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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 mid-1990s all results had polynomial time-space 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 space-time 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. 

Sequence mappability is an important task in genome resequencing. In the (k, m)-mappability problem, for a given sequenceTof lengthn, the goal is to compute a table whoseith entry is the number of indices$$j \ne i$$$j\ne i$such that the length-msubstrings ofTstarting at positionsiandjhave at mostkmismatches. Previous works on this problem focused on heuristics computing a rough approximation of the result or on the case of$$k=1$$$k=1$. We present several efficient algorithms for the general case of the problem. Our main result is an algorithm that, for$$k=O(1)$$$k=O\left(1\right)$, works in$$O(n)$$$O\left(n\right)$space and, with high probability, in$$O(n \cdot \min \{m^k,\log ^k n\})$$$O(n·min\{m^k,{log}^{k}n\})$time. Our algorithm requires a careful adaptation of thek-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 all-pairs Hamming distance problem introduced by Crochemore et al. [WABI 2017]. We further develop$$O(n^2)$$$O\left(n^2\right)$-time algorithms to computeall(k, m)-mappability tables for a fixedmand all$$k\in \{0,\ldots ,m\}$$$k\in \{0,\dots ,m\}$or a fixedkand all$$m\in \{k,\ldots ,n\}$$$m\in \{k,\dots ,n\}$. Finally, we show that, for$$k,m = \Theta (\log n)$$$k,m=\Theta (logn)$, the (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.