John Iacono

Dynamic Optimality—Almost

We present an O(lg lg n)-competitive online binary search tree, improving upon the best previous (trivial) competitive ratio of O(lg n). This is the first major progress on Sleator and Tarjans dynamic optimality conjecture of 1985 that O(1)-competitive binary search trees exist.

Improved Upper Bounds for Pairing Heaps

Pairing heaps are shown to have constant amortized time insert and zero amortized time meld, thus improving the previous O(log n) amortized time bound on these operations. It is also shown that pairing heaps have a distribution sensitive behavior whereby the cost to perform an extract-min on an element x is O(log min(n, k)) where k is the number of heap operations performed since xs insertion. Fredman has observed that pairing heaps can be used to merge sorted lists of varying sized optimally, within constant factors. Utilizing the distribution sensitive behavior of pairing heap, an alternative method the employs pairing heaps for optimal list merging is derived.

Space-efficient planar convex hull algorithms

A space-efficient algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. We describe four space-efficient algorithms for computing the convex hull of a planar point set.

The cost of cache-oblivious searching

Tight bounds on the cost of cache-oblivious searching are proved. It is shown that no cache-oblivious search structure can guarantee that a search performs fewer than lg e log/sub B/N block transfers between any two levels of the memory hierarchy. This lower bound holds even if all of the block sizes are limited to be powers of 2. A modified version of the van Emde Boas layout is proposed, whose expected block transfers between any two levels of the memory hierarchy arbitrarily close to [lg e + O(lg lg B/ lgB)] logB N + O(1). This factor approaches lg e /spl ap/ 1.443 as B increases. The expectation is taken over the random placement of the first element of the structure in memory. As searching in the disk access model (DAM) can be performed in log/sub B/N + 1 block transfers, this result shows a separation between the 2-level DAM and cache-oblivious memory-hierarchy models. By extending the DAM model to k levels, multilevel memory hierarchies can be modeled. It is shown that as k grows, the search costs of the optimal k-level DAM search structure and of the optimal cache-oblivious search structure rapidly converge. This demonstrates that for a multilevel memory hierarchy, a simple cache-oblivious structure almost replicates the performance of an optimal parameterized k-level DAM structure.

A locality-preserving cache-oblivious dynamic dictionary

This paper presents a simple dictionary structure designed for a hierarchical memory. The proposed data structure is <i>cache oblivious</i> and <i>locality preserving.</i> A cache-oblivious data structure has memory performance optimized for all levels of the memory hierarchy even though it has no memory-hierarchy-specific parameterization. A locality-preserving dictionary maintains elements of similar key values stored close together for fast access to ranges of data with consecutive keys.The data structure presented here is a simplification of the cache-oblivious B-tree of Bender, Demaine, and Farach-Colton. Like the cache-oblivious B-tree, this structure supports search operations using only <i>O</i>(log<inf><i>B</i></inf><i>N</i>) block operations at a level of the memory hierarchy with block size <i>B.</i> Insertion and deletion operations use <i>O</i>(log<inf><i>B</i></inf><i>N</i> + log² <i>N/B</i>) amortized block transfers. Finally, the data structure returns all <i>k</i> data items in a given search range using <i>O</i>(log<inf><i>B</i></inf><i>N + k/B</i>) block operations.This data structure was implemented and its performance was evaluated on a simulated memory hierarchy. This paper presents the results of this simulation for various combinations of block and memory sizes.

A unified access bound on comparison-based dynamic dictionaries

We present a dynamic comparison-based search structure that supports insertions, deletions, and searches within the unified bound. The unified bound specifies that it is quick to access an element that is near a recently accessed element. More precisely, if w(y) distinct elements have been accessed since the last access to element y, and d(x,y) denotes the rank distance between x and y among the current set of elements, then the amortized cost to access element x is O(minylog[w(y)+d(x,y)+2]). This property generalizes the working-set and dynamic-finger properties of splay trees.

Expected asymptotically optimal planar point location

Given a fixed distribution of point location queries among the triangles in a triangulation of the plane, a data structure is presented that achieves, within constant multiplicative factors, the entropy bound on the expected point location query time. The data structure is a simple variation of Kirkpatricks classic planar point location structure [D.G. Kirkpatrick, SIAM J. Comput. 12 (1) (1983) 28-35], and has linear construction costs and space requirements.

Necklaces, convolutions, and X+Y

We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓp norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p=1, p even, and p=∞. For p even, we reduce the problem to standard convolution, while for p=∞ and p=1, we reduce the problem to (min,+) convolution and

Output-Sensitive Algorithms for Computing Nearest-Neighbour Decision Boundaries

(\operatorname {median},+)

Encoding 2D range maximum queries

convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X+Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X+Y matrix. All of our algorithms run in o(n2) time, whereas the obvious algorithms for these problems run in Θ(n2) time.