A Priority Queue With the Working-Set Property

Abstract

A new priority queue is introduced, for which the cost to insert a new item is constant and the cost to delete the item x with the minimum value is O( log kx), where kx, is the number of items that are inserted after x and are still in the heap when x is deleted. We achieve the above bounds in both the amortized case and the worst case.

A priority queue with the time-finger property

“…Our priority queue builds on the priority queue in [6], which supports insert in constant time and delete-min within the working-set bound. The advantage of the priority queue in [6] over those in [2,9,10] is that it satisfies the stronger workingset property, in which elements that are deleted do not count towards the working set.…”

“…The advantage of the priority queue in [6] over those in [2,9,10] is that it satisfies the stronger workingset property, in which elements that are deleted do not count towards the working set. Next we outline the structure of this priority queue.…”

“…The priority queue in [6] is comprised of heap-ordered (2, 3)-binomial trees. As defined in [6], the subtrees of the root of a (2, 3)-binomial tree of rank r are (2, 3)-binomial trees; there are one or two children having ranks 0, 1, .…”

“…Also priority queues have been designed and analyzed in the context of distribution-sensitivity [2,6,10,12]. In the comparison model, it is easy to observe that a priority queue with constant insertion time cannot have the sequential-access property and hence cannot as well have the dynamic-finger property; for otherwise, a sequence of insertions followed by a sequence of minimum-deletions would list the elements in sorted order in linear time (contradicting the Ω[n log n) lower bound for comparison-based sorting).…”

“…Funnel-heaps [2] are I/O-efficient heaps for which it takes O (log(min{i x , n})) time to delete the minimum x, where i x is the number of insertions performed since x's insertion. Elmasry [6] gave a priority queue supporting the deletion of the minimum x in O (log w x ) worst-case time, where w x is the number of elements inserted after the insertion of x and are still present in the priority queue when x is deleted (note that w x i x o x ); we briefly review this priority queue in Section 2. None of the aforementioned priority queues supports delete (as introduced, they only support delete-min) within the working-set bound.…”