To implement a queue using only stacks, two stacks are needed. This is because a queue follows the FIFO (First-In-First-Out) principle, while a stack follows the LIFO (Last-In-First-Out) principle, so two stacks can be used to reverse the order of elements to simulate queue behavior.

How two stacks implement a queue:

• One stack (let's call it stack1) is used for enqueue (push) operations.
• The other stack (stack2) is used for dequeue (pop) operations.
• When dequeueing, if stack2 is empty, elements from stack1 are popped and pushed onto stack2, reversing their order so the oldest element is on top.
• Then, popping from stack2 gives the front of the queue.

This approach ensures that the first element pushed into the queue is the first one popped out, mimicking queue behavior.

Summary:

• Number of stacks required: 2
• Reason: One stack to hold incoming elements, the other to reverse order for correct dequeue.
• No other data structures needed: Only stacks are used, no arrays or linked lists are required.

This is a standard and well-known method for implementing a queue using stacks

Additional notes on complexity:

• Enqueue operation can be O(1) or O(N) depending on implementation.
• Dequeue operation is generally amortized O(1) when using two stacks.

Thus, two stacks are both necessary and sufficient to implement a queue when no other data structure is available.