The 3SUM problem is to decide, given a set of n real numbers, whether any three sum to zero. It is widely conjectured that a trivial Opn 2 q-time algorithm is optimal and over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies Ωpn 2 q lower bounds on numerous problems in computational geometry and a variant of the conjecture implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures.In this paper we refute the 3SUM conjecture. We prove that the decision tree complexity of 3SUM is Opn 3{2 ?log nq and give two subquadratic 3SUM algorithms, a deterministic one running in Opn 2 {plog n{ log log nq 2{3 q time and a randomized one running in Opn 2 plog log nq 2 { log nq time with high probability. Our results lead directly to improved bounds for k-variate linear degeneracy testing for all odd k ě 3. The problem is to decide, given a linear function f px 1 , . . . , x k q " α 0`ř 1ďiďk α i x i and a set A Ă R, whether 0 P f pA k q. We show the decision tree complexity of this problem is Opn k{2 ? log nq. Finally, we give a subcubic algorithm for a generalization of the pmin,`q-product over realvalued matrices and apply it to the problem of finding zero-weight triangles in weighted graphs. We give a depth-Opn 5{2 ?log nq decision tree for this problem, as well as an algorithm running in time Opn 3 plog log nq 2 { log nq.

We study regular expression membership testing: Given a regular expression of size m and a string of size n, decide whether the string is in the language described by the regular expression. Its classic O(nm) algorithm is one of the big success stories of the 70s, which allowed pattern matching to develop into the standard tool that it is today.Many special cases of pattern matching have been studied that can be solved faster than in quadratic time. However, a systematic study of tractable cases was made possible only recently, with the first conditional lower bounds reported by Backurs and Indyk [FOCS'16]. Restricted to any "type" of homogeneous regular expressions of depth 2 or 3, they either presented a nearlinear time algorithm or a quadratic conditional lower bound, with one exception known as the Word Break problem.In this paper we complete their work as follows:• We present two almost-linear time algorithms that generalize all known almost-linear time algorithms for special cases of regular expression membership testing. * Theorem 3. For any δ > 0, if 4-Clique has no O(n 3+δ ) algorithm, then Word Break has no O(n 1+δ/3 ) algorithm for n = m.We remark that this situation of having matching conditional lower bounds only for combinatorial algorithms is not uncommon, see, e.g., Sliding Window Hamming Distance [6].New Almost-Linear Time Algorithms. We establish two more types for which the membership problem is in almost-linear time. Theorem 4. We design a deterministicÕ(n) + O(m) algorithm for | + •+-membership and an expected time n 1+o(1) + O(m) algorithm for | + •|-membership. These algorithms also work for t-membership for any subsequence t of | + •+ or | + •|, respectively.This generalizes all previously known almost-linear time algorithms for any t-membership problem, as all such types t are proper subsequences of |+•+ or |+•|. Moreover, no further generalization of our algorithms is possible, as shown below.

This paper studies the ε-approximate range emptiness problem, where the task is to represent a set S of n points from {0, . . . , U − 1} and answer emptiness queries of the form "[a; b] ∩ S = ∅ ?" with a probability of false positives allowed. This generalizes the functionality of Bloom filters from single point queries to any interval length L. Setting the false positive rate to ε/L and performing L queries, Bloom filters yield a solution to this problem with space O(n lg(L/ε)) bits, false positive probability bounded by ε for intervals of length up to L, using query time O(L lg(L/ε)). Our first contribution is to show that the space/error trade-off cannot be improved asymptotically: Any data structure for answering approximate range emptiness queries on intervals of length up to L with false positive probability ε, must use space Ω(n lg(L/ε)) − O(n) bits. On the positive side we show that the query time can be improved greatly, to constant time, while matching our space lower bound up to a lower order additive term. This result is achieved through a succinct data structure for (non-approximate 1d) range emptiness/reporting queries, which may be of independent interest. *

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High resolution Digital Elevation models, such as the (Big) grid terrain model of Denmark with more than 200 billion measurements, is a basic requirement for water flow modelling and flood risk analysis. However, a large number of modifications often need to be made to even very accurate terrain models, such as the Danish model, before they can be used in realistic flow modeling. These modifications include removal of bridges, which otherwise will act as dams in flow modeling, and inclusion of culverts that transport water underneath roads. In fact, the danish model is accompanied by a detailed set of hydrological corrections for the digital elevation model. However, producing these hydrological corrections is a very slow an expensive process, since it is to a large extent done manually and often with local input. This also means that corrections can be of varying quality. In this paper we propose a new algorithmic apporach based on machine learning and convolutional neural networks for automatically detecting hydrological corrections for such large terrain data. Our model is able to detect most hydrological corrections known for the danish model and quite a few more that should have been included in the original list.