Tight Conditional Lower Bounds for Longest Common Increasing Subsequence

Duraj, Lech; Künnemann, Marvin; Polak, Adam

doi:10.1007/s00453-018-0485-7

Cited by 6 publications

(7 citation statements)

References 47 publications

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“…For Longest Common Increasing Subsequence, our lower bound was already shown by Duraj et al [24]. We refer to Duraj [23], Duraj et al [24] for a broader literature review on Longest Common Subsequence. Proof.…”

Section: Proof Let (I Ovsupporting

confidence: 67%

“…Using various carefully crafted reductions, Bringmann and Künnemann [15] show parameterized running time lower bounds (under SETH) for Longest Common Subsequence with respect to seven different parameters. In a similar fashion, Duraj et al [24] show that solving Longest Observation 1.1 (folklore). If a problem P admits an O( β n γ )-time algorithm 1 , then it admits for every λ > 0 an O(…”

Section: Related Workmentioning

confidence: 86%

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Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions

Heeger¹,

Nichterlein²,

Niedermeier³

2023

Preprint

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We provide a general framework to exclude parameterized running times of the form O( β + n γ ) for problems that have polynomial running time lower bounds under hypotheses from fine-grained complexity. Our framework is based on cross-compositions from parameterized complexity. We (conditionally) exclude running times of the form O( γ/(γ−1)−ε + n γ ) for any 1 < γ < 2 and ε > 0 for the following problems:Longest Common (Increasing) Subsequence: Given two length-n strings over an alphabet Σ (over N) and ∈ N, is there a common (increasing) subsequence of length in both strings? Discrete Fréchet Distance: Given two lists of n points each and k ∈ N , is the Fréchet distance of the lists at most k? Here is the maximum number of points which one list is ahead of the other list in an optimum traversal. Planar Motion Planning: Given a set of n non-intersecting axis-parallel line segment obstacles in the plane and a line segment robot (called rod), can the rod be moved to a specified target without touching any obstacles? Here is the maximum number of segments any segment has in its vicinity. Moreover, we exclude running times O( 2γ/(γ−1)−ε + n γ ) for any 1 < γ < 3 and ε > 0 for: Negative Triangle: Given an edge-weighted graph with n vertices, is there a triangle whose sum of edge-weights is negative? Here is the order of a maximum connected component. Triangle Collection: Given a vertex-colored graph with n vertices, is there for each triple of colors a triangle whose vertices have these three colors? Here is the order of a maximum connected component. 2nd Shortest Path: Given an n-vertex edge-weighted directed graph, two vertices s and t, and k ∈ N, has the second longest s-t-path length at most k? Here is the directed feedback vertex set number.Except for 2nd Shortest Path all these running time bounds are tight, that is, algorithms with running time O( γ/(γ−1) + n γ ) for any 1 < γ < 2 and O( 2γ/(γ−1) + n γ ) for any 1 < γ < 3, respectively, are known. Our running time lower bounds also imply lower bounds on kernelization algorithms for these problems.

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Section: Proof Let (I Ovsupporting

confidence: 67%

Section: Related Workmentioning

confidence: 86%

Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions

Heeger¹,

Nichterlein²,

Niedermeier³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Actually, we will construct, for every integer k, two sequences of size Θ k • 2 k with k • 2 2k significant pairs. To do that, we borrow a construction from [13]. In section 3.2 of that paper there is a definition -for every integer k -of two integer sequences A k , B k , each being a concatenation of 2 k blocks α i k or β j k :…”

Section: B Lower Bound For Significant Pairsmentioning

confidence: 99%

“…The "obvious" dynamic programming algorithm for LCIS is O n 3 , the first O n 2 -time algorithm was given in [31], and possibly the simplest one was explicitly stated in [32]. A conditional lower bound was proven in [13]: it turns out that, as for LCS, any O n 2−ε -time algorithm for LCIS would refute the Strong Exponential Time Hypothesis. The proof is based, like the one in [1], on a reduction from the Orthogonal Vectors problem (introduced in [28]), but the reduction itself needs a quite different gadget construction.…”

Section: Introductionmentioning

confidence: 99%

A sub-quadratic algorithm for the longest common increasing subsequence problem

Duraj

2019

Preprint

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The Longest Common Increasing Subsequence problem (LCIS) is a natural variant of the celebrated Longest Common Subsequence (LCS) problem. For LCIS, as well as for LCS, there is an O n 2 -time algorithm and a SETH-based conditional lower bound of O n 2−ε . For LCS, there is also the Masek-Paterson O n 2 / log n -time algorithm, which does not seem to adapt to LCIS in any obvious way. Hence, a natural question arises: does any (slightly) sub-quadratic algorithm exist for the Longest Common Increasing Subsequence problem? We answer this question positively, presenting a O n 2 / log a n -time algorithm for a = 1 6 − o (1). The algorithm is not based on memorizing small chunks of data (often used for logarithmic speedups, including the "Four Russians Trick" in LCS), but rather utilizes a new technique, bounding the number of significant symbol matches between the two sequences.

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“…While this is less obvious than for LCS, LCIS can be also solved in O(n 2 ) time [11] (and in linear space [9]), and it can be proved that a strongly subquadratic algorithm would refute SETH [6] (although faster algorithms are known for some special cases [7]). However, as opposed to LCS, the usual "Four Russians" approach, that roughly consists in partitioning the DP table into blocks of size log n × log n, doesn't seem directly applicable to LCIS.…”

Section: Introductionmentioning

confidence: 99%

A Faster Subquadratic Algorithm for the Longest Common Increasing Subsequence Problem

Agrawal¹,

Gawrychowski²

2020

Preprint

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The Longest Common Increasing Subsequence (LCIS) is a variant of the classical Longest Common Subsequence (LCS), in which we additionally require the common subsequence to be strictly increasing. While the well-known "Four Russians " technique can be used to find LCS in subquadratic time, it does not seem applicable to LCIS. Recently, Duraj [STACS 2020] used a completely different method based on the combinatorial properties of LCIS to design an O(n 2 (log log n) 2 / log 1/6 n) time algorithm. We show that an approach based on exploiting tabulation can be used to construct an asymptotically faster O(n 2 log log n/ √ log n) time algorithm. As our solution avoids using the specific combinatorial properties of LCIS, it can be also adapted for the Longest Common Weakly Increasing Subsequence (LCWIS).

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Tight Conditional Lower Bounds for Longest Common Increasing Subsequence

Cited by 6 publications

References 47 publications

Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions

Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions

A sub-quadratic algorithm for the longest common increasing subsequence problem

A Faster Subquadratic Algorithm for the Longest Common Increasing Subsequence Problem

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