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Fredman, Sedgewick, Sleator and Tarjan proposed the pairing heap as a self-adjusting, streamlined version of the Fibonacci heap. It provably supports all priority queue operations in logarithmic time and is known to be extremely efficient in practice. However despite its simplicity and empirical superiority, the pairing heap is one of the few popular data structures whose basic complexity remains open. In this paper we prove that pairing heaps support the deletemin operation in optimal logarithmic time and all other operations (insert, meld, and decreasekey) in time O(22 √(log log n)). This result gives the first sub-logarithmic time bound for decreasekey and comes close to the lower bound of Ω(log log n) established by Fredman. Pairing heaps have a well known but poorly understood relationship to splay trees and, to date, the transfer of ideas has flowed in one direction: from splaying to pairing. One contribution of this paper is a new analysis that reasons explicitly with information-theoretic measures. Whether these ideas could contribute to the analysis of splay trees is an open question.