Stack with Minimum

Design a SpecialStack that supports push(x), pop(), peek(), and getMin() in O(1) time.

• push(x) → add element x
• pop() → remove top element
• peek() → return top element without removing; -1 if empty
• getMin() → return minimum element; -1 if empty

All operations run in O(1).

 Example: 

Input: operations[] = [push(2), push(3), peek(), pop(), getMin(), push(1), getMin()]
Output: [3, 2, 1]
Explanation:
push(2): Stack is [2]
push(3): Stack is [2, 3]
peek(): Top element is 3
pop(): Removes 3, stack is [2]
getMin(): Minimum element is 2
push(1): Stack is [2, 1]
getMin(): Minimum element is 1

Input: operations[] = [push(10), getMin(), push(5), getMin(), pop()]
Output: [10, 5]
Explanation:
push(10): Stack is [10]
getMin(): Minimum element is 10
push(5): Stack is [10, 5]
getMin(): Minimum element is 5
pop(): Removes 5, stack is [10]

Using an Auxiliary Stack - O(1) Time and O(n) Space

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)

• push x to the first stack (the stack with actual elements) 
• compare x with the top element of the second stack (the auxiliary stack). Let the top element be y. 
• If x is smaller than y then push x to the auxiliary stack. 
• If x is greater than y then push y to the auxiliary stack.

Pop()

• pop the top element from the auxiliary stack. 
• pop the top element from the actual stack

int getMin()

• Return the top element of the auxiliary stack.

#include <iostream>
#include <stack>
using namespace std;

class SpecialStack {
    stack<int> st; 
    stack<int> minStack; 

public:
    void push(int x) {
        st.push(x);
        
        // If the minStack is empty or the new element is smaller than 
        // the top of minStack, push it onto minStack
        if (minStack.empty() || x <= minStack.top()) {
            minStack.push(x);
        } else {
            
            // Otherwise, push the top element of minStack 
            // again to keep the minimum unchanged
            minStack.push(minStack.top());
        }
    }

    // Pop the top element from the stack
    void pop() {
        if (st.empty()) {
            return;
        }
        
        // Pop from both stacks
        st.pop();
        minStack.pop();
    }

    // Return the top element of the stack without removing it
    int peek() {
        if (st.empty()) {
            return -1;
        }
        return st.top();
    }

    // Get the minimum element in the stack
    int getMin() {
        if (minStack.empty()) {
            return -1;
        }
        return minStack.top();
    }
};

int main() {
    SpecialStack st;
    
    st.push(18);
    st.push(19);
    st.push(29);
    st.push(15);
    st.push(16);

    cout << st.getMin() << endl;

    return 0;
}

import java.util.Stack;

class SpecialStack {
    Stack<Integer> st = new Stack<>();
    Stack<Integer> minSt = new Stack<>();

    public void push(int x) {
        st.push(x);
        
        // If the minSt is empty or the new element is smaller than 
        // the top of minSt, push it onto minSt
        if (minSt.isEmpty() || x <= minSt.peek()) {
            minSt.push(x);
        } else {
        
            // Otherwise, push the top element of minSt 
            // again to keep the minimum unchanged
            minSt.push(minSt.peek());
        }
    }

    // Pop the top element from the stack
    public void pop() {
        if (st.isEmpty()) {
            return;
        }
        // Pop from both stacks
        st.pop();
        minSt.pop();
    }

    // Return the top element of the stack without removing it
    public int peek() {
        if (st.isEmpty()) {
            return -1;
        }
        return st.peek();
    }

    // Get the minimum element in the stack
    public int getMin() {
        if (minSt.isEmpty()) {
            return -1;
        }
        return minSt.peek();
    }
}

public class GfG {
    public static void main(String[] args) {
        SpecialStack st = new SpecialStack();
        
        st.push(18);
        st.push(19);
        st.push(29);
        st.push(15);
        st.push(16);

        System.out.println(st.getMin());
    }
}

class SpecialStack:
    def __init__(self):
        self.st = []
        self.minSt = []

    def push(self, x):
        self.st.append(x)
        
        # If minSt is empty or new element is smaller than 
        # the top of minSt, push it
        if not self.minSt or x <= self.minSt[-1]:
            self.minSt.append(x)
        else:
        
            # Otherwise, repeat the top of minSt 
            self.minSt.append(self.minSt[-1])

    # Pop the top element
    def pop(self):
        if not self.st:
            return
        self.st.pop()
        self.minSt.pop()

    # Return top element
    def peek(self):
        if not self.st:
            return -1
        return self.st[-1]

    # Get the minimum element
    def getMin(self):
        if not self.minSt:
            return -1
        return self.minSt[-1]


if __name__ == '__main__':
    st = SpecialStack()
    
    st.push(18)
    st.push(19)
    st.push(29)
    st.push(15)
    st.push(16)
    
    print(st.getMin())

using System;
using System.Collections.Generic;

class SpecialStack {
    Stack<int> st = new Stack<int>();
    Stack<int> minSt = new Stack<int>();

    public void push(int x) {
        st.Push(x);
        
        // If minSt is empty or new element is smaller than 
        // the top of minSt, push it
        if (minSt.Count == 0 || x <= minSt.Peek()) {
            minSt.Push(x);
        } else {
        
            // Otherwise, repeat the top of minSt 
            minSt.Push(minSt.Peek());
        }
    }

    // Pop the top element
    public void pop() {
        if (st.Count == 0) {
            return;
        }
        st.Pop();
        minSt.Pop();
    }

    // Return top element
    public int peek() {
        if (st.Count == 0) {
            return -1;
        }
        return st.Peek();
    }

    // Get the minimum element
    public int getMin() {
        if (minSt.Count == 0) {
            return -1;
        }
        return minSt.Peek();
    }
}

class GfG {
    static void Main() {
        SpecialStack st = new SpecialStack();
        
        st.push(18);
        st.push(19);
        st.push(29);
        st.push(15);
        st.push(16);

        Console.WriteLine(st.getMin());
    }
}

class SpecialStack {
    constructor() {
        this.st = [];
        this.minStack = [];
    }

    push(x) {
        this.st.push(x);
        
        // If the minStack is empty or the new element is smaller than 
        // the top of minStack, push it onto minStack
        if (this.minStack.length === 0 ||
                x <= this.minStack[this.minStack.length - 1]) {
            this.minStack.push(x);
        } else {
        
            // Otherwise, push the top element of minStack 
            // again to keep the minimum unchanged
            this.minStack.push(this.minStack[this.minStack.length - 1]);
        }
    }

    // Pop the top element from the stack
    pop() {
        if (this.st.length === 0) {
            return;
        }
        // Pop from both stacks
        this.st.pop();
        this.minStack.pop();
    }

    // Return the top element of the stack without removing it
    peek() {
        if (this.st.length === 0) {
            return -1;
        }
        return this.st[this.st.length - 1];
    }

    // Get the minimum element in the stack
    getMin() {
        if (this.minStack.length === 0) {
            return -1;
        }
        return this.minStack[this.minStack.length - 1];
    }
}

// Driver Code
const st = new SpecialStack();

st.push(18);
st.push(19);
st.push(29);
st.push(15);
st.push(16);

console.log(st.getMin());

15

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)

Auxiliary Space: O(n)

Using a Pair in Stack - O(1) Time and O(n) Space

This approach uses a stack where each element is stored as a pair: the element itself and the minimum value up to that point. When an element is pushed, the minimum is updated. The getMin() function directly accesses the minimum value from the top of the stack in constant time, ensuring that both push(), pop(), and getMin() operations are O(1). This approach efficiently tracks the minimum value without needing to traverse the stack.

#include <iostream>
#include <stack>
using namespace std;

class SpecialStack {
private:
    stack<pair<int, int>> st;

public:
    SpecialStack() {
    }
    
    // Add an element to the top of stack
    void push(int x) {
        int newMin = st.empty() ? x : min(x, st.top().second);
        st.push({x, newMin});
    }
    
    // Remove the top element from the stack
    void pop() {
        if (!st.empty()) {
            st.pop();
        }
    }
    
    // Return top element of the stack
    int peek() {
        if (st.empty()) {
            return -1;
        }
        return st.top().first;
    }
    
    // Find minimum element of the stack
    int getMin() {
        if (st.empty()) {
            return -1;
        }
        return st.top().second;
    }
};

int main() {
    SpecialStack st;
    
    st.push(2);
    st.push(3);
    cout << st.peek() << " ";
    st.pop();
    cout << st.getMin() << " ";
    st.push(1);
    cout << st.getMin() << " ";
}

import java.util.Stack;

class SpecialStack {
    private Stack<int[]> st;

    public SpecialStack() {
        st = new Stack<>();
    }

    // Add an element to the top of Stack
    public void push(int x) {
        int newMin = st.isEmpty() ? x : Math.min(x, st.peek()[1]);
        st.push(new int[]{x, newMin});
    }

    // Remove the top element from the Stack
    public void pop() {
        if (!st.isEmpty()) {
            st.pop();
        }
    }

    // Returns top element of the Stack
    public int peek() {
        return st.isEmpty() ? -1 : st.peek()[0];
    }

    // Finds minimum element of Stack
    public int getMin() {
        return st.isEmpty() ? -1 : st.peek()[1];
    }

    public static void main(String[] args) {
        SpecialStack st = new SpecialStack();

        // Function calls
        st.push(2);
        st.push(3);
        System.out.print(st.peek() + " ");
        st.pop();
        System.out.print(st.getMin() + " ");
        st.push(1);
        System.out.print(st.getMin() + " ");
    }
}

class SpecialStack:
    def __init__(self):
        self.st = []

    # Add an element to the top of Stack
    def push(self, x):
        newMin = x if not self.st else min(x, self.st[-1][1])
        self.st.append((x, newMin))

    # Remove the top element from the Stack
    def pop(self):
        if self.st:
            self.st.pop()

    # Returns top element of the Stack
    def peek(self):
        return -1 if not self.st else self.st[-1][0]

    # Finds minimum element of Stack
    def getMin(self):
        return -1 if not self.st else self.st[-1][1]

if __name__ == "__main__":
    st = SpecialStack()

    st.push(2)
    st.push(3)
    print(st.peek(), end=" ")
    st.pop()
    print(st.getMin(), end=" ")
    st.push(1)
    print(st.getMin(), end=" ")

using System;
using System.Collections.Generic;

class SpecialStack {
    private Stack<(int, int)> st;

    public SpecialStack() {
        st = new Stack<(int, int)>();
    }

    // Add an element to the top of Stack
    public void push(int x) {
        int newMin = st.Count == 0 ? x : Math.Min(x, st.Peek().Item2);
        st.Push((x, newMin));
    }

    // Remove the top element from the Stack
    public void pop() {
        if (st.Count > 0) {
            st.Pop();
        }
    }

    // Returns top element of the Stack
    public int peek() {
        return st.Count == 0 ? -1 : st.Peek().Item1;
    }

    // Finds minimum element of Stack
    public int getMin() {
        return st.Count == 0 ? -1 : st.Peek().Item2;
    }

    public static void Main() {
        SpecialStack st = new SpecialStack();

        st.push(2);
        st.push(3);
        Console.Write(st.peek() + " ");
        st.pop();
        Console.Write(st.getMin() + " ");
        st.push(1);
        Console.Write(st.getMin() + " ");
    }
}

class SpecialStack {
    constructor() {
        this.st = [];
    }

    // Add an element to the top of Stack
    push(x) {
        let newMin = this.st.length === 0 ? x : 
                    Math.min(x, this.st[this.st.length - 1][1]);
        this.st.push([x, newMin]);
    }

    // Remove the top element from the Stack
    pop() {
        if (this.st.length > 0) {
            this.st.pop();
        }
    }

    // Returns top element of the Stack
    peek() {
        return this.st.length === 0 ? -1 : this.st[this.st.length - 1][0];
    }

    // Finds minimum element of Stack
    getMin() {
        return this.st.length === 0 ? -1 : this.st[this.st.length - 1][1];
    }
}

// Driver Code
const st = new SpecialStack();
st.push(2);
st.push(3);
console.log(st.peek() + " ");
st.pop();
console.log(st.getMin() + " ");
st.push(1);
console.log(st.getMin() + " ");

3 2 1 

Without Extra Space- O(1) Time and O(1) Space

The idea is to use a variable minEle to track the minimum element in the stack. Instead of storing the actual value of minEle in the stack, we store a modified value when pushing an element smaller than minEle.

Push(x)

• If the stack is empty, push x and set minEle = x.
• If x >= minEle, push x normally.
• If x < minEle, push 2*x - minEle and update minEle = x (this encodes the previous min).

Pop()

• Remove the top element.
• If the removed element is >= minEle, no change in minEle.
• If the removed element is < minEle, update minEle = 2*minEle - top (decoding the previous min).

Peek()

• Returns minEle if the top is modified (encoded) or top otherwise.

getMin()

• Returns minEle, the current minimum in O(1) time.

How this approach works?

When the element to be inserted is less than minEle, we insert "2x - minEle". The important thing to note is, that 2x - minEle will always be less than x (proved below), i.e., new minEle and while popping out this element we will see that something unusual has happened as the popped element is less than the minEle. So we will be updating minEle.

How 2*x - minEle is less than x in push()? 

x < minEle which means x - minEle < 0 

// Adding x on both sides
x - minEle + x < 0 + x 
2*x - minEle < x 
We can conclude 2*x - minEle < new minEle 

While popping out, if we find the element(y) less than the current minEle, we find the new minEle = 2*minEle - y

How previous minimum element, prevMinEle is, 2*minEle - y
in pop() is y the popped element?

 // We pushed y as 2x - prevMinEle. Here 
// prevMinEle is minEle before y was inserted

y = 2*x - prevMinEle  

// Value of minEle was made equal to x
minEle = x

new minEle = 2 * minEle - y 
                   = 2*x - (2*x - prevMinEle)
                   = prevMinEle // This is what we wanted

#include <iostream>
#include <stack>
using namespace std;

class SpecialStack {
private:
    stack<int> st;
    int minEle;
    
public:
    SpecialStack() {
        minEle = -1;
    }
    
    // Add an element to the top of stack
    void push(int x) {
        if (st.empty()) {
            minEle = x;
            st.push(x);
        }
        // If new number is less than minEle
        else if (x < minEle) {
            st.push(2 * x - minEle);
            minEle = x;
        }
        else {
            st.push(x);
        }
    }
    
    // Remove the top element from the stack
    void pop() {
        if (st.empty()) return;
        
        int top = st.top();
        st.pop();
        
        // Minimum will change if min element is removed
        if (top < minEle) {
            minEle = 2 * minEle - top;
        }
    }
    
    // Return top element of the stack
    int peek() {
        if (st.empty()) return -1;

        int top = st.top();
        return (minEle > top) ? minEle : top;
    }
    
    // Return minimum element of the stack
    int getMin() {
        if (st.empty()) return -1;
        return minEle;
    }
};

int main() {
    SpecialStack st;
    
    st.push(2);
    st.push(3);
    cout << st.peek() << " ";
    st.pop();
    cout << st.getMin() << " ";
    st.push(1);
    cout << st.getMin() << " ";
}

import java.util.Stack;

class SpecialStack {
    private Stack<Integer> st;
    private int minEle;

    public SpecialStack() {
        st = new Stack<>();
        minEle = -1;
    }

    // Add an element to the top of stack
    public void push(int x) {
        if (st.isEmpty()) {
            minEle = x;
            st.push(x);
        }
        // If new number is less than minEle
        else if (x < minEle) {
            st.push(2 * x - minEle);
            minEle = x;
        } else {
            st.push(x);
        }
    }

    // Remove the top element from the stack
    public void pop() {
        if (st.isEmpty()) return;

        int top = st.pop();

        // Minimum will change if min element is removed
        if (top < minEle) {
            minEle = 2 * minEle - top;
        }
    }

    // Return top element of the stack
    public int peek() {
        if (st.isEmpty()) return -1;

        int top = st.peek();

        // If minEle > top, minEle stores value of top
        return (minEle > top) ? minEle : top;
    }

    // Return minimum element of the stack
    public int getMin() {
        if (st.isEmpty()) return -1;

        // variable minEle stores the minimum element
        return minEle;
    }

    public static void main(String[] args) {
        SpecialStack st = new SpecialStack();

        st.push(2);
        st.push(3);
        System.out.print(st.peek() + " ");
        st.pop();
        System.out.print(st.getMin() + " ");
        st.push(1);
        System.out.print(st.getMin() + " ");
    }
}

class SpecialStack:
    def __init__(self):
        self.st = []
        self.minEle = -1

    # Add an element to the top of stack
    def push(self, x):
        if not self.st:
            self.minEle = x
            self.st.append(x)
       
        # If new number is less than minEle
        elif x < self.minEle:
            self.st.append(2 * x - self.minEle)
            self.minEle = x
        else:
            self.st.append(x)

    # Remove the top element from the stack
    def pop(self):
        if not self.st:
            return

        top = self.st.pop()

        # Minimum will change if min element is removed
        if top < self.minEle:
            self.minEle = 2 * self.minEle - top

    # Return top element of the stack
    def peek(self):
        if not self.st:
            return -1

        top = self.st[-1]

        # If minEle > top, minEle stores value of top
        return self.minEle if self.minEle > top else top

    # Return minimum element of the stack
    def getMin(self):
        if not self.st:
            return -1
        return self.minEle

if __name__ == '__main__':
    st = SpecialStack()
    
    st.push(2)
    st.push(3)
    print(st.peek(), end=" ")
    st.pop()
    print(st.getMin(), end=" ")
    st.push(1)
    print(st.getMin(), end=" ")

using System;
using System.Collections.Generic;

class SpecialStack {
    private Stack<int> st;
    private int minEle;

    public SpecialStack() {
        st = new Stack<int>();
        minEle = -1;
    }

    // Add an element to the top of stack
    public void push(int x) {
        if (st.Count == 0) {
            minEle = x;
            st.Push(x);
        }
        // If new number is less than minEle
        else if (x < minEle) {
            st.Push(2 * x - minEle);
            minEle = x;
        } else {
            st.Push(x);
        }
    }

    // Remove the top element from the stack
    public void pop() {
        if (st.Count == 0) return;

        int top = st.Pop();

        // Minimum will change if min element is removed
        if (top < minEle) {
            minEle = 2 * minEle - top;
        }
    }

    // Return top element of the stack
    public int peek() {
        if (st.Count == 0) return -1;

        int top = st.Peek();

        // If minEle > top, minEle stores value of top
        return (minEle > top) ? minEle : top;
    }

    // Return minimum element of the stack
    public int getMin() {
        if (st.Count == 0) return -1;
        return minEle;
    }

    static void Main() {
        SpecialStack st = new SpecialStack();

        // Function calls
        st.push(2);
        st.push(3);
        Console.Write(st.peek() + " ");
        st.pop();
        Console.Write(st.getMin() + " ");
        st.push(1);
        Console.Write(st.getMin() + " ");
    }
}

class SpecialStack {
    constructor() {
        this.st = [];
        this.minEle = -1;
    }

    // Add an element to the top of stack
    push(x) {
        if (this.st.length === 0) {
            this.minEle = x;
            this.st.push(x);
        }
        // If new number is less than minEle
        else if (x < this.minEle) {
            this.st.push(2 * x - this.minEle);
            this.minEle = x;
        } else {
            this.st.push(x);
        }
    }

    // Remove the top element from the stack
    pop() {
        if (this.st.length === 0) return;

        let top = this.st.pop();

        // Minimum will change if min element is removed
        if (top < this.minEle) {
            this.minEle = 2 * this.minEle - top;
        }
    }

    // Returns top element of the stack
    peek() {
        if (this.st.length === 0) return -1;

        let top = this.st[this.st.length - 1];

        // If minEle > top, minEle stores value of top
        return this.minEle > top ? this.minEle : top;
    }

    // Returns minimum element of the stack
    getMin() {
        if (this.st.length === 0) return -1;
        return this.minEle;
    }
}

// Driver Code
let st = new SpecialStack();
st.push(2);
st.push(3);
console.log(st.peek(), " ");
st.pop();
console.log(st.getMin(), " ");
st.push(1);
console.log(st.getMin(), " ");

3 2 1