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Randomized Complexity Lower Bound for Arrangements and Polyhedra

D. Grigoriev
1999 Discrete & Computational Geometry  
The complexity lower bound (log N) for randomized computation trees is proved for recognizing an arrangement or a polyhedron with N faces.  ...  This provides in particular, the randomized lower bound (n log n) for the DISTINCTNESS problem and generalizes 11] where the randomized lower bound (n 2 ) was ascertained for the KNAPSACK problem.  ...  Relying on this lower bound on the multiplicative complexity, the proof of the complexity lower bound (log N) for RCT recognizing an arrangement or a polyhedron with N faces (see the theorem in section  ... 
doi:10.1007/pl00009425 fatcat:bptptvesw5f27kp62ds3lltp2e

A lower bound for the discrepancy of a random point set

Benjamin Doerr
2014 Journal of Complexity  
Consequently, the expected star discrepancy of a random point set is of order √(s/N).  ...  We show that there is a constant K > 0 such that for all N, s ∈, s < N, the point set consisting of N points chosen uniformly at random in the s-dimensional unit cube [0,1]^s with probability at least  ...  Surprisingly, not even a lower bound for the discrepancy of a random point set is known.  ... 
doi:10.1016/j.jco.2013.06.001 fatcat:4bn4yx4w4bccdmfuhooc6skdnm

Lower bounds for randomized and quantum query complexity using Kolmogorov arguments [article]

Sophie Laplante, Frederic Magniez
2003 arXiv   pre-print
We prove a very general lower bound technique for quantum and randomized query complexity, that is easy to prove as well as to apply.  ...  As an immediate consequence of our main theorem, adversary methods can only prove lower bounds for boolean functions f in O((√(n C_0(f)),√(n C_1(f)))), where C_0, C_1 is the certificate complexity, and  ...  Introduction Overview In this paper, we study lower bounds for randomized and quantum query complexity.  ... 
arXiv:quant-ph/0311189v1 fatcat:24r2oyo6kbbwpc4fpgwlkz4fcq

Lower Bounds on the Randomized Communication Complexity of Read-Once Functions

Nikos Leonardos, Michael Saks
2010 Computational Complexity  
We prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae.  ...  In this paper we use information theory methods to prove lower bounds that hold for any read-once formula.  ...  We also thank the anonymous referees of the 24th IEEE Conference on Computational Complexity and the Computational Complexity journal for their corrections and useful suggestions.  ... 
doi:10.1007/s00037-010-0292-2 fatcat:hiy5kdsiibbldhp73h3owfzgu4

Improved lower bounds on the randomized complexity of graph properties

Amit Chakrabarti, Subhash Khot
2007 Random structures & algorithms (Print)  
We prove a lower bound of (n 4/3 log 1/3 n) on the randomized decision tree complexity of any nontrivial monotone n-vertex graph property, and of any nontrivial monotone bipartite graph property with bipartitions  ...  This improves the previous best bound of (n 4/3 ) due to Hajnal [Haj91].  ...  Returning to randomized complexity (the focus of this paper), the first nonlinear lower bound on C R (P), for general P ∈ P n , was an (n log 1/12 n) bound proven by Yao [Yao87] .  ... 
doi:10.1002/rsa.20164 fatcat:pmyrmhdenzhxhaqetkcnjxrkte

Lower bounds for the complexity of linear functionals in the randomized setting

Erich Novak, Henryk Woźniakowski
2011 Journal of Complexity  
We also present lower bounds for reproducing kernels that are not decomposable but have a decomposable part. However, in this case it is not clear if the lower bounds are sharp.  ...  Hinrichs [3] recently studied multivariate integration defined over reproducing kernel Hilbert spaces in the randomized setting and for the normalized error criterion.  ...  We must admit that after we completed the paper [5] on lower bounds in the worst case setting, we started to work on lower bounds in the randomized setting around the year 2002.  ... 
doi:10.1016/j.jco.2010.08.002 fatcat:uaoinntnvjchro2r6wmnb2vq6a

Improved Lower Bounds on the Randomized Complexity of Graph Properties [chapter]

Amit Chakrabarti, Subhash Khot
2001 Lecture Notes in Computer Science  
We prove a lower bound of (n 4/3 log 1/3 n) on the randomized decision tree complexity of any nontrivial monotone n-vertex graph property, and of any nontrivial monotone bipartite graph property with bipartitions  ...  This improves the previous best bound of (n 4/3 ) due to Hajnal [Haj91].  ...  Returning to randomized complexity (the focus of this paper), the first nonlinear lower bound on C R (P), for general P ∈ P n , was an (n log 1/12 n) bound proven by Yao [Yao87] .  ... 
doi:10.1007/3-540-48224-5_24 fatcat:mlml6mbvdfcvrd3h5d5a6z7qaq

Lower bounds in communication complexity based on factorization norms

Nati Linial, Adi Shraibman
2009 Random structures & algorithms (Print)  
We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory.  ...  Among our new results we extend some known lower bounds to the realm of quantum communication complexity with entanglement.  ...  Introduction We study lower bounds for randomized and quantum communication complexity.  ... 
doi:10.1002/rsa.20232 fatcat:ec546rrdsneyhmt4j3gij7d6xi

A lower bound for randomized algebraic decision trees

Dima Grigoriev, Marek Karpinski, Friedhelm Meyer auf der Heide, Roman Smolensky
1996 Computational Complexity  
As an application, among other things, we derive, for the first time, an f~(n 2) randomized lower bound for the Knapsack Problem, and an ft(n logn) randomized lower bound for the Element Distinctness Problem  ...  We prove the first nontrivial (and superlinear) lower bounds on the depth of randomized algebraic decision trees (with two-sided error) for problems being finite unions of hyperplanes and intersections  ...  [] Lower bound on the complexity of deterministic decision trees For d -DT T' which recognizes either an arrangement S or a polyhedron S +, we can give the similar complexity lower bound ~(log N)  ... 
doi:10.1007/bf01270387 fatcat:5mckzdtgl5futpdwltgc6qcuo4

An Improved Lower Bound for the Randomized Decision Tree Complexity of Recursive Majority, [chapter]

Nikos Leonardos
2013 Lecture Notes in Computer Science  
In 2011, Magniez, Nayak, Santha, and Xiao, improved the lower bound to Ω (5/2) d and the upper bound to O(2.64946 d ).  ...  We prove that the randomized decision tree complexity of the recursive majority-of-three is Ω(2.55 d ), where d is the depth of the recursion.  ...  We prove a lower bound on the randomized decision tree complexity of the recursive majority-ofthree function.  ... 
doi:10.1007/978-3-642-39206-1_59 fatcat:sjvfccrhjncx5ogtvetuzuv75a

Nondeterministic Circuit Lower Bounds from Mildly De-randomizing Arthur-Merlin Games

Baris Aydinlioglu, Dieter van Melkebeek
2012 2012 IEEE 27th Conference on Computational Complexity  
Our equivalence result represents a full analogue of a similar result by Impagliazzo et al. in the deterministic setting: Randomized polynomial-time decision procedures can be simulated in NSUBEXP (the  ...  INTRODUCTION The power of randomness constitutes a central topic in complexity theory.  ...  As the circuit lower bounds seem plausible, even at the high end, the hardness versus randomness tradeoffs have fueled the conjecture that prBPP can be fully derandomized to P, and prAM to NP.  ... 
doi:10.1109/ccc.2012.32 dblp:conf/coco/AydinliogluM12 fatcat:b4dgncxwyrhgrmziowvmkssihm

Lower Bounds for Randomized and Quantum Query Complexity Using Kolmogorov Arguments

Sophie Laplante, Frédéric Magniez
2008 SIAM journal on computing (Print)  
We prove a very general lower bound technique for quantum and randomized query complexity, that is easy to prove as well as to apply.  ...  We also derive a general form of the ad hoc weighted method used by Høyer, Neerbek and Shi to give a quantum lower bound on ordered search and sorting.  ...  Introduction Overview In this paper, we study lower bounds for randomized and quantum query complexity.  ... 
doi:10.1137/050639090 fatcat:p64in7p24vcwthswopb25hhj5u

Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes

Maurice Jansen, Rahul Santhanam, Marc Herbstritt
2012 Symposium on Theoretical Aspects of Computer Science  
We prove the following lower bound and randomness-to-hardness results: 1.  ...  The third item improves a lower bound due to Santhanam [11] .  ...  However, it doesn't tell us whether these will be Boolean lower bounds or arithmetic lower bounds.  ... 
doi:10.4230/lipics.stacs.2012.519 dblp:conf/stacs/JansenS12 fatcat:qe34yv6bfnekzl2jzrmc7eg5p4

Lower bounds on random-self-reducibility

J. Feigenbaum, S. Kannan, N. Nisan
Proceedings Fifth Annual Structure in Complexity Theory Conference  
Such functions are fundamental in many areas of theoretical computer science, including lower bounds, pseudorandom number-generators, interactive proof systems, zeroknowledge, instance-hiding, program-checking  ...  For example, we show unconditionally that random boolean functions do not have random-selfreductions, even of a quite general nature.  ...  random-self-reducibility would nd more applications in complexity theory and in practice.  ... 
doi:10.1109/sct.1990.113959 dblp:conf/coco/FeigenbaumKN90 fatcat:vwerkcayyvec5g5npfebebkg2a

Lower bounds for randomized and quantum query complexity using Kolmogorov arguments

S. Laplante, F. Magniez
Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004.  
We prove a very general lower bound technique for quantum and randomized query complexity, that is easy to prove as well as to apply.  ...  We also derive a general form of the ad hoc weighted method used by Høyer, Neerbek and Shi to give a quantum lower bound on ordered search and sorting.  ...  Introduction Overview In this paper, we study lower bounds for randomized and quantum query complexity.  ... 
doi:10.1109/ccc.2004.1313852 dblp:conf/coco/LaplanteM04 fatcat:7mlg5ylqubgilhtnofmo46mhu4
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