This paper resolves one of the longest standing basic problems in the streaming computational model. Namely, optimal construction of quantile sketches. An e approximate quantile sketch receives a stream of items x1,·,xn and allows one to approximate the rank of any query item up to additive error e n with probability at least 1-δ.The rank of a query x is the number of stream items such that xi ≤ x. The minimal sketch size required for this task is trivially at least 1/e.Felber and Ostrovsky obtain a O((1/e)log(1/e)) space sketch for a fixed δ.Without restrictions on the nature of the stream or the ratio between e and n, no better upper or lower bounds were known to date. This paper obtains an O((1/e)log log (1/δ)) space sketch and a matching lower bound. This resolves the open problem and proves a qualitative gap between randomized and deterministic quantile sketching for which an Ω((1/e)log(1/e)) lower bound is known. One of our contributions is a novel representation and modification of the widely used merge-and-reduce construction. This modification allows for an analysis which is both tight and extremely simple. The same technique was reported, in private communications, to be useful for improving other sketching objectives and geometric coreset constructions.

Optimal Quantile Approximation in Streams

Selection hyper-heuristics (HHs) are randomised search methodologies which choose and execute heuristics during the optimisation process from a set of low-level heuristics. A machine learning mecha...

Simple Hyper-Heuristics Control the Neighbourhood Size of Randomised Local Search Optimally for LeadingOnes.

This paper resolves one of the longest standing basic problems in the streaming computational model. Namely, optimal construction of quantile sketches. An $\varepsilon$ approximate quantile sketch receives a stream of items $x_1,\ldots,x_n$ and allows one to approximate the rank of any query up to additive error $\varepsilon n$ with probability at least $1-\delta$. The rank of a query $x$ is the number of stream items such that $x_i \le x$. The minimal sketch size required for this task is trivially at least $1/\varepsilon$. Felber and Ostrovsky obtain a $O((1/\varepsilon)\log(1/\varepsilon))$ space sketch for a fixed $\delta$. To date, no better upper or lower bounds were known even for randomly permuted streams or for approximating a specific quantile, e.g.,\ the median. This paper obtains an $O((1/\varepsilon)\log \log (1/\delta))$ space sketch and a matching lower bound. This resolves the open problem and proves a qualitative gap between randomized and deterministic quantile sketching. One of our contributions is a novel representation and modification of the widely used merge-and-reduce construction. This subtle modification allows for an analysis which is both tight and extremely simple. Similar techniques should be useful for improving other sketching objectives and geometric coreset constructions.

It is shown that the online binary search tree data structure GreedyASS performs asymptotically as well on a sufficiently long sequence of searches as any static binary search tree where each search begins from the previous search (rather than the root). This bound is known to be equivalent to assigning each item i in the search tree a positive weight wi and bounding the search cost of an item in the search sequence s1, . . ., sm by O(1 + log[EQUATION]) amortized. This result is the strongest finger-type bound to be proven for binary search trees. By setting the weights to be equal, one observes that our bound implies the dynamic finger bound. Compared to the previous proof of the dynamic finger bound for Splay trees, our result is significantly shorter, stronger, simpler, and has reasonable constants.

https://epubs.siam.org/doi/pdf/10.1137/1.9781611974331.ch49

Weighted dynamic finger in binary search trees

The Lempel-Ziv parsing of a string (LZ77 for short) is one of the most important and widely-used algorithmic tools in data compression and string processing. We show that the Lempel-Ziv parsing of a string of length n on an alphabet of size σ can be computed in O(n log log σ) time (O(n) time if we allow randomization) using O(n log σ) bits of working space; that is, using space proportional to that of the input string in bits. The previous fastest algorithm using O(n log σ) space takes O(n(log σ + log log n)) time. We also consider the important rightmost variant of the problem, where the goal is to associate with each phrase of the parsing its most recent occurrence in the input string. We solve this problem in O(n(1 + (log σ/[EQUATION])) time, using the same working space as above. The previous best solution for rightmost parsing uses O(n(1 + log σ/log log n)) time and O(n log n) space. As a bonus, in our solution for rightmost parsing we provide a faster construction method for efficient 2D orthogonal range reporting, which is of independent interest.

Range predecessor and lempel-ziv parsing

The Query Complexity of Finding a Hidden Permutation -- Bounds for Scheduling Jobs on Grid Processors -- Quake Heaps: A Simple Alternative to Fibonacci Heaps -- Variations on Instant Insanity -- A Simple Linear-Space Data Structure for Constant-Time Range -- Closing a Long-Standing Complexity Gap for Selection: V3(42) = 50 -- Frugal Streaming for Estimating Quantiles -- From Time to Space: Fast Algorithms That Yield Small and Fast Data -- Computing (and life) Is all About Tradeoffs: A Small Sample of Some Computational Tradeoffs -- A History of Distribution-Sensitive Data Structures -- A Survey on Priority Queues -- On Generalized Comparison-Based Sorting Problems -- A Survey of the Game “Lights Out!” -- Random Access to High-Order Entropy Compressed Text -- Succinct and Implicit Data Structures for Computational Geometry -- In Pursuit of the Dynamic Optimality Conjecture -- A Survey of Algorithms and Models for List Update -- Orthogonal Range Searching for Text Indexing -- A Survey of Data Structures in the Bitprobe Model -- Succinct Representations of Ordinal Trees -- Array Range Queries -- Indexes for Document Retrieval with Relevance.

Alfredo Viola

Papers

Space-Efficient Data Structures, Streams, and Algorithms Papers in Honor of J. Ian Munro, on the Occasion of His 66th Birthday

Space-Efficient Data Structures, Streams, and Algorithms