Double-ended priority queue

In computer science, a double-ended priority queue (DEPQ) or double-ended heap is a data structure similar to a priority queue or heap, but allows for efficient removal of both the maximum and minimum, according to some ordering on the keys (items) stored in the structure. Every element in a DEPQ has a priority or value. In a DEPQ, it is possible to remove the elements in both ascending as well as descending order.

A double-ended priority queue features the follow operations:

If an operation is to be performed on two elements having the same priority, then the element inserted first is chosen. Also, the priority of any element can be changed once it has been inserted in the DEPQ.

Double-ended priority queues can be built from balanced binary search trees (where the minimum and maximum elements are the leftmost and rightmost leaves, respectively), or using specialized data structures like min-max heap and pairing heap.

Generic methods of arriving at double-ended priority queues from normal priority queues are:

In this method two different priority queues for min and max are maintained. The same elements in both the PQs are shown with the help of correspondence pointers.
Here, the minimum and maximum elements are values contained in the root nodes of min heap and max heap respectively.

Half the elements are in the min PQ and the other half in the max PQ. Each element in the min PQ has a one-to-one correspondence with an element in max PQ. If the number of elements in the DEPQ is odd, one of the elements is retained in a buffer. Priority of every element in the min PQ will be less than or equal to the corresponding element in the max PQ.

In this method only the leaf elements of the min and max PQ form corresponding one-to-one pairs. It is not necessary for non-leaf elements to be in a one-to-one correspondence pair.

Apart from the above-mentioned correspondence methods, DEPQ's can be obtained efficiently using interval heaps. An interval heap is like an embedded min-max heap in which each node contains two elements. It is a complete binary tree in which:

Depending on the number of elements, two cases are possible -

Depending on the number of elements already present in the interval heap, following cases are possible:

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