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We give a variation of the pairing heaps for which the time bounds for all the operations match the lower bound proved by Fredman for a family of similar self-adjusting heaps. Namely, our heap structure requires O(1) for insert and findmin, O(log n) for delete-min, and O(log log n) for decreasekey and meld (all the bounds are in the amortized sense except for find-min).1 Introduction A self-adjusting data structure is a structure that does not maintain structural information (like size or height) within its nodes, still can adjust itself to perform efficiently and theoretically compete with other structures. Avoiding the restrictions governing other structures and the necessity of maintaining structural information, selfadjusting structures showed practical superiority over their counterparts.Following splay trees [12], a self-adjusting alternative to balanced trees with many other interesting properties, the next move was a self-adjusting heap. Fredman et al. [6] introduced the pairing heaps as a selfadjusting alternative to Fibonacci heaps. Despite the ingenuity of their structure and proofs, they were able to illustrate an amortized O(log n) bound for various heap operations. Lagging behind the Fibonacci heaps, the asymptotic time bounds for various heap operations except for delete-min were to be improved.The pairing heap [6] is a heap-ordered general tree. The values in the heap are stored one value per node. The basic operation on a pairing heap is the linking operation in which two trees are combined by linking the root with the larger key value to the other as its leftmost child. The following operations are defined for the standard implementation of the pairing heaps:

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.

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We give a new, simple proof for the sequential access theorem for splay trees. For an n-node splay tree, our bound on the number of rotations is 4:5n, with a smaller constant than the bound of 10:8n concluded by Tarjan. We extend our proof to prove the deque conjecture for output-restricted deques. Our proofs provide additional insights into the workings of splay trees.