#### Approach #1: Two Stacks [Accepted]

Intuition and Algorithm

A regular stack already supports the first 3 operations, so we focus on the last two.

For peekMax, we remember the largest value we've seen on the side. For example if we add [2, 1, 5, 3, 9], we'll remember [2, 2, 5, 5, 9]. This works seamlessly with pop operations, and also it's easy to compute: it's just the maximum of the element we are adding and the previous maximum.

For popMax, we know what the current maximum (peekMax) is. We can pop until we find that maximum, then push the popped elements back on the stack.

Our implementation in Python will showcase extending the list class.

Complexity Analysis

• Time Complexity: for the popMax operation, and for the other operations, where is the number of operations performed.

• Space Complexity: , the maximum size of the stack.

#### Approach #2: Double Linked List + TreeMap [Accepted]

Intuition

Using structures like Array or Stack will never let us popMax quickly. We turn our attention to tree and linked-list structures that have a lower time complexity for removal, with the aim of making popMax faster than time complexity.

Say we have a double linked list as our "stack". This reduces the problem to finding which node to remove, since we can remove nodes in time.

We can use a TreeMap mapping values to a list of nodes to answer this question. TreeMap can find the largest value, insert values, and delete values, all in time.

Algorithm

Let's store the stack as a double linked list dll, and store a map from value to a List of Node.

• When we MaxStack.push(x), we add a node to our dll, and add or update our entry map.get(x).add(node).

• When we MaxStack.pop(), we find the value val = dll.pop(), and remove the node from our map, deleting the entry if it was the last one.

• When we MaxStack.popMax(), we use the map to find the relevant node to unlink, and return it's value.

The above operations are more clear given that we have a working DoubleLinkedList class. The implementation provided uses head and tail sentinels to simplify the relevant DoubleLinkedList operations.

A Python implementation was not included for this approach because there is no analog to TreeMap available.

Complexity Analysis

• Time Complexity: for all operations except peek which is , where is the number of operations performed. Most operations involving TreeMap are .

• Space Complexity: , the size of the data structures used.

Analysis written by: @awice.