Conditional Lower Bounds for Space/Time Tradeoffs

Goldstein, Israel; Kopelowitz, Tsvi; Lewenstein, Moshe; Porat, Ely

doi:10.1007/978-3-319-62127-2_36

Cited by 24 publications

(59 citation statements)

References 36 publications

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“…In the Set Disjointness problem, we are given a collection of m sets S 1 , S 2 , … , S m of total size N from some universe U for preprocessing in order to answer queries on the emptiness of the intersection of some two query sets from the collection. Goldstein et al [52] demonstrated conditional hardness of Set Disjointness with regard to its space-query time tradeoff. Specifically, Goldstein et al state the following conjecture.…”

Section: A Lower Bound Based On Set Disjointnessmentioning

confidence: 99%

Dynamic and Internal Longest Common Substring

et al. 2020

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Given two strings S and T, each of length at most n, the longest common substring (LCS) problem is to find a longest substring common to S and T. This is a classical problem in computer science with an $$\mathcal {O}(n)$$ O ( n ) -time solution. In the fully dynamic setting, edit operations are allowed in either of the two strings, and the problem is to find an LCS after each edit. We present the first solution to the fully dynamic LCS problem requiring sublinear time in n per edit operation. In particular, we show how to find an LCS after each edit operation in $$\tilde{\mathcal {O}}(n^{2/3})$$ O ~ ( n 2 / 3 ) time, after $$\tilde{\mathcal {O}}(n)$$ O ~ ( n ) -time and space preprocessing. This line of research has been recently initiated in a somewhat restricted dynamic variant by Amir et al. [SPIRE 2017]. More specifically, the authors presented an $$\tilde{\mathcal {O}}(n)$$ O ~ ( n ) -sized data structure that returns an LCS of the two strings after a single edit operation (that is reverted afterwards) in $$\tilde{\mathcal {O}}(1)$$ O ~ ( 1 ) time. At CPM 2018, three papers (Abedin et al., Funakoshi et al., and Urabe et al.) studied analogously restricted dynamic variants of problems on strings; specifically, computing the longest palindrome and the Lyndon factorization of a string after a single edit operation. We develop dynamic sublinear-time algorithms for both of these problems as well. We also consider internal LCS queries, that is, queries in which we are to return an LCS of a pair of substrings of S and T. We show that answering such queries is hard in general and propose efficient data structures for several restricted cases.

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Section: A Lower Bound Based On Set Disjointnessmentioning

confidence: 99%

Dynamic and Internal Longest Common Substring

et al. 2020

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“…SETH is one of the most fruitful conjectures in the Fine-Grained Complexity. There are numerous conditional lower bounds based on it for problems in P among different areas, including: dynamic data structures [61,7,45,55,3,46,41], computational geometry [24,37,75,67], pattern matching [8,22,21,25,26], graph algorithms [66,40,9,56]. See [72] for a recent survey on SETH-based lower bounds (and more).…”

Section: Related Workmentioning

confidence: 99%

Untitled

Chen¹

2020

Theory of Comput.

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In this paper we study the (Bichromatic) Maximum Inner Product Problem (Max-IP), in which we are given sets A and B of vectors, and the goal is to find a ∈ A and b ∈ B maximizing inner product a • b. Max-IP is a basic question and serves as the base problem in the recent breakthrough of [Abboud et al., FOCS 2017] on hardness of approximation for polynomial-time problems. It is also used (implicitly) in the argument for hardness of exact 2-Furthest Pair (and other important problems in computational geometry) in poly-loglog dimensions in [Williams, SODA 2018]. We have three main results regarding this problem. * Supported by an Akamai Fellowship.

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“…• Fine-Grained Time/Space Tradeoffs for Data Structures. Goldstein et al [93] define various data structure variants of 3-SUM, BMM and Directed Reachability, formulate novel conjectures and show consequences for the time/space tradeoffs for various data structure problems. Fine-grained complexity is a growing field and we hope that its ideas will spread to many other parts of TCS and beyond.…”

Section: Further Applications Of Fine-grained Complexitymentioning

confidence: 99%

On Some Fine-Grained Questions in Algorithms and Complexity

Williams

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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In recent years, a new "fine-grained" theory of computational hardness has been developed, based on "fine-grained reductions" that focus on exact running times for problems. Mimicking NP-hardness, the approach is to (1) select a key problem X that for some function t, is conjectured to not be solvable by any O(t(n) 1−ε ) time algorithm for ε > 0, and (2) reduce X in a fine-grained way to many important problems, thus giving tight conditional time lower bounds for them. This approach has led to the discovery of many meaningful relationships between problems, and to equivalence classes.The main key problems used to base hardness on have been: the 3-SUM problem, the CNF-SAT problem (based on the Strong Exponential Time Hypothesis (SETH)) and the All Pairs Shortest Paths Problem. Research on SETH-based lower bounds has flourished in particular in recent years showing that the classical algorithms are optimal for problems such as Approximate Diameter, Edit Distance, Frechet Distance and Longest Common Subsequence.This paper surveys the current progress in this area, and highlights some exciting new developments. IntroductionArguably the main goal of the theory of algorithms is to study the worst case time complexity of fundamental computational problems. When considering a problem P , we fix a computational model, such as a Random Access Machine (RAM) or a Turing machine (TM). Then we strive to develop an efficient algorithm that solves P and to prove that for a (hopefully slow growing) function t(n), the algorithm solves P on instances of size n in O(t(n)) time in that computational model. The gold standard for the running time t(n) is linear time, O(n); to solve most problems, one needs to at least read the input, and so linear time is necessary. The theory of algorithms has developed a wide variety of techniques. These have yielded near-linear time algorithms for many diverse problems. For instance, it is known since the 1960s and 70s (e.g. [143,144,145,99]) that Depth-First Search (DFS) and Breadth-First Search (BFS) run in linear time in graphs, and that using these techniques one can obtain linear time algorithms (on a RAM) for many interesting graph problems: Single-Source Shortest paths, Topological Sort of a Directed Acyclic Graph, Strongly Connected Components, Testing Graph Planarity etc. More recent work has shown that even more complex problems such as Approximate Max Flow, Maximum Bipartite Matching, Linear Systems on Structured Matrices, and many others, admit close to linear time algorithms, by combining combinatorial and linear algebraic techniques (see e.g. [140,64,141,116,117,67,68,65,66,113]).Nevertheless, for most problems of interest, the fastest known algorithms run much slower than linear time. This is perhaps not too surprising. Time hierarchy theorems show that for most computational models, for any computable function t(n) ≥ n, there exist problems that are solvable in O(t(n)) time but are NOT solvable in O(t(n) 1−ε ) time for ε > 0 (this was first proven for TMs [95], see [124] for more).Ti...

show abstract

Conditional Lower Bounds for Space/Time Tradeoffs

Cited by 24 publications

References 36 publications

Dynamic and Internal Longest Common Substring

Dynamic and Internal Longest Common Substring

Untitled

On Some Fine-Grained Questions in Algorithms and Complexity

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