Improved Bounds for Multipass Pairing Heaps and Path-Balanced Binary Search Trees

We revisit multipass pairing heaps and path-balanced binary search trees (BSTs), two classical algorithms for data structure maintenance. The pairing heap is a simple and efficient "self-adjusting" heap, introduced in 1986 by Fredman, Sedgewick, Sleator, and Tarjan. In the multipass variant (one of the original pairing heap variants described by Fredman et al.) the minimum item is extracted via repeated pairing rounds in which neighboring siblings are linked. Path-balanced BSTs, proposed by Sleator (cf. Subramanian, 1996), are a natural alternative to Splay trees (Sleator and Tarjan, 1983). In a path-balanced BST, whenever an item is accessed, the search path leading to that item is re-arranged into a balanced tree. Despite their simplicity, both algorithms turned out to be difficult to analyse. Fredman et al. showed that operations in multipass pairing heaps take amortized O(log n * log log n / log log log n) time. For searching in path-balanced BSTs, Balasubramanian and Raman showed in 1995 the same amortized time bound of O(log n * log log n / log log log n), using a different argument. In this paper we show an explicit connection between the two algorithms and improve both bounds to O(log n * 2^{log^* n} * log^* n), respectively O(log n * 2^{log^* n} * (log^* n)^2), where log^* denotes the slowly growing iterated logarithm function. These are the first improvements in more than three, resp. two decades, approaching the information-theoretic lower bound of Omega(log n)

Smooth heaps and a dual view of self-adjusting data structures

We present a new connection between self-adjusting binary search trees (BSTs) and heaps, two fundamental, extensively studied, and practically relevant families of data structures. Roughly speaking, we map an arbitrary heap algorithm within a natural model, to a corresponding BST algorithm with the same cost on a dual sequence of operations (i.e. the same sequence with the roles of time and key-space switched). This is the first general transformation between the two families of data structures. There is a rich theory of dynamic optimality for BSTs (i.e. the theory of competitiveness between BST algorithms). The lack of an analogous theory for heaps has been noted in the literature. Through our connection, we transfer all instance-specific lower bounds known for BSTs to a general model of heaps, initiating a theory of dynamic optimality for heaps. On the algorithmic side, we obtain a new, simple and efficient heap algorithm, which we call the smooth heap. We show the smooth heap to be the heap-counterpart of Greedy, the BST algorithm with the strongest proven and conjectured properties from the literature, widely believed to be instance-optimal. Assuming the optimality of Greedy, the smooth heap is also optimal within our model of heap algorithms. As corollaries of results known for Greedy, we obtain instance-specific upper bounds for the smooth heap, with applications in adaptive sorting. Intriguingly, the smooth heap, although derived from a non-practical BST algorithm, is simple and easy to implement (e.g. it stores no auxiliary data besides the keys and tree pointers). It can be seen as a variation on the popular pairing heap data structure, extending it with a "power-of-two-choices" type of heuristic.Comment: Presented at STOC 2018, light revision, additional figure