Space-bounded communication complexity

Brody, Joshua; Chen, Shiteng; Papakonstantinou, Periklis A.; Song, Hao; Sun, Xiaoming

doi:10.1145/2422436.2422456

Cited by 5 publications

(3 citation statements)

References 26 publications

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“…5 for each 1 ≤ t ≤ l do 6 Pick b (t) ∈ F 7 q , using 7 log q bits. 7 Pick c (t) ∈ {0, 1} log q , using log q bits. 8 Pick d (t) ∈ F 2 q , using 2 log q bits.…”

Section: Algorithm 4: ρ-Biased Q-bit Randomness Generatormentioning

confidence: 99%

Small Memory Robust Simulation of Client-Server Interactive Protocols over Oblivious Noisy Channels

Chan¹,

Liang²,

Polychroniadou³

et al. 2020

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

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We revisit the problem of low-memory robust simulation of interactive protocols over noisy channels. Haeupler [FOCS 2014] considered robust simulation of two-party interactive protocols over oblivious, as well as adaptive, noisy channels. Since the simulation does not need to have fixed communication pattern, the achieved communication rates can circumvent the lower bound proved by Kol and Raz [STOC 2013]. However, a drawback of this approach is that each party needs to remember the whole history of the simulated transcript. In a subsequent manuscript, Haeupler and Resch considered low-memory simulation. The idea was to view the original protocol as a computational DAG and only the identities of the nodes are saved (as opposed to the whole transcript history) for backtracking to reduce memory usage.In this paper, we consider low-memory robust simulation of more general client-server interactive protocols, in which a leader communicates with other members/servers, who do not communicate among themselves; this setting can be applied to information-theoretic multi-server Private Information Retrieval (PIR) schemes. We propose an information-theoretic technique that converts any correct PIR protocol that assumes reliable channels, into a protocol which is both correct and private in the presence of a noisy channel while keeping the space complexity to a minimum. Despite the huge attention that PIR protocols have received in the literature, the existing works assume that the parties communicate using noiseless channels.Moreover, we observe that the approach of Haeupler and Resch to just save the nodes in the aforementioned DAG without taking the transcript history into account will lead to a correctness issue even for oblivious corruptions. We resolve this issue by saving hashes of prefixes of past transcripts. Departing from the DAG representation also allows us to accommodate scenarios where a party can simulate its part of the protocol without any extra knowledge (such as the DAG representation of the whole protocol). In the the two-party setting, our simulation has the same dependence on the error rate as in the work of Haeupler, and in the client-server setting it also depends on the number of servers. Furthermore, since our approach does not remember the complete transcript history, our current technique can defend only against oblivious corruptions.1. If protocol Π is run where the communication channel between Alice and each Bob has oblivious error rate , then, except with probability δ, Π correctly simulates Π.2. In Π , for each (noisy) communication channel, the number of bits transmitted is at most3. If party X has a memory usage of M X bits and is involved in κ (= 1 for each Bob, or m for Alice) communication channels in Π, then its memory usage (in bits) in Π is at mostPaper Organization. While we are trying to give the precise results of this paper in the above description as soon as possible, readers unfamiliar with the formal definitions and settings can first refer to Section 2. The research backgroun...

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“…5 for each 1 ≤ t ≤ l do 6 Pick b (t) ∈ F 7 q , using 7 log q bits. 7 Pick c (t) ∈ {0, 1} log q , using log q bits. 8 Pick d (t) ∈ F 2 q , using 2 log q bits.…”

Section: Algorithm 4: ρ-Biased Q-bit Randomness Generatormentioning

confidence: 99%

Small Memory Robust Simulation of Client-Server Interactive Protocols over Oblivious Noisy Channels

Chan¹,

Liang²,

Polychroniadou³

et al. 2020

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

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“…The communication model however is still restricted to a referee scenario. Another line of work studies bounded-memory versions of communication [5,7,19]. Several models have been proposed that share the same structure.…”

Section: Introductionmentioning

confidence: 99%

Streaming Communication Protocols

Boczkowski

Kerenidis

Magniez

2018

ACM Trans. Comput. Theory

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We define the Streaming Communication model that combines the main aspects of communication complexity and streaming. We consider two agents that want to compute some function that depends on inputs that are distributed to each agent. The inputs arrive as data streams and each agent has a bounded memory. Agents are allowed to communicate with each other and also update their memory based on the input bit they read and the previous message they received.We provide tight tradeoffs between the necessary resources, i.e. communication and memory, for some of the canonical problems from communication complexity by proving a strong general lower bound technique. Second, we analyze the Approximate Matching problem and show that the complexity of this problem (i.e. the achievable approximation ratio) in the one-way variant of our model is strictly different both from the streaming complexity and the one-way communication complexity thereof. * This work has been partially supported by the ERC project QCC and the French ANR Blanc project RDAM † Π +1 errs. First, when the protocol g +1 x +1 , y +1 = a +1 and the protocol exits (and hence outputs 1 − a +1 ); second, when the protocol completes the simulation of Π and Π errs. We can upper bound the error probability as(2)We know that Pr[Π errs] ≤ ε and hence it remains to show that Pr[Π +1 exits when g +1 (x +1 , y +1 ) = a +1 ] ≤ δ. First, recall that for all possible inputs (x +1 , y +1 ) with g +1 (x +1 , y +1 ) = a +1 , the expected number of communication rounds on block + 1 in Π is bounded by R +1 . Using the results in section 2.4, the protocol Π on block +1 can be simulated by Alice and Bob in the usual communication model with a multiplicative overhead S+2 log t +1 +1. Hence, the expected communication cost of step 2 is R +1 (S + 2 log t +1 + 1).The communication cost at step 3 is equal to S, and so, the total expected cost of Π +1 , when g +1 (x +1 , y +1 ) = a +1 , is smaller than R +1 (S + 2 log t +1 + 1) + S = δA +1 . Thus the probability the protocol exits because the transcript reached A +1 in this case is smaller than δ by Markov inequality. Hence we established that Π +1 is a protocol for g +1 with error ε + δ using ≤ A +1 bits in the worst case over inputs. This concludes the proof.

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“…Garden-hose complexity can be viewed as a natural measure of space, in a situation where two players with private inputs compute a Boolean function cooperatively. Space-bounded communication complexity has been investigated before [2,7,9] (usually for problems with many outputs), and recently Brody et al [3] have studied a related model of space bounded communication complexity for Boolean functions (see also [17]). In this context the garden-hose model can be viewed as a memoryless model of communication that is also reversible.…”

Section: Introduction 1background: the Modelmentioning

confidence: 99%

New Bounds for the Garden-Hose Model

Klauck¹,

Podder²

2014

Preprint

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We show new results about the garden-hose model. Our main results include improved lower bounds based on non-deterministic communication complexity (leading to the previously unknown Θ(n) bounds for Inner Product mod 2 and Disjointness), as well as an O(n • log 3 n) upper bound for the Distributed Majority function (previously conjectured to have quadratic complexity). We show an efficient simulation of formulae made of AND, OR, XOR gates in the garden-hose model, which implies that lower bounds on the garden-hose complexity GH(f ) of the order Ω(n 2+ ) will be hard to obtain for explicit functions. Furthermore we study a time-bounded variant of the model, in which even modest savings in time can lead to exponential lower bounds on the size of garden-hose protocols. * This work is funded by the Singapore Ministry of Education (partly through the Academic Research Fund Tier 3 MOE2012-T3-1-009) and by the Singapore National Research Foundation.1 It should be mentioned that even though Alice and Bob choose to not communicate in any other way, their intentions are not hostile and neither will deviate from a previously agreed upon protocol.

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Space-bounded communication complexity

Cited by 5 publications

References 26 publications

Small Memory Robust Simulation of Client-Server Interactive Protocols over Oblivious Noisy Channels

Small Memory Robust Simulation of Client-Server Interactive Protocols over Oblivious Noisy Channels

Streaming Communication Protocols

New Bounds for the Garden-Hose Model

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