  The pairing heap is a classical heap data structure introduced in 1986 by
Fredman, Sedgewick, Sleator, and Tarjan. It is remarkable both for its
simplicity and for its excellent performance in practice. The "magic" of
pairing heaps lies in the restructuring that happens after the deletion of the
smallest item. The resulting collection of trees is consolidated in two rounds:
a left-to-right pairing round, followed by a right-to-left accumulation round.
Fredman et al. showed, via an elegant correspondence to splay trees, that in a
pairing heap of size $n$ all operations take $O(\log{n})$ amortized time. They
also proposed an arguably more natural variant, where both pairing and
accumulation are performed in a combined left-to-right round (called the
forward variant of pairing heaps). The analogy to splaying breaks down in this
case, and the analysis of the forward variant was left open.
  In this paper we show that inserting an item and deleting the minimum in a
forward-variant pairing heap both take amortized time $O(\log{n} \cdot
4^{\sqrt{\log{n}}} )$. This is the first improvement over the $O(\sqrt{n})$
bound showed by Fredman et al. three decades ago. Our analysis relies on a new
potential function that tracks parent-child rank-differences in the heap.
