In computer science, a queap is a priority queue data structure. The data structure allows insertions and deletions of arbitrary elements, as well as retrieval of the highest-priority element. Each deletion takes amortized time logarithmic in the number of items that have been in the structure for a longer time than the removed item. Insertions take constant amortized time.
The data structure consists of a doubly linked list and a 2–4 tree data structure, each modified to keep track of its minimum-priority element. The basic operation of the structure is to keep newly inserted elements in the doubly linked list, until a deletion would remove one of the list items, at which point they are all moved into the 2–4 tree. The 2–4 tree stores its elements in insertion order, rather than the more conventional priority-sorted order.
Both the data structure and its name were devised by John Iacono and Stefan Langerman.[1]
A queap is a priority queue that inserts elements in O(1) amortized time, and removes the minimum element in O(log(k + 2)) if there are k items that have been in the heap for a longer time than the element to be extracted. The queap has a property called the queueish property: the time to search for element x is O(lg q(x)) where q(x) is equal to n - 1 - w(x) and w(x) is the number of distinct items that has been accessed by operations such as searching, inserting, or deleting. q(x) is defined as how many elements have not been accessed since x's last access. Indeed, the queueish property is the complement of the splay tree working set property: the time to search for element x is O(lg w(x)).
A queap can be represented by two data structures: a doubly linked list and a modified version of 2–4 tree. The doubly linked list, L, is used for a series of insert and locate-min operations. The queap keeps a pointer to the minimum element stored in the list. To add element x to list l, the element x is added to the end of the list and a bit variable in element x is set to one. This operation is done to determine if the element is either in the list or in a 2–4 tree.
A 2–4 tree is used when a delete operation occurs. If the item x is already in tree T, the item is removed using the 2–4 tree delete operation. Otherwise, the item x is in list L (done by checking if the bit variable is set). All the elements stored in list L are then added to the 2–4 tree, setting the bit variable of each element to zero. x is then removed from T.
A queap uses only the 2–4 tree structure properties, not a search tree. The modified 2–4 tree structure is as follows. Suppose list L has the following set of elements:
x1,x2,x3,...,xk
hv
x0
cv
T-Tv-\{r\}
hv
| c | |
| x0 |
An application of queaps includes a unique set of high priority events and extraction of the highest priority event for processing.
Let minL be a pointer that points to the minimum element in the doubly linked list L,
| c | |
| x0 |
New(Q): Initializes a new empty queap.
Initialize an empty doubly linked list L and 2–4 tree T. Set k and n to zero.
Insert(Q, x): Add the element x to queap Q.
Insert the element x in list L. Set the bit in element x to one to demonstrate that the element is in the list L. Update the minL pointer if x is the smallest element in the list. Increment n by 1.
Minimum(Q): Retrieve a pointer to the smallest element from queap Q.
If key(minL) < key(
| c | |
| x0 |
| c | |
| x0 |
Delete(Q, x): Remove element x from queap Q.
If the bit of the element x is set to one, the element is stored in list L. Add all the elements from L to T, setting the bit of each element to zero. Each element is added to the parent of the right most child of T using the insert operation of the 2–4 tree. L becomes empty. Update
hv
x0
cv
If the bit of the element x is set to zero, x is a leaf of T. Delete x using the 2–4 tree delete operation. Starting from node x, walk in T to node
x0
hv
cv
DeleteMin(Q): Delete and return the smallest element from queap Q.
Invoke the Minimum(Q) operation. The operation returns min. Invoke the Delete(Q, min) operation. Return min.
CleanUp(Q): Delete all the elements in list L and tree T.
Starting from the first element in list L, traverse the list, deleting each node.
Starting from the root of the tree T, traverse the tree using the post-order traversal algorithm, deleting each node in the tree.
The running time is analyzed using the amortized analysis. The potential function for queap Q will be
\phi(Q)=c|L|
Q=(T,L)
Insert(Q, x): The cost of the operation is O(1). The size of list L grows by one, the potential increases by some constant c.
Minimum(Q): The operation does not alter the data structure so the amortized cost is equal to its actual cost, O(1).
Delete(Q, x): There are two cases.
If x is in tree T, then the amortized cost is not modified. The delete operation is O(1) amortized 2–4 tree. Since x was removed from the tree,
hv
cv
O(lgq(x))
If x is in list L, then all the elements from L are inserted in T. This has a cost of
a|L|
hv
cv
2a|L|
x0
hv
cv
2a|L|+O(lgq(x))
c>2a
O(lgq(x))
O(lgq(x))
A small Java implementation of a queap: