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We consider the problem of preprocessing a weighted directed planar graph in order to quickly answer exact distance queries. The main tension in this problem is between space S and query time Q , and since the mid1990s all results had polynomial timespace tradeoffs, e.g., Q = ~ Θ( n/√ S ) or Q = ~Θ( n 5/2 /S 3/2 ). In this article we show that there is no polynomial tradeoff between time and space and that it is possible to simultaneously achieve almost optimal space n 1+ o (1) and almost optimal query time n o (1) . More precisely, we achieve the following spacetime tradeoffs: n 1+ o (1) space and log 2+ o (1) n query time, n log 2+ o (1) n space and n o (1) query time, n 4/3+ o (1) space and log 1+ o (1) n query time. We reduce a distance query to a variety of point location problems in additively weighted Voronoi diagrams and develop new algorithms for the point location problem itself using several partially persistent dynamic tree data structures.more » « less

Abstract Sequence mappability is an important task in genome resequencing. In the (
k ,m )mappability problem, for a given sequenceT of lengthn , the goal is to compute a table whosei th entry is the number of indices such that the length$$j \ne i$$ $j\ne i$m substrings ofT starting at positionsi andj have at mostk mismatches. Previous works on this problem focused on heuristics computing a rough approximation of the result or on the case of . We present several efficient algorithms for the general case of the problem. Our main result is an algorithm that, for$$k=1$$ $k=1$ , works in$$k=O(1)$$ $k=O\left(1\right)$ space and, with high probability, in$$O(n)$$ $O\left(n\right)$ time. Our algorithm requires a careful adaptation of the$$O(n \cdot \min \{m^k,\log ^k n\})$$ $O(n\xb7min\{{m}^{k},{log}^{k}n\})$k errata trees of Cole et al. [STOC 2004] to avoid multiple counting of pairs of substrings. Our technique can also be applied to solve the allpairs Hamming distance problem introduced by Crochemore et al. [WABI 2017]. We further develop time algorithms to compute$$O(n^2)$$ $O\left({n}^{2}\right)$all (k ,m )mappability tables for a fixedm and all or a fixed$$k\in \{0,\ldots ,m\}$$ $k\in \{0,\dots ,m\}$k and all . Finally, we show that, for$$m\in \{k,\ldots ,n\}$$ $m\in \{k,\dots ,n\}$ , the ($$k,m = \Theta (\log n)$$ $k,m=\Theta (logn)$k ,m )mappability problem cannot be solved in strongly subquadratic time unless the Strong Exponential Time Hypothesis fails. This is an improved and extended version of a paper presented at SPIRE 2018.