set x equal to temp
- insert x into stack
When the operation Pop(x) is done, we follow the given below algorithm:
- If stack is not empty
- set x equal to topmost element
- if x is less than minimum
- set minimum equal to 2*minimum – x
- set x equal to (x+minimum)/2
- return x
When getMin() is called, we return the element stored in variable, minimum:
Implementation:
// Cpp program to retrieve original elements of the // from a Stack which returns the minimum element // in O(1) time and O(1) space #include<bits/stdc++.h> using namespace std;
// Node class which contains data // and pointer to next node class Node {
public :
int data;
Node* next;
Node()
{
data = -1;
next = NULL;
}
Node( int d)
{
data = d;
next = NULL;
}
void setData( int data)
{
this ->data = data;
}
void setNext(Node* next)
{
this ->next = next;
}
Node* getNext()
{
return next;
}
int getData()
{
return data;
}
}; class Stack{
public :
Node* top;
// Stores minimum element of the stack
int minimum;
// Function to push an element
void push(Node* node)
{
int current=node->getData();
if (top == NULL)
{
top=node;
minimum=current;
}
else {
if (current < minimum)
{
node->setData(2*current - minimum);
minimum = current;
}
node->setNext(top);
top=node;
}
}
// Retrieves topmost element
Node* pop()
{
Node* node = top;
if (node!=NULL)
{
int current=node->getData();
if (current<minimum)
{
minimum = 2*minimum - current;
node->setData((current + minimum) / 2);
}
top = top->getNext();
}
return node;
}
// Retrieves topmost element without
// popping it from the stack
Node* peek()
{
Node* node = NULL;
if (top != NULL) {
node = new Node();
int current = top->getData();
node->setData(current < minimum ? minimum : current);
}
return node;
}
// Function to print all elements in the stack
void printAll()
{
Node* ptr = top;
int min = minimum;
if (ptr != NULL) { // if stack is not empty
while ( true ) {
int val = ptr->getData();
if (val < min) {
min = 2 * min - val;
val = (val + min) / 2;
}
cout << val << " " ;
ptr = ptr->getNext();
if (ptr == NULL)
break ;
}
cout << '\n' ;
}
else
cout << "Empty!\n" ;
}
// Returns minimum of Stack
int getMin()
{
return minimum;
}
bool isEmpty()
{
return top == NULL;
}
}; int main()
{ Stack* stack = new Stack();
Node* node;
stack->push( new Node(5));
stack->push( new Node(3));
stack->push( new Node(4));
// Calls the method to print the stack
cout << "Elements in the stack are:\n" ;
stack->printAll();
// Print current minimum element if stack is
// not empty
if (stack->isEmpty())
cout << "Empty Stack!\n" ;
else
cout << "Minimum: " << stack->getMin() << '\n' ;
// Push new elements into the stack
stack->push( new Node(1));
stack->push( new Node(2));
// Printing the stack
cout << "Stack after adding new elements:\n" ;
stack->printAll();
// Print current minimum element if stack is
// not empty
if (stack->isEmpty())
cout << "Empty Stack!\n" ;
else
cout << "Minimum: " << stack->getMin() << '\n' ;
// Pop elements from the stack
node = stack->pop();
cout << "Element Popped: " ;
if (node == NULL)
cout << "Empty!\n" ;
else
cout << node->getData() << '\n' ;
node = stack->pop();
cout << "Element Popped: " ;
if (node == NULL)
cout << "Empty!\n" ;
else
cout << node->getData() << '\n' ;
// Printing stack after popping elements
cout << "Stack after removing top two elements:\n" ;
stack->printAll();
// Printing current Minimum element in the stack
if (stack->isEmpty())
cout << "Empty Stack!\n" ;
else
cout << "Minimum: " << stack->getMin() << '\n' ;
// Printing top element of the stack
node = stack->peek();
cout << "Top: " ;
if (node == NULL)
cout << "Empty!\n" ;
else
cout << node->getData() << '\n' ;
return 0;
} |
// Java program to retrieve original elements of the // from a Stack which returns the minimum element // in O(1) time and O(1) space class Stack {
Node top;
// Stores minimum element of the stack
int minimum;
// Function to push an element
void push(Node node)
{
int current = node.getData();
if (top == null ) {
top = node;
minimum = current;
}
else {
if (current < minimum) {
node.setData( 2 * current - minimum);
minimum = current;
}
node.setNext(top);
top = node;
}
}
// Retrieves topmost element
Node pop()
{
Node node = top;
if (node != null ) {
int current = node.getData();
if (current < minimum) {
minimum = 2 * minimum - current;
node.setData((current + minimum) / 2 );
}
top = top.getNext();
}
return node;
}
// Retrieves topmost element without
// popping it from the stack
Node peek()
{
Node node = null ;
if (top != null ) {
node = new Node();
int current = top.getData();
node.setData(current < minimum ? minimum : current);
}
return node;
}
// Function to print all elements in the stack
void printAll()
{
Node ptr = top;
int min = minimum;
if (ptr != null ) { // if stack is not empty
while ( true ) {
int val = ptr.getData();
if (val < min) {
min = 2 * min - val;
val = (val + min) / 2 ;
}
System.out.print(val + " " );
ptr = ptr.getNext();
if (ptr == null )
break ;
}
System.out.println();
}
else
System.out.println( "Empty!" );
}
// Returns minimum of Stack
int getMin()
{
return minimum;
}
boolean isEmpty()
{
return top == null ;
}
} // Node class which contains data // and pointer to next node class Node {
int data;
Node next;
Node()
{
data = - 1 ;
next = null ;
}
Node( int d)
{
data = d;
next = null ;
}
void setData( int data)
{
this .data = data;
}
void setNext(Node next)
{
this .next = next;
}
Node getNext()
{
return next;
}
int getData()
{
return data;
}
} // Driver Code public class Main {
public static void main(String[] args)
{
// Create a new stack
Stack stack = new Stack();
Node node;
// Push the element into the stack
stack.push( new Node( 5 ));
stack.push( new Node( 3 ));
stack.push( new Node( 4 ));
// Calls the method to print the stack
System.out.println( "Elements in the stack are:" );
stack.printAll();
// Print current minimum element if stack is
// not empty
System.out.println(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Push new elements into the stack
stack.push( new Node( 1 ));
stack.push( new Node( 2 ));
// Printing the stack
System.out.println( "\nStack after adding new elements:" );
stack.printAll();
// Print current minimum element if stack is
// not empty
System.out.println(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Pop elements from the stack
node = stack.pop();
System.out.println( "\nElement Popped: "
+ (node == null ? "Empty!" : node.getData()));
node = stack.pop();
System.out.println( "Element Popped: "
+ (node == null ? "Empty!" : node.getData()));
// Printing stack after popping elements
System.out.println( "\nStack after removing top two elements:" );
stack.printAll();
// Printing current Minimum element in the stack
System.out.println(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Printing top element of the stack
node = stack.peek();
System.out.println( "\nTop: " + (node == null ?
"\nEmpty!" : node.getData()));
}
} |
""" Python program to retrieve original elements of the from a Stack which returns the minimum element in O(1) time and O(1) space """ # Class to make a Node class Node:
# Constructor which assign argument to node's value
def __init__( self , value):
self .value = value
self . next = None
# This method returns the string
# representation of the object.
def __str__( self ):
return "Node({})" . format ( self .value)
# __repr__ is same as __str__
__repr__ = __str__
class Stack:
# Stack Constructor initialise
# top of stack and counter.
def __init__( self ):
self .top = None
self .count = 0
self .minimum = None
# This method returns the string
# representation of the object (stack).
def __str__( self ):
temp = self .top
m = self .minimum
out = []
if temp is None :
print ( "Empty Stack" )
else :
while not temp is None :
val = temp.value
if val < m:
m = ( 2 * m) - val
val = ( val + m ) / 2
out.append( str ( int (val)))
temp = temp. next
out = ' ' .join(out)
return (out)
# __repr__ is same as __str__
__repr__ = __str__
# This method is used to get minimum element of stack
def getMin( self ):
if self .top is None :
return "Stack is empty"
else :
return self .minimum
# Method to check if Stack is Empty or not
def isEmpty( self ):
# If top equals to None then stack is empty
if self .top = = None :
return True
else :
# If top not equal to None then stack is empty
return False
# This method returns length of stack
def __len__( self ):
self .count = 0
tempNode = self .top
while tempNode:
tempNode = tempNode. next
self .count + = 1
return self .count
# This method returns top of stack
def peek( self ):
if self .top is None :
print ( "Stack is empty" )
else :
if self .top.value < self .minimum:
return self .minimum
else :
return self .top.value
# This method is used to add node to stack
def push( self ,value):
if self .top is None :
self .top = Node(value)
self .minimum = value
else :
new_node = Node(value)
if value < self .minimum:
temp = ( 2 * value) - self .minimum
new_node.value = temp
self .minimum = value
new_node. next = self .top
self .top = new_node
# This method is used to pop top of stack
def pop( self ):
new_node = self .top
if self .top is None :
print ( "Stack is empty" )
else :
removedNode = new_node.value
if removedNode < self .minimum:
self .minimum = ( ( 2 * self .minimum ) - removedNode )
new_node.value = ( (removedNode + self .minimum) / 2 )
self .top = self .top. next
return int (new_node.value)
# Driver program to test above class stack = Stack()
stack.push( 5 )
stack.push( 3 )
stack.push( 4 )
print ( "Elements in the stack are:" )
print (stack)
print ( "Minimum: {}" . format ( stack.getMin() ) )
stack.push( 1 )
stack.push( 2 )
print ( "Stack after adding new elements:" )
print (stack)
print ( "Minimum:{}" . format ( stack.getMin() ) )
print ( "Element Popped: {}" . format (stack.pop()))
print ( "Element Popped: {}" . format (stack.pop()))
print ( "Stack after removing top two elements: " )
print (stack)
print ( "Minimum: {}" . format ( stack.getMin() ) )
print ( "Top: {}" . format ( stack.peek() ) )
# This code is contributed by Blinkii |
// C# program to retrieve original elements of the // from a Stack which returns the minimum element // in O(1) time and O(1) space using System;
public class Stack
{ Node top;
// Stores minimum element of the stack
public int minimum;
// Function to push an element
public void push(Node node)
{
int current = node.getData();
if (top == null )
{
top = node;
minimum = current;
}
else
{
if (current < minimum)
{
node.setData(2 * current - minimum);
minimum = current;
}
node.setNext(top);
top = node;
}
}
// Retrieves topmost element
public Node pop()
{
Node node = top;
if (node != null )
{
int current = node.getData();
if (current < minimum)
{
minimum = 2 * minimum - current;
node.setData((current + minimum) / 2);
}
top = top.getNext();
}
return node;
}
// Retrieves topmost element without
// popping it from the stack
public Node peek()
{
Node node = null ;
if (top != null )
{
node = new Node();
int current = top.getData();
node.setData(current < minimum ? minimum : current);
}
return node;
}
// Function to print all elements in the stack
public void printAll()
{
Node ptr = top;
int min = minimum;
if (ptr != null )
{
// if stack is not empty
while ( true )
{
int val = ptr.getData();
if (val < min)
{
min = 2 * min - val;
val = (val + min) / 2;
}
Console.Write(val + " " );
ptr = ptr.getNext();
if (ptr == null )
break ;
}
Console.WriteLine();
}
else
Console.WriteLine( "Empty!" );
}
// Returns minimum of Stack
public int getMin()
{
return minimum;
}
public bool isEmpty()
{
return top == null ;
}
} // Node class which contains data // and pointer to next node public class Node
{ public int data;
public Node next;
public Node()
{
data = -1;
next = null ;
}
public Node( int d)
{
data = d;
next = null ;
}
public void setData( int data)
{
this .data = data;
}
public void setNext(Node next)
{
this .next = next;
}
public Node getNext()
{
return next;
}
public int getData()
{
return data;
}
} // Driver Code public class MainClass
{ public static void Main(String[] args)
{
// Create a new stack
Stack stack = new Stack();
Node node;
// Push the element into the stack
stack.push( new Node(5));
stack.push( new Node(3));
stack.push( new Node(4));
// Calls the method to print the stack
Console.WriteLine( "Elements in the stack are:" );
stack.printAll();
// Print current minimum element if stack is
// not empty
Console.WriteLine(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Push new elements into the stack
stack.push( new Node(1));
stack.push( new Node(2));
// Printing the stack
Console.WriteLine( "\nStack after adding new elements:" );
stack.printAll();
// Print current minimum element if stack is
// not empty
Console.WriteLine(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Pop elements from the stack
node = stack.pop();
if (node == null )
Console.WriteLine( "\nElement Popped: " + "Empty!" );
else
Console.WriteLine( "\nElement Popped: " +node.getData());
node = stack.pop();
if (node == null )
Console.WriteLine( "Element Popped: " + "Empty!" );
else
Console.WriteLine( "Element Popped: " +node.getData());
// Printing stack after popping elements
Console.WriteLine( "\nStack after removing top two elements:" );
stack.printAll();
// Printing current Minimum element in the stack
Console.WriteLine(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Printing top element of the stack
node = stack.peek();
if (node == null )
Console.WriteLine( "\nTop: " + "\nEmpty!" );
else
Console.WriteLine( "\nTop: " +node.getData());
}
} // This code is contributed by Rajput-Ji |
// Class to make a Node class Node { // Constructor which assign argument to node's value
constructor(value) {
this .value = value;
this .next = null ;
}
// This method returns the string
// representation of the object.
toString() {
return `Node(${ this .value})`;
}
} class Stack { // Stack Constructor initialise
// top of stack and counter.
constructor() {
this .top = null ;
this .count = 0;
this .minimum = null ;
}
// This method returns the string
// representation of the object (stack).
toString() {
let temp = this .top;
let m = this .minimum;
let out = [];
if (temp === null ) {
console.log( "Empty Stack" );
} else {
while (temp !== null ) {
let val = temp.value;
if (val < m) {
m = 2 * m - val;
val = (val + m) / 2;
}
out.push(Math.floor(val));
temp = temp.next;
}
out = out.join( " " );
return out;
}
}
// This method is used to get minimum element of stack
getMin() {
if ( this .top === null ) {
return "Stack is empty" ;
} else {
return this .minimum;
}
}
// Method to check if Stack is Empty or not
isEmpty() {
// If top equals to None then stack is empty
if ( this .top === null ) {
return true ;
} else {
// If top not equal to None then stack is not empty
return false ;
}
}
// This method returns length of stack
length() {
this .count = 0;
let tempNode = this .top;
while (tempNode) {
tempNode = tempNode.next;
this .count++;
}
return this .count;
}
// This method returns top of stack
peek() {
if ( this .top === null ) {
console.log( "Stack is empty" );
} else {
if ( this .top.value < this .minimum) {
return this .minimum;
} else {
return this .top.value;
}
}
}
// This method is used to add node to stack
push(value) {
if ( this .top === null ) {
this .top = new Node(value);
this .minimum = value;
} else {
let new_node = new Node(value);
if (value < this .minimum) {
let temp = 2 * value - this .minimum;
new_node.value = temp;
this .minimum = value;
}
new_node.next = this .top;
this .top = new_node;
}
}
// This method is used to pop top of stack
pop() {
let new_node = this .top;
if ( this .top === null ) {
console.log( "Stack is empty" );
} else {
let removedNode = new_node.value;
if (removedNode < this .minimum) {
this .minimum = 2 * this .minimum - removedNode;
new_node.value = (removedNode + this .minimum) / 2;
}
this .top = this .top.next;
return Math.floor(new_node.value);
}
}
} // Driver program to test above class let stack = new Stack();
stack.push(5); stack.push(3); stack.push(4); console.log( "Elements in the stack are:" );
console.log(stack.toString()); console.log(`Minimum: ${stack.getMin()}`); stack.push(1); stack.push(2); console.log( "Stack after adding new elements:" );
console.log(stack.toString()); console.log(`Minimum: ${stack.getMin()}`); console.log(`Element Popped: ${stack.pop()}`); console.log(`Element Popped: ${stack.pop()}`); console.log( "Stack after removing top two elements: " );
console.log(stack.toString()); console.log(`Minimum: ${stack.getMin()}`); console.log(`Top: ${stack.peek()}`); // This code is contributed by sdeadityasharma |
Elements in the stack are: 4 3 5 Minimum: 3 Stack after adding new elements: 2 1 4 3 5 Minimum: 1 Element Popped: 2 Element Popped: 1 Stack after removing top two elements: 4 3 5 Minimum: 3 Top: 4