Descriptive Complexity, Lower Bounds and Linear Time

1999 ◽  
pp. 9-28 ◽  
Author(s):  
Thomas Schwentick
Keyword(s):  
Lower Bounds ◽  
Linear Time ◽  
Descriptive Complexity ◽  
Complexity Lower Bounds

scholarly journals On Gap-Based Lower Bounding Techniques for Best-Arm Identification

Entropy ◽  
2020 ◽  
Vol 22 (7) ◽  
pp. 788
Author(s):  
Lan V. Truong ◽  
Jonathan Scarlett
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Sample Complexity ◽  
Bandit Problem ◽  
Lower Bounding ◽  
Complexity Lower Bounds

In this paper, we consider techniques for establishing lower bounds on the number of arm pulls for best-arm identification in the multi-armed bandit problem. While a recent divergence-based approach was shown to provide improvements over an older gap-based approach, we show that the latter can be refined to match the former (up to constant factors) in many cases of interest under Bernoulli rewards, including the case that the rewards are bounded away from zero and one. Together with existing upper bounds, this indicates that the divergence-based and gap-based approaches are both effective for establishing sample complexity lower bounds for best-arm identification.

Mixing, Communication Complexity and Conjectures of Gowers and Viola

2016 ◽  
Vol 26 (4) ◽  
pp. 628-640 ◽  
Author(s):  
ANER SHALEV
Keyword(s):  
Simple Group ◽  
Lower Bounds ◽  
Finite Simple Group ◽  
Decision Problem ◽  
Communication Complexity ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Quantitative Estimates ◽  
Complexity Lower Bounds ◽  
Bounded Rank

We study the distribution of products of conjugacy classes in finite simple groups, obtaining effective two-step mixing results, which give rise to an approximation to a conjecture of Thompson.Our results, combined with work of Gowers and Viola, also lead to the solution of recent conjectures they posed on interleaved products and related complexity lower bounds, extending their work on the groups SL(2,q) to all (non-abelian) finite simple groups.In particular it follows that, ifGis a finite simple group, andA,B⊆Gtfort⩾ 2 are subsets of fixed positive densities, then, asa= (a1, . . .,at) ∈Aandb= (b1, . . .,bt) ∈Bare chosen uniformly, the interleaved producta•b:=a1b1. . .atbtis almost uniform onG(with quantitative estimates) with respect to the ℓ∞-norm.It also follows that the communication complexity of an old decision problem related to interleaved products ofa,b∈Gtis at least Ω(tlog |G|) whenGis a finite simple group of Lie type of bounded rank, and at least Ω(tlog log |G|) whenGis any finite simple group. Both these bounds are best possible.

scholarly journals Improved Approximation for Fréchet Distance on c-Packed Curves Matching Conditional Lower Bounds

2017 ◽  
Vol 27 (01n02) ◽  
pp. 85-119 ◽  
Author(s):  
Karl Bringmann ◽  
Marvin Künnemann
Keyword(s):  
Lower Bounds ◽  
Time Complexity ◽  
Linear Time ◽  
High Dimensions ◽  
Fréchet Distance ◽  
Worst Case ◽  
One Dimensional ◽  
Dimension Formula ◽  
Frechet Distance ◽  
Special Case

The Fréchet distance is a well studied and very popular measure of similarity of two curves. The best known algorithms have quadratic time complexity, which has recently been shown to be optimal assuming the Strong Exponential Time Hypothesis (SETH) [Bringmann, FOCS'14]. To overcome the worst-case quadratic time barrier, restricted classes of curves have been studied that attempt to capture realistic input curves. The most popular such class are [Formula: see text]-packed curves, for which the Fréchet distance has a [Formula: see text]-approximation in time [Formula: see text] [Driemel et al., DCG'12]. In dimension [Formula: see text] this cannot be improved to [Formula: see text] for any [Formula: see text] unless SETH fails [Bringmann, FOCS'14]. In this paper, exploiting properties that prevent stronger lower bounds, we present an improved algorithm with time complexity [Formula: see text]. This improves upon the algorithm by Driemel et al. for any [Formula: see text]. Moreover, our algorithm's dependence on [Formula: see text], [Formula: see text] and [Formula: see text] is optimal in high dimensions apart from lower order factors, unless SETH fails. Our main new ingredients are as follows: For filling the classical free-space diagram we project short subcurves onto a line, which yields one-dimensional separated curves with roughly the same pairwise distances between vertices. Then we tackle this special case in near-linear time by carefully extending a greedy algorithm for the Fréchet distance of one-dimensional separated curves.

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