Design and Implement Special Stack Data Structure | Added Space Optimized Version - GeeksforGeeks

21 Feb, 2022
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 the 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 the 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.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 the auxiliary stack.
We can see that all the 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
The following is the implementation for SpecialStack class. In the below implementation, SpecialStack inherits from Stack and has one Stack object min which works as an auxiliary stack.
C++
Java
Python3
#include <iostream>#include <stdlib.h> using namespace std; /* A simple stack class withbasic stack functionalities */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 checkif the stack is empty */bool Stack::isEmpty(){    if (top == -1)        return true;    return false;} /* Stack's member method to checkif the stack is full */bool Stack::isFull(){    if (top == max - 1)        return true;    return false;} /* Stack's member method to removean element from it */int Stack::pop(){    if (isEmpty()) {        cout << "Stack Underflow";        abort();    }    int x = arr[top];    top--;    return x;} /* Stack's member method to insertan element to it */void Stack::push(int x){    if (isFull()) {        cout << "Stack Overflow";        abort();    }    top++;    arr[top] = x;} /* A class that supports all the stackoperations and one additional  operation getMin() that returns theminimum element from stack at  any time.  This class inherits fromthe stack class and uses an  auxiliary stack that holds minimumelements */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 alsoupdated 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 toremove 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;}
// Java implementation of SpecialStack// Note : here we use Stack class for// Stack implementation import java.util.Stack; /* A class that supports all thestack operations and one additionaloperation getMin() that returnsthe minimum element from stack atany time. This class inherits fromthe stack class and uses anauxiliary stack that holds minimum elements */ class SpecialStack extends Stack<Integer> {    Stack<Integer> min = new Stack<>();     /* SpecialStack's member method toinsert an element to it. This method    makes sure that the min stack isalso updated with appropriate minimum    values */    void push(int x)    {        if (isEmpty() == true) {            super.push(x);            min.push(x);        }        else {            super.push(x);            int y = min.pop();            min.push(y);            if (x < y)                min.push(x);            else                min.push(y);        }    }     /* SpecialStack's member method toinsert an element to it. This method    makes sure that the min stack isalso updated with appropriate minimum    values */    public Integer pop()    {        int x = super.pop();        min.pop();        return x;    }     /* SpecialStack's member method to getminimum element from it. */    int getMin()    {        int x = min.pop();        min.push(x);        return x;    }     /* Driver program to test SpecialStackmethods */    public static void main(String[] args)    {        SpecialStack s = new SpecialStack();        s.push(10);        s.push(20);        s.push(30);        System.out.println(s.getMin());        s.push(5);        System.out.println(s.getMin());    }}// This code is contributed by Sumit Ghosh
# Python3 program for the# above approach# A simple stack class with # basic stack functionalitiesclass stack:   def __init__(self):     self.array = []    self.top = -1    self.max = 100    # Stack's member method to check  # if the stack is empty  def isEmpty(self):     if self.top == -1:      return True    else:      return False    # Stack's member method to check  # if the stack is full   def isFull(self):          if self.top == self.max - 1:      return True    else:      return False     # Stack's member method to  # insert an element to it     def push(self, data):     if self.isFull():      print('Stack OverFlow')      return    else:      self.top += 1      self.array.append(data)       # Stack's member method to  # remove an element from it  def pop(self):     if self.isEmpty():      print('Stack UnderFlow')      return    else:      self.top -= 1      return self.array.pop() # 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# auxiliary stack that holds# minimum elements class SpecialStack(stack):   def __init__(self):    super().__init__()    self.Min = stack()    # 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  def push(self, x):     if self.isEmpty():      super().push(x)      self.Min.push(x)    else:      super().push(x)      y = self.Min.pop()      self.Min.push(y)      if x <= y:        self.Min.push(x)      else:        self.Min.push(y)    # SpecialStack's member method to   # remove an element from it. This  # method removes top element from  # min stack also.  def pop(self):     x = super().pop()    self.Min.pop()    return x    # SpecialStack's member method  # to get minimum element from it.  def getmin(self):     x = self.Min.pop()    self.Min.push(x)    return x # Driver codeif __name__ == '__main__':     s = SpecialStack()  s.push(10)  s.push(20)  s.push(30)  print(s.getmin())  s.push(5)  print(s.getmin()) # This code is contributed by rachitkatiyar99
10
5
Complexity Analysis: 
Time Complexity: For insert operation: O(1) (As insertion ‘push’ in a stack takes constant time)For delete operation: O(1) (As deletion ‘pop’ in a stack takes constant time)For ‘Get Min’ operation: O(1) (As we have used an auxiliary stack which has it’s top as the minimum element)
For insert operation: O(1) (As insertion ‘push’ in a stack takes constant time)For delete operation: O(1) (As deletion ‘pop’ in a stack takes constant time)For ‘Get Min’ operation: O(1) (As we have used an auxiliary stack which has it’s top as the minimum element)
For insert operation: O(1) (As insertion ‘push’ in a stack takes constant time)
For delete operation: O(1) (As deletion ‘pop’ in a stack takes constant time)
For ‘Get Min’ operation: O(1) (As we have used an auxiliary stack which has it’s top as the minimum element)
Auxiliary Space: O(n). Use of auxiliary stack for storing values.
Space Optimized Version The above approach can be optimized. We can limit the number of elements in the auxiliary stack. We can push only when the incoming element of the main stack is smaller than or equal to the top of the auxiliary stack. Similarly during pop, if the pop-off element equal to the top of the auxiliary stack, remove the top element of the auxiliary stack. Following is the modified implementation of push() and pop(). 
C++
Java
Python3
C#
Javascript
/* 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;}
/* SpecialStack's member method toinsert an element to it. This methodmakes sure that the min stack isalso updated with appropriate minimumvalues */ void push(int x){    if (isEmpty() == true) {        super.push(x);        min.push(x);    }    else {        super.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. */public Integer pop(){    int x = super.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;} // This code is contributed by Sumit Ghosh
''' SpecialStack's member method toinsert an element to it. This methodmakes sure that the min stack isalso updated with appropriate minimumvalues ''' def push(x):    if (isEmpty() == True):        super.append(x);        min.append(x);         else:        super.append(x);        y = min.pop();        min.append(y);         ''' push only when the incoming           element of main stack is smaller        than or equal to top of auxiliary stack '''        if (x <= y):            min.append(x);       ''' SpecialStack's member method to   remove an element from it. This method   removes top element from min stack also. '''def pop():    x = super.pop();    y = min.pop();     ''' Push the popped element y back       only if it is not equal to x '''    if (y != x):        min.append(y);    return x;  # This code contributed by umadevi9616
/* SpecialStack's member method toinsert an element to it. This methodmakes sure that the min stack isalso updated with appropriate minimumvalues */ void push(int x){    if (min.Count==0) {        super.Push(x);        min.Push(x);    }    else {        super.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. */public int pop(){    int x = super.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;} // This code is contributed by umadevi9616
<script>/* SpecialStack's member method toinsert an element to it. This methodmakes sure that the min stack isalso updated with appropriate minimumvalues */ function push(x){    if (isEmpty() == true) {        super.push(x);        min.push(x);    }    else {        super.push(x);        var 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. */ function pop(){    var x = super.pop();    var 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;} // This code is contributed by umadevi9616</script>
Complexity Analysis:  
Time Complexity: For Insert operation: O(1) (As insertion ‘push’ in a stack takes constant time)For Delete operation: O(1) (As deletion ‘pop’ in a stack takes constant time)For ‘Get Min’ operation: O(1) (As we have used an auxiliary which has it’s top as the minimum element)
For Insert operation: O(1) (As insertion ‘push’ in a stack takes constant time)For Delete operation: O(1) (As deletion ‘pop’ in a stack takes constant time)For ‘Get Min’ operation: O(1) (As we have used an auxiliary which has it’s top as the minimum element)
For Insert operation: O(1) (As insertion ‘push’ in a stack takes constant time)
For Delete operation: O(1) (As deletion ‘pop’ in a stack takes constant time)
For ‘Get Min’ operation: O(1) (As we have used an auxiliary which has it’s top as the minimum element)
Auxiliary Space: O(n). The complexity in the worst case is the same as above but in other cases, it will take slightly less space than the above approach as repetition is neglected.
Further optimized O(1) time complexity and O(1) space complexity solution :The above approach can be optimized further and the solution can be made to work in O(1) time complexity and O(1) space complexity. The idea is to store min element found till current insertion) along with all the elements as a reminder of a DUMMY_VALUE, and the actual element as a multiple of the DUMMY_VALUE.For example, while pushing an element ‘e’ into the stack, store it as (e * DUMMY_VALUE + minFoundSoFar), this way we know what was the minimum value present in the stack at the time ‘e’ was being inserted.
To pop the actual value just return e/DUMMY_VALUE and set the new minimum as (minFoundSoFar % DUMMY_VALUE).
Note: Following method will fail if we try to insert DUMMY_VALUE in the stack, so we have to make our selection of DUMMY_VALUE carefully.Let’s say the following elements are being inserted in the stack – 3 2 6 1 8 5
d is dummy value.
s is wrapper stack
top is top element of the stack
min is the minimum value at that instant when the elements were inserted/removed
The following steps shows the current state of the above variables at any instant – 
s.push(3);min=3 //updated min as stack here is empty s = {3*d + 3}top = (3*d + 3)/d = 3  s.push(2);min = 2 //updated min as min > current elements = {3*d + 3-> 2*d + 2}top = (2*d + 2)/d = 2 s.push(6);min = 2s = {3*d + 3-> 2*d + 2-> 6*d + 2}top = (6*d + 2)/d = 6 s.push(1);min = 1 //updated min as min > current elements = {3*d + 3-> 2*d + 2-> 6*d + 2 -> 1*d + 1}top = (1*d + 1)/d = 1 s.push(8);min = 1s = {3*d + 3-> 2*d + 2-> 6*d + 2 -> 1*d + 1 -> 8*d + 1}top = (8*d + 1)/d = 8 s.push(5);min = 1s = {3*d + 3-> 2*d + 2-> 6*d + 2 -> 1*d + 1 -> 8*d + 1 -> 5*d + 1}top = (5*d + 1)/d = 5 s.pop();s = {3*d + 3 -> 2*d + 2 -> 6*d + 2 -> 1*d + 1 -> 8*d + 1 -> 5*d + 1}top = (5*d + 1)/d = 5min = (8*d + 1)%d = 1 // min is always remainder of the second top element in stack. s.pop();s = {3*d + 3 -> 2*d + 2-> 6*d + 2 -> 1*d + 1 -> 8*d + 1}top = (8*d + 1)/d = 8min = (1*d + 1)%d = 1 s.pop()s = {3*d + 3 -> 2*d + 2-> 6*d + 2 -> 1*d + 1} top = (1*d + 1)/d = 1min = (6*d + 2)%d = 2 s.pop()s = {3*d + 3-> 2*d + 2-> 6*d + 2}top = (6*d + 2)/d = 6min = (2*d + 2)%d = 2 s.pop()s = {3*d + 3-> 2*d + 2}top = (2*d + 2)/d = 2min = (3*d + 3)%d = 3 s.pop()s = {3*d + 3}top = (3*d + 3)/d = 3min  = -1 // since stack is now empty
s.push(3);min=3 //updated min as stack here is empty s = {3*d + 3}top = (3*d + 3)/d = 3  
s.push(2);min = 2 //updated min as min > current elements = {3*d + 3-> 2*d + 2}top = (2*d + 2)/d = 2 
s.push(6);min = 2s = {3*d + 3-> 2*d + 2-> 6*d + 2}top = (6*d + 2)/d = 6 
s.push(1);min = 1 //updated min as min > current elements = {3*d + 3-> 2*d + 2-> 6*d + 2 -> 1*d + 1}top = (1*d + 1)/d = 1 
s.push(8);min = 1s = {3*d + 3-> 2*d + 2-> 6*d + 2 -> 1*d + 1 -> 8*d + 1}top = (8*d + 1)/d = 8 
s.push(5);min = 1s = {3*d + 3-> 2*d + 2-> 6*d + 2 -> 1*d + 1 -> 8*d + 1 -> 5*d + 1}top = (5*d + 1)/d = 5 
s.pop();s = {3*d + 3 -> 2*d + 2 -> 6*d + 2 -> 1*d + 1 -> 8*d + 1 -> 5*d + 1}top = (5*d + 1)/d = 5min = (8*d + 1)%d = 1 // min is always remainder of the second top element in stack. 
s.pop();s = {3*d + 3 -> 2*d + 2-> 6*d + 2 -> 1*d + 1 -> 8*d + 1}top = (8*d + 1)/d = 8min = (1*d + 1)%d = 1 
s.pop()s = {3*d + 3 -> 2*d + 2-> 6*d + 2 -> 1*d + 1} top = (1*d + 1)/d = 1min = (6*d + 2)%d = 2 
s.pop()s = {3*d + 3-> 2*d + 2-> 6*d + 2}top = (6*d + 2)/d = 6min = (2*d + 2)%d = 2 
s.pop()s = {3*d + 3-> 2*d + 2}top = (2*d + 2)/d = 2min = (3*d + 3)%d = 3 
s.pop()s = {3*d + 3}top = (3*d + 3)/d = 3min  = -1 // since stack is now empty
C++
Java
C#
#include <iostream>#include <stack>#include <vector>using namespace std; /* A special stack having peek() pop() and * push() along with additional getMin() that * returns minimum value in a stack without * using extra space and all operations in O(1) * time.. ???? */class SpecialStack{         // Sentinel value for min    int min = -1;          // DEMO_VALUE    static const int demoVal = 9999;    stack<int> st; public:     void getMin()    {        cout << "min is: " << min << endl;    }     void push(int val)    {                 // If stack is empty OR current element        // is less than min, update min.        if (st.empty() || val < min)        {            min = val;        }                 // Encode the current value with        // demoVal, combine with min and        // insert into stack        st.push(val * demoVal + min);        cout << "pushed: " << val << endl;    }     int pop()    {           // if stack is empty return -1;        if ( st.empty() ) {          cout << "stack underflow" << endl ;          return -1;        }               int val = st.top();        st.pop();                 // If stack is empty, there would        // be no min value present, so        // make min as -1        if (!st.empty())            min = st.top() % demoVal;        else            min = -1;                     cout << "popped: " << val / demoVal << endl;                 // Decode actual value from        // encoded value        return val / demoVal;    }     int peek()    {                 // Decode actual value        // from encoded value        return st.top() / demoVal;    }}; // Driver Codeint main(){    SpecialStack s;     vector<int> arr = { 3, 2, 6, 1, 8, 5, 5, 5, 5 };     for(int i = 0; i < arr.size(); i++)    {        s.push(arr[i]);        s.getMin();    }         cout << endl;    for(int i = 0; i < arr.size(); i++)    {        s.pop();        s.getMin();    }    return 0;} // This code is contributed by scisaif
import java.util.Stack; /* A special stack having peek() pop() and push() along with * additional getMin() that returns minimum value in a stack * without using extra space and all operations in O(1) * time..  * */public class SpecialStack {     int min = -1; // sentinel value for min    static int demoVal = 9999; // DEMO_VALUE    Stack<Integer> st = new Stack<Integer>();     void getMin() { System.out.println("min is: " + min); }     void push(int val)    {        // if stack is empty OR current element is less than        // min, update min..        if (st.isEmpty() || val < min) {            min = val;        }         st.push(val * demoVal                + min); // encode the current value with                        // demoVal, combine with min and                        // insert into stack        System.out.println("pushed: " + val);    }     int pop()    {           // if stack is empty return -1;        if (st.isEmpty() ) {             System.out.println("stack underflow");               return -1;          }               int val = st.pop();         if (!st.isEmpty()) // if stack is empty, there would                           // be no min value present, so                           // make min as -1            min = st.peek() % demoVal;        else            min = -1;        System.out.println("popped: " + val / demoVal);        return val / demoVal; // decode actual value from                              // encoded value    }     int peek()    {        return st.peek() / demoVal; // decode actual value                                    // from encoded value    }     // Driver Code    public static void main(String[] args)    {        SpecialStack s = new SpecialStack();         int[] arr = { 3, 2, 6, 1, 8, 5, 5, 5, 5 };         for (int i = 0; i < arr.length; i++) {            s.push(arr[i]);            s.getMin();        }        System.out.println();        for (int i = 0; i < arr.length; i++) {            s.pop();            s.getMin();        }    }}
using System;using System.Collections.Generic;  /* A special stack having peek() pop() and push() along with * additional getMin() that returns minimum value in a stack * without using extra space and all operations in O(1) * time..  * */public class SpecialStack {     int min = -1; // sentinel value for min    static int demoVal = 9999; // DEMO_VALUE    Stack<int> st = new Stack<int>();     void getMin() { Console.WriteLine("min is: " + min); }     void push(int val)    {        // if stack is empty OR current element is less than        // min, update min..        if (st.Count==0 || val < min) {            min = val;        }         st.Push(val * demoVal                + min); // encode the current value with                        // demoVal, combine with min and                        // insert into stack        Console.WriteLine("pushed: " + val);    }     int pop()    {           // if stack is empty return -1;        if (st.Count==0 ) {             Console.WriteLine("stack underflow");               return -1;          }               int val = st.Pop();         if (st.Count!=0) // if stack is empty, there would                           // be no min value present, so                           // make min as -1            min = st.Peek() % demoVal;        else            min = -1;        Console.WriteLine("popped: " + val / demoVal);        return val / demoVal; // decode actual value from                              // encoded value    }     int peek()    {        return st.Peek() / demoVal; // decode actual value                                    // from encoded value    }     // Driver Code    public static void Main(String[] args)    {        SpecialStack s = new SpecialStack();         int[] arr = { 3, 2, 6, 1, 8, 5, 5, 5, 5 };         for (int i = 0; i < arr.Length; i++) {            s.push(arr[i]);            s.getMin();        }        Console.WriteLine();        for (int i = 0; i < arr.Length; i++) {            s.pop();            s.getMin();        }    }} // This code is contributed by gauravrajput1
pushed: 3
min is: 3
pushed: 2
min is: 2
pushed: 6
min is: 2
pushed: 1
min is: 1
pushed: 8
min is: 1
pushed: 5
min is: 1
pushed: 5
min is: 1
pushed: 5
min is: 1
pushed: 5
min is: 1

popped: 5
min is: 1
popped: 5
min is: 1
popped: 5
min is: 1
popped: 5
min is: 1
popped: 8
min is: 1
popped: 1
min is: 2
popped: 6
min is: 2
popped: 2
min is: 3
popped: 3
min is: -1
Complexity Analysis:  
For push() operation: O(1) (As insertion ‘push’ in a stack takes constant time)For pop() operation: O(1) (As pop operation in a stack takes constant time)
For ‘Get Min’ operation: O(1) (As we have maintained min variable throughout the code)
Auxiliary Space: O(1). No extra space is used.
https://youtu.be/1Ld7gbW1oHcDesign a stack that supports getMin() in O(1) time and O(1) extra spaceThanks to @Venki, @swarup, and @Jing Huang for their inputs.Please write comments if you find the above code incorrect, or find other ways to solve the same problem.
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