Min Stack
Design a stack that supports push, pop, top, and retrieving the minimum element in O(1) time.
Input: operations separated by semicolons. Output results of top, getMin operations.
Example:
push,-2;push,0;push,-3;getMin;pop;top;getMin
-3 0 -2
- We start with an empty stack and apply the operations in sequence: push -2, push 0, push -3.
- The
getMinoperation returns the current minimum element, which is -3, so the first output is −3. - We then
popthe top element (-3), and thetopoperation returns the new top element, which is 0. - After the
topoperation, anothergetMinoperation is performed, returning the new minimum element, which is -2.
Constraints:
- -2^31 <= val <= 2^31 - 1
- pop, top, getMin always called on non-empty stacks
Background Knowledge
The Min Stack problem involves designing a data structure that supports standard stack operations like push, pop, and top, along with an additional operation to retrieve the minimum element in constant time, O(1). To approach this problem, it's essential to understand the basics of stacks and how they can be implemented using arrays or linked lists. A stack is a Last-In-First-Out (LIFO) data structure, meaning the last element added to the stack will be the first one to be removed.
Key concepts to grasp include the time complexity of various operations on stacks, such as push, pop, and accessing the top element, all of which are typically O(1) for a basic stack implementation. However, finding the minimum element in a standard stack would require iterating through all elements, resulting in a time complexity of O(n). To achieve O(1) time complexity for retrieving the minimum element, we need to consider additional data structures or modifications to the standard stack implementation.
Understanding auxiliary data structures like additional stacks or arrays can also be beneficial. These can help keep track of the minimum element at each step, ensuring that the minimum can be retrieved in constant time. Familiarity with trade-offs between time and space complexity is also important, as optimizing for one often affects the other.
Algorithm/Approach
The general approach to solving the Min Stack problem involves using a combination of data structures to maintain the minimum element alongside the main stack. This can be achieved by using an additional stack or array that keeps track of the minimum elements seen so far. The key idea is to update this auxiliary data structure whenever an element is pushed or popped from the main stack, ensuring that it always contains the current minimum element.
Step-by-Step Strategy
To implement the solution:
- Design a main stack to store the elements.
- Create an auxiliary data structure (e.g., another stack) to keep track of the minimum elements.
- When pushing an element onto the main stack, compare it with the current minimum (if any) and update the auxiliary data structure accordingly.
- When popping an element from the main stack, update the auxiliary data structure if the popped element was the current minimum.
- Implement the top operation to return the top element of the main stack.
- Implement the getMin operation to return the current minimum element from the auxiliary data structure.
Common Pitfalls
- Failing to update the auxiliary data structure correctly during push and pop operations.
- Not handling edge cases, such as an empty stack or pushing/popping the minimum element.
- Incorrectly implementing the top or getMin operations.
Time & Space Complexity
- Time Complexity: O(1) for push, pop, top, and getMin operations, assuming the auxiliary data structure is updated correctly.
- Space Complexity: O(n), where n is the number of elements in the stack, because we are using an auxiliary data structure to keep track of the minimum elements.