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Design a stack to retrieve original elements and return the minimum element in O(1) time and O(1) space

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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: 

  1. If stack is empty 
    1. insert x into the stack
    2. make minimum equal to x.
  2. If stack is not empty 
    1. if x is less than minimum 
      1. set temp equal to 2*x-minimum
      2. set minimum equal to x
      3. set x equal to temp
    2. insert x into stack

When the operation Pop(x) is done, we follow the given below algorithm:  

  1. If stack is not empty 
    1. set x equal to topmost element
    2. if x is less than minimum 
      1. set minimum equal to 2*minimum – x
      2. set x equal to (x+minimum)/2
    3. return x

When getMin() is called, we return the element stored in variable, minimum:  

Implementation:

C++




// 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




// 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()));
    }
}


Python3




"""
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#




// 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


Javascript




// 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


Output

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


Last Updated : 01 Mar, 2023
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