716. Max Stack

Design a max stack that supports push, pop, top, peekMax and popMax.

push(x) -- Push element x onto stack.
pop() -- Remove the element on top of the stack and return it.
top() -- Get the element on the top.
peekMax() -- Retrieve the maximum element in the stack.
popMax() -- Retrieve the maximum element in the stack, and remove it. If you find more than one maximum elements, only remove the top-most one.
Example 1:
MaxStack stack = new MaxStack();
stack.push(5); 
stack.push(1);
stack.push(5);
stack.top(); -> 5
stack.popMax(); -> 5
stack.top(); -> 1
stack.peekMax(); -> 5
stack.pop(); -> 1
stack.top(); -> 5

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 O(N)O(N) 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 O(1)O(1) 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 O(log N)O(logN) 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 DoubleLinkedListoperations.

class MaxStack {
    TreeMap<Integer, List<Node>> map;
    DoubleLinkedList dll;

    public MaxStack() {
        map = new TreeMap();
        dll = new DoubleLinkedList();
    }

    public void push(int x) {
        Node node = dll.add(x);
        if(!map.containsKey(x))
            map.put(x, new ArrayList<Node>());
        map.get(x).add(node);
    }

    public int pop() {
        int val = dll.pop();
        List<Node> L = map.get(val);
        L.remove(L.size() - 1);
        if (L.isEmpty()) map.remove(val);
        return val;
    }

    public int top() {
        return dll.peek();
    }

    public int peekMax() {
        return map.lastKey();
    }

    public int popMax() {
        int max = peekMax();
        List<Node> L = map.get(max);
        Node node = L.remove(L.size() - 1);
        dll.unlink(node);
        if (L.isEmpty()) map.remove(max);
        return max;
    }
}

class DoubleLinkedList {
    Node head, tail;

    public DoubleLinkedList() {
        head = new Node(0);
        tail = new Node(0);
        head.next = tail;
        tail.prev = head;
    }

    public Node add(int val) {
        Node x = new Node(val);
        x.next = tail;
        x.prev = tail.prev;
        tail.prev = tail.prev.next = x;
        return x;
    }

    public int pop() {
        return unlink(tail.prev).val;
    }

    public int peek() {
        return tail.prev.val;
    }

    public Node unlink(Node node) {
        node.prev.next = node.next;
        node.next.prev = node.prev;
        return node;
    }
}

class Node {
    int val;
    Node prev, next;
    public Node(int v) {val = v;}
}

　　

Complexity Analysis

• Time Complexity: O(log N)O(logN) for all operations except peek which is O(1)O(1), where NN is the number of operations performed. Most operations involving TreeMap are O(log N)O(logN).

• Space Complexity: O(N)O(N), the size of the data structures used.