We consider several well-studied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms:1. Is the 3SUM problem on n numbers in O(n 2−ε ) time for some ε > 0?Conjecture 1 (No truly subquadratic 3SUM). In the Word RAM model with words of O(log n) bits, 1 any algorithm requires n 2−o(1) time in expectation to determine whether a set S ⊂ {−n 3 , . . . , n 3 } of |S| = n integers contains three distinct elements a, b, c ∈ S with a + b = c.Conjecture 2 (No truly subcubic APSP). There is a constant c, such that in the Word RAM model with words of O(log n) bits, any algorithm requires n 3−o(1) time in expectation to compute the distances between every pair of vertices in an n node graph with edge weights in {1, . . . , n c }.Conjecture 3 (Strong Exponential Time Hypothesis (SETH)). For every ε > 0, there exists a k, such that SAT on k-CNF formulas on n variables cannot be solved in O * (2 (1−ε)n ) time 1 .Conjecture 4 (No almost linear time triangle). There is a constant δ > 0, such that in the Word RAM model with words of O(log n) bits, any algorithm requires m 1+δ−o(1) time in expectation to detect whether an m edge graph contains a triangle."Conjecture" 5 (No truly subcubic combinatorial BMM). In the Word RAM model with words of O(log n) bits, any combinatorial algorithm requires n 3−o(1) time in expectation to compute the Boolean product of two n × n matrices. 2 This paper is the first study that relates the complexity of any dynamic problem to the exact complexity of Boolean Satisfiability (via the SETH). Our lower bounds hold even for randomized fully dynamic algorithms with (expected) amortized update times. Most of our results also hold for partially dynamic (incremental and decremental) algorithms with worst-case time bounds.Interestingly, many of our lower bounds (those based on the SETH) hold even when one allows arbitrary polynomial preprocessing time, and achieve essentially optimal guarantees. These are the first lower bounds of this nature.Most of our lower bounds also hold in the setting when one knows the list of updates and queries in advance, i.e. in the lookahead model. This is of interest since many dynamic problems can be solved faster given sufficient lookahead, e.g. graph transitive closure [83] and matrix rank [59].Organization. In Section 2 we discuss our results and the prior work on the problems we address. In Section 3 we describe our techniques. In Section 4 we give an overview of the prior work on the conjectures. In Section 5 we give a formal statement of the theorems we prove. The problems we consider are summarized in Table 1 and the results are summarized in Table 2. In section 6 we define some useful notation and prove reductions between dynamic problems. In section 7 we prove lower bounds based on Conjecture 3 (SETH). In section 8 we prove lower bounds based on Conjectures 4 and 5 (Triangle and BMM). In section 9 we prove lower bounds based on Conjecture 2 (APSP). And fi...

Two important similarity measures between sequences are the longest common subsequence (LCS) and the dynamic time warping distance (DTWD). The computations of these measures for two given sequences are central tasks in a variety of applications. Simple dynamic programming algorithms solve these tasks in O(n 2 ) time, and despite an extensive amount of research, no algorithms with significantly better worst case upper bounds are known.In this paper, we show that an O(n 2−ε ) time algorithm, for some ε > 0, for computing the LCS or the DTWD of two sequences of length n over a constant size alphabet, refutes the popular Strong Exponential Time Hypothesis (SETH). Moreover, we show that computing the LCS of k strings over an alphabet of size O(k) cannot be done in O(n k−ε ) time, for any ε > 0, under SETH. Finally, we also address the time complexity of approximating the DTWD of two strings in truly subquadratic time.

The radius and diameter are fundamental graph parameters, with several natural definitions for directed graphs. Each definition is well-motivated in a variety of applications. All versions of diameter and radius can be solved via solving all-pairs shortest paths (APSP), followed by a fast postprocessing step. However, solving APSP on n-node graphs requires Ω(n 2 ) time even in sparse graphs. We study the question: when can diameter and radius in sparse graphs be solved in truly subquadratic time, and when is such an algorithm unlikely? Motivated by our conditional lower bounds on computing these measures exactly in truly subquadratic time, we search for approximation and fixed parameter subquadratic algorithms, and alternatively, for reasons why they do not exist.We find that:• Most versions of Diameter and Radius can be solved in truly subquadratic time with optimal approximation guarantees, under plausible assumptions. For example, there is a 2-approximation algorithm for directed Radius with one-way distances that runs inÕ(m √ n) time, while a (2 − δ)-approximation algorithm in O(n 2−ε ) time is considered unlikely.• On graphs with treewidth k, we can solve all versions in 2 O(k log k) n 1+o(1) time. We show that these algorithms are near optimal since even a (3/2 − δ)-approximation algorithm that runs in time 2 o(k) n 2−ε would refute plausible assumptions.Two conceptual contributions of this work that we hope will incite future work are: the introduction of a Fixed Parameter Tractability in P framework, and the statement of a differently-quantified variant of the Orthogonal Vectors Conjecture, which we call the Hitting Set Conjecture. * The full version of the paper can be found at: http://arxiv.org/abs/1506. 01799. A.A and V