Subtree Isomorphism Revisited

Abboud, Amir; Bačkurs, Artūrs; Hansen, Thomas Dueholm; Williams, Virginia Vassilevska; Zamir, Or

doi:10.1137/1.9781611974331.ch88

Cited by 10 publications

(7 citation statements)

References 58 publications

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“…Let S(N, ℓ) denote the space required by a data structure for histogram indexing on N -length string over alphabet size ℓ and let Q(N, ℓ) denote the query time for the same parameters. Assuming the strong 3SUM-Indexing conjecture and following our reduction, we have that S(N, ℓ) = O(n 2−Ω(1) ) and Q(N, ℓ) = O(n 1−α−Ω (1) ). Plugging in the value of n in terms of N we get the required lower bound.…”

Section: Static Problemsmentioning

confidence: 85%

“…Examples of such hard problems include the well-known 3SUM problem, the fundamental APSP problem, (combinatorial) Boolean matrix multiplication, etc. Recently, conditional time lower bounds have been proven based on the conjectured hardness of these problems for graph algorithms [4,42], edit distance [13], longest common subsequence (LCS) [3,15], dynamic algorithms [5,36], jumbled indexing [11], and many other problems [1,2,6,7,14,25,31,34,40].…”

Section: Introductionmentioning

confidence: 99%

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Conditional Lower Bounds for Space/Time Tradeoffs

Goldstein

Kopelowitz

Lewenstein

et al. 2017

Lecture Notes in Computer Science

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In recent years much effort has been concentrated towards achieving polynomial time lower bounds on algorithms for solving various well-known problems. A useful technique for showing such lower bounds is to prove them conditionally based on well-studied hardness assumptions such as 3SUM, APSP, SETH, etc. This line of research helps to obtain a better understanding of the complexity inside P. A related question asks to prove conditional space lower bounds on data structures that are constructed to solve certain algorithmic tasks after an initial preprocessing stage. This question received little attention in previous research even though it has potential strong impact. In this paper we address this question and show that surprisingly many of the well-studied hard problems that are known to have conditional polynomial time lower bounds are also hard when concerning space. This hardness is shown as a tradeoff between the space consumed by the data structure and the time needed to answer queries. The tradeoff may be either smooth or admit one or more singularity points. We reveal interesting connections between different space hardness conjectures and present matching upper bounds. We also apply these hardness conjectures to both static and dynamic problems and prove their conditional space hardness. We believe that this novel framework of polynomial space conjectures can play an important role in expressing polynomial space lower bounds of many important algorithmic problems. Moreover, it seems that it can also help in achieving a better understanding of the hardness of their corresponding problems in terms of time

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Section: Static Problemsmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Conditional Lower Bounds for Space/Time Tradeoffs

Goldstein

Kopelowitz

Lewenstein

et al. 2017

Lecture Notes in Computer Science

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“…To clarify what we mean, consider the following "Direct-OR" version of Subset Sum: Given N different and independent instances of Subset Sum, each on n numbers and each with a different target T i ≤ T , decide whether any of them is a YES instance. It is natural to expect the time complexity of this problem to be (N T ) 1−o (1) , but how do we formally argue that this is the case? If we could assume that this holds, it would be a very useful tool for conditional lower bounds (as we show in Section 1.3).…”

Section: A Direct-or Theorem For Subset Summentioning

confidence: 99%

“…Since the work of Cygan et al [48], SETH has enjoyed great success as a basis for lower bounds in Parameterized Complexity [87] and for problems within P [112]. Some of the most fundamental problems on strings (e.g., [9,17,2,35,18,34]), graphs (e.g., [86,104,6,58]), curves (e.g., [32]), vectors [110,111,19,29] and trees [1] have been shown to be SETH-hard: a small improvement to the running time of these problems would refute SETH. Despite the remarkable quantity and diversity of these results, we are yet to see a (tight) reduction from SAT to any problem like Subset Sum, where the complexity comes from the hardness of analyzing a search space defined by addition of numbers.…”

Section: Introductionmentioning

confidence: 99%

SETH-Based Lower Bounds for Subset Sum and Bicriteria Path

Abboud

Bringmann

Hermelin

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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Subset Sum and k-SAT are two of the most extensively studied problems in computer science, and conjectures about their hardness are among the cornerstones of fine-grained complexity.An important open problem in this area is to base the hardness of one of these problems on the other.Our main result is a tight reduction from k-SAT to Subset Sum on dense instances, proving that Bellman's 1962 pseudo-polynomial O * (T )-time algorithm for Subset Sum on n numbers and target T cannot be improved to time T 1−ε · 2 o(n) for any ε > 0, unless the Strong Exponential Time Hypothesis (SETH) fails.As a corollary, we prove a "Direct-OR" theorem for Subset Sum under SETH, offering a new tool for proving conditional lower bounds: It is now possible to assume that deciding whether one out of N given instances of Subset Sum is a YES instance requires time (N T ) 1−o(1) . As an application of this corollary, we prove a tight SETH-based lower bound for the classical Bicriteria s, t-Path problem, which is extensively studied in Operations Research. We separate its complexity from that of Subset Sum: On graphs with m edges and edge lengths bounded by L, we show that the O(Lm) pseudopolynomial time algorithm by Joksch from 1966 cannot be improved toÕ(L + m), in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).Subset Sum. Subset Sum is one of the most fundamental problems in computer science. Its most basic form is the following: given n integers x 1 , . . . , x n ∈ N, and a target value T ∈ N, decide whether there is a subset of the numbers that sums to T . The two most classical algorithms for the problem are the pseudo-polynomial O(T n) algorithm using dynamic programming [28], and the O(2 n/2 · poly(n, log T )) algorithm via "meet-in-the-middle" [67]. A central open question in Exact Algorithms [114] is whether faster algorithms exist, e.g., can we combine the two approaches to get a T 1/2 · n O(1) time algorithm? Such a bound was recently found in a Merlin-Arthur setting [94].The status of Subset Sum as a major problem has been established due to many applications, deep connections to other fields, and educational value. The O(T n) algorithm from 1957 is an illuminating example of dynamic programming that is taught in most undergraduate algorithms courses, and the NP-hardness proof (from Karp's original paper [78]) is a prominent example of a reduction to a problem on numbers. Interestingly, one of the earliest cryptosystems by Merkle and Hellman was based on Subset Sum [92], and was later extended to a host of Knapsack-type cryptosystems 4 (see [106,31,97,46,69] and the references therein).The version of Subset Sum where we ask for k numbers that sum to zero (the k-SUM problem) is conjectured to have n ⌈k/2⌉±o(1) time complexity. Most famously, the k = 3 case is the 3-SUM conjecture highlighted in the seminal work of Gajentaan and Overmars [57]. It has been shown that this problem lies at the core and captures the difficulty of dozens of problems in computational geometry. Searching in Google Scholar for "3su...

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“…While a considerable share of the recent conditional lower bounds is on string pattern matching problems [3,4,7,10,15,19,20,28,29,32,45], the only conditional lower bound for a tree pattern matching problem is the recent SODA'16 quadratic lower bound for exact pattern matching [2] (the problem of deciding whether one tree is a subtree of another). We solve the main remaining open problem in tree pattern matching, and one of the last remaining classic dynamic programming problems.…”

Section: Our Resultsmentioning

confidence: 99%

Tree Edit Distance Cannot be Computed in Strongly Subcubic Time (unless APSP can)

Bringmann¹,

Gawrychowski²,

Mozes³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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The edit distance between two rooted ordered trees with n nodes labeled from an alphabet Σ is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as inserting new nodes. Tree edit distance is a well known generalization of string edit distance. The fastest known algorithm for tree edit distance runs in cubic O(n 3 ) time and is based on a similar dynamic programming solution as string edit distance. In this paper we show that a truly subcubic O(n 3−ε ) time algorithm for tree edit distance is unlikely: For |Σ| = Ω(n), a truly subcubic algorithm for tree edit distance implies a truly subcubic algorithm for the all pairs shortest paths problem. For |Σ| = O(1), a truly subcubic algorithm for tree edit distance implies an O(n k−ε ) algorithm for finding a maximum weight k-clique.Thus, while in terms of upper bounds string edit distance and tree edit distance are highly related, in terms of lower bounds string edit distance exhibits the hardness of the strong exponential time hypothesis [Backurs, Indyk STOC'15] whereas tree edit distance exhibits the hardness of all pairs shortest paths. Our result provides a matching conditional lower bound for one of the last remaining classic dynamic programming problems. *

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Subtree Isomorphism Revisited

Cited by 10 publications

References 58 publications

Conditional Lower Bounds for Space/Time Tradeoffs

Conditional Lower Bounds for Space/Time Tradeoffs

SETH-Based Lower Bounds for Subset Sum and Bicriteria Path

Tree Edit Distance Cannot be Computed in Strongly Subcubic Time (unless APSP can)

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