Given an n-vertex planar embedded digraph G with non-negative edge weights and a face f of G, Klein presented a data structure with O(n log n) space and preprocessing time which can answer any query (u, v) for the shortest path distance in G from u to v or from v to u in O(log n) time, provided u is on f . This data structure is a key tool in a number of state-of-the-art algorithms and data structures for planar graphs.Klein's data structure relies on dynamic trees and the persistence technique as well as a highly non-trivial interaction between primal shortest path trees and their duals. The construction of our data structure follows a completely different and in our opinion very simple divide-and-conquer approach that solely relies on Single-Source Shortest Path computations and contractions in the primal graph. Our space and preprocessing time bound is O(n log |f |) and query time is O(log |f |) which is an improvement over Klein's data structure when f has small size.
Stochastic sample-based estimators are among the most fundamental and universally applied tools in statistics. Such estimators are particularly important when processing huge amounts of data, where we need to be able to answer a wide range of statistical queries reliably, yet cannot afford to store the data in its full length.
In many applications we need the sampling to be coordinated which is typically attained using hashing. In previous work, a common strategy to obtain reliable sample-based estimators that work within certain error bounds with high probability has been to design one that works with constant probability, and then boost the probability by taking the median over
r
independent repetitions. Aamand et al. (STOC'20) recently proposed a fast and practical hashing scheme with
strong concentration bounds
, Tabulation-1Permutation, the first of its kind. In this paper, we demonstrate that using such a hash family for the sampling, we achieve the same high probability bounds without any need for repetitions. Using the same space, this saves a factor
r
in time, and simplifies the overall algorithms.
We validate our approach experimentally on both real and synthetic data. We compare Tabulation-1Permutation with other hash functions such as strongly universal hash functions and various other hash functions such as MurmurHash3 and BLAKE3, both with and without resorting to repetitions. We see that if we want reliability in terms of small error probabilities, then Tabulation-1Permutation is significantly faster.
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