Double ended priority queue

A double ended priority queue supports operations of both max heap (a max priority queue) and min heap (a min priority queue). The following operations are expected from double ended priority queue. 

1. getMax() : Returns maximum element.
2. getMin() : Returns minimum element.
3. deleteMax() : Deletes maximum element.
4. deleteMin() : Deletes minimum element.
5. size() : Returns count of elements.
6. isEmpty() : Returns true if the queue is empty.

We can try different data structure like Linked List. In case of linked list, if we maintain elements in sorted order, then time complexity of all operations become O(1) except the operation insert() which takes O(n) time. 

We can try two heaps (min heap and max heap). We maintain a pointer of every max heap element in min heap. To get minimum element, we simply return root. To get maximum element, we return root of max heap. To insert an element, we insert in min heap and max heap both. The main idea is to maintain one to one correspondence, so that deleteMin() and deleteMax() can be done in O(Log n) time. 

1. getMax() : O(1)
2. getMin() : O(1)
3. deleteMax() : O(Log n)
4. deleteMin() : O(Log n)
5. size() : O(1)
6. isEmpty() : O(1)

Another solution is to use self balancing binary search tree. A self balancing BST is implemented as set in C++ and TreeSet in Java.

1. getMax() : O(1)
2. getMin() : O(1)
3. deleteMax() : O(Log n)
4. deleteMin() : O(Log n)
5. size() : O(1)
6. isEmpty() : O(1)

Below is the implementation of above approach: 

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Comparison of Heap and BST solutions 
Heap based solution requires O(n) extra space for an extra heap. BST based solution does not require extra space. The advantage of heap based solution is cache friendly.