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Lifting randomized query complexity to randomized communication complexity [article]

Anurag Anshu, Naresh B. Goud, Rahul Jain, Srijita Kundu, Priyanka Mukhopadhyay
2018 arXiv   pre-print
query complexity of f (randomized communication complexity of f∘ g^n) with worst case error 1/3.  ...  ( n))), where f∘ g^n represents the composition of f and g^n, M_g is the sign matrix for g, disc(M_g) is the discrepancy of M_g under the uniform distribution and R_1/3(f) (R_1/3(f∘ g^n)) denotes the randomized  ...  Acknowledgement This work is supported by the Singapore Ministry of Education and the National Research Foundation, also through the Tier 3 Grant "Random numbers from quantum processes" MOE2012-T3-1-009  ... 
arXiv:1703.07521v5 fatcat:epjpdkhmdrcuxmcruxqdtmryxq

Query-To-Communication Lifting for BPP Using Inner Product

Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, Toniann Pitassi, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
We prove a new query-to-communication lifting for randomized protocols, with inner product as gadget. This allows us to use a much smaller gadget, leading to a more efficient lifting.  ...  The only query-to-communication lifting result for randomized protocols, due to Göös, Pitassi and Watson [13], used the much larger indexing gadget.  ...  Lifting theorems are theorems that relate the communication complexity of f • g n to the query complexity of f and the communication complexity of g.  ... 
doi:10.4230/lipics.icalp.2019.35 dblp:conf/icalp/ChattopadhyayFK19 fatcat:xp3myxq5ijchfeif36bssemuqy

A ZPP^NP[1] Lifting Theorem

Thomas Watson, Michael Wagner
2019 Symposium on Theoretical Aspects of Computer Science  
The complexity class ZPP NP[1] (corresponding to zero-error randomized algorithms with access to one NP oracle query) is known to have a number of curious properties.  ...  We further explore this class in the settings of time complexity, query complexity, and communication complexity.  ...  This idea has led to an ongoing project: prove lifting theorems for the query/communication analogues of various classical complexity classes.  ... 
doi:10.4230/lipics.stacs.2019.59 dblp:conf/stacs/Watson19 fatcat:qgy2vy3sujhx7hqt3qqbzrbple

Randomized query complexity of sabotaged and composed functions [article]

Shalev Ben-David, Robin Kothari
2016 arXiv   pre-print
Using this technique we also show implications for lifting theorems in communication complexity.  ...  We prove this using a new lower bound measure for randomized query complexity we call randomized sabotage complexity, RS(f).  ...  Lifting theorems Using randomized sabotage complexity we are also able to show a relationship between lifting theorems in communication complexity.  ... 
arXiv:1605.09071v1 fatcat:qwkq6dzu5fdorhklwfbdaotalm

A note on the tight example in On the randomised query complexity of composition [article]

Yaqiao Li
2019 arXiv   pre-print
We make two observations regarding a recent tight example for a composition theorem for randomized query complexity: (1) it implies general randomized query-to-communication lifting is not always true  ...  Example 1 implies that such a general randomized query-to-communication lifting is not true for the socalled gap Hamming distance gadget if one allows the outer f to be a relation.  ...  The above discussion does not, however, rule out the possibility that if we insist f to be a boolean function then the randomized query-to-communication lifting might still be true for GHD n .  ... 
arXiv:1809.01118v2 fatcat:tp6tjlmaanbw7hpccioerryigu

Quantum Distinguishing Complexity, Zero-Error Algorithms, and Statistical Zero Knowledge

Shalev Ben-David, Robin Kothari, Michael Wagner
2019 Theory of Quantum Computation, Communication, and Cryptography  
Lastly, we show implications for lifting theorems in communication complexity.  ...  We show that a general lifting theorem for either zero-error quantum query complexity or for QSZK would imply a general lifting theorem for bounded-error quantum query complexity.  ...  theorems Background Lifting theorems are results that relate communication complexity measures to query complexity measures.  ... 
doi:10.4230/lipics.tqc.2019.2 dblp:conf/tqc/Ben-DavidK19 fatcat:7m7noj75jnhz7ao5hop6pyzgyu

Query-to-Communication Lifting Using Low-Discrepancy Gadgets [article]

Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, Toniann Pitassi
2021 arXiv   pre-print
Lifting theorems are theorems that relate the query complexity of a function f:{0,1}^n→{0,1} to the communication complexity of the composed function f ∘ g^n, for some "gadget" g:{0,1}^b×{0,1}^b→{0,1}.  ...  Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in the recent years.  ...  The authors would also like to thank anonymous referees for comments that improved the presentation of this work.  ... 
arXiv:1904.13056v4 fatcat:saf7os36i5gjviuglms5lp4aqe

One-way Communication and Non-adaptive Decision Tree [article]

Swagato Sanyal
2017 Electronic colloquium on computational complexity  
Let R → cc, (•) and D → cc (•) denote the randomized -error and deterministic one-way communication complexity respectively. Let D → dt (•) denote the deterministic non-adaptive query complexity.  ...  In this work we bound the one-way communication complexity of f in terms of the non-adaptive query complexity of f , from below.  ...  Our second result relates the deterministic one-way communication complexity of f to the deterministic non-adaptive query complexity of f , where f is an arbitrary partial Boolean function.  ... 
dblp:journals/eccc/Sanyal17 fatcat:6ixo3bozovf55ba6br3nimddv4

Quantum distinguishing complexity, zero-error algorithms, and statistical zero knowledge [article]

Shalev Ben-David, Robin Kothari
2019 arXiv   pre-print
Lastly, we show implications for lifting theorems in communication complexity.  ...  We show that a general lifting theorem for either zero-error quantum query complexity or for QSZK would imply a general lifting theorem for bounded-error quantum query complexity.  ...  Recently, lifting theorems have been shown for R and R 0 (with the corresponding communication complexity measures being the obvious ones: randomized communication with bounded error and randomized communication  ... 
arXiv:1902.03660v1 fatcat:ax7hxx6xe5hsffxhnshsrxepyy

Query-to-Communication Lifting for BPP [article]

Mika Göös, Toniann Pitassi, Thomas Watson
2017 arXiv   pre-print
In particular, this means that many query complexity separations involving randomized models (e.g., classical vs. quantum) automatically imply analogous separations in communication complexity.  ...  For any n-bit boolean function f, we show that the randomized communication complexity of the composed function f∘ g^n, where g is an index gadget, is characterized by the randomized decision tree complexity  ...  Acknowledgements Thanks to Shalev Ben-David and Robin Kothari for quantum references. Thanks to Anurag Anshu, Rahul Jain, Raghu Meka, Aviad Rubinstein, and Henry Yuen for discussions.  ... 
arXiv:1703.07666v1 fatcat:am5teisncfbq3bl5hoaqpqxube

Nondeterministic and Randomized Boolean Hierarchies in Communication Complexity

Toniann Pitassi, Morgan Shirley, Thomas Watson, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming  
In particular, we prove that the Randomized Boolean Hierarchy does not collapse, and we prove a query-to-communication lifting theorem for all levels of the Nondeterministic Boolean Hierarchy and use it  ...  We investigate the power of randomness in two-party communication complexity.  ...  The "moreover" part is straightforward to verify: the levels assigned to conjunctions in L C come from the levels assigned to rectangles in L R (namely {0, . . . , q}), in the same order (which is non-increasing  ... 
doi:10.4230/lipics.icalp.2020.92 dblp:conf/icalp/PitassiS020 fatcat:rotvneekpnfhhfn7e65mu2tqky

Separations in Communication Complexity Using Cheat Sheets and Information Complexity

Anurag Anshu, Aleksandrs Belovs, Shalev Ben-David, Mika Goos, Rahul Jain, Robin Kothari, Troy Lee, Miklos Santha
2016 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)  
Our main technical results are randomized communication and information complexity lower bounds for a family of functions, called lookup functions, that generalize and port the cheat sheet framework to  ...  Our results are the communication analogues of separations in query complexity proved using the recent cheat sheet framework of Aaronson, Ben-David, and Kothari (STOC 2016).  ...  R.J would like to thank Ankit Garg for helpful discussions.  ... 
doi:10.1109/focs.2016.66 dblp:conf/focs/AnshuBBG0KLS16 fatcat:srja2xc2efchxpx53hhm27roam

On Query-to-Communication Lifting for Adversary Bounds [article]

Anurag Anshu, Shalev Ben-David, Srijita Kundu
2020 arXiv   pre-print
We show that the classical adversary bound lifts to a lower bound on randomized communication complexity with a constant-sized gadget.  ...  strongest lifting theorems for randomized communication complexity using a constant-sized gadget. 2.  ...  We thank Mika G\"o\"os for helpful discussions regarding critical block sensitivity and its lifting theorem.  ... 
arXiv:2012.03415v1 fatcat:2mrce6ljvfcvvprisdmvoxmcem

Simulation Methods in Communication Lower Bounds, Revisited [article]

Guangxu Yang, Jiapeng Zhang
2022 Electronic colloquium on computational complexity  
(CCC 2005), this implies the randomized communication complexity of f • g n is lower bounded by Ω(bs(f )).  ...  The notion of lifting theorems is a generic method to lift hardness of one-party functions to two-party lower bounds in communication model.  ...  Remembering our result, that is Corr (f • g n ) = Ω(bs(f )) We can chose the gadget g carefully to lift the result P dt (f ) ≤ deg(f )bs(f ) [Mid04] in query complexity to communication complexity.  ... 
dblp:journals/eccc/YangZ22 fatcat:rqfifjxv4faybefbooljm2ktni

Spatial Isolation Implies Zero Knowledge Even in a Quantum World [article]

Alessandro Chiesa, Michael A. Forbes, Tom Gur, Nicholas Spooner
2018 arXiv   pre-print
Our proof consists of constructing a zero knowledge interactive PCP with a strong algebraic structure, and then lifting it to the MIP* model.  ...  Zero knowledge plays a central role in cryptography and complexity. The seminal work of Ben-Or et al.  ...  to include his proof of Theorem 7.2.  ... 
arXiv:1803.01519v1 fatcat:qpwt35jmt5fa3n3mpgor5oougi
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