Theory and applications of hashing: report from Dagstuhl Seminar 17181

This report documents the program and the topics discussed of the 4-day Dagstuhl Seminar 17181 “Theory and Applications of Hashing”, which took place May 1–5, 2017. Four long and eighteen short talks covered a wide and diverse range of topics within the theme of the workshop. The program left sufficient space for informal discussions among the 40 participants

Compression with graphical constraints: An interactive browser

Abstract—We study the problem of searching for a given element in a set of objects using a membership oracle. The membership oracle, given a subset of objects A, and a target object t, determines whether A contains t or not. The goal is to find the target object with the minimum number of questions asked from the oracle. This problem is known to be strongly related to lossless source compression. In fact, the optimum strategy is provided by Hufmman coding with the average number of questions very close to the entropy H(P) of the object set. The membership oracle aims at modelling interactive methods (i.e., incorporate human feedback) has many real life applica-tions. Due to practical constraints imposed by such applications not every subset A of objects can be queried. It is known that in general finding the optimum strategy with such constrains is NP-complete. Given this negative result we restrict attention to the cases represented by graphical models: graph G whose nodes are the database objects is given, and the queries are restricted to be those subsets A that are connected in G. We show that when G itself is connected, there is a search algorithm that finds the target in 4H(P) + 2 queries on the average. Since entropy is the trivial lower bound, our algorithm performs within a constant gap from the optimum strategy. I

Reordering Rows for Better Compression: Beyond the Lexicographic Order

Sorting database tables before compressing them improves the compression rate. Can we do better than the lexicographical order? For minimizing the number of runs in a run-length encoding compression scheme, the best approaches to row-ordering are derived from traveling salesman heuristics, although there is a significant trade-off between running time and compression. A new heuristic, Multiple Lists, which is a variant on Nearest Neighbor that trades off compression for a major running-time speedup, is a good option for very large tables. However, for some compression schemes, it is more important to generate long runs rather than few runs. For this case, another novel heuristic, Vortex, is promising. We find that we can improve run-length encoding up to a factor of 3 whereas we can improve prefix coding by up to 80%: these gains are on top of the gains due to lexicographically sorting the table. We prove that the new row reordering is optimal (within 10%) at minimizing the runs of identical values within columns, in a few cases.Comment: to appear in ACM TOD

Net and Prune: A Linear Time Algorithm for Euclidean Distance Problems

We provide a general framework for getting expected linear time constant factor approximations (and in many cases FPTAS's) to several well known problems in Computational Geometry, such as $k$-center clustering and farthest nearest neighbor. The new approach is robust to variations in the input problem, and yet it is simple, elegant and practical. In particular, many of these well studied problems which fit easily into our framework, either previously had no linear time approximation algorithm, or required rather involved algorithms and analysis. A short list of the problems we consider include farthest nearest neighbor, $k$-center clustering, smallest disk enclosing $k$ points, $k$th largest distance, $k$th smallest $m$-nearest neighbor distance, $k$th heaviest edge in the MST and other spanning forest type problems, problems involving upward closed set systems, and more. Finally, we show how to extend our framework such that the linear running time bound holds with high probability

Non-Mergeable Sketching for Cardinality Estimation

Cardinality estimation is perhaps the simplest non-trivial statistical problem that can be solved via sketching. Industrially-deployed sketches like HyperLogLog, MinHash, and PCSA are mergeable, which means that large data sets can be sketched in a distributed environment, and then merged into a single sketch of the whole data set. In the last decade a variety of sketches have been developed that are non-mergeable, but attractive for other reasons. They are simpler, their cardinality estimates are strictly unbiased, and they have substantially lower variance. We evaluate sketching schemes on a reasonably level playing field, in terms of their memory-variance product (MVP). E.g., a sketch that occupies 5m bits and whose relative variance is 2/m (standard error ?{2/m}) has an MVP of 10. Our contributions are as follows. - Cohen [Edith Cohen, 2015] and Ting [Daniel Ting, 2014] independently discovered what we call the {Martingale transform} for converting a mergeable sketch into a non-mergeable sketch. We present a simpler way to analyze the limiting MVP of Martingale-type sketches. - Pettie and Wang proved that the Fishmonger sketch [Seth Pettie and Dingyu Wang, 2021] has the best MVP, H?/I? ? 1.98, among a class of mergeable sketches called "linearizable" sketches. (H? and I? are precisely defined constants.) We prove that the Martingale transform is optimal in the non-mergeable world, and that Martingale Fishmonger in particular is optimal among linearizable sketches, with an MVP of H?/2 ? 1.63. E.g., this is circumstantial evidence that to achieve 1% standard error, we cannot do better than a 2 kilobyte sketch. - Martingale Fishmonger is neither simple nor practical. We develop a new mergeable sketch called Curtain that strikes a nice balance between simplicity and efficiency, and prove that Martingale Curtain has limiting MVP? 2.31. It can be updated with O(1) memory accesses and it has lower empirical variance than Martingale LogLog, a practical non-mergeable version of HyperLogLog